# CV Reference Manual

- Image Processing
  - Gradients, Edges and Corners
  - Sampling, Interpolation and Geometrical Transforms
  - Morphological Operations
  - Filters and Color Conversion
  - Pyramids and the Applications
  - Connected Components
  - Image and Contour Moments
  - Special Image Transforms
  - Histograms
  - Matching
- Structural Analysis
  - Contour Processing
  - Computational Geometry
  - Planar Subdivisions
- Motion Analysis and Object Tracking
  - Accumulation of Background Statistics
  - Motion Templates
  - Object Tracking
  - Optical Flow
  - Estimators
- Pattern Recognition
  - Object Detection
- Camera Calibration and 3D Reconstruction
  - Camera Calibration
  - Pose Estimation
  - Epipolar Geometry
- Alphabetical List of Functions
- Bibliography

# Image Processing

Note:
The chapter describes functions for image processing and analysis. Most of the functions work with 2d arrays of pixels. We refer the arrays as "images" however they do not neccesserily have to be IplImage's, they may be CvMat's or CvMatND's as well.

## Gradients, Edges and Corners

### Sobel

Calculates first, second, third or mixed image derivatives using extended Sobel operator

    void cvSobel( const CvArr* src, CvArr* dst, int xorder, int yorder, int aperture_size=3 );

src
  Source image.
dst
  Destination image.
xorder
  Order of the derivative x .
yorder
  Order of the derivative y .
aperture_size
  Size of the extended Sobel kernel, must be 1, 3, 5 or 7. In all cases except 1, aperture_size ×aperture_size separable kernel will be used to calculate the derivative. For `aperture_size`=1 3x1 or 1x3 kernel is used (Gaussian smoothing is not done). There is also special value `CV_SCHARR` (=-1) that corresponds to 3x3 Scharr filter that may give more accurate results than 3x3 Sobel. Scharr aperture is:

      | -3 0  3|
      |-10 0 10|
      | -3 0  3|

  for x-derivative or transposed for y-derivative.

The function cvSobel calculates the image derivative by convolving the image with the appropriate kernel:

    dst(x,y) = dxorder+yodersrc/dxxorder•dyyorder |(x,y)

The Sobel operators combine Gaussian smoothing and differentiation so the result is more or less robust to the noise. Most often, the function is called with (xorder=1, yorder=0, aperture_size=3) or (xorder=0, yorder=1, aperture_size=3) to calculate first x- or y- image derivative. The first case corresponds to

      |-1  0  1|
      |-2  0  2|
      |-1  0  1|

kernel and the second one corresponds to

      |-1 -2 -1|
      | 0  0  0|
      | 1  2  1|
    or
      | 1  2  1|
      | 0  0  0|
      |-1 -2 -1|

kernel, depending on the image origin (`origin` field of `IplImage` structure). No scaling is done, so the destination image usually has larger by absolute value numbers than the source image. To avoid overflow, the function requires 16-bit destination image if the source image is 8-bit. The result can be converted back to 8-bit using cvConvertScale or cvConvertScaleAbs functions. Besides 8-bit images the function can process 32-bit floating-point images. Both source and destination must be single-channel images of equal size or ROI size.

### Laplace

Calculates Laplacian of the image

    void cvLaplace( const CvArr* src, CvArr* dst, int aperture_size=3 );

src
  Source image.
dst
  Destination image.
aperture_size
  Aperture size (it has the same meaning as in cvSobel).

The function cvLaplace calculates Laplacian of the source image by summing second x- and y- derivatives calculated using Sobel operator:

    dst(x,y) = d2src/dx2 + d2src/dy2

Specifying `aperture_size`=1 gives the fastest variant that is equal to convolving the image with the following kernel:

    |0  1  0|
    |1 -4  1|
    |0  1  0|

Similar to cvSobel function, no scaling is done and the same combinations of input and output formats are supported.

### Canny

Implements Canny algorithm for edge detection

    void cvCanny( const CvArr* image, CvArr* edges, double threshold1,
                  double threshold2, int aperture_size=3 );

image
  Input image.
edges
  Image to store the edges found by the function.
threshold1
  The first threshold.
threshold2
  The second threshold.
aperture_size
  Aperture parameter for Sobel operator (see cvSobel).

The function cvCanny finds the edges on the input image `image` and marks them in the output image `edges` using the Canny algorithm. The smallest of `threshold1` and `threshold2` is used for edge linking, the largest - to find initial segments of strong edges.

### PreCornerDetect

Calculates feature map for corner detection

    void cvPreCornerDetect( const CvArr* image, CvArr* corners, int aperture_size=3 );

image
  Input image.
corners
  Image to store the corner candidates.
aperture_size
  Aperture parameter for Sobel operator (see cvSobel).

The function cvPreCornerDetect calculates the function Dx2Dyy+Dy2Dxx - 2DxDyDxy where D? denotes one of the first image derivatives and D?? denotes a second image derivative. The corners can be found as local maximums of the function:

    // assuming that the image is floating-point
    IplImage* corners = cvCloneImage(image);
    IplImage* dilated_corners = cvCloneImage(image);
    IplImage* corner_mask = cvCreateImage( cvGetSize(image), 8, 1 );
    cvPreCornerDetect( image, corners, 3 );
    cvDilate( corners, dilated_corners, 0, 1 );
    cvSubS( corners, dilated_corners, corners );
    cvCmpS( corners, 0, corner_mask, CV_CMP_GE );
    cvReleaseImage( &corners );
    cvReleaseImage( &dilated_corners );

### CornerEigenValsAndVecs

Calculates eigenvalues and eigenvectors of image blocks for corner detection

    void cvCornerEigenValsAndVecs( const CvArr* image, CvArr* eigenvv,
                                   int block_size, int aperture_size=3 );

image
  Input image.
eigenvv
  Image to store the results. It must be 6 times wider than the input image.
block_size
  Neighborhood size (see discussion).
aperture_size
  Aperture parameter for Sobel operator (see cvSobel).

For every pixel the function cvCornerEigenValsAndVecs considers `block_size` × `block_size` neigborhood S(p). It calcualtes covariation matrix of derivatives over the neigborhood as:

        | sumS(p)(dI/dx)2   sumS(p)(dI/dx•dI/dy)|
    M = |                                 |
        | sumS(p)(dI/dx•dI/dy)  sumS(p)(dI/dy)2 |

After that it finds eigenvectors and eigenvalues of the matrix and stores them into destination image in form (λ1, λ2, x1, y1, x2, y2), where
λ1, λ2 - eigenvalues of `M`; not sorted
(x1, y1) - eigenvector corresponding to λ1
(x2, y2) - eigenvector corresponding to λ2

### CornerMinEigenVal

Calculates minimal eigenvalue of gradient matrices for corner detection

    void cvCornerMinEigenVal( const CvArr* image, CvArr* eigenval, int block_size, int aperture_size=3 );

image
  Input image.
eigenval
  Image to store the minimal eigen values. Should have the same size as `image`
block_size
  Neighborhood size (see discussion of cvCornerEigenValsAndVecs).
aperture_size
  Aperture parameter for Sobel operator (see cvSobel). format. In the case of floating-point input format this parameter is the number of the fixed float filter used for differencing.

The function cvCornerMinEigenVal is similar to cvCornerEigenValsAndVecs but it calculates and stores only the minimal eigen value of derivative covariation matrix for every pixel, i.e. min(λ1, λ2) in terms of the previous function.

### FindCornerSubPix

Refines corner locations

    void cvFindCornerSubPix( const CvArr* image, CvPoint2D32f* corners,
                             int count, CvSize win, CvSize zero_zone,
                             CvTermCriteria criteria );

image
  Input image.
corners
  Initial coordinates of the input corners and refined coordinates on output.
count
  Number of corners.
win
  Half sizes of the search window. For example, if `win`=(5,5) then 5*2+1 × 5*2+1 = 11 × 11 search window is used.
zero_zone
  Half size of the dead region in the middle of the search zone over which the summation in formulae below is not done. It is used sometimes to avoid possible singularities of the autocorrelation matrix. The value of (-1,-1) indicates that there is no such size.
criteria
  Criteria for termination of the iterative process of corner refinement. That is, the process of corner position refinement stops either after certain number of iteration or when a required accuracy is achieved. The `criteria` may specify either of or both the maximum number of iteration and the required accuracy.

The function cvFindCornerSubPix iterates to find the sub-pixel accurate location of corners, or radial saddle points, as shown in on the picture below.

Sub-pixel accurate corner locator is based on the observation that every vector from the center `q` to a point `p` located within a neighborhood of `q` is orthogonal to the image gradient at `p` subject to image and measurement noise. Consider the expression:

    εi=DIpiT•(q-pi)

where `DIpi` is the image gradient at the one of the points `pi` in a neighborhood of `q`. The value of `q` is to be found such that `εi` is minimized. A system of equations may be set up with `εi`' set to zero:

    sumi(DIpi•DIpiT)•q - sumi(DIpi•DIpiT•pi) = 0

where the gradients are summed within a neighborhood ("search window") of `q`. Calling the first gradient term `G` and the second gradient term `b` gives:

    q=G-1•b

The algorithm sets the center of the neighborhood window at this new center `q` and then iterates until the center keeps within a set threshold.

### GoodFeaturesToTrack

Determines strong corners on image

    void cvGoodFeaturesToTrack( const CvArr* image, CvArr* eig_image, CvArr* temp_image,
                                CvPoint2D32f* corners, int* corner_count,
                                double quality_level, double min_distance,
                                const CvArr* mask=NULL );

image
  The source 8-bit or floating-point 32-bit, single-channel image.
eig_image
  Temporary floating-point 32-bit image of the same size as `image`.
temp_image
  Another temporary image of the same size and same format as `eig_image`.
corners
  Output parameter. Detected corners.
corner_count
  Output parameter. Number of detected corners.
quality_level
  Multiplier for the maxmin eigenvalue; specifies minimal accepted quality of image corners.
min_distance
  Limit, specifying minimum possible distance between returned corners; Euclidian distance is used.
mask
  Region of interest. The function selects points either in the specified region or in the whole image if the mask is NULL.

The function cvGoodFeaturesToTrack finds corners with big eigenvalues in the image. The function first calculates the minimal eigenvalue for every source image pixel using cvCornerMinEigenVal function and stores them in `eig_image`. Then it performs non-maxima suppression (only local maxima in 3x3 neighborhood remain). The next step is rejecting the corners with the minimal eigenvalue less than `quality_level`•max(`eig_image`(x,y)). Finally, the function ensures that all the corners found are distanced enough from one another by considering the corners (the most strongest corners are considered first) and checking that the distance between the newly considered feature and the features considered earlier is larger than `min_distance`. So, the function removes the features than are too close to the stronger features.

## Sampling, Interpolation and Geometrical Transforms

### InitLineIterator

Initializes line iterator

    int cvInitLineIterator( const CvArr* image, CvPoint pt1, CvPoint pt2,
                            CvLineIterator* line_iterator, int connectivity=8 );

image
  Image to sample the line from.
pt1
  Starting the line point.
pt2
  Ending the line point.
line_iterator
  Pointer to the line iterator state structure.
connectivity
  The scanned line connectivity, 4 or 8.

The function cvInitLineIterator initializes the line iterator and returns the number of pixels between two end points. Both points must be inside the image. After the iterator has been initialized, all the points on the raster line that connects the two ending points may be retrieved by successive calls of `CV_NEXT_LINE_POINT` point. The points on the line are calculated one by one using 4-connected or 8-connected Bresenham algorithm.

#### Example. Using line iterator to calculate sum of pixel values along the color line

        CvScalar sum_line_pixels( IplImage* image, CvPoint pt1, CvPoint pt2 )
        {
            CvLineIterator iterator;
            int blue_sum = 0, green_sum = 0, red_sum = 0;
            int count = cvInitLineIterator( image, pt1, pt2, &iterator, 8 );

            for( int i = 0; i < count; i++ ){
                blue_sum += iterator.ptr[0];
                green_sum += iterator.ptr[1];
                red_sum += iterator.ptr[2];
                CV_NEXT_LINE_POINT(iterator);

                /* print the pixel coordinates: demonstrates how to calculate the coordinates */
                {
                int offset, x, y;
                /* assume that ROI is not set, otherwise need to take it into account. */
                offset = iterator.ptr - (uchar*)(image->imageData);
                y = offset/image->widthStep;
                x = (offset - y*image->widthStep)/(3*sizeof(uchar) /* size of pixel */);
                printf("(%d,%d)\n", x, y );
                }
            }
            return cvScalar( blue_sum, green_sum, red_sum );
        }

### SampleLine

Reads raster line to buffer

    int cvSampleLine( const CvArr* image, CvPoint pt1, CvPoint pt2,
                      void* buffer, int connectivity=8 );

image
  Image to sample the line from.
pt1
  Starting the line point.
pt2
  Ending the line point.
buffer
  Buffer to store the line points; must have enough size to store max( |`pt2.x`-`pt1.x`|+1, |`pt2.y`-`pt1.y`|+1 ) points in case of 8-connected line and |`pt2.x`-`pt1.x`|+|`pt2.y`-`pt1.y`|+1 in case of 4-connected line.
connectivity
  The line connectivity, 4 or 8.

The function cvSampleLine implements a particular case of application of line iterators. The function reads all the image points lying on the line between `pt1` and `pt2`, including the ending points, and stores them into the buffer.

### GetRectSubPix

Retrieves pixel rectangle from image with sub-pixel accuracy

    void cvGetRectSubPix( const CvArr* src, CvArr* dst, CvPoint2D32f center );

src
  Source image.
dst
  Extracted rectangle.
center
  Floating point coordinates of the extracted rectangle center within the source image. The center must be inside the image.

The function cvGetRectSubPix extracts pixels from `src`:

    dst(x, y) = src(x + center.x - (width(dst)-1)*0.5, y + center.y - (height(dst)-1)*0.5)

where the values of pixels at non-integer coordinates are retrieved using bilinear interpolation. Every channel of multiple-channel images is processed independently. Whereas the rectangle center must be inside the image, the whole rectangle may be partially occluded. In this case, the replication border mode is used to get pixel values beyond the image boundaries.

### GetQuadrangleSubPix

Retrieves pixel quadrangle from image with sub-pixel accuracy

    void cvGetQuadrangleSubPix( const CvArr* src, CvArr* dst, const CvMat* map_matrix,
                                int fill_outliers=0, CvScalar fill_value=cvScalarAll(0) );

src
  Source image.
dst
  Extracted quadrangle.
map_matrix
  The transformation 3 × 2 matrix [`A`|`b`] (see the discussion).
fill_outliers
  The flag indicating whether to interpolate values of pixel taken from outside of the source image using replication mode (`fill_outliers`=0) or set them a fixed value (`fill_outliers`=1).
fill_value
  The fixed value to set the outlier pixels to if `fill_outliers`=1.

The function cvGetQuadrangleSubPix extracts pixels from `src` at sub-pixel accuracy and stores them to `dst` as follows:

    dst(x+width(dst)/2, y+height(dst)/2)= src( A11x+A12y+b1, A21x+A22y+b2),

    where A and b are taken from map_matrix
                 | A11 A12  b1 |
    map_matrix = |            |
                 | A21 A22  b2 |

where the values of pixels at non-integer coordinates A•(x,y)T+b are retrieved using bilinear interpolation. Every channel of multiple-channel images is processed independently.

#### Example. Using cvGetQuadrangleSubPix for image rotation.

    #include "cv.h"
    #include "highgui.h"
    #include "math.h"

    int main( int argc, char** argv )
    {
        IplImage* src;
        /* the first command line parameter must be image file name */
        if( argc==2 && (src = cvLoadImage(argv[1], -1))!=0)
        {
            IplImage* dst = cvCloneImage( src );
            int delta = 1;
            int angle = 0;

            cvNamedWindow( "src", 1 );
            cvShowImage( "src", src );

            for(;;)
            {
                float m[6];
                double factor = (cos(angle*CV_PI/180.) + 1.1)*3;
                CvMat M = cvMat( 2, 3, CV_32F, m );
                int w = src->width;
                int h = src->height;

                m[0] = (float)(factor*cos(-angle*2*CV_PI/180.));
                m[1] = (float)(factor*sin(-angle*2*CV_PI/180.));
                m[2] = w*0.5f;
                m[3] = -m[1];
                m[4] = m[0];
                m[5] = h*0.5f;

                cvGetQuadrangleSubPix( src, dst, &M, 1, cvScalarAll(0));

                cvNamedWindow( "dst", 1 );
                cvShowImage( "dst", dst );

                if( cvWaitKey(5) == 27 )
                    break;

                angle = (angle + delta) % 360;
            }
        }
        return 0;
    }

### Resize

Resizes image

    void cvResize( const CvArr* src, CvArr* dst, int interpolation=CV_INTER_LINEAR );

src
  Source image.
dst
  Destination image.
interpolation
  Interpolation method:
  - CV_INTER_NN - nearest-neigbor interpolation,
  - CV_INTER_LINEAR - bilinear interpolation (used by default)
  - CV_INTER_AREA - resampling using pixel area relation. It is preferred method for image decimation that gives moire-free results. In case of zooming it is similar to `CV_INTER_NN` method.
  - CV_INTER_CUBIC - bicubic interpolation.

The function cvResize resizes image `src` so that it fits exactly to `dst`. If ROI is set, the function consideres the ROI as supported as usual.

### WarpAffine

Applies affine transformation to the image

    void cvWarpAffine( const CvArr* src, CvArr* dst, const CvMat* map_matrix,
                       int flags=CV_INTER_LINEAR+CV_WARP_FILL_OUTLIERS,
                       CvScalar fillval=cvScalarAll(0) );

src
  Source image.
dst
  Destination image.
map_matrix
  2×3 transformation matrix.
flags
  A combination of interpolation method and the following optional flags:
  - CV_WARP_FILL_OUTLIERS - fill all the destimation image pixels. If some of them correspond to outliers in the source image, they are set to `fillval`.
  - CV_WARP_INVERSE_MAP - indicates that `matrix` is inverse transform from destination image to source and, thus, can be used directly for pixel interpolation. Otherwise, the function finds the inverse transform from `map_matrix`.
fillval
  A value used to fill outliers.

The function cvWarpAffine transforms source image using the specified matrix:

    dst(x',y')<-src(x,y)
    (x',y')T=map_matrix•(x,y,1)T+b if CV_WARP_INVERSE_MAP is not set,
    (x, y)T=map_matrix•(x',y&apos,1)T+b otherwise

The function is similar to cvGetQuadrangleSubPix but they are not exactly the same. cvWarpAffine requires input and output image have the same data type, has larger overhead (so it is not quite suitable for small images) and can leave part of destination image unchanged. While cvGetQuadrangleSubPix may extract quadrangles from 8-bit images into floating-point buffer, has smaller overhead and always changes the whole destination image content.

To transform a sparse set of points, use cvTransform function from cxcore.

### 2DRotationMatrix

Calculates affine matrix of 2d rotation

    CvMat* cv2DRotationMatrix( CvPoint2D32f center, double angle,
                               double scale, CvMat* map_matrix );

center
  Center of the rotation in the source image.
angle
  The rotation angle in degrees. Positive values mean couter-clockwise rotation (the coordiate origin is assumed at top-left corner).
scale
  Isotropic scale factor.
map_matrix
  Pointer to the destination 2×3 matrix.

The function cv2DRotationMatrix calculates matrix:

    [  α  β  |  (1-α)*center.x - β*center.y ]
    [ -β  α  |  β*center.x + (1-α)*center.y ]

    where α=scale*cos(angle), β=scale*sin(angle)

The transformation maps the rotation center to itself. If this is not the purpose, the shift should be adjusted.

### WarpPerspective

Applies perspective transformation to the image

    void cvWarpPerspective( const CvArr* src, CvArr* dst, const CvMat* map_matrix,
                            int flags=CV_INTER_LINEAR+CV_WARP_FILL_OUTLIERS,
                            CvScalar fillval=cvScalarAll(0) );

src
  Source image.
dst
  Destination image.
map_matrix
  3×3 transformation matrix.
flags
  A combination of interpolation method and the following optional flags:
  - CV_WARP_FILL_OUTLIERS - fill all the destimation image pixels. If some of them correspond to outliers in the source image, they are set to `fillval`.
  - CV_WARP_INVERSE_MAP - indicates that `matrix` is inverse transform from destination image to source and, thus, can be used directly for pixel interpolation. Otherwise, the function finds the inverse transform from `map_matrix`.
fillval
  A value used to fill outliers.

The function cvWarpPerspective transforms source image using the specified matrix:

    dst(x',y')<-src(x,y)
    (tx',ty',t)T=map_matrix•(x,y,1)T+b if CV_WARP_INVERSE_MAP is not set,
    (tx, ty, t)T=map_matrix•(x',y&apos,1)T+b otherwise

For a sparse set of points use cvPerspectiveTransform function from cxcore.

### WarpPerspectiveQMatrix

Calculates perspective transform from 4 corresponding points

    CvMat* cvWarpPerspectiveQMatrix( const CvPoint2D32f* src,
                                     const CvPoint2D32f* dst,
                                     CvMat* map_matrix );

src
  Coordinates of 4 quadrangle vertices in the source image.
dst
  Coordinates of the 4 corresponding quadrangle vertices in the destination image.
map_matrix
  Pointer to the destination 3×3 matrix.

The function cvWarpPerspectiveQMatrix calculates matrix of perspective transform such that:

    (tix'i,tiy'i,ti)T=matrix•(xi,yi,1)T

where `dst(i)=(x'i,y'i), src(i)=(xi,yi), i=0..3`.

## Morphological Operations

### CreateStructuringElementEx

Creates structuring element

    IplConvKernel* cvCreateStructuringElementEx( int cols, int rows, int anchor_x, int anchor_y,
                                                 int shape, int* values=NULL );

cols
  Number of columns in the structuring element.
rows
  Number of rows in the structuring element.
anchor_x
  Relative horizontal offset of the anchor point.
anchor_y
  Relative vertical offset of the anchor point.
shape
  Shape of the structuring element; may have the following values:
  - `CV_SHAPE_RECT`, a rectangular element;
  - `CV_SHAPE_CROSS`, a cross-shaped element;
  - `CV_SHAPE_ELLIPSE`, an elliptic element;
  - `CV_SHAPE_CUSTOM`, a user-defined element. In this case the parameter `values` specifies the mask, that is, which neighbors of the pixel must be considered.
values
  Pointer to the structuring element data, a plane array, representing row-by-row scanning of the element matrix. Non-zero values indicate points that belong to the element. If the pointer is `NULL`, then all values are considered non-zero, that is, the element is of a rectangular shape. This parameter is considered only if the shape is `CV_SHAPE_CUSTOM` .

