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details vigra/gaussians.hxx VIGRA

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 1.3.2, Jan 27 2005 )                                    */
00008 /*    You may use, modify, and distribute this software according       */
00009 /*    to the terms stated in the LICENSE file included in               */
00010 /*    the VIGRA distribution.                                           */
00011 /*                                                                      */
00012 /*    The VIGRA Website is                                              */
00013 /*        http://kogs-www.informatik.uni-hamburg.de/~koethe/vigra/      */
00014 /*    Please direct questions, bug reports, and contributions to        */
00015 /*        koethe@informatik.uni-hamburg.de                              */
00016 /*                                                                      */
00017 /*  THIS SOFTWARE IS PROVIDED AS IS AND WITHOUT ANY EXPRESS OR          */
00018 /*  IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED      */
00019 /*  WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. */
00020 /*                                                                      */
00021 /************************************************************************/
00022 
00023 #ifndef VIGRA_GAUSSIANS_HXX
00024 #define VIGRA_GAUSSIANS_HXX
00025 
00026 #include <cmath>
00027 #include "vigra/config.hxx"
00028 #include "vigra/mathutil.hxx"
00029 #include "vigra/array_vector.hxx"
00030 
00031 namespace vigra {
00032 
00033 #if 0
00034 /** \addtogroup MathFunctions Mathematical Functions
00035 */
00036 //@{
00037 #endif
00038 /*! The Gaussian function and its derivatives.
00039 
00040     Implemented as a unary functor. Since it supports the <tt>radius()</tt> function
00041     it can also be used as a kernel in \ref resamplingConvolveImage().
00042 
00043     <b>\#include</b> "<a href="gaussians_8hxx-source.html">vigra/gaussians.hxx</a>"<br>
00044     Namespace: vigra
00045 
00046     \ingroup MathFunctions
00047 */
00048 template <class T = double>
00049 class Gaussian
00050 {
00051   public:
00052   
00053         /** the value type if used as a kernel in \ref resamplingConvolveImage().
00054         */
00055     typedef T            value_type;  
00056         /** the functor's argument type
00057         */
00058     typedef T            argument_type;  
00059         /** the functor's result type
00060         */
00061     typedef T            result_type; 
00062     
00063         /** Create functor for the given standard deviation <tt>sigma</tt> and 
00064             derivative order <i>n</i>. The functor then realizes the function
00065              
00066             \f[ f_{\sigma,n}(x)=\frac{\partial^n}{\partial x^n}
00067                  \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{x^2}{2\sigma^2}}
00068             \f]
00069              
00070             Precondition:
00071             \code
00072             sigma > 0.0
00073             \endcode
00074         */
00075     explicit Gaussian(T sigma = 1.0, unsigned int derivativeOrder = 0)
00076     : sigma_(sigma),
00077       sigma2_(-0.5 / sigma / sigma),
00078       norm_(0.0),
00079       order_(derivativeOrder),
00080       hermitePolynomial_(derivativeOrder / 2 + 1)
00081     {
00082         vigra_precondition(sigma_ > 0.0,
00083             "Gaussian::Gaussian(): sigma > 0 required.");
00084         switch(order_)
00085         {
00086             case 1:
00087             case 2:
00088                 norm_ = -1.0 / (VIGRA_CSTD::sqrt(2.0 * M_PI) * sq(sigma) * sigma);
00089                 break;
00090             case 3:
00091                 norm_ = 1.0 / (VIGRA_CSTD::sqrt(2.0 * M_PI) * sq(sigma) * sq(sigma) * sigma);
00092                 break;
00093             default:
00094                 norm_ = 1.0 / VIGRA_CSTD::sqrt(2.0 * M_PI) / sigma;
00095         }
00096         calculateHermitePolynomial();
00097     }
00098 
00099         /** Function (functor) call.
00100         */
00101     result_type operator()(argument_type x) const;
00102 
00103         /** Get the standard deviation of the Gaussian.
00104         */
