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vigra/gaussians.hxx VIGRA

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

© Ullrich Köthe (ullrich.koethe@iwr.uni-heidelberg.de)
Heidelberg Collaboratory for Image Processing, University of Heidelberg, Germany

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