The function cv CreateStructuringElementEx allocates and fills the structure `IplConvKernel`, which can be used as a structuring element in the morphological operations.

### ReleaseStructuringElement

Deletes structuring element

    void cvReleaseStructuringElement( IplConvKernel** element );

element
  Pointer to the deleted structuring element.

The function cvReleaseStructuringElement releases the structure `IplConvKernel` that is no longer needed. If `*element` is `NULL`, the function has no effect.

### Erode

Erodes image by using arbitrary structuring element

    void cvErode( const CvArr* src, CvArr* dst, IplConvKernel* element=NULL, int iterations=1 );

src
  Source image.
dst
  Destination image.
element
  Structuring element used for erosion. If it is `NULL`, a 3×3 rectangular structuring element is used.
iterations
  Number of times erosion is applied.

The function cvErode erodes the source image using the specified structuring element that determines the shape of a pixel neighborhood over which the minimum is taken:

    dst=erode(src,element):  dst(x,y)=min((x',y') in element))src(x+x',y+y')

The function supports the in-place mode. Erosion can be applied several (`iterations`) times. In case of color image each channel is processed independently.

### Dilate

Dilates image by using arbitrary structuring element

    void cvDilate( const CvArr* src, CvArr* dst, IplConvKernel* element=NULL, int iterations=1 );

src
  Source image.
dst
  Destination image.
element
  Structuring element used for erosion. If it is `NULL`, a 3×3 rectangular structuring element is used.
iterations
  Number of times erosion is applied.

The function cvDilate dilates the source image using the specified structuring element that determines the shape of a pixel neighborhood over which the maximum is taken:

    dst=dilate(src,element):  dst(x,y)=max((x',y') in element))src(x+x',y+y')

The function supports the in-place mode. Dilation can be applied several (`iterations`) times. In case of color image each channel is processed independently.

### MorphologyEx

Performs advanced morphological transformations

    void cvMorphologyEx( const CvArr* src, CvArr* dst, CvArr* temp,
                         IplConvKernel* element, int operation, int iterations=1 );

src
  Source image.
dst
  Destination image.
temp
  Temporary image, required in some cases.
element
  Structuring element.
operation
  Type of morphological operation, one of:
  `CV_MOP_OPEN` - opening
  `CV_MOP_CLOSE` - closing
  `CV_MOP_GRADIENT` - morphological gradient
  `CV_MOP_TOPHAT` - "top hat"
  `CV_MOP_BLACKHAT` - "black hat"
iterations
  Number of times erosion and dilation are applied.

The function cvMorphologyEx can perform advanced morphological transformations using erosion and dilation as basic operations.

    Opening:
    dst=open(src,element)=dilate(erode(src,element),element)

    Closing:
    dst=close(src,element)=erode(dilate(src,element),element)

    Morphological gradient:
    dst=morph_grad(src,element)=dilate(src,element)-erode(src,element)

    "Top hat":
    dst=tophat(src,element)=src-open(src,element)

    "Black hat":
    dst=blackhat(src,element)=close(src,element)-src

The temporary image `temp` is required for morphological gradient and, in case of in-place operation, for "top hat" and "black hat".

## Filters and Color Conversion

### Smooth

Smooths the image in one of several ways

    void cvSmooth( const CvArr* src, CvArr* dst,
                   int smoothtype=CV_GAUSSIAN,
                   int param1=3, int param2=0, double param3=0 );

src
  The source image.
dst
  The destination image.
smoothtype
  Type of the smoothing:
  - CV_BLUR_NO_SCALE (simple blur with no scaling) - summation over a pixel `param1`×`param2` neighborhood. If the neighborhood size may vary, one may precompute integral image with cvIntegral function.
  - CV_BLUR (simple blur) - summation over a pixel `param1`×`param2` neighborhood with subsequent scaling by 1/(`param1`•`param2`).
  - CV_GAUSSIAN (gaussian blur) - convolving image with `param1`×`param2` Gaussian kernel.
  - CV_MEDIAN (median blur) - finding median of `param1`×`param1` neighborhood (i.e. the neighborhood is square).
  - CV_BILATERAL (bilateral filter) - applying bilateral 3x3 filtering with color sigma=`param1` and space sigma=`param2`. Information about bilateral filtering can be found at http://www.dai.ed.ac.uk/CVonline/LOCAL_COPIES/MANDUCHI1/Bilateral_Filtering.html
param1
  The first parameter of smoothing operation.
param2
  The second parameter of smoothing operation. In case of simple scaled/non-scaled and Gaussian blur if `param2` is zero, it is set to `param1`.
param3
  In case of Gaussian parameter this parameter may specify Gaussian sigma (standard deviation). If it is zero, it is calculated from the kernel size:

                    sigma = (n/2 - 1)*0.3 + 0.8, where n=param1 for horizontal kernel,
                                                       n=param2 for vertical kernel.

  Using standard sigma for small kernels (3×3 to 7×7) gives better speed. If `param3` is not zero, while `param1` and `param2` are zeros, the kernel size is calculated from the sigma (to provide accurate enough operation).

The function cvSmooth smooths image using one of several methods. Every of the methods has some features and restrictions listed below

Blur with no scaling works with single-channel images only and supports accumulation of 8-bit to 16-bit format (similar to cvSobel and cvLaplace) and 32-bit floating point to 32-bit floating-point format.

Simple blur and Gaussian blur support 1- or 3-channel, 8-bit and 32-bit floating point images. These two methods can process images in-place.

Median and bilateral filters work with 1- or 3-channel 8-bit images and can not process images in-place.

### Filter2D

Convolves the image with the kernel

    void cvFilter2D( const CvArr* src, CvArr* dst,
                     const CvMat* kernel,
                     CvPoint anchor=cvPoint(-1,-1));
    #define cvConvolve2D cvFilter2D

src
  The source image.
dst
  The destination image.
kernel
  Convolution kernel, single-channel floating point matrix. If you want to apply different kernels to different channels, split the image using cvSplit into separate color planes and process them individually.
anchor
  The anchor of the kernel that indicates the relative position of a filtered point within the kernel. The anchor shoud lie within the kernel. The special default value (-1,-1) means that it is at the kernel center.

The function cvFilter2D applies arbitrary linear filter to the image. In-place operation is supported. When the aperture is partially outside the image, the function interpolates outlier pixel values from the nearest pixels that is inside the image.

### Integral

Calculates integral images

    void cvIntegral( const CvArr* image, CvArr* sum, CvArr* sqsum=NULL, CvArr* tilted_sum=NULL );

image
  The source image, `W`×`H`, single-channel, 8-bit, or floating-point (32f or 64f).
sum
  The integral image, `W+1`×`H+1`, single-channel, 32-bit integer or double precision floating-point (64f).
sqsum
  The integral image for squared pixel values, `W+1`×`H+1`, single-channel, double precision floating-point (64f).
tilted_sum
  The integral for the image rotated by 45 degrees, `W+1`×`H+1`, single-channel, the same data type as `sum`.

The function cvIntegral calculates one or more integral images for the source image as following:

    sum(X,Y)=sumx<X,y<Yimage(x,y)

    sqsum(X,Y)=sumx<X,y<Yimage(x,y)2

    tilted_sum(X,Y)=sumy<Y,abs(x-X)<yimage(x,y)

Using these integral images, one may calculate sum, mean, standard deviation over arbitrary pixel up-right or rotated rectangle in O(1), for example:

    sumx1<=x<x2,y1<=y<y2image(x,y)=sum(x2,y2)-sum(x1,y2)-sum(x2,y1)+sum(x1,x1)

It makes possible to do a fast blurring or fast block correlation with variable window size etc.

### CvtColor

Converts image from one color space to another

    void cvCvtColor( const CvArr* src, CvArr* dst, int code );

src
  The source 8-bit or floating-point image.
dst
  The destination 8-bit or floating-point image.
code
  Color conversion operation that can be specifed using CV_<src_color_space>2<dst_color_space> constants (see below).

The function cvCvtColor converts input image from one color space to another. The function ignores `colorModel` and `channelSeq` fields of `IplImage` header, so the source image color space should be specified correctly (including order of the channels in case of RGB space, e.g. BGR means 24-bit format with B0 G0 R0 B1 G1 R1 ... layout, whereas RGB means 24-format with R0 G0 B0 R1 G1 B1 ... layout). The function can do the following transformations:

- Transformations within RGB space like adding/removing alpha channel, reversing the channel order, conversion to/from 16-bit RGB color (Rx5:Gx6:Rx5) color, 15-bit RGB color as well as conversion to/from grayscale using:

      RGB[A]->Gray: Y=0.212671*R + 0.715160*G + 0.072169*B + 0*A
      Gray->RGB[A]: R=Y G=Y B=Y A=0

  All the possible combinations of input and output format are allowed here.

- RGB<=>XYZ (CV_BGR2XYZ, CV_RGB2XYZ, CV_XYZ2BGR, CV_XYZ2RGB):

      |X|   |0.412411  0.357585  0.180454| |R|
      |Y| = |0.212649  0.715169  0.072182|*|G|
      |Z|   |0.019332  0.119195  0.950390| |B|

      |R|   | 3.240479  -1.53715  -0.498535| |X|
      |G| = |-0.969256   1.875991  0.041556|*|Y|
      |B|   | 0.055648  -0.204043  1.057311| |Z|

- RGB<=>YCrCb (CV_BGR2YCrCb, CV_RGB2YCrCb, CV_YCrCb2BGR, CV_YCrCb2RGB)

      Y=0.299*R + 0.587*G + 0.114*B
      Cr=(R-Y)*0.713 + 128
      Cb=(B-Y)*0.564 + 128

      R=Y + 1.403*(Cr - 128)
      G=Y - 0.344*(Cr - 128) - 0.714*(Cb - 128)
      B=Y + 1.773*(Cb - 128)

- RGB=>HSV (CV_BGR2HSV,CV_RGB2HSV)

      V=max(R,G,B)
      S=(V-min(R,G,B))*255/V   if V!=0, 0 otherwise

             (G - B)*60/S,  if V=R
      H= 180+(B - R)*60/S,  if V=G
         240+(R - G)*60/S,  if V=B

      if H<0 then H=H+360

  The hue values calcualted using the above formulae vary from 0° to 360° so they are divided by 2 to fit into 8 bits.

- RGB=>Lab (CV_BGR2Lab, CV_RGB2Lab)

      |X|   |0.433910  0.376220  0.189860| |R/255|
      |Y| = |0.212649  0.715169  0.072182|*|G/255|
      |Z|   |0.017756  0.109478  0.872915| |B/255|

      L = 116*Y1/3      for Y>0.008856
      L = 903.3*Y      for Y<=0.008856

      a = 500*(f(X)-f(Y))
      b = 200*(f(Y)-f(Z))
      where f(t)=t1/3              for t>0.008856
            f(t)=7.787*t+16/116   for t<=0.008856

  The above formulae have been taken from http://www.cica.indiana.edu/cica/faq/color_spaces/color.spaces.html

- Bayer=>RGB (CV_BayerBG2BGR, CV_BayerGB2BGR, CV_BayerRG2BGR, CV_BayerGR2BGR,
  CV_BayerBG2RGB, CV_BayerRG2BGR, CV_BayerGB2RGB, CV_BayerGR2BGR,
  CV_BayerRG2RGB, CV_BayerBG2BGR, CV_BayerGR2RGB, CV_BayerGB2BGR)

  Bayer pattern is widely used in CCD and CMOS cameras. It allows to get color picture out of a single plane where R,G and B pixels (sensors of a particular component) are interleaved like this:

  R | G | R | G | R
  G | B | G | B | G
  R | G | R | G | R
  G | B | G | B | G
  R | G | R | G | R
  G | B | G | B | G

  The output RGB components of a pixel are interpolated from 1, 2 or 4 neighbors of the pixel having the same color. There are several modifications of the above pattern that can be achieved by shifting the pattern one pixel left and/or one pixel up. The two letters C1 and C2 in the conversion constants CV_BayerC1C22{BGR|RGB} indicate the particular pattern type - these are components from the second row, second and third columns, respectively. For example, the above pattern has very popular "BG" type.

### Threshold

Applies fixed-level threshold to array elements

    void cvThreshold( const CvArr* src, CvArr* dst, double threshold,
                      double max_value, int threshold_type );

src
  Source array (single-channel, 8-bit of 32-bit floating point).
dst
  Destination array; must be either the same type as `src` or 8-bit.
threshold
  Threshold value.
max_value
  Maximum value to use with `CV_THRESH_BINARY` and `CV_THRESH_BINARY_INV` thresholding types.
threshold_type
  Thresholding type (see the discussion)

The function cvThreshold applies fixed-level thresholding to single-channel array. The function is typically used to get bi-level (binary) image out of grayscale image (cvCmpS could be also used for this purpose) or for removing a noise, i.e. filtering out pixels with too small or too large values. There are several types of thresholding the function supports that are determined by `threshold_type`:

    threshold_type=CV_THRESH_BINARY:
    dst(x,y) = max_value, if src(x,y)>threshold
               0, otherwise

    threshold_type=CV_THRESH_BINARY_INV:
    dst(x,y) = 0, if src(x,y)>threshold
               max_value, otherwise

    threshold_type=CV_THRESH_TRUNC:
    dst(x,y) = threshold, if src(x,y)>threshold
               src(x,y), otherwise

    threshold_type=CV_THRESH_TOZERO:
    dst(x,y) = src(x,y), if (x,y)>threshold
               0, otherwise

    threshold_type=CV_THRESH_TOZERO_INV:
    dst(x,y) = 0, if src(x,y)>threshold
               src(x,y), otherwise

And this is the visual description of thresholding types:

### AdaptiveThreshold

Applies adaptive threshold to array

    void cvAdaptiveThreshold( const CvArr* src, CvArr* dst, double max_value,
                              int adaptive_method=CV_ADAPTIVE_THRESH_MEAN_C,
                              int threshold_type=CV_THRESH_BINARY,
                              int block_size=3, double param1=5 );

src
  Source image.
dst
  Destination image.
max_value
  Maximum value that is used with `CV_THRESH_BINARY` and `CV_THRESH_BINARY_INV`.
adaptive_method
  Adaptive thresholding algorithm to use: `CV_ADAPTIVE_THRESH_MEAN_C` or `CV_ADAPTIVE_THRESH_GAUSSIAN_C` (see the discussion).
threshold_type
  Thresholding type; must be one of
  - `CV_THRESH_BINARY,`
  - `CV_THRESH_BINARY_INV`
block_size
  The size of a pixel neighborhood that is used to calculate a threshold value for the pixel: 3, 5, 7, ...
param1
  The method-dependent parameter. For the methods `CV_ADAPTIVE_THRESH_MEAN_C` and `CV_ADAPTIVE_THRESH_GAUSSIAN_C` it is a constant subtracted from mean or weighted mean (see the discussion), though it may be negative.

The function cvAdaptiveThreshold transforms grayscale image to binary image according to the formulae:

    threshold_type=CV_THRESH_BINARY:
    dst(x,y) = max_value, if src(x,y)>T(x,y)
               0, otherwise

    threshold_type=CV_THRESH_BINARY_INV:
    dst(x,y) = 0, if src(x,y)>T(x,y)
               max_value, otherwise

where TI is a threshold calculated individually for each pixel.

For the method `CV_ADAPTIVE_THRESH_MEAN_C` it is a mean of `block_size` × `block_size` pixel neighborhood, subtracted by `param1`.

For the method `CV_ADAPTIVE_THRESH_GAUSSIAN_C` it is a weighted sum (gaussian) of `block_size` × `block_size` pixel neighborhood, subtracted by `param1`.

## Pyramids and the Applications

### PyrDown

Downsamples image

    void cvPyrDown( const CvArr* src, CvArr* dst, int filter=CV_GAUSSIAN_5x5 );

src
  The source image.
dst
  The destination image, should have 2x smaller width and height than the source.
filter
  Type of the filter used for convolution; only `CV_GAUSSIAN_5x5` is currently supported.

The function cvPyrDown performs downsampling step of Gaussian pyramid decomposition. First it convolves source image with the specified filter and then downsamples the image by rejecting even rows and columns.

### PyrUp

Upsamples image

    void cvPyrUp( const CvArr* src, CvArr* dst, int filter=CV_GAUSSIAN_5x5 );

src
  The source image.
dst
  The destination image, should have 2x smaller width and height than the source.
filter
  Type of the filter used for convolution; only `CV_GAUSSIAN_5x5` is currently supported.

The function cvPyrUp performs up-sampling step of Gaussian pyramid decomposition. First it upsamples the source image by injecting even zero rows and columns and then convolves result with the specified filter multiplied by 4 for interpolation. So the destination image is four times larger than the source image.

### PyrSegmentation

Implements image segmentation by pyramids

    void cvPyrSegmentation( IplImage* src, IplImage* dst,
                            CvMemStorage* storage, CvSeq** comp,
                            int level, double threshold1, double threshold2 );

src
  The source image.
dst
  The destination image.
storage
  Storage; stores the resulting sequence of connected components.
comp
  Pointer to the output sequence of the segmented components.
level
  Maximum level of the pyramid for the segmentation.
threshold1
  Error threshold for establishing the links.
threshold2
  Error threshold for the segments clustering.

The function cvPyrSegmentation implements image segmentation by pyramids. The pyramid builds up to the level `level`. The links between any pixel `a` on level `i` and its candidate father pixel `b` on the adjacent level are established if

`p(c(a),c(b))<threshold1`. After the connected components are defined, they are joined into several clusters. Any two segments A and B belong to the same cluster, if
`p(c(A),c(B))<threshold2`. The input image has only one channel, then
`p(c¹,c²)=|c¹-c²|`. If the input image has three channels (red, green and blue), then
`p(c¹,c²)=0,3·(c¹r-c²r)+0,59·(c¹g-c²g)+0,11·(c¹b-c²b)` . There may be more than one connected component per a cluster.
The images `src` and `dst` should be 8-bit single-channel or 3-channel images or equal size

## Connected Components

### CvConnectedComp

Connected component

        typedef struct CvConnectedComp
        {
            double area; /* area of the segmented component */
            float value; /* gray scale value of the segmented component */
            CvRect rect; /* ROI of the segmented component */
        } CvConnectedComp;

### FloodFill

Fills a connected component with given color

    void cvFloodFill( CvArr* image, CvPoint seed_point, CvScalar new_val,
                      CvScalar lo_diff=cvScalarAll(0), CvScalar up_diff=cvScalarAll(0),
                      CvConnectedComp* comp=NULL, int flags=4, CvArr* mask=NULL );
    #define CV_FLOODFILL_FIXED_RANGE (1 << 16)
    #define CV_FLOODFILL_MASK_ONLY   (1 << 17)

image
  Input 1- or 3-channel, 8-bit or floating-point image. It is modified by the function unless CV_FLOODFILL_MASK_ONLY flag is set (see below).
seed_point
  The starting point.
new_val
  New value of repainted domain pixels.
lo_diff
  Maximal lower brightness/color difference between the currently observed pixel and one of its neighbor belong to the component or seed pixel to add the pixel to component. In case of 8-bit color images it is packed value.
up_diff
  Maximal upper brightness/color difference between the currently observed pixel and one of its neighbor belong to the component or seed pixel to add the pixel to component. In case of 8-bit color images it is packed value.
comp
  Pointer to structure the function fills with the information about the repainted domain.
flags
  The operation flags. Lower bits contain connectivity value, 4 (by default) or 8, used within the function. Connectivity determines which neighbors of a pixel are considered. Upper bits can be 0 or combination of the following flags:
  - CV_FLOODFILL_FIXED_RANGE - if set the difference between the current pixel and seed pixel is considered, otherwise difference between neighbor pixels is considered (the range is floating).
  - CV_FLOODFILL_MASK_ONLY - if set, the function does not fill the image (`new_val` is ignored), but the fills mask (that must be non-NULL in this case).
mask
  Operation mask, should be singe-channel 8-bit image, 2 pixels wider and 2 pixels taller than `image`. If not NULL, the function uses and updates the mask, so user takes responsibility of initializing `mask` content. Floodfilling can't go across non-zero pixels in the mask, for example, an edge detector output can be used as a mask to stop filling at edges. Or it is possible to use the same mask in multiple calls to the function to make sure the filled area do not overlap. Note: because mask is larger than the filled image, pixel in `mask` that corresponds to `(x,y)` pixel in `image` will have coordinates `(x+1,y+1)`.

The function cvFloodFill fills a connected component starting from the seed point with the specified color. The connectivity is determined by the closeness of pixel values. The pixel at `(x, y)` is considered to belong to the repainted domain if:

    src(x',y')-lo_diff<=src(x,y)<=src(x',y')+up_diff,     grayscale image, floating range
    src(seed.x,seed.y)-lo<=src(x,y)<=src(seed.x,seed.y)+up_diff, grayscale image, fixed range

    src(x',y')r-lo_diffr<=src(x,y)r<=src(x',y')r+up_diffr and
    src(x',y')g-lo_diffg<=src(x,y)g<=src(x',y')g+up_diffg and
    src(x',y')b-lo_diffb<=src(x,y)b<=src(x',y')b+up_diffb, color image, floating range

    src(seed.x,seed.y)r-lo_diffr<=src(x,y)r<=src(seed.x,seed.y)r+up_diffr and
    src(seed.x,seed.y)g-lo_diffg<=src(x,y)g<=src(seed.x,seed.y)g+up_diffg and
    src(seed.x,seed.y)b-lo_diffb<=src(x,y)b<=src(seed.x,seed.y)b+up_diffb, color image, fixed range

where `src(x',y')` is value of one of pixel neighbors. That is, to be added to the connected component, a pixel's color/brightness should be close enough to:

- color/brightness of one of its neighbors that are already referred to the connected component in case of floating range
- color/brightness of the seed point in case of fixed range.

