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# vigra/gaussians.hxx

    00001 /************************************************************************/
    00002 /*                                                                      */
    00003 /*               Copyright 1998-2004 by Ullrich Koethe                  */
    00004 /*       Cognitive Systems Group, University of Hamburg, Germany        */
    00005 /*                                                                      */
    00006 /*    This file is part of the VIGRA computer vision library.           */
    00007 /*    ( Version 1.6.0, Aug 13 2008 )                                    */
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    00011 /*        ullrich.koethe@iwr.uni-heidelberg.de    or                    */
    00012 /*        vigra@informatik.uni-hamburg.de                               */
    00013 /*                                                                      */
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    00022 /*                                                                      */
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    00025 /*    Software.                                                         */
    00026 /*                                                                      */
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    00035 /*                                                                      */
    00036 /************************************************************************/
    00037
    00038 #ifndef VIGRA_GAUSSIANS_HXX
    00039 #define VIGRA_GAUSSIANS_HXX
    00040
    00041 #include <cmath>
    00042 #include "config.hxx"
    00043 #include "mathutil.hxx"
    00044 #include "array_vector.hxx"
    00045 #include "error.hxx"
    00046
    00047 namespace vigra {
    00048
    00049 #if 0
    00050 /** \addtogroup MathFunctions Mathematical Functions
    00051 */
    00052 //@{
    00053 #endif
    00054 /*! The Gaussian function and its derivatives.
    00055
    00056     Implemented as a unary functor. Since it supports the <tt>radius()</tt> function
    00057     it can also be used as a kernel in \ref resamplingConvolveImage().
    00058
    00059     <b>\#include</b> <<a href="gaussians_8hxx-source.html">vigra/gaussians.hxx</a>><br>
    00060     Namespace: vigra
    00061
    00062     \ingroup MathFunctions
    00063 */
    00064 template <class T = double>
    00065 class Gaussian
    00066 {
    00067   public:
    00068
    00069         /** the value type if used as a kernel in \ref resamplingConvolveImage().
    00070         */
    00071     typedef T            value_type;
    00072         /** the functor's argument type
    00073         */
    00074     typedef T            argument_type;
    00075         /** the functor's result type
    00076         */
    00077     typedef T            result_type;
    00078
    00079         /** Create functor for the given standard deviation <tt>sigma</tt> and
    00080             derivative order <i>n</i>. The functor then realizes the function
    00081
    00082             \f[ f_{\sigma,n}(x)=\frac{\partial^n}{\partial x^n}
    00083                  \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{x^2}{2\sigma^2}}
    00084             \f]
    00085
    00086             Precondition:
    00087             \code
    00088             sigma > 0.0
    00089             \endcode
    00090         */
    00091     explicit Gaussian(T sigma = 1.0, unsigned int derivativeOrder = 0)
    00092     : sigma_(sigma),
    00093       sigma2_(-0.5 / sigma / sigma),
    00094       norm_(0.0),
    00095       order_(derivativeOrder),
    00096       hermitePolynomial_(derivativeOrder / 2 + 1)
    00097     {
    00098         vigra_precondition(sigma_ > 0.0,
    00099             "Gaussian::Gaussian(): sigma > 0 required.");
    00100         switch(order_)
    00101         {
    00102             case 1:
    00103             case 2:
    00104                 norm_ = -1.0 / (VIGRA_CSTD::sqrt(2.0 * M_PI) * sq(sigma) * sigma);
    00105                 break;
    00106             case 3:
    00107                 norm_ = 1.0 / (VIGRA_CSTD::sqrt(2.0 * M_PI) * sq(sigma) * sq(sigma) * sigma);
    00108                 break;
    00109             default:
    00110                 norm_ = 1.0 / VIGRA_CSTD::sqrt(2.0 * M_PI) / sigma;
    00111         }
    00112         calculateHermitePolynomial();
    00113     }
    00114
    00115         /** Function (functor) call.
    00116         */
    00117     result_type operator()(argument_type x) const;
    00118
    00119         /** Get the standard deviation of the Gaussian.
    00120         */
    00121     value_type sigma() const
    00122         { return sigma_; }
    00123
    00124         /** Get the derivative order of the Gaussian.
    00125         */
    00126     unsigned int derivativeOrder() const
    00127         { return order_; }
    00128
    00129         /** Get the required filter radius for a discrete approximation of the Gaussian.
