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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 /* The VIGRA Website is */ 00008 /* http://kogs-www.informatik.uni-hamburg.de/~koethe/vigra/ */ 00009 /* Please direct questions, bug reports, and contributions to */ 00010 /* ullrich.koethe@iwr.uni-heidelberg.de or */ 00011 /* vigra@informatik.uni-hamburg.de */ 00012 /* */ 00013 /* Permission is hereby granted, free of charge, to any person */ 00014 /* obtaining a copy of this software and associated documentation */ 00015 /* files (the "Software"), to deal in the Software without */ 00016 /* restriction, including without limitation the rights to use, */ 00017 /* copy, modify, merge, publish, distribute, sublicense, and/or */ 00018 /* sell copies of the Software, and to permit persons to whom the */ 00019 /* Software is furnished to do so, subject to the following */ 00020 /* conditions: */ 00021 /* */ 00022 /* The above copyright notice and this permission notice shall be */ 00023 /* included in all copies or substantial portions of the */ 00024 /* Software. */ 00025 /* */ 00026 /* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND */ 00027 /* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES */ 00028 /* OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND */ 00029 /* NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT */ 00030 /* HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, */ 00031 /* WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING */ 00032 /* FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR */ 00033 /* OTHER DEALINGS IN THE SOFTWARE. */ 00034 /* */ 00035 /************************************************************************/ 00036 00037 #ifndef VIGRA_GAUSSIANS_HXX 00038 #define VIGRA_GAUSSIANS_HXX 00039 00040 #include <cmath> 00041 #include "config.hxx" 00042 #include "mathutil.hxx" 00043 #include "array_vector.hxx" 00044 #include "error.hxx" 00045 00046 namespace vigra { 00047 00048 #if 0 00049 /** \addtogroup MathFunctions Mathematical Functions 00050 */ 00051 //@{ 00052 #endif 00053 /*! The Gaussian function and its derivatives. 00054 00055 Implemented as a unary functor. Since it supports the <tt>radius()</tt> function 00056 it can also be used as a kernel in \ref resamplingConvolveImage(). 00057 00058 <b>\#include</b> <<a href="gaussians_8hxx-source.html">vigra/gaussians.hxx</a>><br> 00059 Namespace: vigra 00060 00061 \ingroup MathFunctions 00062 */ 00063 template <class T = double> 00064 class Gaussian 00065 { 00066 public: 00067 00068 /** the value type if used as a kernel in \ref resamplingConvolveImage(). 00069 */ 00070 typedef T value_type; 00071 /** the functor's argument type 00072 */ 00073 typedef T argument_type; 00074 /** the functor's result type 00075 */ 00076 typedef T result_type; 00077 00078 /** Create functor for the given standard deviation <tt>sigma</tt> and 00079 derivative order <i>n</i>. The functor then realizes the function 00080 00081 \f[ f_{\sigma,n}(x)=\frac{\partial^n}{\partial x^n} 00082 \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{x^2}{2\sigma^2}} 00083 \f] 00084 00085 Precondition: 00086 \code 00087 sigma > 0.0 00088 \endcode 00089 */ 00090 explicit Gaussian(T sigma = 1.0, unsigned int derivativeOrder = 0) 00091 : sigma_(sigma), 00092 sigma2_(-0.5 / sigma / sigma), 00093 norm_(0.0), 00094 order_(derivativeOrder), 00095 hermitePolynomial_(derivativeOrder / 2 + 1) 00096 { 00097 vigra_precondition(sigma_ > 0.0, 00098 "Gaussian::Gaussian(): sigma > 0 required."); 00099 switch(order_) 00100 { 00101 case 1: 00102 case 2: 00103 norm_ = -1.0 / (VIGRA_CSTD::sqrt(2.0 * M_PI) * sq(sigma) * sigma); 00104 break; 00105 case 3: 00106 norm_ = 1.0 / (VIGRA_CSTD::sqrt(2.0 * M_PI) * sq(sigma) * sq(sigma) * sigma); 00107 break; 00108 default: 00109 norm_ = 1.0 / VIGRA_CSTD::sqrt(2.0 * M_PI) / sigma; 00110 } 00111 calculateHermitePolynomial(); 00112 } 00113 00114 /** Function (functor) call. 00115 */ 00116 result_type operator()(argument_type x) const; 00117 00118 /** Get the standard deviation of the Gaussian. 