### FindContours

Finds contours in binary image

    int cvFindContours( CvArr* image, CvMemStorage* storage, CvSeq** first_contour,
                        int header_size=sizeof(CvContour), int mode=CV_RETR_LIST,
                        int method=CV_CHAIN_APPROX_SIMPLE, CvPoint offset=cvPoint(0,0) );

image
  The source 8-bit single channel image. Non-zero pixels are treated as 1's, zero pixels remain 0's - that is image treated as `binary`. To get such a binary image from grayscale, one may use cvThreshold, cvAdaptiveThreshold or cvCanny. The function modifies the source image content.
storage
  Container of the retrieved contours.
first_contour
  Output parameter, will contain the pointer to the first outer contour.
header_size
  Size of the sequence header, >=sizeof(CvChain) if `method`=CV_CHAIN_CODE, and >=sizeof(CvContour) otherwise.
mode
  Retrieval mode.
  - `CV_RETR_EXTERNAL` - retrive only the extreme outer contours
  - `CV_RETR_LIST` - retrieve all the contours and puts them in the list
  - `CV_RETR_CCOMP` - retrieve all the contours and organizes them into two-level hierarchy: top level are external boundaries of the components, second level are bounda boundaries of the holes
  - `CV_RETR_TREE` - retrieve all the contours and reconstructs the full hierarchy of nested contours
method
  Approximation method (for all the modes, except `CV_RETR_RUNS`, which uses built-in approximation).
  - `CV_CHAIN_CODE` - output contours in the Freeman chain code. All other methods output polygons (sequences of vertices).
  - `CV_CHAIN_APPROX_NONE` - translate all the points from the chain code into points;
  - `CV_CHAIN_APPROX_SIMPLE` - compress horizontal, vertical, and diagonal segments, that is, the function leaves only their ending points;
  - `CV_CHAIN_APPROX_TC89_L1,
    CV_CHAIN_APPROX_TC89_KCOS
    ` - apply one of the flavors of Teh-Chin chain approximation algorithm.
  - `CV_LINK_RUNS` - use completely different contour retrieval algorithm via linking of horizontal segments of 1's. Only `CV_RETR_LIST` retrieval mode can be used with this method.
offset
  Offset, by which every contour point is shifted. This is useful if the contours are extracted from the image ROI and then they should be analyzed in the whole image context.

The function cvFindContours retrieves contours from the binary image and returns the number of retrieved contours. The pointer `first_contour` is filled by the function. It will contain pointer to the first most outer contour or NULL if no contours is detected (if the image is completely black). Other contours may be reached from `first_contour` using `h_next` and `v_next` links. The sample in cvDrawContours discussion shows how to use contours for connected component detection. Contours can be also used for shape analysis and object recognition - see `squares` sample in CVPR 2001 tutorial course located at SourceForge site.

### StartFindContours

Initializes contour scanning process

    CvContourScanner cvStartFindContours( CvArr* image, CvMemStorage* storage,
                                          int header_size=sizeof(CvContour),
                                          int mode=CV_RETR_LIST,
                                          int method=CV_CHAIN_APPROX_SIMPLE,
                                          CvPoint offset=cvPoint(0,0) );

image
  The source 8-bit single channel binary image.
storage
  Container of the retrieved contours.
header_size
  Size of the sequence header, >=sizeof(CvChain) if `method`=CV_CHAIN_CODE, and >=sizeof(CvContour) otherwise.
mode
  Retrieval mode; see cvFindContours.
method
  Approximation method. It has the same meaning as in cvFindContours, but CV_LINK_RUNS can not be used here.
offset
  ROI offset; see cvFindContours.

The function cvStartFindContours initializes and returns pointer to the contour scanner. The scanner is used further in cvFindNextContour to retrieve the rest of contours.

### FindNextContour

Finds next contour in the image

    CvSeq* cvFindNextContour( CvContourScanner scanner );

scanner
  Contour scanner initialized by the function cvStartFindContours .

The function cvFindNextContour locates and retrieves the next contour in the image and returns pointer to it. The function returns NULL, if there is no more contours.

### SubstituteContour

Replaces retrieved contour

    void cvSubstituteContour( CvContourScanner scanner, CvSeq* new_contour );

scanner
  Contour scanner initialized by the function cvStartFindContours .
new_contour
  Substituting contour.

The function cvSubstituteContour replaces the retrieved contour, that was returned from the preceding call of the function cvFindNextContour and stored inside the contour scanner state, with the user-specified contour. The contour is inserted into the resulting structure, list, two-level hierarchy, or tree, depending on the retrieval mode. If the parameter `new_contour`=NULL, the retrieved contour is not included into the resulting structure, nor all of its children that might be added to this structure later.

### EndFindContours

Finishes scanning process

    CvSeq* cvEndFindContours( CvContourScanner* scanner );

scanner
  Pointer to the contour scanner.

The function cvEndFindContours finishes the scanning process and returns the pointer to the first contour on the highest level.

## Image and Contour moments

### Moments

Calculates all moments up to third order of a polygon or rasterized shape

    void cvMoments( const CvArr* arr, CvMoments* moments, int binary=0 );

arr
  Image (1-channel or 3-channel with COI set) or polygon (CvSeq of points or a vector of points).
moments
  Pointer to returned moment state structure.
binary
  (For images only) If the flag is non-zero, all the zero pixel values are treated as zeroes, all the others are treated as 1's.

The function cvMoments calculates spatial and central moments up to the third order and writes them to `moments`. The moments may be used then to calculate gravity center of the shape, its area, main axises and various shape characeteristics including 7 Hu invariants.

### GetSpatialMoment

Retrieves spatial moment from moment state structure

    double cvGetSpatialMoment( CvMoments* moments, int x_order, int y_order );

moments
  The moment state, calculated by cvMoments.
x_order
  x order of the retrieved moment, `x_order` >= 0.
y_order
  y order of the retrieved moment, `y_order` >= 0 and `x_order` + `y_order` <= 3.

The function cvGetSpatialMoment retrieves the spatial moment, which in case of image moments is defined as:

    Mx_order,y_order=sumx,y(I(x,y)•xx_order•yy_order)

where `I(x,y)` is the intensity of the pixel `(x, y)`.

### GetCentralMoment

Retrieves central moment from moment state structure

    double cvGetCentralMoment( CvMoments* moments, int x_order, int y_order );

moments
  Pointer to the moment state structure.
x_order
  x order of the retrieved moment, `x_order` >= 0.
y_order
  y order of the retrieved moment, `y_order` >= 0 and `x_order` + `y_order` <= 3.

The function cvGetCentralMoment retrieves the central moment, which in case of image moments is defined as:

    μx_order,y_order=sumx,y(I(x,y)•(x-xc)x_order•(y-yc)y_order),

where `xc=M10/M00, yc=M01/M00` - coordinates of the gravity center

### GetNormalizedCentralMoment

Retrieves normalized central moment from moment state structure

    double cvGetNormalizedCentralMoment( CvMoments* moments, int x_order, int y_order );

moments
  Pointer to the moment state structure.
x_order
  x order of the retrieved moment, `x_order` >= 0.
y_order
  y order of the retrieved moment, `y_order` >= 0 and `x_order` + `y_order` <= 3.

The function cvGetNormalizedCentralMoment retrieves the normalized central moment:

    ηx_order,y_order= μx_order,y_order/M00((y_order+x_order)/2+1)

### GetHuMoments

Calculates seven Hu invariants

    void cvGetHuMoments( CvMoments* moments, CvHuMoments* hu_moments );

moments
  Pointer to the moment state structure.
hu_moments
  Pointer to Hu moments structure.

The function cvGetHuMoments calculates seven Hu invariants that are defined as:

     h1=η20+η02

     h2=(η20-η02)²+4η11²

     h3=(η30-3η12)²+ (3η21-η03)²

     h4=(η30+η12)²+ (η21+η03)²

     h5=(η30-3η12)(η30+η12)[(η30+η12)²-3(η21+η03)²]+(3η21-η03)(η21+η03)[3(η30+η12)²-(η21+η03)²]

     h6=(η20-η02)[(η30+η12)²- (η21+η03)²]+4η11(η30+η12)(η21+η03)

     h7=(3η21-η03)(η21+η03)[3(η30+η12)²-(η21+η03)²]-(η30-3η12)(η21+η03)[3(η30+η12)²-(η21+η03)²]

These values are proved to be invariants to the image scale, rotation, and reflection except the seventh one, whose sign is changed by reflection.

## Special Image Transforms

### HoughLines

Finds lines in binary image using Hough transform

    CvSeq* cvHoughLines2( CvArr* image, void* line_storage, int method,
                          double rho, double theta, int threshold,
                          double param1=0, double param2=0 );

image
  Source 8-bit single-channel (binary) image. It may be modified by the function.
line_storage
  The storage for the lines detected. It can be a memory storage (in this case a sequence of lines is created in the storage and returned by the function) or single row/single column matrix (CvMat*) of a particular type (see below) to which the lines' parameters are written. The matrix header is modified by the function so its `cols`/`rows` will contain a number of lines detected. If `line_storage` is a matrix and the actual number of lines exceeds the matrix size, the maximum possible number of lines is returned (the lines are not sorted by length, confidence or any other criteria).
method
  The Hough transform variant, one of:
  - `CV_HOUGH_STANDARD` - classical or standard Hough transform. Every line is represented by two floating-point numbers (ρ, θ), where ρ is a distance between (0,0) point and the line, and θ is the angle between x-axis and the normal to the line. Thus, the matrix must be (the created sequence will be) of CV_32FC2 type.
  - `CV_HOUGH_PROBABILISTIC` - probabilistic Hough transform (more efficient in case if picture contains a few long linear segments). It returns line segments rather than the whole lines. Every segment is represented by starting and ending points, and the matrix must be (the created sequence will be) of CV_32SC4 type.
  - `CV_HOUGH_MULTI_SCALE` - multi-scale variant of classical Hough transform. The lines are encoded the same way as in CV_HOUGH_STANDARD.
rho
  Distance resolution in pixel-related units.
theta
  Angle resolution measured in radians.
threshold
  Threshold parameter. A line is returned by the function if the corresponding accumulator value is greater than `threshold`.
param1
  The first method-dependent parameter:
  - For classical Hough transform it is not used (0).
  - For probabilistic Hough transform it is the minimum line length.
  - For multi-scale Hough transform it is divisor for distance resolution `rho`. (The coarse distance resolution will be `rho` and the accurate resolution will be (`rho` / `param1`)).
param2
  The second method-dependent parameter:
  - For classical Hough transform it is not used (0).
  - For probabilistic Hough transform it is the maximum gap between line segments lieing on the same line to treat them as the single line segment (i.e. to join them).
  - For multi-scale Hough transform it is divisor for angle resolution `theta`. (The coarse angle resolution will be `theta` and the accurate resolution will be (`theta` / `param2`)).

The function cvHoughLines2 implements a few variants of Hough transform for line detection.

#### Example. Detecting lines with Hough transform.

    /* This is a standalone program. Pass an image name as a first parameter of the program.
       Switch between standard and probabilistic Hough transform by changing "#if 1" to "#if 0" and back */
    #include <cv.h>
    #include <highgui.h>
    #include <math.h>

    int main(int argc, char** argv)
    {
        IplImage* src;
        if( argc == 2 && (src=cvLoadImage(argv[1], 0))!= 0)
        {
            IplImage* dst = cvCreateImage( cvGetSize(src), 8, 1 );
            IplImage* color_dst = cvCreateImage( cvGetSize(src), 8, 3 );
            CvMemStorage* storage = cvCreateMemStorage(0);
            CvSeq* lines = 0;
            int i;
            cvCanny( src, dst, 50, 200, 3 );
            cvCvtColor( dst, color_dst, CV_GRAY2BGR );
    #if 1
            lines = cvHoughLines2( dst, storage, CV_HOUGH_STANDARD, 1, CV_PI/180, 150, 0, 0 );

            for( i = 0; i < lines->total; i++ )
            {
                float* line = (float*)cvGetSeqElem(lines,i);
                float rho = line[0];
                float theta = line[1];
                CvPoint pt1, pt2;
                double a = cos(theta), b = sin(theta);
                if( fabs(a) < 0.001 )
                {
                    pt1.x = pt2.x = cvRound(rho);
                    pt1.y = 0;
                    pt2.y = color_dst->height;
                }
                else if( fabs(b) < 0.001 )
                {
                    pt1.y = pt2.y = cvRound(rho);
                    pt1.x = 0;
                    pt2.x = color_dst->width;
                }
                else
                {
                    pt1.x = 0;
                    pt1.y = cvRound(rho/b);
                    pt2.x = cvRound(rho/a);
                    pt2.y = 0;
                }
                cvLine( color_dst, pt1, pt2, CV_RGB(255,0,0), 3, 8 );
            }
    #else
            lines = cvHoughLines2( dst, storage, CV_HOUGH_PROBABILISTIC, 1, CV_PI/180, 80, 30, 10 );
            for( i = 0; i < lines->total; i++ )
            {
                CvPoint* line = (CvPoint*)cvGetSeqElem(lines,i);
                cvLine( color_dst, line[0], line[1], CV_RGB(255,0,0), 3, 8 );
            }
    #endif
            cvNamedWindow( "Source", 1 );
            cvShowImage( "Source", src );

            cvNamedWindow( "Hough", 1 );
            cvShowImage( "Hough", color_dst );

            cvWaitKey(0);
        }
    }

This is the sample picture the function parameters have been tuned for:

And this is the output of the above program in case of probabilistic Hough transform ("#if 0" case):

### DistTransform

Calculates distance to closest zero pixel for all non-zero pixels of source image

    void cvDistTransform( const CvArr* src, CvArr* dst, int distance_type=CV_DIST_L2,
                          int mask_size=3, const float* mask=NULL );

src
  Source 8-bit single-channel (binary) image.
dst
  Output image with calculated distances (32-bit floating-point, single-channel).
distance_type
  Type of distance; can be `CV_DIST_L1, CV_DIST_L2, CV_DIST_C` or `CV_DIST_USER`.
mask_size
  Size of distance transform mask; can be 3 or 5. In case of `CV_DIST_L1` or `CV_DIST_C` the parameter is forced to 3, because 3×3 mask gives the same result as 5×5 yet it is faster.
mask
  User-defined mask in case of user-defined distance, it consists of 2 numbers (horizontal/vertical shift cost, diagonal shift cost) in case of 3×3 mask and 3 numbers (horizontal/vertical shift cost, diagonal shift cost, knight's move cost) in case of 5×5 mask.

The function cvDistTransform calculates the approximated distance from every binary image pixel to the nearest zero pixel. For zero pixels the function sets the zero distance, for others it finds the shortest path consisting of basic shifts: horizontal, vertical, diagonal or knight's move (the latest is available for 5×5 mask). The overal distance is calculated as a sum of these basic distances. Because the distance function should be symmetric, all the horizontal and vertical shifts must have the same cost (that is denoted as `a`), all the diagonal shifts must have the same cost (denoted `b`), and all knight's moves' must have the same cost (denoted `c`). For `CV_DIST_C` and `CV_DIST_L1` types the distance is calculated precisely, whereas for `CV_DIST_L2` (Euclidian distance) the distance can be calculated only with some relative error (5×5 mask gives more accurate results), OpenCV uses the values suggested in [Borgefors86]:

    CV_DIST_C (3×3):
    a=1, b=1

    CV_DIST_L1 (3×3):
    a=1, b=2

    CV_DIST_L2 (3×3):
    a=0.955, b=1.3693

    CV_DIST_L2 (5×5):
    a=1, b=1.4, c=2.1969

And below are samples of distance field (black (0) pixel is in the middle of white square) in case of user-defined distance:

#### User-defined 3×3 mask (a=1, b=1.5)

4.5 | 4 | 3.5 | 3 | 3.5 | 4 | 4.5
4 | 3 | 2.5 | 2 | 2.5 | 3 | 4
3.5 | 2.5 | 1.5 | 1 | 1.5 | 2.5 | 3.5
3 | 2 | 1 | 0 | 1 | 2 | 3
3.5 | 2.5 | 1.5 | 1 | 1.5 | 2.5 | 3.5
4 | 3 | 2.5 | 2 | 2.5 | 3 | 4
4.5 | 4 | 3.5 | 3 | 3.5 | 4 | 4.5

#### User-defined 5×5 mask (a=1, b=1.5, c=2)

4.5 | 3.5 | 3 | 3 | 3 | 3.5 | 4.5
3.5 | 3 | 2 | 2 | 2 | 3 | 3.5
3 | 2 | 1.5 | 1 | 1.5 | 2 | 3
3 | 2 | 1 | 0 | 1 | 2 | 3
3 | 2 | 1.5 | 1 | 1.5 | 2 | 3
3.5 | 3 | 2 | 2 | 2 | 3 | 3.5
4 | 3.5 | 3 | 3 | 3 | 3.5 | 4

Typically, for fast coarse distance estimation CV_DIST_L2, 3×3 mask is used, and for more accurate distance estimation CV_DIST_L2, 5×5 mask is used.

## Histograms

### CvHistogram

Muti-dimensional histogram

        typedef struct CvHistogram
        {
            int header_size; /* header's size */
            CvHistType type; /* type of histogram */
            int flags; /* histogram's flags */
            int c_dims; /* histogram's dimension */
            int dims[CV_HIST_MAX_DIM]; /* every dimension size */
            int mdims[CV_HIST_MAX_DIM]; /* coefficients for fast access to element */
            /* &m[a,b,c] = m + a*mdims[0] + b*mdims[1] + c*mdims[2] */
            float* thresh[CV_HIST_MAX_DIM]; /* bin boundaries arrays for every dimension */
            float* array; /* all the histogram data, expanded into the single row */
        struct CvNode* root; /* root of balanced tree storing histogram bins */
            CvSet* set; /* pointer to memory storage (for the balanced tree) */
            int* chdims[CV_HIST_MAX_DIM]; /* cache data for fast calculating */
        } CvHistogram;

### CreateHist

Creates histogram

    CvHistogram* cvCreateHist( int dims, int* sizes, int type,
                               float** ranges=NULL, int uniform=1 );

dims
  Number of histogram dimensions.
sizes
  Array of histogram dimension sizes.
type
  Histogram representation format: `CV_HIST_ARRAY` means that histogram data is represented as an multi-dimensional dense array CvMatND; `CV_HIST_TREE` means that histogram data is represented as a multi-dimensional sparse array CvSparseMat.
ranges
  Array of ranges for histogram bins. Its meaning depends on the `uniform` parameter value. The ranges are used for when histogram is calculated or backprojected to determine, which histogram bin corresponds to which value/tuple of values from the input image[s].
uniform
  Uniformity flag; if not 0, the histogram has evenly spaced bins and for every `0<=i<cDims` `ranges[i]` is array of two numbers: lower and upper boundaries for the i-th histogram dimension. The whole range [lower,upper] is split then into `dims[i]` equal parts to determine `i-th` input tuple value ranges for every histogram bin. And if `uniform=0`, then `i-th` element of `ranges` array contains `dims[i]+1` elements: `lower0, upper0, lower1, upper1 == lower2, ..., upperdims[i]-1`, where `lowerj` and `upperj` are lower and upper boundaries of `i-th` input tuple value for `j-th` bin, respectively. In either case, the input values that are beyond the specified range for a histogram bin, are not counted by cvCalcHist and filled with 0 by cvCalcBackProject.

The function cvCreateHist creates a histogram of the specified size and returns the pointer to the created histogram. If the array `ranges` is 0, the histogram bin ranges must be specified later via the function cvSetHistBinRanges, though cvCalcHist and cvCalcBackProject may process 8-bit images without setting bin ranges, they assume equally spaced in 0..255 bins.

### SetHistBinRanges

Sets bounds of histogram bins

    void cvSetHistBinRanges( CvHistogram* hist, float** ranges, int uniform=1 );

hist
  Histogram.
ranges
  Array of bin ranges arrays, see cvCreateHist.
uniform
  Uniformity flag, see cvCreateHist.

The function cvSetHistBinRanges is a stand-alone function for setting bin ranges in the histogram. For more detailed description of the parameters `ranges` and `uniform` see cvCalcHist function, that can initialize the ranges as well. Ranges for histogram bins must be set before the histogram is calculated or backproject of the histogram is calculated.

### ReleaseHist

Releases histogram

    void cvReleaseHist( CvHistogram** hist );

hist
  Double pointer to the released histogram.

The function cvReleaseHist releases the histogram (header and the data). The pointer to histogram is cleared by the function. If `*hist` pointer is already `NULL`, the function does nothing.

### ClearHist

Clears histogram

    void cvClearHist( CvHistogram* hist );

hist
  Histogram.

The function cvClearHist sets all histogram bins to 0 in case of dense histogram and removes all histogram bins in case of sparse array.

### MakeHistHeaderForArray

Makes a histogram out of array

    CvHistogram*  cvMakeHistHeaderForArray( int dims, int* sizes, CvHistogram* hist,
                                            float* data, float** ranges=NULL, int uniform=1 );

dims
  Number of histogram dimensions.
sizes
  Array of histogram dimension sizes.
hist
  The histogram header initialized by the function.
data
  Array that will be used to store histogram bins.
ranges
  Histogram bin ranges, see cvCreateHist.
uniform
  Uniformity flag, see cvCreateHist.

The function cvMakeHistHeaderForArray initializes the histogram, which header and bins are allocated by user. No cvReleaseHist need to be called afterwards. Only dense histograms can be initialized this way. The function returns `hist`.

### QueryHistValue_1D

Queries value of histogram bin

    #define cvQueryHistValue_1D( hist, idx0 ) \
        cvGetReal1D( (hist)->bins, (idx0) )
    #define cvQueryHistValue_2D( hist, idx0, idx1 ) \
        cvGetReal2D( (hist)->bins, (idx0), (idx1) )
    #define cvQueryHistValue_3D( hist, idx0, idx1, idx2 ) \
        cvGetReal3D( (hist)->bins, (idx0), (idx1), (idx2) )
    #define cvQueryHistValue_nD( hist, idx ) \
        cvGetRealND( (hist)->bins, (idx) )

hist
  Histogram.
idx0, idx1, idx2, idx3
  Indices of the bin.
idx
  Array of indices

The macros cvQueryHistValue_*D return the value of the specified bin of 1D, 2D, 3D or N-D histogram. In case of sparse histogram the function returns 0, if the bin is not present in the histogram, and no new bin is created.