00105     value_type sigma() const
00106         { return sigma_; }
00107     
00108         /** Get the derivative order of the Gaussian.
00109         */
00110     unsigned int derivativeOrder() const
00111         { return order_; }
00112     
00113         /** Get the required filter radius for a discrete approximation of the Gaussian.
00114             The radius is given as a multiple of the Gaussian's standard deviation 
00115             (default: <tt>sigma * (3 + 1/2 * derivativeOrder()</tt> -- the second term
00116             accounts for the fact that the derivatives of the Gaussian become wider 
00117             with increasing order). The result is rounded to the next higher integer.
00118         */
00119     double radius(double sigmaMultiple = 3.0) const
00120         { return VIGRA_CSTD::ceil(sigma_ * (sigmaMultiple + 0.5 * derivativeOrder())); }
00121 
00122   private:
00123     void calculateHermitePolynomial();    
00124     T horner(T x) const;
00125     
00126     T sigma_, sigma2_, norm_;
00127     unsigned int order_;
00128     ArrayVector<T> hermitePolynomial_;
00129 };
00130 
00131 template <class T>
00132 typename Gaussian<T>::result_type 
00133 Gaussian<T>::operator()(argument_type x) const
00134 {
00135     T x2 = x * x;
00136     T g  = norm_ * VIGRA_CSTD::exp(x2 * sigma2_);
00137     switch(order_)
00138     {
00139         case 0:
00140             return g;
00141         case 1:
00142             return x * g;
00143         case 2:
00144             return (1.0 - sq(x / sigma_)) * g;
00145         case 3:
00146             return (3.0 - sq(x / sigma_)) * x * g;
00147         default:
00148             return order_ % 2 == 0 ?
00149                        g * horner(x2)
00150                      : x * g * horner(x2);
00151     }
00152 }
00153 
00154 template <class T>
00155 T Gaussian<T>::horner(T x) const
00156 {
00157     int i = order_ / 2;
00158     T res = hermitePolynomial_[i];
00159     for(--i; i >= 0; --i)
00160         res = x * res + hermitePolynomial_[i];
00161     return res;
00162 }
00163     
00164 template <class T>
00165 void Gaussian<T>::calculateHermitePolynomial()
00166 {
00167     if(order_ == 0)
00168     {
00169         hermitePolynomial_[0] = 1.0;
00170     }
00171     else if(order_ == 1)
00172     {
00173         hermitePolynomial_[0] = -1.0 / sigma_ / sigma_;
00174     }
00175     else
00176     {
00177         // calculate Hermite polynomial for requested derivative 
00178         // recursively according to
00179         //     (0)
00180         //    h   (x) = 1
00181         //
00182         //     (1)
00183         //    h   (x) = -x / s^2
00184         //
00185         //     (n+1)                        (n)           (n-1)
00186         //    h     (x) = -1 / s^2 * [ x * h   (x) + n * h     (x) ]
00187         //
00188         T s2 = -1.0 / sigma_ / sigma_;
00189         ArrayVector<T> hn(3*order_+3, 0.0);
00190         typename ArrayVector<T>::iterator hn0 = hn.begin(),
00191                                           hn1 = hn0 + order_+1,
00192                                           hn2 = hn1 + order_+1,
00193                                           ht;
00194         hn2[0] = 1.0;
00195         hn1[1] = s2;
00196         for(unsigned int i = 2; i <= order_; ++i)
00197         {
00198             hn0[0] = s2 * (i-1) * hn2[0];
00199             for(unsigned int j = 1; j <= i; ++j)
00200                 hn0[j] = s2 * (hn1[j-1] + (i-1) * hn2[j]);
00201             ht = hn2;
00202             hn2 = hn1;
00203             hn1 = hn0;
00204             hn0 = ht;
00205         }
00206         // keep only non-zero coefficients of the polynomial
00207         for(unsigned int i = 0; i < hermitePolynomial_.size(); ++i)
00208             hermitePolynomial_[i] = order_ % 2 == 0 ?
00209                                          hn1[2*i]
00210                                        : hn1[2*i+1];
00211     }
00212 }
00213 
00214 
00215 ////@}
00216 
00217 } // namespace vigra
00218 
00219 
00220 #endif /* VIGRA_GAUSSIANS_HXX */

© Ullrich Köthe (koethe@informatik.uni-hamburg.de)
Cognitive Systems Group, University of Hamburg, Germany

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VIGRA 1.3.2 (27 Jan 2005)