00119 */ 00120 value_type sigma() const 00121 { return sigma_; } 00122 00123 /** Get the derivative order of the Gaussian. 00124 */ 00125 unsigned int derivativeOrder() const 00126 { return order_; } 00127 00128 /** Get the required filter radius for a discrete approximation of the Gaussian. 00129 The radius is given as a multiple of the Gaussian's standard deviation 00130 (default: <tt>sigma * (3 + 1/2 * derivativeOrder()</tt> -- the second term 00131 accounts for the fact that the derivatives of the Gaussian become wider 00132 with increasing order). The result is rounded to the next higher integer. 00133 */ 00134 double radius(double sigmaMultiple = 3.0) const 00135 { return VIGRA_CSTD::ceil(sigma_ * (sigmaMultiple + 0.5 * derivativeOrder())); } 00136 00137 private: 00138 void calculateHermitePolynomial(); 00139 T horner(T x) const; 00140 00141 T sigma_, sigma2_, norm_; 00142 unsigned int order_; 00143 ArrayVector<T> hermitePolynomial_; 00144 }; 00145 00146 template <class T> 00147 typename Gaussian<T>::result_type 00148 Gaussian<T>::operator()(argument_type x) const 00149 { 00150 T x2 = x * x; 00151 T g = norm_ * VIGRA_CSTD::exp(x2 * sigma2_); 00152 switch(order_) 00153 { 00154 case 0: 00155 return g; 00156 case 1: 00157 return x * g; 00158 case 2: 00159 return (1.0 - sq(x / sigma_)) * g; 00160 case 3: 00161 return (3.0 - sq(x / sigma_)) * x * g; 00162 default: 00163 return order_ % 2 == 0 ? 00164 g * horner(x2) 00165 : x * g * horner(x2); 00166 } 00167 } 00168 00169 template <class T> 00170 T Gaussian<T>::horner(T x) const 00171 { 00172 int i = order_ / 2; 00173 T res = hermitePolynomial_[i]; 00174 for(--i; i >= 0; --i) 00175 res = x * res + hermitePolynomial_[i]; 00176 return res; 00177 } 00178 00179 template <class T> 00180 void Gaussian<T>::calculateHermitePolynomial() 00181 { 00182 if(order_ == 0) 00183 { 00184 hermitePolynomial_[0] = 1.0; 00185 } 00186 else if(order_ == 1) 00187 { 00188 hermitePolynomial_[0] = -1.0 / sigma_ / sigma_; 00189 } 00190 else 00191 { 00192 // calculate Hermite polynomial for requested derivative 00193 // recursively according to 00194 // (0) 00195 // h (x) = 1 00196 // 00197 // (1) 00198 // h (x) = -x / s^2 00199 // 00200 // (n+1) (n) (n-1) 00201 // h (x) = -1 / s^2 * [ x * h (x) + n * h (x) ] 00202 // 00203 T s2 = -1.0 / sigma_ / sigma_; 00204 ArrayVector<T> hn(3*order_+3, 0.0); 00205 typename ArrayVector<T>::iterator hn0 = hn.begin(), 00206 hn1 = hn0 + order_+1, 00207 hn2 = hn1 + order_+1, 00208 ht; 00209 hn2[0] = 1.0; 00210 hn1[1] = s2; 00211 for(unsigned int i = 2; i <= order_; ++i) 00212 { 00213 hn0[0] = s2 * (i-1) * hn2[0]; 00214 for(unsigned int j = 1; j <= i; ++j) 00215 hn0[j] = s2 * (hn1[j-1] + (i-1) * hn2[j]); 00216 ht = hn2; 00217 hn2 = hn1; 00218 hn1 = hn0; 00219 hn0 = ht; 00220 } 00221 // keep only non-zero coefficients of the polynomial 00222 for(unsigned int i = 0; i < hermitePolynomial_.size(); ++i) 00223 hermitePolynomial_[i] = order_ % 2 == 0 ? 00224 hn1[2*i] 00225 : hn1[2*i+1]; 00226 } 00227 } 00228 00229 00230 ////@} 00231 00232 } // namespace vigra 00233 00234 00235 #endif /* VIGRA_GAUSSIANS_HXX */
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