### GetHistValue_1D

Returns pointer to histogram bin

    #define cvGetHistValue_1D( hist, idx0 ) \
        ((float*)(cvPtr1D( (hist)->bins, (idx0), 0 ))
    #define cvGetHistValue_2D( hist, idx0, idx1 ) \
        ((float*)(cvPtr2D( (hist)->bins, (idx0), (idx1), 0 ))
    #define cvGetHistValue_3D( hist, idx0, idx1, idx2 ) \
        ((float*)(cvPtr3D( (hist)->bins, (idx0), (idx1), (idx2), 0 ))
    #define cvGetHistValue_nD( hist, idx ) \
        ((float*)(cvPtrND( (hist)->bins, (idx), 0 ))

hist
  Histogram.
idx0, idx1, idx2, idx3
  Indices of the bin.
idx
  Array of indices

The macros cvGetHistValue_*D return pointer to the specified bin of 1D, 2D, 3D or N-D histogram. In case of sparse histogram the function creates a new bin and sets it to 0, unless it exists already.

### GetMinMaxHistValue

Finds minimum and maximum histogram bins

    void cvGetMinMaxHistValue( const CvHistogram* hist,
                               float* min_value, float* max_value,
                               int* min_idx=NULL, int* max_idx=NULL );

hist
  Histogram.
min_value
  Pointer to the minimum value of the histogram
max_value
  Pointer to the maximum value of the histogram
min_idx
  Pointer to the array of coordinates for minimum
max_idx
  Pointer to the array of coordinates for maximum

The function cvGetMinMaxHistValue finds the minimum and maximum histogram bins and their positions. Any of output arguments is optional. Among several extremums with the same value the ones with minimum index (in lexicographical order) In case of several maximums or minimums the earliest in lexicographical order extrema locations are returned.

### NormalizeHist

Normalizes histogram

    void cvNormalizeHist( CvHistogram* hist, double factor );

hist
  Pointer to the histogram.
factor
  Normalization factor.

The function cvNormalizeHist normalizes the histogram bins by scaling them, such that the sum of the bins becomes equal to `factor`.

### ThreshHist

Thresholds histogram

    void cvThreshHist( CvHistogram* hist, double threshold );

hist
  Pointer to the histogram.
threshold
  Threshold level.

The function cvThreshHist clears histogram bins that are below the specified threshold.

### CompareHist

Compares two dense histograms

    double cvCompareHist( const CvHistogram* hist1, const CvHistogram* hist2, int method );

hist1
  The first dense histogram.
hist2
  The second dense histogram.
method
  Comparison method, one of:
  - CV_COMP_CORREL
  - CV_COMP_CHISQR
  - CV_COMP_INTERSECT

The function cvCompareHist compares two dense histograms using the specified method as following (`H1` denotes the first histogram, `H2` - the second):

    Correlation (method=CV_COMP_CORREL):
    d(H1,H2)=sumI(H'1(I)•H'2(I))/sqrt(sumI[H'1(I)2]•sumI[H'2(I)2])
    where
    H'k(I)=Hk(I)-1/N•sumJHk(J) (N=number of histogram bins)

    Chi-Square (method=CV_COMP_CHISQR):
    d(H1,H2)=sumI[(H1(I)-H2(I))/(H1(I)+H2(I))]

    Intersection (method=CV_COMP_INTERSECT):
    d(H1,H2)=sumImax(H1(I),H2(I))

The function returns `d(H1,H2)` value.

To compare sparse histogram or more general sparse configurations of weighted points, consider using cvCalcEMD function.

### CopyHist

Copies histogram

    void cvCopyHist( const CvHistogram* src, CvHistogram** dst );

src
  Source histogram.
dst
  Pointer to destination histogram.

The function cvCopyHist makes a copy of the histogram. If the second histogram pointer `*dst` is NULL, a new histogram of the same size as `src` is created. Otherwise, both histograms must have equal types and sizes. Then the function copies the source histogram bins values to destination histogram and sets the same as `src`'s value ranges.

### CalcHist

Calculates histogram of image(s)

    void cvCalcHist( IplImage** image, CvHistogram* hist,
                     int accumulate=0, const CvArr* mask=NULL );

image
  Source images (though, you may pass CvMat** as well).
hist
  Pointer to the histogram.
accumulate
  Accumulation flag. If it is set, the histogram is not cleared in the beginning. This feature allows user to compute a single histogram from several images, or to update the histogram online.
mask
  The operation mask, determines what pixels of the source images are counted.

The function cvCalcHist calculates the histogram of one or more single-channel images. The elements of a tuple that is used to increment a histogram bin are taken at the same location from the corresponding input images.

#### Sample. Calculating and displaying 2D Hue-Saturation histogram of a color image

    #include <cv.h>
    #include <highgui.h>

    int main( int argc, char** argv )
    {
        IplImage* src;
        if( argc == 2 && (src=cvLoadImage(argv[1], 1))!= 0)
        {
            IplImage* h_plane = cvCreateImage( cvGetSize(src), 8, 1 );
            IplImage* s_plane = cvCreateImage( cvGetSize(src), 8, 1 );
            IplImage* v_plane = cvCreateImage( cvGetSize(src), 8, 1 );
            IplImage* planes[] = { h_plane, s_plane };
            IplImage* hsv = cvCreateImage( cvGetSize(src), 8, 3 );
            int h_bins = 30, s_bins = 32;
            int hist_size[] = {h_bins, s_bins};
            float h_ranges[] = { 0, 180 }; /* hue varies from 0 (~0°red) to 180 (~360°red again) */
            float s_ranges[] = { 0, 255 }; /* saturation varies from 0 (black-gray-white) to 255 (pure spectrum color) */
            float* ranges[] = { h_ranges, s_ranges };
            int scale = 10;
            IplImage* hist_img = cvCreateImage( cvSize(h_bins*scale,s_bins*scale), 8, 3 );
            CvHistogram* hist;
            float max_value = 0;
            int h, s;

            cvCvtColor( src, hsv, CV_BGR2HSV );
            cvCvtPixToPlane( hsv, h_plane, s_plane, v_plane, 0 );
            hist = cvCreateHist( 2, hist_size, CV_HIST_ARRAY, ranges, 1 );
            cvCalcHist( planes, hist, 0, 0 );
            cvGetMinMaxHistValue( hist, 0, &max_value, 0, 0 );
            cvZero( hist_img );

            for( h = 0; h < h_bins; h++ )
            {
                for( s = 0; s < s_bins; s++ )
                {
                    float bin_val = cvQueryHistValue_2D( hist, h, s );
                    int intensity = cvRound(bin_val*255/max_value);
                    cvRectangle( hist_img, cvPoint( h*scale, s*scale ),
                                 cvPoint( (h+1)*scale - 1, (s+1)*scale - 1),
                                 CV_RGB(intensity,intensity,intensity), /* graw a grayscale histogram.
                                                                           if you have idea how to do it
                                                                           nicer let us know */
                                 CV_FILLED );
                }
            }

            cvNamedWindow( "Source", 1 );
            cvShowImage( "Source", src );

            cvNamedWindow( "H-S Histogram", 1 );
            cvShowImage( "H-S Histogram", hist_img );

            cvWaitKey(0);
        }
    }

### CalcBackProject

Calculates back projection

    void cvCalcBackProject( IplImage** image, CvArr* back_project, const CvHistogram* hist );

image
  Source images (though you may pass CvMat** as well).
back_project
  Destination back projection image of the same type as the source images.
hist
  Histogram.

The function cvCalcBackProject calculates the back project of the histogram. For each tuple of pixels at the same position of all input single-channel images the function puts the value of the histogram bin, corresponding to the tuple, to the destination image. In terms of statistics, the value of each output image pixel is probability of the observed tuple given the distribution (histogram). For example, to find a red object in the picture, one may do the following:

1. Calculate a hue histogram for the red object assuming the image contains only this object. The histogram is likely to have a strong maximum, corresponding to red color.
2. Calculate back projection of a hue plane of input image where the object is searched, using the histogram. Threshold the image.
3. Find connected components in the resulting picture and choose the right component using some additional criteria, for example, the largest connected component.

That is the approximate algorithm of Camshift color object tracker, except for the 3rd step, instead of which CAMSHIFT algorithm is used to locate the object on the back projection given the previous object position.

### CalcBackProjectPatch

Locates a template within image by histogram comparison

    void cvCalcBackProjectPatch( IplImage** image, CvArr* dst,
                                 CvSize patch_size, CvHistogram* hist,
                                 int method, float factor );

image
  Source images (though, you may pass CvMat** as well)
dst
  Destination image.
patch_size
  Size of patch slid though the source image.
hist
  Histogram
method
  Compasion method, passed to cvCompareHist (see description of that function).
factor
  Normalization factor for histograms, will affect normalization scale of destination image, pass 1. if unsure.

The function cvCalcBackProjectPatch calculates back projection by comparing histograms of the source image patches with the given histogram. Taking measurement results from some image at each location over ROI creates an array `image`. These results might be one or more of hue, `x` derivative, `y` derivative, Laplacian filter, oriented Gabor filter, etc. Each measurement output is collected into its own separate image. The `image` image array is a collection of these measurement images. A multi-dimensional histogram `hist` is constructed by sampling from the `image` image array. The final histogram is normalized. The `hist` histogram has as many dimensions as the number of elements in `image` array.

Each new image is measured and then converted into an `image` image array over a chosen ROI. Histograms are taken from this `image` image in an area covered by a "patch" with anchor at center as shown in the picture below. The histogram is normalized using the parameter `norm_factor` so that it may be compared with `hist`. The calculated histogram is compared to the model histogram; `hist` uses the function cvCompareHist with the comparison method=`method`). The resulting output is placed at the location corresponding to the patch anchor in the probability image `dst`. This process is repeated as the patch is slid over the ROI. Iterative histogram update by subtracting trailing pixels covered by the patch and adding newly covered pixels to the histogram can save a lot of operations, though it is not implemented yet.

#### Back Project Calculation by Patches

### CalcProbDensity

Divides one histogram by another

    void  cvCalcProbDensity( const CvHistogram* hist1, const CvHistogram* hist2,
                             CvHistogram* dst_hist, double scale=255 );

hist1
  first histogram (the divisor).
hist2
  second histogram.
dst_hist
  destination histogram.
scale
  scale factor for the destination histogram.

The function cvCalcProbDensity calculates the object probability density from the two histograms as:

    dist_hist(I)=0      if hist1(I)==0
                 scale  if hist1(I)!=0 && hist2(I)>hist1(I)
                 hist2(I)*scale/hist1(I) if hist1(I)!=0 && hist2(I)<=hist1(I)

So the destination histogram bins are within less than scale.

## Matching

### MatchTemplate

Compares template against overlapped image regions

    void cvMatchTemplate( const CvArr* image, const CvArr* templ,
                          CvArr* result, int method );

image
  Image where the search is running. It should be single-chanel, 8-bit or 32-bit floating-point.
templ
  Searched template; must be not greater than the source image and the same data type as the image.
result
  A map of comparison results; single-channel 32-bit floating-point. If `image` is `W`×`H` and `templ` is `w`×`h` then `result` must be `W-w+1`×`H-h+1`.
method
  Specifies the way the template must be compared with image regions (see below).

The function cvMatchTemplate is similiar to cvCalcBackProjectPatch. It slids through `image`, compares overlapped patches of size `w`×`h` against `templ` using the specified method and stores the comparison results to `result`. Here are the formulae for the different comparison methods one may use (`I` denotes image, `T` - template, `R` - result. The summation is done over template and/or the image patch: `x'=0..w-1, y'=0..h-1`):

    method=CV_TM_SQDIFF:
    R(x,y)=sumx',y'[T(x',y')-I(x+x',y+y')]2

    method=CV_TM_SQDIFF_NORMED:
    R(x,y)=sumx',y'[T(x',y')-I(x+x',y+y')]2/sqrt[sumx',y'T(x',y')2•sumx',y'I(x+x',y+y')2]

    method=CV_TM_CCORR:
    R(x,y)=sumx',y'[T(x',y')•I(x+x',y+y')]

    method=CV_TM_CCORR_NORMED:
    R(x,y)=sumx',y'[T(x',y')•I(x+x',y+y')]/sqrt[sumx',y'T(x',y')2•sumx',y'I(x+x',y+y')2]

    method=CV_TM_CCOEFF:
    R(x,y)=sumx',y'[T'(x',y')•I'(x+x',y+y')],

    where T'(x',y')=T(x',y') - 1/(w•h)•sumx",y"T(x",y") (mean template brightness=>0)
          I'(x+x',y+y')=I(x+x',y+y') - 1/(w•h)•sumx",y"I(x+x",y+y") (mean patch brightness=>0)

    method=CV_TM_CCOEFF_NORMED:
    R(x,y)=sumx',y'[T'(x',y')•I'(x+x',y+y')]/sqrt[sumx',y'T'(x',y')2•sumx',y'I'(x+x',y+y')2]

After the function finishes comparison, the best matches can be found as global minimums (CV_TM_SQDIFF*) or maximums (CV_TM_CCORR* and CV_TM_CCOEFF*) using cvMinMaxLoc function.

### MatchShapes

Compares two shapes

    double cvMatchShapes( const void* object1, const void* object2,
                          int method, double parameter=0 );

object1
  First contour or grayscale image
object2
  Second contour or grayscale image
method
  Comparison method, one of CV_CONTOUR_MATCH_I1, CV_CONTOURS_MATCH_I2 or CV_CONTOURS_MATCH_I3.
parameter
  Method-specific parameter (is not used now).

The function cvMatchShapes compares two shapes. The 3 implemented methods all use Hu moments (see cvGetHuMoments) (`A` ~ `object1`, `B` - `object2`):

    method=CV_CONTOUR_MATCH_I1:
    I1(A,B)=sumi=1..7abs(1/mAi - 1/mBi)

    method=CV_CONTOUR_MATCH_I2:
    I2(A,B)=sumi=1..7abs(mAi - mBi)

    method=CV_CONTOUR_MATCH_I3:
    I3(A,B)=sumi=1..7abs(mAi - mBi)/abs(mAi)

    where
    mAi=sign(hAi)•log(hAi),
    mBi=sign(hBi)•log(hBi),
    hAi, hBi - Hu moments of A and B, respectively.

### CalcEMD2

Computes "minimal work" distance between two weighted point configurations

    float cvCalcEMD2( const CvArr* signature1, const CvArr* signature2, int distance_type,
                      CvDistanceFunction distance_func=NULL, const CvArr* cost_matrix=NULL,
                      CvArr* flow=NULL, float* lower_bound=NULL, void* userdata=NULL );
    typedef float (*CvDistanceFunction)(const float* f1, const float* f2, void* userdata);

signature1
  First signature, `size1`×`dims+1` floating-point matrix. Each row stores the point weight followed by the point coordinates. The matrix is allowed to have a single column (weights only) if the user-defined cost matrix is used.
signature2
  Second signature of the same format as `signature1`, though the number of rows may be different. The total weights may be different, in this case an extra "dummy" point is added to either `signature1` or `signature2`.
distance_type
  Metrics used; `CV_DIST_L1, CV_DIST_L2`, and `CV_DIST_C` stand for one of the standard metrics; `CV_DIST_USER` means that a user-defined function `distance_func` or pre-calculated `cost_matrix` is used.
distance_func
  The user-defined distance function. It takes coordinates of two points and returns the distance between the points.
cost_matrix
  The user-defined `size1`×`size2` cost matrix. At least one of `cost_matrix` and `distance_func` must be NULL. Also, if a cost matrix is used, lower boundary (see below) can not be calculated, because it needs a metric function.
flow
  The resultant `size1`×`size2` flow matrix: `flowij` is a flow from i-th point of `signature1` to j-th point of `signature2`
lower_bound
  Optional input/output parameter: lower boundary of distance between the two signatures that is a distance between mass centers. The lower boundary may not be calculated if the user-defined cost matrix is used, the total weights of point configurations are not equal, or there is the signatures consist of weights only (i.e. the signature matrices have a single column). User must initialize `*lower_bound`. If the calculated distance between mass centers is greater or equal to `*lower_bound` (it means that the signatures are far enough) the function does not calculate EMD. In any case `*lower_bound` is set to the calculated distance between mass centers on return. Thus, if user wants to calculate both distance between mass centers and EMD, `*lower_bound` should be set to 0.
userdata
  Pointer to optional data that is passed into the user-defined distance function.

The function cvCalcEMD2 computes earth mover distance and/or a lower boundary of the distance between the two weighted point configurations. One of the application desctibed in [RubnerSept98] is multi-dimensional histogram comparison for image retrieval. EMD is a transportation problem that is solved using some modification of simplex algorithm, thus the complexity is exponential in the worst case, though, it is much faster in average. In case of a real metric the lower boundary can be calculated even faster (using linear-time algorithm) and it can be used to determine roughly whether the two signatures are far enough so that they cannot relate to the same object.

# Structural Analysis

## Contour Processing Functions

### ApproxChains

Approximates Freeman chain(s) with polygonal curve

    CvSeq* cvApproxChains( CvSeq* src_seq, CvMemStorage* storage,
                           int method=CV_CHAIN_APPROX_SIMPLE,
                           double parameter=0, int minimal_perimeter=0, int recursive=0 );

src_seq
  Pointer to the chain that can refer to other chains.
storage
  Storage location for the resulting polylines.
method
  Approximation method (see the description of the function cvFindContours).
parameter
  Method parameter (not used now).
minimal_perimeter
  Approximates only those contours whose perimeters are not less than `minimal_perimeter`. Other chains are removed from the resulting structure.
recursive
  If not 0, the function approximates all chains that access can be obtained to from `src_seq` by `h_next` or `v_next links`. If 0, the single chain is approximated.

This is a stand-alone approximation routine. The function cvApproxChains works exactly in the same way as cvFindContours with the corresponding approximation flag. The function returns pointer to the first resultant contour. Other approximated contours, if any, can be accessed via `v_next` or `h_next` fields of the returned structure.

### StartReadChainPoints

Initializes chain reader

    void cvStartReadChainPoints( CvChain* chain, CvChainPtReader* reader );

chain Pointer to chain. reader Chain reader state.

The function cvStartReadChainPoints initializes a special reader (see Dynamic Data Structures for more information on sets and sequences).

### ReadChainPoint

Gets next chain point

    CvPoint cvReadChainPoint( CvChainPtReader* reader );

reader
  Chain reader state.

The function cvReadChainPoint returns the current chain point and updates the reader position.

### ApproxPoly

Approximates polygonal curve(s) with desired precision

    CvSeq* cvApproxPoly( const void* src_seq, int header_size, CvMemStorage* storage,
                         int method, double parameter, int parameter2=0 );

src_seq
  Sequence of array of points.
header_size
  Header size of approximated curve[s].
storage
  Container for approximated contours. If it is NULL, the input sequences' storage is used.
method
  Approximation method; only `CV_POLY_APPROX_DP` is supported, that corresponds to Douglas-Peucker algorithm.
parameter
  Method-specific parameter; in case of `CV_POLY_APPROX_DP` it is a desired approximation accuracy.
parameter2
  If case if `src_seq` is sequence it means whether the single sequence should be approximated or all sequences on the same level or below `src_seq` (see cvFindContours for description of hierarchical contour structures). And if `src_seq` is array (CvMat*) of points, the parameter specifies whether the curve is closed (`parameter2`!=0) or not (`parameter2`=0).

The function cvApproxPoly approximates one or more curves and returns the approximation result[s]. In case of multiple curves approximation the resultant tree will have the same structure as the input one (1:1 correspondence).

### BoundingRect

Calculates up-right bounding rectangle of point set

    CvRect cvBoundingRect( CvArr* points, int update=0 );

points
  2D point set, either a sequence or vector (`CvMat`) of points.
update
  The update flag. Here is list of possible combination of the flag values and type of `contour`:
  - update=0, contour ~ CvContour*: the bounding rectangle is not calculated, but it is taken from `rect` field of the contour header.
  - update=1, contour ~ CvContour*: the bounding rectangle is calculated and written to `rect` field of the contour header.
  - update=0, contour ~ CvSeq* or CvMat*: the bounding rectangle is calculated and returned.
  - update=1, contour ~ CvSeq* or CvMat*: runtime error is raised.

The function cvBoundingRect returns the up-right bounding rectangle for 2d point set.

### ContourArea

Calculates area of the whole contour or contour section

    double cvContourArea( const CvArr* contour, CvSlice slice=CV_WHOLE_SEQ );

contour
  Contour (sequence or array of vertices).
slice
  Starting and ending points of the contour section of interest, by default area of the whole contour is calculated.

The function cvContourArea calculates area of the whole contour or contour section. In the latter case the total area bounded by the contour arc and the chord connecting the 2 selected points is calculated as shown on the picture below:

NOTE: Orientation of the contour affects the area sign, thus the function may return `negative` result. Use `fabs()` function from C runtime to get the absolute value of area.

### ArcLength

Calculates contour perimeter or curve length

    double cvArcLength( const void* curve, CvSlice slice=CV_WHOLE_SEQ, int is_closed=-1 );

curve
  Sequence or array of the curve points.
slice
  Starting and ending points of the curve, by default the whole curve length is calculated.
is_closed
  Indicates whether the curve is closed or not. There are 3 cases:
  - is_closed=0 - the curve is assumed to be unclosed.
  - is_closed>0 - the curve is assumed to be closed.
  - is_closed<0 - if curve is sequence, the flag CV_SEQ_FLAG_CLOSED of ((CvSeq*)curve)->flags is checked to determine if the curve is closed or not, otherwise (curve is represented by array (CvMat*) of points) it is assumed to be unclosed.

The function cvArcLength calculates length or curve as sum of lengths of segments between subsequent points

### CreateContourTree

Creates hierarchical representation of contour

    CvContourTree* cvCreateContourTree( const CvSeq* contour, CvMemStorage* storage, double threshold );

contour
  Input contour.
storage
  Container for output tree.
threshold
  Approximation accuracy.