    00130             The radius is given as a multiple of the Gaussian's standard deviation
    00131             (default: <tt>sigma * (3 + 1/2 * derivativeOrder()</tt> -- the second term
    00132             accounts for the fact that the derivatives of the Gaussian become wider
    00133             with increasing order). The result is rounded to the next higher integer.
    00134         */
    00135     double radius(double sigmaMultiple = 3.0) const
    00136         { return VIGRA_CSTD::ceil(sigma_ * (sigmaMultiple + 0.5 * derivativeOrder())); }
    00137
    00138   private:
    00139     void calculateHermitePolynomial();
    00140     T horner(T x) const;
    00141
    00142     T sigma_, sigma2_, norm_;
    00143     unsigned int order_;
    00144     ArrayVector<T> hermitePolynomial_;
    00145 };
    00146
    00147 template <class T>
    00148 typename Gaussian<T>::result_type
    00149 Gaussian<T>::operator()(argument_type x) const
    00150 {
    00151     T x2 = x * x;
    00152     T g  = norm_ * VIGRA_CSTD::exp(x2 * sigma2_);
    00153     switch(order_)
    00154     {
    00155         case 0:
    00156             return g;
    00157         case 1:
    00158             return x * g;
    00159         case 2:
    00160             return (1.0 - sq(x / sigma_)) * g;
    00161         case 3:
    00162             return (3.0 - sq(x / sigma_)) * x * g;
    00163         default:
    00164             return order_ % 2 == 0 ?
    00165                        g * horner(x2)
    00166                      : x * g * horner(x2);
    00167     }
    00168 }
    00169
    00170 template <class T>
    00171 T Gaussian<T>::horner(T x) const
    00172 {
    00173     int i = order_ / 2;
    00174     T res = hermitePolynomial_[i];
    00175     for(--i; i >= 0; --i)
    00176         res = x * res + hermitePolynomial_[i];
    00177     return res;
    00178 }
    00179
    00180 template <class T>
    00181 void Gaussian<T>::calculateHermitePolynomial()
    00182 {
    00183     if(order_ == 0)
    00184     {
    00185         hermitePolynomial_[0] = 1.0;
    00186     }
    00187     else if(order_ == 1)
    00188     {
    00189         hermitePolynomial_[0] = -1.0 / sigma_ / sigma_;
    00190     }
    00191     else
    00192     {
    00193         // calculate Hermite polynomial for requested derivative
    00194         // recursively according to
    00195         //     (0)
    00196         //    h   (x) = 1
    00197         //
    00198         //     (1)
    00199         //    h   (x) = -x / s^2
    00200         //
    00201         //     (n+1)                        (n)           (n-1)
    00202         //    h     (x) = -1 / s^2 * [ x * h   (x) + n * h     (x) ]
    00203         //
    00204         T s2 = -1.0 / sigma_ / sigma_;
    00205         ArrayVector<T> hn(3*order_+3, 0.0);
    00206         typename ArrayVector<T>::iterator hn0 = hn.begin(),
    00207                                           hn1 = hn0 + order_+1,
    00208                                           hn2 = hn1 + order_+1,
    00209                                           ht;
    00210         hn2[0] = 1.0;
    00211         hn1[1] = s2;
    00212         for(unsigned int i = 2; i <= order_; ++i)
    00213         {
    00214             hn0[0] = s2 * (i-1) * hn2[0];
    00215             for(unsigned int j = 1; j <= i; ++j)
    00216                 hn0[j] = s2 * (hn1[j-1] + (i-1) * hn2[j]);
    00217             ht = hn2;
    00218             hn2 = hn1;
    00219             hn1 = hn0;
    00220             hn0 = ht;
    00221         }
    00222         // keep only non-zero coefficients of the polynomial
    00223         for(unsigned int i = 0; i < hermitePolynomial_.size(); ++i)
    00224             hermitePolynomial_[i] = order_ % 2 == 0 ?
    00225                                          hn1[2*i]
    00226                                        : hn1[2*i+1];
    00227     }
    00228 }
    00229
    00230
    00231 ////@}
    00232
    00233 } // namespace vigra
    00234
    00235
    00236 #endif /* VIGRA_GAUSSIANS_HXX */

© Ullrich Köthe (ullrich.koethe@iwr.uni-heidelberg.de) Heidelberg Collaboratory for Image Processing, University of Heidelberg, Germany | html generated using doxygen and Python VIGRA 1.6.0 (13 Aug 2008)