The function cvCreateContourTree creates binary tree representation for the input `contour` and returns the pointer to its root. If the parameter `threshold` is less than or equal to 0, the function creates full binary tree representation. If the threshold is greater than 0, the function creates representation with the precision `threshold`: if the vertices with the interceptive area of its base line are less than `threshold`, the tree should not be built any further. The function returns the created tree.

### ContourFromContourTree

Restores contour from tree

    CvSeq* cvContourFromContourTree( const CvContourTree* tree, CvMemStorage* storage,
                                     CvTermCriteria criteria );

tree
  Contour tree.
storage
  Container for the reconstructed contour.
criteria
  Criteria, where to stop reconstruction.

The function cvContourFromContourTree restores the contour from its binary tree representation. The parameter `criteria` determines the accuracy and/or the number of tree levels used for reconstruction, so it is possible to build approximated contour. The function returns reconstructed contour.

### MatchContourTrees

Compares two contours using their tree representations

    double cvMatchContourTrees( const CvContourTree* tree1, const CvContourTree* tree2,
                                int method, double threshold );

tree1
  First contour tree.
tree2
  Second contour tree.
method
  Similarity measure, only `CV_CONTOUR_TREES_MATCH_I1` is supported.
threshold
  Similarity threshold.

The function cvMatchContourTrees calculates the value of the matching measure for two contour trees. The similarity measure is calculated level by level from the binary tree roots. If at the certain level difference between contours becomes less than `threshold`, the reconstruction process is interrupted and the current difference is returned.

## Computational Geometry

### MaxRect

Finds bounding rectangle for two given rectangles

    CvRect cvMaxRect( const CvRect* rect1, const CvRect* rect2 );

rect1
  First rectangle
rect2
  Second rectangle

The function cvMaxRect finds minimum area rectangle that contains both input rectangles inside:

### CvBox2D

Rotated 2D box

    typedef struct CvBox2D
    {
        CvPoint2D32f center;  /* center of the box */
        CvSize2D32f  size;    /* box width and length */
        float angle;          /* angle between the horizontal axis
                                 and the first side (i.e. length) in radians */
    }
    CvBox2D;

### BoxPoints

Finds box vertices

    void cvBoxPoints( CvBox2D box, CvPoint2D32f pt[4] );

box
  Box
pt
  Array of vertices

The function cvBoxPoints calculates vertices of the input 2d box. Here is the function code:

    void cvBoxPoints( CvBox2D box, CvPoint2D32f pt[4] )
    {
        float a = (float)cos(box.angle)*0.5f;
        float b = (float)sin(box.angle)*0.5f;

        pt[0].x = box.center.x - a*box.size.height - b*box.size.width;
        pt[0].y = box.center.y + b*box.size.height - a*box.size.width;
        pt[1].x = box.center.x + a*box.size.height - b*box.size.width;
        pt[1].y = box.center.y - b*box.size.height - a*box.size.width;
        pt[2].x = 2*box.center.x - pt[0].x;
        pt[2].y = 2*box.center.y - pt[0].y;
        pt[3].x = 2*box.center.x - pt[1].x;
        pt[3].y = 2*box.center.y - pt[1].y;
    }

### FitEllipse

Fits ellipse to set of 2D points

    CvBox2D cvFitEllipse2( const CvArr* points );

points
  Sequence or array of points.

The function cvFitEllipse calculates ellipse that fits best (in least-squares sense) to a set of 2D points. The meaning of the returned structure fields is similar to those in cvEllipse except that `size` stores the full lengths of the ellipse axises, not half-lengths

### FitLine

Fits line to 2D or 3D point set

    void  cvFitLine( const CvArr* points, int dist_type, double param,
                     double reps, double aeps, float* line );

points
  Sequence or array of 2D or 3D points with 32-bit integer or floating-point coordinates.
dist_type
  The distance used for fitting (see the discussion).
param
  Numerical parameter (`C`) for some types of distances, if 0 then some optimal value is chosen.
reps, aeps
  Sufficient accuracy for radius (distance between the coordinate origin and the line) and angle, respectively, 0.01 would be a good defaults for both. is used.
line
  The output line parameters. In case of 2d fitting it is array of 4 floats `(vx, vy, x0, y0)` where `(vx, vy)` is a normalized vector collinear to the line and `(x0, y0)` is some point on the line. In case of 3D fitting it is array of 6 floats `(vx, vy, vz, x0, y0, z0)` where `(vx, vy, vz)` is a normalized vector collinear to the line and `(x0, y0, z0)` is some point on the line.

The function cvFitLine fits line to 2D or 3D point set by minimizing sumiρ(ri), where ri is distance between i-th point and the line and ρ(r) is a distance function, one of:

    dist_type=CV_DIST_L2 (L2):
    ρ(r)=r2/2 (the simplest and the fastest least-squares method)

    dist_type=CV_DIST_L1 (L1):
    ρ(r)=r

    dist_type=CV_DIST_L12 (L1-L2):
    ρ(r)=2•[sqrt(1+r2/2) - 1]

    dist_type=CV_DIST_FAIR (Fair):
    ρ(r)=C2•[r/C - log(1 + r/C)],  C=1.3998

    dist_type=CV_DIST_WELSCH (Welsch):
    ρ(r)=C2/2•[1 - exp(-(r/C)2)],  C=2.9846

    dist_type=CV_DIST_HUBER (Huber):
    ρ(r)= r2/2,   if r < C
          C•(r-C/2),   otherwise;   C=1.345

### ConvexHull2

Finds convex hull of point set

    CvSeq* cvConvexHull2( const CvArr* input, void* hull_storage=NULL,
                          int orientation=CV_CLOCKWISE, int return_points=0 );

points
  Sequence or array of 2D points with 32-bit integer or floating-point coordinates.
hull_storage
  The destination array (CvMat*) or memory storage (CvMemStorage*) that will store the convex hull. If it is array, it should be 1d and have the same number of elements as the input array/sequence. On output the header is modified so to truncate the array downto the hull size.
orientation
  Desired orientation of convex hull: `CV_CLOCKWISE` or `CV_COUNTER_CLOCKWISE`.
return_points
  If non-zero, the points themselves will be stored in the hull instead of indices if `hull_storage` is array, or pointers if `hull_storage` is memory storage.

The function cvConvexHull2 finds convex hull of 2D point set using Sklansky's algorithm. If `hull_storage` is memory storage, the function creates a sequence containing the hull points or pointers to them, depending on `return_points` value and returns the sequence on output.

#### Example. Building convex hull for a sequence or array of points

    #include "cv.h"
    #include "highgui.h"
    #include <stdlib.h>

    #define ARRAY  0 /* switch between array/sequence method by replacing 0<=>1 */

    void main( int argc, char** argv )
    {
        IplImage* img = cvCreateImage( cvSize( 500, 500 ), 8, 3 );
        cvNamedWindow( "hull", 1 );

    #if !ARRAY
            CvMemStorage* storage = cvCreateMemStorage();
    #endif

        for(;;)
        {
            int i, count = rand()%100 + 1, hullcount;
            CvPoint pt0;
    #if !ARRAY
            CvSeq* ptseq = cvCreateSeq( CV_SEQ_KIND_GENERIC|CV_32SC2, sizeof(CvContour),
                                         sizeof(CvPoint), storage );
            CvSeq* hull;

            for( i = 0; i < count; i++ )
            {
                pt0.x = rand() % (img->width/2) + img->width/4;
                pt0.y = rand() % (img->height/2) + img->height/4;
                cvSeqPush( ptseq, &pt0 );
            }
            hull = cvConvexHull2( ptseq, 0, CV_CLOCKWISE, 0 );
            hullcount = hull->total;
    #else
            CvPoint* points = (CvPoint*)malloc( count * sizeof(points[0]));
            int* hull = (int*)malloc( count * sizeof(hull[0]));
            CvMat point_mat = cvMat( 1, count, CV_32SC2, points );
            CvMat hull_mat = cvMat( 1, count, CV_32SC1, hull );

            for( i = 0; i < count; i++ )
            {
                pt0.x = rand() % (img->width/2) + img->width/4;
                pt0.y = rand() % (img->height/2) + img->height/4;
                points[i] = pt0;
            }
            cvConvexHull2( &point_mat, &hull_mat, CV_CLOCKWISE, 0 );
            hullcount = hull_mat.cols;
    #endif
            cvZero( img );
            for( i = 0; i < count; i++ )
            {
    #if !ARRAY
                pt0 = *CV_GET_SEQ_ELEM( CvPoint, ptseq, i );
    #else
                pt0 = points[i];
    #endif
                cvCircle( img, pt0, 2, CV_RGB( 255, 0, 0 ), CV_FILLED );
            }

    #if !ARRAY
            pt0 = **CV_GET_SEQ_ELEM( CvPoint*, hull, hullcount - 1 );
    #else
            pt0 = points[hull[hullcount-1]];
    #endif

            for( i = 0; i < hullcount; i++ )
            {
    #if !ARRAY
                CvPoint pt = **CV_GET_SEQ_ELEM( CvPoint*, hull, i );
    #else
                CvPoint pt = points[hull[i]];
    #endif
                cvLine( img, pt0, pt, CV_RGB( 0, 255, 0 ));
                pt0 = pt;
            }

            cvShowImage( "hull", img );

            int key = cvWaitKey(0);
            if( key == 27 ) // 'ESC'
                break;

    #if !ARRAY
            cvClearMemStorage( storage );
    #else
            free( points );
            free( hull );
    #endif
        }
    }

### CheckContourConvexity

Tests contour convex

    int cvCheckContourConvexity( const CvArr* contour );

contour
  Tested contour (sequence or array of points).

The function cvCheckContourConvexity tests whether the input contour is convex or not. The contour must be simple, i.e. without self-intersections.

### CvConvexityDefect

Structure describing a single contour convexity detect

    typedef struct CvConvexityDefect
    {
        CvPoint* start; /* point of the contour where the defect begins */
        CvPoint* end; /* point of the contour where the defect ends */
        CvPoint* depth_point; /* the farthest from the convex hull point within the defect */
        float depth; /* distance between the farthest point and the convex hull */
    } CvConvexityDefect;

#### Picture. Convexity defects of hand contour.

### ConvexityDefects

Finds convexity defects of contour

    CvSeq* cvConvexityDefects( const CvArr* contour, const CvArr* convexhull,
                               CvMemStorage* storage=NULL );

contour
  Input contour.
convexhull
  Convex hull obtained using cvConvexHull2 that should contain pointers or indices to the contour points, not the hull points themselves, i.e. `return_points` parameter in cvConvexHull2 should be 0.
storage
  Container for output sequence of convexity defects. If it is NULL, contour or hull (in that order) storage is used.

The function cvConvexityDefects finds all convexity defects of the input contour and returns a sequence of the CvConvexityDefect structures.

### MinAreaRect2

Finds circumscribed rectangle of minimal area for given 2D point set

    CvBox2D  cvMinAreaRect2( const CvArr* points, CvMemStorage* storage=NULL );

points
  Sequence or array of points.
storage
  Optional temporary memory storage.

The function cvMinAreaRect2 finds a circumscribed rectangle of the minimal area for 2D point set by building convex hull for the set and applying rotating calipers technique to the hull.

#### Picture. Minimal-area bounding rectangle for contour

### MinEnclosingCircle

Finds circumscribed circle of minimal area for given 2D point set

    int cvMinEnclosingCircle( const CvArr* points, CvPoint2D32f* center, float* radius );

points
  Sequence or array of 2D points.
center
  Output parameter. The center of the enclosing circle.
radius
  Output parameter. The radius of the enclosing circle.

The function cvMinEnclosingCircle finds the minimal circumscribed circle for 2D point set using iterative algorithm. It returns nonzero if the resultant circle contains all the input points and zero otherwise (i.e. algorithm failed).

### CalcPGH

Calculates pair-wise geometrical histogram for contour

    void cvCalcPGH( const CvSeq* contour, CvHistogram* hist );

contour
  Input contour. Currently, only integer point coordinates are allowed.
hist
  Calculated histogram; must be two-dimensional.

The function cvCalcPGH calculates 2D pair-wise geometrical histogram (PGH), described in [Iivarinen97], for the contour. The algorithm considers every pair of the contour edges. The angle between the edges and the minimum/maximum distances are determined for every pair. To do this each of the edges in turn is taken as the base, while the function loops through all the other edges. When the base edge and any other edge are considered, the minimum and maximum distances from the points on the non-base edge and line of the base edge are selected. The angle between the edges defines the row of the histogram in which all the bins that correspond to the distance between the calculated minimum and maximum distances are incremented (that is, the histogram is transposed relatively to [Iivarninen97] definition). The histogram can be used for contour matching.

## Planar Subdivisions

### CvSubdiv2D

Planar subdivision

    #define CV_SUBDIV2D_FIELDS()    \
        CV_GRAPH_FIELDS()           \
        int  quad_edges;            \
        int  is_geometry_valid;     \
        CvSubdiv2DEdge recent_edge; \
        CvPoint2D32f  topleft;      \
        CvPoint2D32f  bottomright;

    typedef struct CvSubdiv2D
    {
        CV_SUBDIV2D_FIELDS()
    }
    CvSubdiv2D;

Planar subdivision is a subdivision of a plane into a set of non-overlapped regions (facets) that cover the whole plane. The above structure describes a subdivision built on 2d point set, where the points are linked together and form a planar graph, which, together with a few edges connecting exterior subdivision points (namely, convex hull points) with infinity, subdivides a plane into facets by its edges.

For every subdivision there exists dual subdivision there facets and points (subdivision vertices) swap their roles, that is, a facet is treated as a vertex (called virtual point below) of dual subdivision and the original subdivision vertices become facets. On the picture below original subdivision is marked with solid lines and dual subdivision with dot lines

OpenCV subdivides plane into triangles using Delaunay's algorithm. Subdivision is built iteratively starting from a dummy triangle that includes all the subdivision points for sure. In this case the dual subdivision is Voronoi diagram of input 2d point set. The subdivisions can be used for 3d piece-wise transformation of a plane, morphing, fast location of points on the plane, building special graphs (such as NNG,RNG) etc.

### CvQuadEdge2D

Quad-edge of planar subdivision

    /* one of edges within quad-edge, lower 2 bits is index (0..3)
       and upper bits are quad-edge pointer */
    typedef long CvSubdiv2DEdge;

    /* quad-edge structure fields */
    #define CV_QUADEDGE2D_FIELDS()     \
        int flags;                     \
        struct CvSubdiv2DPoint* pt[4]; \
        CvSubdiv2DEdge  next[4];

    typedef struct CvQuadEdge2D
    {
        CV_QUADEDGE2D_FIELDS()
    }
    CvQuadEdge2D;

Quad-edge is a basic element of subdivision, it contains four edges (e, eRot and reversed e & eRot):

### CvSubdiv2DPoint

Point of original or dual subdivision

    #define CV_SUBDIV2D_POINT_FIELDS()\
        int            flags;      \
        CvSubdiv2DEdge first;      \
        CvPoint2D32f   pt;

    #define CV_SUBDIV2D_VIRTUAL_POINT_FLAG (1 << 30)

    typedef struct CvSubdiv2DPoint
    {
        CV_SUBDIV2D_POINT_FIELDS()
    }
    CvSubdiv2DPoint;

### Subdiv2DGetEdge

Returns one of edges related to given

    CvSubdiv2DEdge  cvSubdiv2DGetEdge( CvSubdiv2DEdge edge, CvNextEdgeType type );
    #define cvSubdiv2DNextEdge( edge ) cvSubdiv2DGetEdge( edge, CV_NEXT_AROUND_ORG )

edge
  Subdivision edge (not a quad-edge)
type
  Specifies, which of related edges to return, one of:
  - CV_NEXT_AROUND_ORG - next around the edge origin (`eOnext` on the picture above if `e` is the input edge)
  - CV_NEXT_AROUND_DST - next around the edge vertex (`eDnext`)
  - CV_PREV_AROUND_ORG - previous around the edge origin (reversed `eRnext`)
  - CV_PREV_AROUND_DST - previous around the edge destination (reversed `eLnext`)
  - CV_NEXT_AROUND_LEFT - next around the left facet (`eLnext`)
  - CV_NEXT_AROUND_RIGHT - next around the right facet (`eRnext`)
  - CV_PREV_AROUND_LEFT - previous around the left facet (reversed `eOnext`)
  - CV_PREV_AROUND_RIGHT - previous around the right facet (reversed `eDnext`)

The function cvSubdiv2DGetEdge returns one the edges related to the input edge.

### Subdiv2DRotateEdge

Returns another edge of the same quad-edge

    CvSubdiv2DEdge  cvSubdiv2DRotateEdge( CvSubdiv2DEdge edge, int rotate );

edge
  Subdivision edge (not a quad-edge)
type
  Specifies, which of edges of the same quad-edge as the input one to return, one of:
  - 0 - the input edge (`e` on the picture above if `e` is the input edge)
  - 1 - the rotated edge (`eRot`)
  - 2 - the reversed edge (reversed `e` (in green))
  - 3 - the reversed rotated edge (reversed `eRot` (in green))

The function cvSubdiv2DRotateEdge returns one the edges of the same quad-edge as the input edge.

### Subdiv2DEdgeOrg

Returns edge origin

    CvSubdiv2DPoint* cvSubdiv2DEdgeOrg( CvSubdiv2DEdge edge );

edge
  Subdivision edge (not a quad-edge)

The function cvSubdiv2DEdgeOrg returns the edge origin. The returned pointer may be NULL if the edge is from dual subdivision and the virtual point coordinates are not calculated yet. The virtual points can be calculated using function cvCalcSubdivVoronoi2D.

### Subdiv2DEdgeDst

Returns edge destination

    CvSubdiv2DPoint* cvSubdiv2DEdgeDst( CvSubdiv2DEdge edge );

edge
  Subdivision edge (not a quad-edge)

The function cvSubdiv2DEdgeDst returns the edge destination. The returned pointer may be NULL if the edge is from dual subdivision and the virtual point coordinates are not calculated yet. The virtual points can be calculated using function cvCalcSubdivVoronoi2D.

### CreateSubdivDelaunay2D

Creates empty Delaunay triangulation

    CvSubdiv2D* cvCreateSubdivDelaunay2D( CvRect rect, CvMemStorage* storage );

rect
  Rectangle that includes all the 2d points that are to be added to subdivision.
storage
  Container for subdivision.

The function cvCreateSubdivDelaunay2D creates an empty Delaunay subdivision, where 2d points can be added further using function cvSubdivDelaunay2DInsert. All the points to be added must be within the specified rectangle, otherwise a runtime error will be raised.

### SubdivDelaunay2DInsert

Inserts a single point to Delaunay triangulation

    CvSubdiv2DPoint*  cvSubdivDelaunay2DInsert( CvSubdiv2D* subdiv, CvPoint2D32f pt);

subdiv
  Delaunay subdivision created by function cvCreateSubdivDelaunay2D.
pt
  Inserted point.

The function cvSubdivDelaunay2DInsert inserts a single point to subdivision and modifies the subdivision topology appropriately. If a points with same coordinates exists already, no new points is added. The function returns pointer to the allocated point. No virtual points coordinates is calculated at this stage.

### Subdiv2DLocate

Inserts a single point to Delaunay triangulation

    CvSubdiv2DPointLocation  cvSubdiv2DLocate( CvSubdiv2D* subdiv, CvPoint2D32f pt,
                                               CvSubdiv2DEdge* edge,
                                               CvSubdiv2DPoint** vertex=NULL );

subdiv
  Delaunay or another subdivision.
pt
  The point to locate.
edge
  The output edge the point falls onto or right to.
vertex
  Optional output vertex double pointer the input point coinsides with.

The function cvSubdiv2DLocate locates input point within subdivision. There are 5 cases:

- point falls into some facet. The function returns CV_PTLOC_INSIDE and `*edge` will contain one of edges of the facet.
- point falls onto the edge. The function returns CV_PTLOC_ON_EDGE and `*edge` will contain this edge.
- point coinsides with one of subdivision vertices. The function returns CV_PTLOC_VERTEX and `*vertex` will contain pointer to the vertex.
- point is outside the subdivsion reference rectangle. The function returns CV_PTLOC_OUTSIDE_RECT and no pointers is filled.
- one of input arguments is invalid. Runtime error is raised or, if silent or "parent" error processing mode is selected, CV_PTLOC_ERROR is returnd.

### FindNearestPoint2D

Finds the closest subdivision vertex to given point

    CvSubdiv2DPoint* cvFindNearestPoint2D( CvSubdiv2D* subdiv, CvPoint2D32f pt );

subdiv
  Delaunay or another subdivision.
pt
  Input point.

The function cvFindNearestPoint2D is another function that locates input point within subdivision. It finds subdivision vertex that is the closest to the input point. It is not necessarily one of vertices of the facet containing the input point, though the facet (located using cvSubdiv2DLocate) is used as a starting point. The function returns pointer to the found subdivision vertex

### CalcSubdivVoronoi2D

Calculates coordinates of Voronoi diagram cells

    void cvCalcSubdivVoronoi2D( CvSubdiv2D* subdiv );

subdiv
  Delaunay subdivision, where all the points are added already.

The function cvCalcSubdivVoronoi2D calculates coordinates of virtual points. All virtual points corresponding to some vertex of original subdivision form (when connected together) a boundary of Voronoi cell of that point.

### ClearSubdivVoronoi2D

Removes all virtual points

    void cvClearSubdivVoronoi2D( CvSubdiv2D* subdiv );

subdiv
  Delaunay subdivision.

The function cvClearSubdivVoronoi2D removes all virtual points. It is called internally in cvCalcSubdivVoronoi2D if the subdivision was modified after previous call to the function.

There are a few other lower-level functions that work with planar subdivisions, see cv.h and the sources. Demo script delaunay.c that builds Delaunay triangulation and Voronoi diagram of random 2d point set can be found at opencv/samples/c.

# Motion Analysis and Object Tracking Reference

## Accumulation of Background Statistics

### Acc

Adds frame to accumulator

    void cvAcc( const CvArr* image, CvArr* sum, const CvArr* mask=NULL );

image
  Input image, 1- or 3-channel, 8-bit or 32-bit floating point. (each channel of multi-channel image is processed independently).
sum
  Accumulator of the same number of channels as input image, 32-bit or 64-bit floating-point.
mask
  Optional operation mask.

The function cvAcc adds the whole image `image` or its selected region to accumulator `sum`:

    sum(x,y)=sum(x,y)+image(x,y) if mask(x,y)!=0

### SquareAcc

Adds the square of source image to accumulator

    void cvSquareAcc( const CvArr* image, CvArr* sqsum, const CvArr* mask=NULL );

image
  Input image, 1- or 3-channel, 8-bit or 32-bit floating point (each channel of multi-channel image is processed independently).
sqsum
  Accumulator of the same number of channels as input image, 32-bit or 64-bit floating-point.
mask
  Optional operation mask.

The function cvSquareAcc adds the input image `image` or its selected region, raised to power 2, to the accumulator `sqsum`:

    sqsum(x,y)=sqsum(x,y)+image(x,y)2 if mask(x,y)!=0

### MultiplyAcc

Adds product of two input images to accumulator

    void cvMultiplyAcc( const CvArr* image1, const CvArr* image2, CvArr* acc, const CvArr* mask=NULL );

image1
  First input image, 1- or 3-channel, 8-bit or 32-bit floating point (each channel of multi-channel image is processed independently).
image2
  Second input image, the same format as the first one.
acc
  Accumulator of the same number of channels as input images, 32-bit or 64-bit floating-point.
mask
  Optional operation mask.

The function cvMultiplyAcc adds product of 2 images or thier selected regions to accumulator `acc`:

    acc(x,y)=acc(x,y) + image1(x,y)•image2(x,y) if mask(x,y)!=0

### RunningAvg

Updates running average

    void cvRunningAvg( const CvArr* image, CvArr* acc, double alpha, const CvArr* mask=NULL );

image
  Input image, 1- or 3-channel, 8-bit or 32-bit floating point (each channel of multi-channel image is processed independently).
acc
  Accumulator of the same number of channels as input image, 32-bit or 64-bit floating-point.
alpha
  Weight of input image.
mask
  Optional operation mask.

The function cvRunningAvg calculates weighted sum of input image `image` and the accumulator `acc` so that `acc` becomes a running average of frame sequence:

    acc(x,y)=(1-α)•acc(x,y) + α•image(x,y) if mask(x,y)!=0

where α (alpha) regulates update speed (how fast accumulator forgets about previous frames).

## Motion Templates

### UpdateMotionHistory

Updates motion history image by moving silhouette

    void cvUpdateMotionHistory( const CvArr* silhouette, CvArr* mhi,
                                double timestamp, double duration );

silhouette
  Silhouette mask that has non-zero pixels where the motion occurs.
mhi
  Motion history image, that is updated by the function (single-channel, 32-bit floating-point)
timestamp
  Current time in milliseconds or other units.
duration
  Maximal duration of motion track in the same units as `timestamp`.

The function cvUpdateMotionHistory updates the motion history image as following:

    mhi(x,y)=timestamp  if silhouette(x,y)!=0
             0          if silhouette(x,y)=0 and mhi(x,y)<timestamp-duration
             mhi(x,y)   otherwise

That is, MHI pixels where motion occurs are set to the current timestamp, while the pixels where motion happened far ago are cleared.

### CalcMotionGradient

Calculates gradient orientation of motion history image

    void cvCalcMotionGradient( const CvArr* mhi, CvArr* mask, CvArr* orientation,
                               double delta1, double delta2, int aperture_size=3 );

mhi
  Motion history image.
mask
  Mask image; marks pixels where motion gradient data is correct. Output parameter.
orientation
  Motion gradient orientation image; contains angles from 0 to ~360°.
delta1, delta2
  The function finds minimum (m(x,y)) and maximum (M(x,y)) mhi values over each pixel (x,y) neihborhood and assumes the gradient is valid only if

      min(delta1,delta2) <= M(x,y)-m(x,y) <= max(delta1,delta2).

aperture_size
  Aperture size of derivative operators used by the function: CV_SCHARR, 1, 3, 5 or 7 (see cvSobel).

The function cvCalcMotionGradient calculates the derivatives `Dx` and `Dy` of `mhi` and then calculates gradient orientation as:

    orientation(x,y)=arctan(Dy(x,y)/Dx(x,y))

where both `Dx(x,y)`' and `Dy(x,y)`' signs are taken into account (as in cvCartToPolar function). After that `mask` is filled to indicate where the orientation is valid (see `delta1` and `delta2` description).

### CalcGlobalOrientation

Calculates global motion orientation of some selected region

    double cvCalcGlobalOrientation( const CvArr* orientation, const CvArr* mask, const CvArr* mhi,
                                    double timestamp, double duration );

orientation
  Motion gradient orientation image; calculated by the function cvCalcMotionGradient.
mask
  Mask image. It may be a conjunction of valid gradient mask, obtained with cvCalcMotionGradient and mask of the region, whose direction needs to be calculated.
mhi
  Motion history image.
timestamp
  Current time in milliseconds or other units, it is better to store time passed to cvUpdateMotionHistory before and reuse it here, because running cvUpdateMotionHistory and cvCalcMotionGradient on large images may take some time.
duration
  Maximal duration of motion track in milliseconds, the same as in cvUpdateMotionHistory.

The function cvCalcGlobalOrientation calculates the general motion direction in the selected region and returns the angle between 0° and 360°. At first the function builds the orientation histogram and finds the basic orientation as a coordinate of the histogram maximum. After that the function calculates the shift relative to the basic orientation as a weighted sum of all orientation vectors: the more recent is the motion, the greater is the weight. The resultant angle is a circular sum of the basic orientation and the shift.

### SegmentMotion

Segments whole motion into separate moving parts

    CvSeq* cvSegmentMotion( const CvArr* mhi, CvArr* seg_mask, CvMemStorage* storage,
                            double timestamp, double seg_thresh );

mhi
  Motion history image.
seg_mask
  Image where the mask found should be stored, single-channel, 32-bit floating-point.
storage
  Memory storage that will contain a sequence of motion connected components.
timestamp
  Current time in milliseconds or other units.
seg_thresh
  Segmentation threshold; recommended to be equal to the interval between motion history "steps" or greater.

The function cvSegmentMotion finds all the motion segments and marks them in `seg_mask` with individual values each (1,2,...). It also returns a sequence of CvConnectedComp structures, one per each motion components. After than the motion direction for every component can be calculated with cvCalcGlobalOrientation using extracted mask of the particular component (using cvCmp)

## Object Tracking

### MeanShift

Finds object center on back projection

    int cvMeanShift( const CvArr* prob_image, CvRect window,
                     CvTermCriteria criteria, CvConnectedComp* comp );

prob_image
  Back projection of object histogram (see cvCalcBackProject).
window
  Initial search window.
criteria
  Criteria applied to determine when the window search should be finished.
comp
  Resultant structure that contains converged search window coordinates (`comp->rect` field) and sum of all pixels inside the window (`comp->area` field).

The function cvMeanShift iterates to find the object center given its back projection and initial position of search window. The iterations are made until the search window center moves by less than the given value and/or until the function has done the maximum number of iterations. The function returns the number of iterations made.

### CamShift

Finds object center, size, and orientation

    int cvCamShift( const CvArr* prob_image, CvRect window, CvTermCriteria criteria,
                    CvConnectedComp* comp, CvBox2D* box=NULL );

prob_image
  Back projection of object histogram (see cvCalcBackProject).
window
  Initial search window.
criteria
  Criteria applied to determine when the window search should be finished.
comp
  Resultant structure that contains converged search window coordinates (`comp->rect` field) and sum of all pixels inside the window (`comp->area` field).
box
  Circumscribed box for the object. If not `NULL`, contains object size and orientation.

The function cvCamShift implements CAMSHIFT object tracking algrorithm ([Bradski98]). First, it finds an object center using cvMeanShift and, after that, calculates the object size and orientation. The function returns number of iterations made within cvMeanShift.

CvCamShiftTracker class declared in cv.hpp implements color object tracker that uses the function.

### SnakeImage

Changes contour position to minimize its energy

    void cvSnakeImage( const IplImage* image, CvPoint* points, int length,
                       float* alpha, float* beta, float* gamma, int coeff_usage,
                       CvSize win, CvTermCriteria criteria, int calc_gradient=1 );

image
  The source image or external energy field.
points
  Contour points (snake).
length
  Number of points in the contour.
alpha
  Weight[s] of continuity energy, single float or array of `length` floats, one per each contour point.
beta
  Weight[s] of curvature energy, similar to `alpha`.
gamma
  Weight[s] of image energy, similar to `alpha`.
coeff_usage
  Variant of usage of the previous three parameters:
  - `CV_VALUE` indicates that each of `alpha, beta, gamma` is a pointer to a single value to be used for all points;
  - `CV_ARRAY` indicates that each of `alpha, beta, gamma` is a pointer to an array of coefficients different for all the points of the snake. All the arrays must have the size equal to the contour size.
win
  Size of neighborhood of every point used to search the minimum, both `win.width` and `win.height` must be odd.
criteria
  Termination criteria.
calc_gradient
  Gradient flag. If not 0, the function calculates gradient magnitude for every image pixel and consideres it as the energy field, otherwise the input image itself is considered.

The function cvSnakeImage updates snake in order to minimize its total energy that is a sum of internal energy that depends on contour shape (the smoother contour is, the smaller internal energy is) and external energy that depends on the energy field and reaches minimum at the local energy extremums that correspond to the image edges in case of image gradient.

The parameter `criteria.epsilon` is used to define the minimal number of points that must be moved during any iteration to keep the iteration process running.

If at some iteration the number of moved points is less than `criteria.epsilon` or the function performed `criteria.max_iter` iterations, the function terminates.

## Optical Flow

### CalcOpticalFlowHS

Calculates optical flow for two images

    void cvCalcOpticalFlowHS( const CvArr* prev, const CvArr* curr, int use_previous,
                              CvArr* velx, CvArr* vely, double lambda,
                              CvTermCriteria criteria );

prev
  First image, 8-bit, single-channel.
curr
  Second image, 8-bit, single-channel.
use_previous
  Uses previous (input) velocity field.
velx
  Horizontal component of the optical flow of the same size as input images, 32-bit floating-point, single-channel.
vely
  Vertical component of the optical flow of the same size as input images, 32-bit floating-point, single-channel.
lambda
  Lagrangian multiplier.
criteria
  Criteria of termination of velocity computing.

The function cvCalcOpticalFlowHS computes flow for every pixel of the first input image using Horn & Schunck algorithm [Horn81].

### CalcOpticalFlowLK

Calculates optical flow for two images

    void cvCalcOpticalFlowLK( const CvArr* prev, const CvArr* curr, CvSize win_size,
                              CvArr* velx, CvArr* vely );

prev
  First image, 8-bit, single-channel.
curr
  Second image, 8-bit, single-channel.
win_size
  Size of the averaging window used for grouping pixels.
velx
  Horizontal component of the optical flow of the same size as input images, 32-bit floating-point, single-channel.
vely
  Vertical component of the optical flow of the same size as input images, 32-bit floating-point, single-channel.

The function cvCalcOpticalFlowLK computes flow for every pixel of the first input image using Lucas & Kanade algorithm [Lucas81].

### CalcOpticalFlowBM

Calculates optical flow for two images by block matching method

    void cvCalcOpticalFlowBM( const CvArr* prev, const CvArr* curr, CvSize block_size,
                              CvSize shift_size, CvSize max_range, int use_previous,
                              CvArr* velx, CvArr* vely );

prev
  First image, 8-bit, single-channel.
curr
  Second image, 8-bit, single-channel.
block_size
  Size of basic blocks that are compared.
shift_size
  Block coordinate increments.
max_range
  Size of the scanned neighborhood in pixels around block.
use_previous
  Uses previous (input) velocity field.
velx
  Horizontal component of the optical flow of
  floor((prev->width - block_size.width)/shiftSize.width) × floor((prev->height - block_size.height)/shiftSize.height) size, 32-bit floating-point, single-channel.
vely
  Vertical component of the optical flow of the same size `velx`, 32-bit floating-point, single-channel.

The function cvCalcOpticalFlowBM calculates optical flow for overlapped blocks `block_size.width×block_size.height` pixels each, thus the velocity fields are smaller than the original images. For every block in `prev` the functions tries to find a similar block in `curr` in some neighborhood of the original block or shifted by (velx(x0,y0),vely(x0,y0)) block as has been calculated by previous function call (if `use_previous=1`)

### CalcOpticalFlowPyrLK

Calculates optical flow for a sparse feature set using iterative Lucas-Kanade method in pyramids

    void cvCalcOpticalFlowPyrLK( const CvArr* prev, const CvArr* curr, CvArr* prev_pyr, CvArr* curr_pyr,
                                 const CvPoint2D32f* prev_features, CvPoint2D32f* curr_features,
                                 int count, CvSize win_size, int level, char* status,
                                 float* track_error, CvTermCriteria criteria, int flags );

prev
  First frame, at time `t`.
curr
  Second frame, at time `t + dt` .
prev_pyr
  Buffer for the pyramid for the first frame. If the pointer is not `NULL` , the buffer must have a sufficient size to store the pyramid from level `1` to level #`level` ; the total size of `(image_width+8)*image_height/3` bytes is sufficient.
curr_pyr
  Similar to `prev_pyr`, used for the second frame.
prev_features
  Array of points for which the flow needs to be found.
curr_features
  Array of 2D points containing calculated new positions of input
features
  in the second image.
count
  Number of feature points.
win_size
  Size of the search window of each pyramid level.
level
  Maximal pyramid level number. If `0` , pyramids are not used (single level), if `1` , two levels are used, etc.
status
  Array. Every element of the array is set to `1` if the flow for the corresponding feature has been found, `0` otherwise.
error
  Array of double numbers containing difference between patches around the original and moved points. Optional parameter; can be `NULL` .
criteria
  Specifies when the iteration process of finding the flow for each point on each pyramid level should be stopped.
flags
  Miscellaneous flags:
  - `CV_LKFLOW_PYR_A_READY` , pyramid for the first frame is precalculated before the call;
  - `CV_LKFLOW_PYR_B_READY` , pyramid for the second frame is precalculated before the call;
  - `CV_LKFLOW_INITIAL_GUESSES` , array B contains initial coordinates of features before the function call.

The function cvCalcOpticalFlowPyrLK implements sparse iterative version of Lucas-Kanade optical flow in pyramids ([Bouguet00]). It calculates coordinates of the feature points on the current video frame given their coordinates on the previous frame. The function finds the coordinates with sub-pixel accuracy.

Both parameters `prev_pyr` and `curr_pyr` comply with the following rules: if the image pointer is 0, the function allocates the buffer internally, calculates the pyramid, and releases the buffer after processing. Otherwise, the function calculates the pyramid and stores it in the buffer unless the flag `CV_LKFLOW_PYR_A[B]_READY` is set. The image should be large enough to fit the Gaussian pyramid data. After the function call both pyramids are calculated and the readiness flag for the corresponding image can be set in the next call (i.e., typically, for all the image pairs except the very first one `CV_LKFLOW_PYR_A_READY` is set).

## Estimators

### CvKalman

Kalman filter state

    typedef struct CvKalman
    {
        int MP;                     /* number of measurement vector dimensions */
        int DP;                     /* number of state vector dimensions */
        int CP;                     /* number of control vector dimensions */

        /* backward compatibility fields */
    #if 1
        float* PosterState;         /* =state_pre->data.fl */
        float* PriorState;          /* =state_post->data.fl */
        float* DynamMatr;           /* =transition_matrix->data.fl */
        float* MeasurementMatr;     /* =measurement_matrix->data.fl */
        float* MNCovariance;        /* =measurement_noise_cov->data.fl */
        float* PNCovariance;        /* =process_noise_cov->data.fl */
        float* KalmGainMatr;        /* =gain->data.fl */
        float* PriorErrorCovariance;/* =error_cov_pre->data.fl */
        float* PosterErrorCovariance;/* =error_cov_post->data.fl */
        float* Temp1;               /* temp1->data.fl */
        float* Temp2;               /* temp2->data.fl */
    #endif

        CvMat* state_pre;           /* predicted state (x'(k)):
                                        x(k)=A*x(k-1)+B*u(k) */
        CvMat* state_post;          /* corrected state (x(k)):
                                        x(k)=x'(k)+K(k)*(z(k)-H*x'(k)) */
        CvMat* transition_matrix;   /* state transition matrix (A) */
        CvMat* control_matrix;      /* control matrix (B)
                                       (it is not used if there is no control)*/
        CvMat* measurement_matrix;  /* measurement matrix (H) */
        CvMat* process_noise_cov;   /* process noise covariance matrix (Q) */
        CvMat* measurement_noise_cov; /* measurement noise covariance matrix (R) */
        CvMat* error_cov_pre;       /* priori error estimate covariance matrix (P'(k)):
                                        P'(k)=A*P(k-1)*At + Q)*/
        CvMat* gain;                /* Kalman gain matrix (K(k)):
                                        K(k)=P'(k)*Ht*inv(H*P'(k)*Ht+R)*/
        CvMat* error_cov_post;      /* posteriori error estimate covariance matrix (P(k)):
                                        P(k)=(I-K(k)*H)*P'(k) */
        CvMat* temp1;               /* temporary matrices */
        CvMat* temp2;
        CvMat* temp3;
        CvMat* temp4;
        CvMat* temp5;
    }
    CvKalman;

The structure CvKalman is used to keep Kalman filter state. It is created by cvCreateKalman function, updated by cvKalmanPredict and cvKalmanCorrect functions and released by cvReleaseKalman functions. Normally, the structure is used for standard Kalman filter (notation and the formulae below are borrowed from the excellent Kalman tutorial [Welch95]):

    xk=A•xk-1+B•uk+wk
    zk=H•xk+vk,

where:

    xk (xk-1) - state of the system at the moment k (k-1)
    zk - measurement of the system state at the moment k
    uk - external control applied at the moment k

    wk and vk are normally-distributed process and measurement noise, respectively:
    p(w) ~ N(0,Q)
    p(v) ~ N(0,R),

    that is,
    Q - process noise covariance matrix, constant or variable,
    R - measurement noise covariance matrix, constant or variable

In case of standard Kalman filter, all the matrices: A, B, H, Q and R are initialized once after CvKalman structure is allocated via cvCreateKalman. However, the same structure and the same functions may be used to simulate extended Kalman filter by linearizing extended Kalman filter equation in the current system state neighborhood, in this case A, B, H (and, probably, Q and R) should be updated on every step.

### CreateKalman

Allocates Kalman filter structure

    CvKalman* cvCreateKalman( int dynam_params, int measure_params, int control_params=0 );

dynam_params
  dimensionality of the state vector
measure_params
  dimensionality of the measurement vector
control_params
  dimensionality of the control vector

The function cvCreateKalman allocates CvKalman and all its matrices and initializes them somehow.

### ReleaseKalman

Deallocates Kalman filter structure

    void cvReleaseKalman( CvKalman** kalman );

kalman
  double pointer to the Kalman filter structure.

The function cvReleaseKalman releases the structure CvKalman and all underlying matrices.

### KalmanPredict

Estimates subsequent model state

    const CvMat* cvKalmanPredict( CvKalman* kalman, const CvMat* control=NULL );
    #define cvKalmanUpdateByTime cvKalmanPredict

kalman
  Kalman filter state.
control
  Control vector (uk), should be NULL iff there is no external control (`control_params`=0).

The function cvKalmanPredict estimates the subsequent stochastic model state by its current state and stores it at `kalman->state_pre`:

        x'k=A•xk+B•uk
        P'k=A•Pk-1*AT + Q,
    where
    x'k is predicted state (kalman->state_pre),
    xk-1 is corrected state on the previous step (kalman->state_post)
                    (should be initialized somehow in the beginning, zero vector by default),
    uk is external control (control parameter),
    P'k is priori error covariance matrix (kalman->error_cov_pre)
    Pk-1 is posteriori error covariance matrix on the previous step (kalman->error_cov_post)
                    (should be initialized somehow in the beginning, identity matrix by default),

The function returns the estimated state.

### KalmanCorrect

Adjusts model state

    const CvMat* cvKalmanCorrect( CvKalman* kalman, const CvMat* measurement );
    #define cvKalmanUpdateByMeasurement cvKalmanCorrect

kalman
  Pointer to the structure to be updated.
measurement
  Pointer to the structure CvMat containing the measurement vector.

The function cvKalmanCorrect adjusts stochastic model state on the basis of the given measurement of the model state:

    Kk=P'k•HT•(H•P'k•HT+R)-1
    xk=x'k+Kk•(zk-H•x'k)
    Pk=(I-Kk•H)•P'k
    where
    zk - given measurement (mesurement parameter)
    Kk - Kalman "gain" matrix.

The function stores adjusted state at `kalman->state_post` and returns it on output.

#### Example. Using Kalman filter to track a rotating point

    #include "cv.h"
    #include "highgui.h"
    #include <math.h>

    int main(int argc, char** argv)
    {
        /* A matrix data */
        const float A[] = { 1, 1, 0, 1 };

        IplImage* img = cvCreateImage( cvSize(500,500), 8, 3 );
        CvKalman* kalman = cvCreateKalman( 2, 1, 0 );
        /* state is (phi, delta_phi) - angle and angle increment */
        CvMat* state = cvCreateMat( 2, 1, CV_32FC1 );
        CvMat* process_noise = cvCreateMat( 2, 1, CV_32FC1 );
        /* only phi (angle) is measured */
        CvMat* measurement = cvCreateMat( 1, 1, CV_32FC1 );
        CvRandState rng;
        int code = -1;

        cvRandInit( &rng, 0, 1, -1, CV_RAND_UNI );

        cvZero( measurement );
        cvNamedWindow( "Kalman", 1 );

        for(;;)
        {
            cvRandSetRange( &rng, 0, 0.1, 0 );
            rng.disttype = CV_RAND_NORMAL;

            cvRand( &rng, state );

            memcpy( kalman->transition_matrix->data.fl, A, sizeof(A));
            cvSetIdentity( kalman->measurement_matrix, cvRealScalar(1) );
            cvSetIdentity( kalman->process_noise_cov, cvRealScalar(1e-5) );
            cvSetIdentity( kalman->measurement_noise_cov, cvRealScalar(1e-1) );
            cvSetIdentity( kalman->error_cov_post, cvRealScalar(1));
            /* choose random initial state */
            cvRand( &rng, kalman->state_post );

            rng.disttype = CV_RAND_NORMAL;

            for(;;)
            {
                #define calc_point(angle)                                      \
                    cvPoint( cvRound(img->width/2 + img->width/3*cos(angle)),  \
                             cvRound(img->height/2 - img->width/3*sin(angle)))

                float state_angle = state->data.fl[0];
                CvPoint state_pt = calc_point(state_angle);

                /* predict point position */
                const CvMat* prediction = cvKalmanPredict( kalman, 0 );
                float predict_angle = prediction->data.fl[0];
                CvPoint predict_pt = calc_point(predict_angle);
                float measurement_angle;
                CvPoint measurement_pt;

                cvRandSetRange( &rng, 0, sqrt(kalman->measurement_noise_cov->data.fl[0]), 0 );
                cvRand( &rng, measurement );

                /* generate measurement */
                cvMatMulAdd( kalman->measurement_matrix, state, measurement, measurement );

                measurement_angle = measurement->data.fl[0];
                measurement_pt = calc_point(measurement_angle);

                /* plot points */
                #define draw_cross( center, color, d )                                 \
                    cvLine( img, cvPoint( center.x - d, center.y - d ),                \
                                 cvPoint( center.x + d, center.y + d ), color, 1, 0 ); \
                    cvLine( img, cvPoint( center.x + d, center.y - d ),                \
                                 cvPoint( center.x - d, center.y + d ), color, 1, 0 )

                cvZero( img );
                draw_cross( state_pt, CV_RGB(255,255,255), 3 );
                draw_cross( measurement_pt, CV_RGB(255,0,0), 3 );
                draw_cross( predict_pt, CV_RGB(0,255,0), 3 );
                cvLine( img, state_pt, predict_pt, CV_RGB(255,255,0), 3, 0 );

                /* adjust Kalman filter state */
                cvKalmanCorrect( kalman, measurement );

                cvRandSetRange( &rng, 0, sqrt(kalman->process_noise_cov->data.fl[0]), 0 );
                cvRand( &rng, process_noise );
                cvMatMulAdd( kalman->transition_matrix, state, process_noise, state );

                cvShowImage( "Kalman", img );
                code = cvWaitKey( 100 );

                if( code > 0 ) /* break current simulation by pressing a key */
                    break;
            }
            if( code == 27 ) /* exit by ESCAPE */
                break;
        }

        return 0;
    }

### CvConDensation

ConDenstation state

        typedef struct CvConDensation
        {
            int MP;     //Dimension of measurement vector
            int DP;     // Dimension of state vector
            float* DynamMatr;       // Matrix of the linear Dynamics system
            float* State;           // Vector of State
            int SamplesNum;         // Number of the Samples
            float** flSamples;      // array of the Sample Vectors
            float** flNewSamples;   // temporary array of the Sample Vectors
            float* flConfidence;    // Confidence for each Sample
            float* flCumulative;    // Cumulative confidence
            float* Temp;            // Temporary vector
            float* RandomSample;    // RandomVector to update sample set
            CvRandState* RandS;     // Array of structures to generate random vectors
        } CvConDensation;

The structure CvConDensation stores CONditional DENSity propagATION tracker state. The information about the algorithm can be found at http://www.dai.ed.ac.uk/CVonline/LOCAL_COPIES/ISARD1/condensation.html

### CreateConDensation

Allocates ConDensation filter structure

    CvConDensation* cvCreateConDensation( int dynam_params, int measure_params, int sample_count );

dynam_params
  Dimension of the state vector.
measure_params
  Dimension of the measurement vector.
sample_count
  Number of samples.

The function cvCreateConDensation creates CvConDensation structure and returns pointer to the structure.

### ReleaseConDensation

Deallocates ConDensation filter structure

    void cvReleaseConDensation( CvConDensation** condens );

condens
  Pointer to the pointer to the structure to be released.

The function cvReleaseConDensation releases the structure CvConDensation (see cvConDensation) and frees all memory previously allocated for the structure.

### ConDensInitSampleSet

Initializes sample set for ConDensation algorithm

    void cvConDensInitSampleSet( CvConDensation* condens, CvMat* lower_bound, CvMat* upper_bound );

condens
  Pointer to a structure to be initialized.
lower_bound
  Vector of the lower boundary for each dimension.
upper_bound
  Vector of the upper boundary for each dimension.

The function cvConDensInitSampleSet fills the samples arrays in the structure CvConDensation with values within specified ranges.

### ConDensUpdateByTime

Estimates subsequent model state

    void cvConDensUpdateByTime( CvConDensation* condens );

condens
  Pointer to the structure to be updated.

The function cvConDensUpdateByTime estimates the subsequent stochastic model state from its current state.

# Pattern Recognition

## Object Detection

The object detector described below has been initially proposed by Paul Viola [Viola01] and improved by Rainer Lienhart [Lienhart02]. First, a classifier (namely a `cascade of boosted classifiers working with haar-like features`) is trained with a few hundreds of sample views of a particular object (i.e., a face or a car), called positive examples, that are scaled to the same size (say, 20x20), and negative examples - arbitrary images of the same size.

After a classifier is trained, it can be applied to a region of interest (of the same size as used during the training) in an input image. The classifier outputs a "1" if the region is likely to show the object (i.e., face/car), and "0" otherwise. To search for the object in the whole image one can move the search window across the image and check every location using the classifier. The classifier is designed so that it can be easily "resized" in order to be able to find the objects of interest at different sizes, which is more efficient than resizing the image itself. So, to find an object of an unknown size in the image the scan procedure should be done several times at different scales.

The word "cascade" in the classifier name means that the resultant classifier consists of several simpler classifiers (`stages`) that are applied subsequently to a region of interest until at some stage the candidate is rejected or all the stages are passed. The word "boosted" means that the classifiers at every stage of the cascade are complex themselves and they are built out of basic classifiers using one of four different `boosting` techniques (weighted voting). Currently Discrete Adaboost, Real Adaboost, Gentle Adaboost and Logitboost are supported. The basic classifiers are decision-tree classifiers with at least 2 leaves. Haar-like features are the input to the basic classifers, and are calculated as described below. The current algorithm uses the following Haar-like features:

The feature used in a particular classifier is specified by its shape (1a, 2b etc.), position within the region of interest and the scale (this scale is not the same as the scale used at the detection stage, though these two scales are multiplied). For example, in case of the third line feature (2c) the response is calculated as the difference between the sum of image pixels under the rectangle covering the whole feature (including the two white stripes and the black stripe in the middle) and the sum of the image pixels under the black stripe multiplied by 3 in order to compensate for the differences in the size of areas. The sums of pixel values over a rectangular regions are calculated rapidly using integral images (see below and cvIntegral description).

To see the object detector at work, have a look at HaarFaceDetect demo.

The following reference is for the detection part only. There is a separate application called `haartraining` that can train a cascade of boosted classifiers from a set of samples. See `opencv/apps/haartraining` for details.

### CvHaarFeature, CvHaarClassifier, CvHaarStageClassifier, CvHaarClassifierCascade

Boosted Haar classifier structures

    #define CV_HAAR_FEATURE_MAX  3

    /* a haar feature consists of 2-3 rectangles with appropriate weights */
    typedef struct CvHaarFeature
    {
        int  tilted;  /* 0 means up-right feature, 1 means 45--rotated feature */

        /* 2-3 rectangles with weights of opposite signs and
           with absolute values inversely proportional to the areas of the rectangles.
           if rect[2].weight !=0, then
           the feature consists of 3 rectangles, otherwise it consists of 2 */
        struct
        {
            CvRect r;
            float weight;
        } rect[CV_HAAR_FEATURE_MAX];
    }
    CvHaarFeature;

    /* a single tree classifier (stump in the simplest case) that returns the response for the feature
       at the particular image location (i.e. pixel sum over subrectangles of the window) and gives out
       a value depending on the responce */
    typedef struct CvHaarClassifier
    {
        int count;  /* number of nodes in the decision tree */

        /* these are "parallel" arrays. Every index i
           corresponds to a node of the decision tree (root has 0-th index).

           left[i] - index of the left child (or negated index if the left child is a leaf)
           right[i] - index of the right child (or negated index if the right child is a leaf)
           threshold[i] - branch threshold. if feature responce is <= threshold, left branch
                          is chosen, otherwise right branch is chosed.
           alpha[i] - output value correponding to the leaf. */
        CvHaarFeature* haar_feature;
        float* threshold;
        int* left;
        int* right;
        float* alpha;
    }
    CvHaarClassifier;

    /* a boosted battery of classifiers(=stage classifier):
       the stage classifier returns 1
       if the sum of the classifiers' responces
       is greater than threshold and 0 otherwise */
    typedef struct CvHaarStageClassifier
    {
        int  count;  /* number of classifiers in the battery */
        float threshold; /* threshold for the boosted classifier */
        CvHaarClassifier* classifier; /* array of classifiers */

        /* these fields are used for organizing trees of stage classifiers,
           rather than just stright cascades */
        int next;
        int child;
        int parent;
    }
    CvHaarStageClassifier;

    typedef struct CvHidHaarClassifierCascade CvHidHaarClassifierCascade;

    /* cascade or tree of stage classifiers */
    typedef struct CvHaarClassifierCascade
    {
        int  flags; /* signature */
        int  count; /* number of stages */
        CvSize orig_window_size; /* original object size (the cascade is trained for) */

        /* these two parameters are set by cvSetImagesForHaarClassifierCascade */
        CvSize real_window_size; /* current object size */
        double scale; /* current scale */
        CvHaarStageClassifier* stage_classifier; /* array of stage classifiers */
        CvHidHaarClassifierCascade* hid_cascade; /* hidden optimized representation of the cascade,
                                                    created by cvSetImagesForHaarClassifierCascade */
    }
    CvHaarClassifierCascade;

All the structures are used for representing a cascaded of boosted Haar classifiers. The cascade has the following hierarchical structure:

        Cascade:
            Stage1:
                Classifier11:
                    Feature11
                Classifier12:
                    Feature12
                ...
            Stage2:
                Classifier21:
                    Feature21
                ...
            ...

The whole hierarchy can be constructed manually or loaded from a file or an embedded base using function cvLoadHaarClassifierCascade.

### cvLoadHaarClassifierCascade

Loads a trained cascade classifier from file or the classifier database embedded in OpenCV

    CvHaarClassifierCascade* cvLoadHaarClassifierCascade(
                             const char* directory,
                             CvSize orig_window_size );

directory
  Name of directory containing the description of a trained cascade classifier.
orig_window_size
  Original size of objects the cascade has been trained on. Note that it is not stored in the cascade and therefore must be specified separately.

The function cvLoadHaarClassifierCascade loads a trained cascade of haar classifiers from a file or the classifier database embedded in OpenCV. The base can be trained using `haartraining` application (see opencv/apps/haartraining for details).

The function is obsolete. Nowadays object detection classifiers are stored in XML or YAML files, rather than in directories. To load cascade from a file, use cvLoad function.

### cvReleaseHaarClassifierCascade

Releases haar classifier cascade

    void cvReleaseHaarClassifierCascade( CvHaarClassifierCascade** cascade );

cascade
  Double pointer to the released cascade. The pointer is cleared by the function.

The function cvReleaseHaarClassifierCascade deallocates the cascade that has been created manually or loaded using cvLoadHaarClassifierCascade or cvLoad.

### cvHaarDetectObjects

Detects objects in the image

    typedef struct CvAvgComp
    {
        CvRect rect; /* bounding rectangle for the object (average rectangle of a group) */
        int neighbors; /* number of neighbor rectangles in the group */
    }
    CvAvgComp;

    CvSeq* cvHaarDetectObjects( const CvArr* image, CvHaarClassifierCascade* cascade,
                                CvMemStorage* storage, double scale_factor=1.1,
                                int min_neighbors=3, int flags=0,
                                CvSize min_size=cvSize(0,0) );

image
  Image to detect objects in.
cascade
  Haar classifier cascade in internal representation.
storage
  Memory storage to store the resultant sequence of the object candidate rectangles.
scale_factor
  The factor by which the search window is scaled between the subsequent scans, for example, 1.1 means increasing window by 10%.
min_neighbors
  Minimum number (minus 1) of neighbor rectangles that makes up an object. All the groups of a smaller number of rectangles than `min_neighbors`-1 are rejected. If `min_neighbors` is 0, the function does not any grouping at all and returns all the detected candidate rectangles, which may be useful if the user wants to apply a customized grouping procedure.
flags
  Mode of operation. Currently the only flag that may be specified is `CV_HAAR_DO_CANNY_PRUNING`. If it is set, the function uses Canny edge detector to reject some image regions that contain too few or too much edges and thus can not contain the searched object. The particular threshold values are tuned for face detection and in this case the pruning speeds up the processing.
min_size
  Minimum window size. By default, it is set to the size of samples the classifier has been trained on (~20×20 for face detection).

The function cvHaarDetectObjects finds rectangular regions in the given image that are likely to contain objects the cascade has been trained for and returns those regions as a sequence of rectangles. The function scans the image several times at different scales (see cvSetImagesForHaarClassifierCascade). Each time it considers overlapping regions in the image and applies the classifiers to the regions using cvRunHaarClassifierCascade. It may also apply some heuristics to reduce number of analyzed regions, such as Canny prunning. After it has proceeded and collected the candidate rectangles (regions that passed the classifier cascade), it groups them and returns a sequence of average rectangles for each large enough group. The default parameters (`scale_factor`=1.1, `min_neighbors`=3, `flags`=0) are tuned for accurate yet slow object detection. For a faster operation on real video images the settings are: `scale_factor`=1.2, `min_neighbors`=2, `flags`=CV_HAAR_DO_CANNY_PRUNING, `min_size`=<minimum possible face size> (for example, ~1/4 to 1/16 of the image area in case of video conferencing).

#### Example. Using cascade of Haar classifiers to find objects (e.g. faces).

    #include "cv.h"
    #include "highgui.h"

    CvHaarClassifierCascade* load_object_detector( const char* cascade_path )
    {
        return (CvHaarClassifierCascade*)cvLoad( cascade_path );
    }

    void detect_and_draw_objects( IplImage* image,
                                  CvHaarClassifierCascade* cascade,
                                  int do_pyramids )
    {
        IplImage* small_image = image;
        CvMemStorage* storage = cvCreateMemStorage(0);
        CvSeq* faces;
        int i, scale = 1;

        /* if the flag is specified, down-scale the input image to get a
           performance boost w/o loosing quality (perhaps) */
        if( do_pyramids )
        {
            small_image = cvCreateImage( cvSize(image->width/2,image->height/2), IPL_DEPTH_8U, 3 );
            cvPyrDown( image, small_image, CV_GAUSSIAN_5x5 );
            scale = 2;
        }

        /* use the fastest variant */
        faces = cvHaarDetectObjects( small_image, cascade, storage, 1.2, 2, CV_HAAR_DO_CANNY_PRUNING );

        /* draw all the rectangles */
        for( i = 0; i < faces->total; i++ )
        {
            /* extract the rectanlges only */
            CvRect face_rect = *(CvRect*)cvGetSeqElem( faces, i, 0 );
            cvRectangle( image, cvPoint(face_rect.x*scale,face_rect.y*scale),
                         cvPoint((face_rect.x+face_rect.width)*scale,
                                 (face_rect.y+face_rect.height)*scale),
                         CV_RGB(255,0,0), 3 );
        }

        if( small_image != image )
            cvReleaseImage( &small_image );
        cvReleaseMemStorage( &storage );
    }

    /* takes image filename and cascade path from the command line */
    int main( int argc, char** argv )
    {
        IplImage* image;
        if( argc==3 && (image = cvLoadImage( argv[1], 1 )) != 0 )
        {
            CvHaarClassifierCascade* cascade = load_object_detector(argv[2]);
            detect_and_draw_objects( image, cascade, 1 );
            cvNamedWindow( "test", 0 );
            cvShowImage( "test", image );
            cvWaitKey(0);
            cvReleaseHaarClassifierCascade( &cascade );
            cvReleaseImage( &image );
        }

        return 0;
    }

### cvSetImagesForHaarClassifierCascade

Assigns images to the hidden cascade

    void cvSetImagesForHaarClassifierCascade( CvHaarClassifierCascade* cascade,
                                              const CvArr* sum, const CvArr* sqsum,
                                              const CvArr* tilted_sum, double scale );

cascade
  Hidden Haar classifier cascade, created by cvCreateHidHaarClassifierCascade.
sum
  Integral (sum) single-channel image of 32-bit integer format. This image as well as the two subsequent images are used for fast feature evaluation and brightness/contrast normalization. They all can be retrieved from input 8-bit or floating point single-channel image using the function cvIntegral.
sqsum
  Square sum single-channel image of 64-bit floating-point format.
tilted_sum
  Tilted sum single-channel image of 32-bit integer format.
scale
  Window scale for the cascade. If `scale`=1, original window size is used (objects of that size are searched) - the same size as specified in cvLoadHaarClassifierCascade (24x24 in case of "<default_face_cascade>"), if `scale`=2, a two times larger window is used (48x48 in case of default face cascade). While this will speed-up search about four times, faces smaller than 48x48 cannot be detected.

The function cvSetImagesForHaarClassifierCascade assigns images and/or window scale to the hidden classifier cascade. If image pointers are NULL, the previously set images are used further (i.e. NULLs mean "do not change images"). Scale parameter has no such a "protection" value, but the previous value can be retrieved by cvGetHaarClassifierCascadeScale function and reused again. The function is used to prepare cascade for detecting object of the particular size in the particular image. The function is called internally by cvHaarDetectObjects, but it can be called by user if there is a need in using lower-level function cvRunHaarClassifierCascade.

### cvRunHaarClassifierCascade

Runs cascade of boosted classifier at given image location

    int cvRunHaarClassifierCascade( CvHaarClassifierCascade* cascade,
                                    CvPoint pt, int start_stage=0 );

cascade
  Haar classifier cascade.
pt
  Top-left corner of the analyzed region. Size of the region is a original window size scaled by the currenly set scale. The current window size may be retrieved using cvGetHaarClassifierCascadeWindowSize function.
start_stage
  Initial zero-based index of the cascade stage to start from. The function assumes that all the previous stages are passed. This feature is used internally by cvHaarDetectObjects for better processor cache utilization.

The function cvRunHaarHaarClassifierCascade runs Haar classifier cascade at a single image location. Before using this function the integral images and the appropriate scale (=> window size) should be set using cvSetImagesForHaarClassifierCascade. The function returns positive value if the analyzed rectangle passed all the classifier stages (it is a candidate) and zero or negative value otherwise.

# Camera Calibration and 3D Reconstruction

## Camera Calibration

### CalibrateCamera

Calibrates camera with single precision

    void cvCalibrateCamera( int image_count, int* point_counts, CvSize image_size,
                            CvPoint2D32f* image_points, CvPoint3D32f* object_points,
                            CvVect32f distortion_coeffs, CvMatr32f camera_matrix,
                            CvVect32f translation_vectors, CvMatr32f rotation_matrixes,
                            int use_intrinsic_guess );

image_count
  Number of the images.
point_counts
  Array of the number of points in each image.
image_size
  Size of the image.
image_points
  Pointer to the images.
object_points
  Pointer to the pattern.
distortion_coeffs
  Array of four distortion coefficients found.
camera_matrix
  Camera matrix found.
translation_vectors
  Array of translate vectors for each pattern position in the image.
rotation_matrixes
  Array of the rotation matrix for each pattern position in the image.
use_intrinsic_guess
  Intrinsic guess. If equal to 1, intrinsic guess is needed.

The function cvCalibrateCamera calculates the camera parameters using information points on the pattern object and pattern object images.

### CalibrateCamera_64d

Calibrates camera with double precision

    void cvCalibrateCamera_64d( int image_count, int* point_counts, CvSize image_size,
                                CvPoint2D64d* image_points, CvPoint3D64d* object_points,
                                CvVect64d distortion_coeffs, CvMatr64d camera_matrix,
                                CvVect64d translation_vectors, CvMatr64d rotation_matrixes,
                                int use_intrinsic_guess );

image_count
  Number of the images.
point_counts
  Array of the number of points in each image.
image_size
  Size of the image.
image_points
  Pointer to the images.
object_points
  Pointer to the pattern.
distortion_coeffs
  Distortion coefficients found.
camera_matrix
  Camera matrix found.
translation_vectors
  Array of the translate vectors for each pattern position on the image.
rotation_matrixes
  Array of the rotation matrix for each pattern position on the image.
use_intrinsic_guess
  Intrinsic guess. If equal to 1, intrinsic guess is needed.

The function cvCalibrateCamera_64d is basically the same as the function cvCalibrateCamera, but uses double precision.

### Rodrigues

Converts rotation matrix to rotation vector and vice versa with single precision

    void  cvRodrigues( CvMat* rotation_matrix, CvMat* rotation_vector,
                       CvMat* jacobian, int conv_type);

rotation_matrix
  Rotation matrix (3x3), 32-bit or 64-bit floating point.
rotation_vector
  Rotation vector (3x1 or 1x3) of the same type as `rotation_matrix`.
jacobian
  Jacobian matrix 3 × 9.
conv_type
  Type of conversion; must be `CV_RODRIGUES_M2V` for converting the matrix to the vector or `CV_RODRIGUES_V2M` for converting the vector to the matrix.

The function cvRodrigues converts the rotation matrix to the rotation vector or vice versa.

### UnDistortOnce

Corrects camera lens distortion

    void cvUnDistortOnce( const CvArr* src, CvArr* dst,
                          const float* intrinsic_matrix,
                          const float* distortion_coeffs,
                          int interpolate=1 );

src
  Source (distorted) image.
dst
  Destination (corrected) image.
intrinsic_matrix
  Matrix of the camera intrinsic parameters (3x3).
distortion_coeffs
  Vector of the four distortion coefficients `k1, k2, p1` and `p2`.
interpolate
  Bilinear interpolation flag.

The function cvUnDistortOnce corrects camera lens distortion in case of a single image. Matrix of the camera intrinsic parameters and distortion coefficients `k1, k2 , p1` , and `p2` must be preliminarily calculated by the function cvCalibrateCamera.

### UnDistortInit

Calculates arrays of distorted points indices and interpolation coefficients

    void cvUnDistortInit( const CvArr* src, CvArr* undistortion_map,
                          const float* intrinsic_matrix,
                          const float* distortion_coeffs,
                          int interpolate=1 );

src
  Artibtrary source (distorted) image, the image size and number of channels do matter.
undistortion_map
  32-bit integer image of the same size as the source image (if `interpolate=0`) or 3 times wider than the source image (if `interpolate=1`).
intrinsic_matrix
  Matrix of the camera intrinsic parameters.
distortion_coeffs
  Vector of the 4 distortion coefficients `k1, k2, p1` and `p2`.
interpolate
  Bilinear interpolation flag.

The function cvUnDistortInit calculates arrays of distorted points indices and interpolation coefficients using known matrix of the camera intrinsic parameters and distortion coefficients. It calculates undistortion map for cvUnDistort.

Matrix of the camera intrinsic parameters and the distortion coefficients may be calculated by cvCalibrateCamera.

### UnDistort

Corrects camera lens distortion

    void cvUnDistort( const CvArr* src, CvArr* dst,
                      const CvArr* undistortion_map, int interpolate=1 );

src
  Source (distorted) image.
dst
  Destination (corrected) image.
undistortion_map
  Undistortion map, pre-calculated by cvUnDistortInit.
interpolate
  Bilinear interpolation flag, the same as in cvUnDistortInit.

The function cvUnDistort corrects camera lens distortion using previously calculated undistortion map. It is faster than cvUnDistortOnce.

### FindChessBoardCornerGuesses

Finds approximate positions of internal corners of the chessboard

    int cvFindChessBoardCornerGuesses( const CvArr* image, CvArr* thresh,
                                       CvMemStorage* storage, CvSize board_size,
                                       CvPoint2D32f* corners, int* corner_count=NULL );

image
  Source chessboard view; must have the depth of `IPL_DEPTH_8U`.
thresh
  Temporary image of the same size and format as the source image.
storage
  Memory storage for intermediate data. If it is NULL, the function creates a temporary memory storage.
board_size
  Number of inner corners per chessboard row and column. The width (the number of columns) must be less or equal to the height (the number of rows).
corners
  Pointer to the corner array found.
corner_count
  Signed value whose absolute value is the number of corners found. A positive number means that a whole chessboard has been found and a negative number means that not all the corners have been found.

The function cvFindChessBoardCornerGuesses attempts to determine whether the input image is a view of the chessboard pattern and locate internal chessboard corners. The function returns non-zero value if all the corners have been found and they have been placed in a certain order (row by row, left to right in every row), otherwise, if the function fails to find all the corners or reorder them, the function returns 0. For example, a simple chessboard has 8 x 8 squares and 7 x 7 internal corners, that is, points, where the squares are tangent. The word "approximate" in the above description means that the corner coordinates found may differ from the actual coordinates by a couple of pixels. To get more precise coordinates, the user may use the function cvFindCornerSubPix.

## Pose Estimation

### FindExtrinsicCameraParams

Finds extrinsic camera parameters for pattern

    void cvFindExtrinsicCameraParams( int point_count, CvSize image_size,
                                      CvPoint2D32f* image_points, CvPoint3D32f* object_points,
                                      CvVect32f focal_length, CvPoint2D32f principal_point,
                                      CvVect32f distortion_coeffs, CvVect32f rotation_vector,
                                      CvVect32f translation_vector );

point_count
  Number of the points.
ImageSize
  Size of the image.
image_points
  Pointer to the image.
object_points
  Pointer to the pattern.
focal_length
  Focal length.
principal_point
  Principal point.
distortion_coeffs
  Distortion coefficients.
rotation_vector
  Rotation vector.
translation_vector
  Translate vector.

The function cvFindExtrinsicCameraParams finds the extrinsic parameters for the pattern.

### FindExtrinsicCameraParams_64d

Finds extrinsic camera parameters for pattern with double precision

    void cvFindExtrinsicCameraParams_64d( int point_count, CvSize image_size,
                                          CvPoint2D64d* image_points, CvPoint3D64d* object_points,
                                          CvVect64d focal_length, CvPoint2D64d principal_point,
                                          CvVect64d distortion_coeffs, CvVect64d rotation_vector,
                                          CvVect64d translation_vector );

point_count
  Number of the points.
ImageSize
  Size of the image.
image_points
  Pointer to the image.
object_points
  Pointer to the pattern.
focal_length
  Focal length.
principal_point
  Principal point.
distortion_coeffs
  Distortion coefficients.
rotation_vector
  Rotation vector.
translation_vector
  Translate vector.

The function cvFindExtrinsicCameraParams_64d finds the extrinsic parameters for the pattern with double precision.

### CreatePOSITObject

Initializes structure containing object information

    CvPOSITObject* cvCreatePOSITObject( CvPoint3D32f* points, int point_count );

points
  Pointer to the points of the 3D object model.
point_count
  Number of object points.

The function cvCreatePOSITObject allocates memory for the object structure and computes the object inverse matrix.

The preprocessed object data is stored in the structure CvPOSITObject, internal for OpenCV, which means that the user cannot directly access the structure data. The user may only create this structure and pass its pointer to the function.

Object is defined as a set of points given in a coordinate system. The function cvPOSIT computes a vector that begins at a camera-related coordinate system center and ends at the `points[0]` of the object.

Once the work with a given object is finished, the function cvReleasePOSITObject must be called to free memory.

### POSIT

Implements POSIT algorithm

    void cvPOSIT( CvPOSITObject* posit_object, CvPoint2D32f* image_points, double focal_length,
                  CvTermCriteria criteria, CvMatr32f rotation_matrix, CvVect32f translation_vector );

posit_object
  Pointer to the object structure.
image_points
  Pointer to the object points projections on the 2D image plane.
focal_length
  Focal length of the camera used.
criteria
  Termination criteria of the iterative POSIT algorithm.
rotation_matrix
  Matrix of rotations.
translation_vector
  Translation vector.

The function cvPOSIT implements POSIT algorithm. Image coordinates are given in a camera-related coordinate system. The focal length may be retrieved using camera calibration functions. At every iteration of the algorithm new perspective projection of estimated pose is computed.

Difference norm between two projections is the maximal distance between corresponding points. The parameter `criteria.epsilon` serves to stop the algorithm if the difference is small.

### ReleasePOSITObject

Deallocates 3D object structure

    void cvReleasePOSITObject( CvPOSITObject** posit_object );

posit_object
  Double pointer to `CvPOSIT` structure.

The function cvReleasePOSITObject releases memory previously allocated by the function cvCreatePOSITObject.

### CalcImageHomography

Calculates homography matrix for oblong planar object (e.g. arm)

    void cvCalcImageHomography( float* line, CvPoint3D32f* center,
                                float* intrinsic, float* homography );

line
  the main object axis direction (vector (dx,dy,dz)).
center
  object center ((cx,cy,cz)).
intrinsic
  intrinsic camera parameters (3x3 matrix).
homography
  output homography matrix (3x3).

The function cvCalcImageHomography calculates the homography matrix for the initial image transformation from image plane to the plane, defined by 3D oblong object line (See Figure 6-10 in OpenCV Guide 3D Reconstruction Chapter).

## Epipolar Geometry

### FindFundamentalMat

Calculates fundamental matrix from corresponding points in two images

    int cvFindFundamentalMat( CvMat* points1,
                              CvMat* points2,
                              CvMat* fundamental_matrix,
                              int    method,
                              double param1,
                              double param2,
                              CvMat* status=0);

points1
  Array of the first image points of 2xN/Nx2 or 3xN/Nx3 size (N is number of points). The point coordinates should be floating-point (single or double precision)
points2
  Array of the second image points of the same size and format as `points1`
fundamental_matrix
  The output fundamental matrix or matrices. Size 3x3 or 9x3 (7-point method can returns up to 3 matrices).
method
  Method for computing fundamental matrix
  CV_FM_7POINT - for 7-point algorithm. Number of points == 7
  CV_FM_8POINT - for 8-point algorithm. Number of points >= 8
  CV_FM_RANSAC - for RANSAC algorithm. Number of points >= 8
  CV_FM_LMEDS - for LMedS algorithm. Number of points >= 8
param1
  The parameter is used for RANSAC or LMedS methods only. It is the maximum distance from point to epipolar line, beyound which the point is considered bad and is not considered in further calculations. Usually it is set to 0.5 or 1.0.
param2
  The parameter is used for RANSAC or LMedS methods only. It denotes the desirable level of confidense the matrix is the correct (up to some precision). It can be set to 0.99 for example.
status
  Array of N elements, every element of which is set to 1 if the point was not rejected during the computation, 0 otherwise. The array is computed only in RANSAC and LMedS methods. For other methods it is set to all 1's. This is the optional parameter.

The epipolar geometry is described by the following equation:

    p2T*F*p1=0,

where `F` is fundamental matrix, `p1` and `p2` are corresponding points on the two images.

The function `FindFundamentalMat` calculates fundamental matrix using one of four methods listed above and returns the number of fundamental matrix found: 0 if the matrix could not be found, 1 or 3 if the matrix or matrices have been found successfully.

The calculated fundamental matrix may be passed further to `ComputeCorrespondEpilines` function that computes coordinates of corresponding epilines on two images.

For 7-point method uses exactly 7 points. It can find 1 or 3 fundamental matrices. It returns number of the matrices found and if there is a room in the destination array to keep all the detected matrices, stores all of them there, otherwise it stores only one of the matrices.

All other methods use 8 or more points and return a single fundamental matrix.

#### Example. Fundamental matrix calculation

    int point_count = 100;
    CvMat* points1;
    CvMat* points2;
    CvMat* status;
    CvMat* fundamental_matrix;

    points1  = cvCreateMat(2,point_count,CV_32F);
    points2  = cvCreateMat(2,point_count,CV_32F);
    status   = cvCreateMat(1,point_count,CV_32F);

    /* Fill the points here ... */

    fundamental_matrix = cvCreateMat(3,3,CV_32F);
    int num = cvFindFundamentalMat(points1,points2,fundamental_matrix,CV_FM_RANSAC,1.0,0.99,status);
    if( num == 1 )
    {
        printf("Fundamental matrix was found\n");
    }
    else
    {
        printf("Fundamental matrix was not found\n");
    }

    /*====== Example of code for three matrixes ======*/
    CvMat* points1;
    CvMat* points2;
    CvMat* fundamental_matrix;

    points1  = cvCreateMat(2,7,CV_32F);
    points2  = cvCreateMat(2,7,CV_32F);

    /* Fill the points here... */

    fundamental_matrix = cvCreateMat(9,3,CV_32F);
    int num = cvFindFundamentalMat(points1,points2,fundamental_matrix,CV_FM_7POINT,0,0,0);
    printf("Found %d matrixes\n",num);

### ComputeCorrespondEpilines

For points in one image of stereo pair computes the corresponding epilines in the other image

    void cvComputeCorrespondEpilines( const CvMat* points,
                                      int which_image,
                                      const CvMat* fundamental_matrix,
                                      CvMat* correspondent_lines);

points
  The input points: 2xN or 3xN array (N number of points)
which_image
  Image index (1 or 2) that contains the `points`
fundamental_matrix
  Fundamental matrix
correspondent_lines
  Computed epilines, 3xN array

The function `ComputeCorrespondEpilines` computes the corresponding epiline for every input point using the basic equation of epipolar line geometry. If the points are located in the first image (`which_image`=1), the corresponding epipolar line can be computed as:

    l2=F*p1

where `F` is the fundamental matrix, `p1` is a point in the first image, `l2` - the corresponding epipolar line in the second image. If the points are located on the second image (`which_image`=2):

    l1=FT*p2

where `p2` is a point in the second image, `l1` is the corresponding epipolar line in the first image

Each epipolar line is represented by 3 coefficients a, b, c:

    a*x + b*y + c = 0

The normalized coefficents (`a2+b2=1`) of every corresponding epipolar line are stored into `correspondent_lines`.

# Alphabetical List of Functions

### 2

2DRotationMatrix |  |

### A

Acc | ApproxChains | ArcLength
AdaptiveThreshold | ApproxPoly |

### B

BoundingRect | BoxPoints |

### C

CalcBackProject | CalibrateCamera | ConvexityDefects
CalcBackProjectPatch | CalibrateCamera_64d | CopyHist
CalcEMD2 | CamShift | CornerEigenValsAndVecs
CalcGlobalOrientation | Canny | CornerMinEigenVal
CalcHist | CheckContourConvexity | CreateConDensation
CalcImageHomography | ClearHist | CreateContourTree
CalcMotionGradient | ClearSubdivVoronoi2D | CreateHist
CalcOpticalFlowBM | CompareHist | CreateKalman
CalcOpticalFlowHS | ComputeCorrespondEpilines | CreatePOSITObject
CalcOpticalFlowLK | ConDensInitSampleSet | CreateStructuringElementEx
CalcOpticalFlowPyrLK | ConDensUpdateByTime | CreateSubdivDelaunay2D
CalcPGH | ContourArea | CvtColor
CalcProbDensity | ContourFromContourTree |
CalcSubdivVoronoi2D | ConvexHull2 |

### D

Dilate | DistTransform |

### E

EndFindContours | Erode |

### F

Filter2D | FindExtrinsicCameraParams | FindNextContour
FindChessBoardCornerGuesses | FindExtrinsicCameraParams_64d | FitEllipse
FindContours | FindFundamentalMat | FitLine2D
FindCornerSubPix | FindNearestPoint2D | FloodFill

### G

GetCentralMoment | GetMinMaxHistValue | GetRectSubPix
GetHistValue_1D | GetNormalizedCentralMoment | GetSpatialMoment
GetHuMoments | GetQuadrangleSubPix | GoodFeaturesToTrack

### H

HaarDetectObjects | HoughLines |

### I

InitLineIterator | Integral |

### K

KalmanCorrect | KalmanPredict |

### L

Laplace | LoadHaarClassifierCascade |

### M

MakeHistHeaderForArray | MaxRect | Moments
MatchContourTrees | MeanShift | MorphologyEx
MatchShapes | MinAreaRect2 | MultiplyAcc
MatchTemplate | MinEnclosingCircle |

### N

NormalizeHist |  |

### P

POSIT | PyrDown | PyrUp
PreCornerDetect | PyrSegmentation |

### Q

QueryHistValue_1D |  |

### R

ReadChainPoint | ReleaseKalman | Rodrigues
ReleaseConDensation | ReleasePOSITObject | RunHaarClassifierCascade
ReleaseHaarClassifierCascade | ReleaseStructuringElement | RunningAvg
ReleaseHist | Resize |

### S

SampleLine | Sobel | Subdiv2DGetEdge
SegmentMotion | SquareAcc | Subdiv2DLocate
SetHistBinRanges | StartFindContours | Subdiv2DRotateEdge
SetImagesForHaarClassifierCascade | StartReadChainPoints | SubdivDelaunay2DInsert
Smooth | Subdiv2DEdgeDst | SubstituteContour
SnakeImage | Subdiv2DEdgeOrg |

### T

ThreshHist | Threshold |

### U

UnDistort | UnDistortOnce |
UnDistortInit | UpdateMotionHistory |

### W

WarpAffine | WarpPerspective | WarpPerspectiveQMatrix

# Bibliography

This bibliography provides a list of publications that might be useful to the Intel î Computer Vision Library users. This list is not complete; it serves only as a starting point.

1. [Borgefors86] Gunilla Borgefors, "Distance Transformations in Digital Images". Computer Vision, Graphics and Image Processing 34, 344-371 (1986).
2. [Bouguet00] Jean-Yves Bouguet. Pyramidal Implementation of the Lucas Kanade Feature Tracker.
   The paper is included into OpenCV distribution (algo_tracking.pdf)
3. [Bradski98] G.R. Bradski. Computer vision face tracking as a component of a perceptual user interface. In Workshop on Applications of Computer Vision, pages 214–219, Princeton, NJ, Oct. 1998.
   Updated version can be found at http://www.intel.com/technology/itj/q21998/articles/art_2.htm.
   Also, it is included into OpenCV distribution (camshift.pdf)
4. [Bradski00] G. Bradski and J. Davis. Motion Segmentation and Pose Recognition with Motion History Gradients. IEEE WACV'00, 2000.
5. [Burt81] P. J. Burt, T. H. Hong, A. Rosenfeld. Segmentation and Estimation of Image Region Properties Through Cooperative Hierarchical Computation. IEEE Tran. On SMC, Vol. 11, N.12, 1981, pp. 802-809.
6. [Canny86] J. Canny. A Computational Approach to Edge Detection, IEEE Trans. on Pattern Analysis and Machine Intelligence, 8(6), pp. 679-698 (1986).
7. [Davis97] J. Davis and Bobick. The Representation and Recognition of Action Using Temporal Templates. MIT Media Lab Technical Report 402, 1997.
8. [DeMenthon92] Daniel F. DeMenthon and Larry S. Davis. Model-Based Object Pose in 25 Lines of Code. In Proceedings of ECCV '92, pp. 335-343, 1992.
9. [Fitzgibbon95] Andrew W. Fitzgibbon, R.B.Fisher. A Buyer's Guide to Conic Fitting. Proc.5th British Machine Vision Conference, Birmingham, pp. 513-522, 1995.
10. [Horn81] Berthold K.P. Horn and Brian G. Schunck. Determining Optical Flow. Artificial Intelligence, 17, pp. 185-203, 1981.
11. [Hu62] M. Hu. Visual Pattern Recognition by Moment Invariants, IRE Transactions on Information Theory, 8:2, pp. 179-187, 1962.
12. [Iivarinen97] Jukka Iivarinen, Markus Peura, Jaakko Srel, and Ari Visa. Comparison of Combined Shape Descriptors for Irregular Objects, 8th British Machine Vision Conference, BMVC'97.
    http://www.cis.hut.fi/research/IA/paper/publications/bmvc97/bmvc97.html
13. [Jahne97] B. Jahne. Digital Image Processing. Springer, New York, 1997.
14. [Lucas81] Lucas, B., and Kanade, T. An Iterative Image Registration Technique with an Application to Stereo Vision, Proc. of 7th International Joint Conference on Artificial Intelligence (IJCAI), pp. 674-679.
15. [Kass88] M. Kass, A. Witkin, and D. Terzopoulos. Snakes: Active Contour Models, International Journal of Computer Vision, pp. 321-331, 1988.
16. [Lienhart02] Rainer Lienhart and Jochen Maydt. An Extended Set of Haar-like Features for Rapid Object Detection. IEEE ICIP 2002, Vol. 1, pp. 900-903, Sep. 2002.
    This paper, as well as the extended technical report, can be retrieved at http://www.lienhart.de/Publications/publications.html
17. [Matas98] J.Matas, C.Galambos, J.Kittler. Progressive Probabilistic Hough Transform. British Machine Vision Conference, 1998.
18. [Rosenfeld73] A. Rosenfeld and E. Johnston. Angle Detection on Digital Curves. IEEE Trans. Computers, 22:875-878, 1973.
19. [RubnerJan98] Y. Rubner. C. Tomasi, L.J. Guibas. Metrics for Distributions with Applications to Image Databases. Proceedings of the 1998 IEEE International Conference on Computer Vision, Bombay, India, January 1998, pp. 59-66.
20. [RubnerSept98] Y. Rubner. C. Tomasi, L.J. Guibas. The Earth Mover's Distance as a Metric for Image Retrieval. Technical Report STAN-CS-TN-98-86, Department of Computer Science, Stanford University, September 1998.
21. [RubnerOct98] Y. Rubner. C. Tomasi. Texture Metrics. Proceeding of the IEEE International Conference on Systems, Man, and Cybernetics, San-Diego, CA, October 1998, pp. 4601-4607. http://robotics.stanford.edu/~rubner/publications.html
22. [Serra82] J. Serra. Image Analysis and Mathematical Morphology. Academic Press, 1982.
23. [Schiele00] Bernt Schiele and James L. Crowley. Recognition without Correspondence Using Multidimensional Receptive Field Histograms. In International Journal of Computer Vision 36 (1), pp. 31-50, January 2000.
24. [Suzuki85] S. Suzuki, K. Abe. Topological Structural Analysis of Digital Binary Images by Border Following. CVGIP, v.30, n.1. 1985, pp. 32-46.
25. [Teh89] C.H. Teh, R.T. Chin. On the Detection of Dominant Points on Digital Curves. - IEEE Tr. PAMI, 1989, v.11, No.8, p. 859-872.
26. [Trucco98] Emanuele Trucco, Alessandro Verri. Introductory Techniques for 3-D Computer Vision. Prentice Hall, Inc., 1998.
27. [Viola01] Paul Viola and Michael J. Jones. Rapid Object Detection using a Boosted Cascade of Simple Features. IEEE CVPR, 2001.
    The paper is available online at http://www.ai.mit.edu/people/viola/
28. [Welch95] Greg Welch, Gary Bishop. An Introduction To the Kalman Filter. Technical Report TR95-041, University of North Carolina at Chapel Hill, 1995.
    Online version is available at http://www.cs.unc.edu/~welch/kalman/kalman_filter/kalman.html
29. [Williams92] D. J. Williams and M. Shah. A Fast Algorithm for Active Contours and Curvature Estimation. CVGIP: Image Understanding, Vol. 55, No. 1, pp. 14-26, Jan., 1992. http://www.cs.ucf.edu/~vision/papers/shah/92/WIS92A.pdf.
30. [Yuille89] A.Y.Yuille, D.S.Cohen, and P.W.Hallinan. Feature Extraction from Faces Using Deformable Templates in CVPR, pp. 104-109, 1989.
31. [Zhang96] Z. Zhang. Parameter Estimation Techniques: A Tutorial with Application to Conic Fitting, Image and Vision Computing Journal, 1996.
32. [Zhang99] Z. Zhang. Flexible Camera Calibration By Viewing a Plane From Unknown Orientations. International Conference on Computer Vision (ICCV'99), Corfu, Greece, pages 666-673, September 1999.
33. [Zhang00] Z. Zhang. A Flexible New Technique for Camera Calibration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1330-1334, 2000.