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/* */
/* Copyright 1998-2004 by Ullrich Koethe */
/* */
/* This file is part of the VIGRA computer vision library. */
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/************************************************************************/
#ifndef VIGRA_GAUSSIANS_HXX
#define VIGRA_GAUSSIANS_HXX
#include <cmath>
#include "config.hxx"
#include "mathutil.hxx"
#include "array_vector.hxx"
#include "error.hxx"
namespace vigra {
#if 0
/** \addtogroup MathFunctions Mathematical Functions
*/
//@{
#endif
/** The Gaussian function and its derivatives.
Implemented as a unary functor. Since it supports the <tt>radius()</tt> function
it can also be used as a kernel in \ref resamplingConvolveImage().
<b>\#include</b> \<vigra/gaussians.hxx\><br>
Namespace: vigra
\ingroup MathFunctions
*/
template <class T = double>
class Gaussian
{
public:
/** the value type if used as a kernel in \ref resamplingConvolveImage().
*/
typedef T value_type;
/** the functor's argument type
*/
typedef T argument_type;
/** the functor's result type
*/
typedef T result_type;
/** Create functor for the given standard deviation <tt>sigma</tt> and
derivative order <i>n</i>. The functor then realizes the function
\f[ f_{\sigma,n}(x)=\frac{\partial^n}{\partial x^n}
\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{x^2}{2\sigma^2}}
\f]
Precondition:
\code
sigma > 0.0
\endcode
*/
explicit Gaussian(T sigma = 1.0, unsigned int derivativeOrder = 0)
: sigma_(sigma),
sigma2_(T(-0.5 / sigma / sigma)),
norm_(0.0),
order_(derivativeOrder),
hermitePolynomial_(derivativeOrder / 2 + 1)
{
vigra_precondition(sigma_ > 0.0,
"Gaussian::Gaussian(): sigma > 0 required.");
switch(order_)
{
case 1:
case 2:
norm_ = T(-1.0 / (VIGRA_CSTD::sqrt(2.0 * M_PI) * sq(sigma) * sigma));
break;
case 3:
norm_ = T(1.0 / (VIGRA_CSTD::sqrt(2.0 * M_PI) * sq(sigma) * sq(sigma) * sigma));
break;
default:
norm_ = T(1.0 / VIGRA_CSTD::sqrt(2.0 * M_PI) / sigma);
}
calculateHermitePolynomial();
}
/** Function (functor) call.
*/
result_type operator()(argument_type x) const;
/** Get the standard deviation of the Gaussian.
*/
value_type sigma() const
{ return sigma_; }
/** Get the derivative order of the Gaussian.
*/
unsigned int derivativeOrder() const
{ return order_; }
/** Get the required filter radius for a discrete approximation of the Gaussian.
The radius is given as a multiple of the Gaussian's standard deviation
(default: <tt>sigma * (3 + 1/2 * derivativeOrder()</tt> -- the second term
accounts for the fact that the derivatives of the Gaussian become wider
with increasing order). The result is rounded to the next higher integer.
*/
double radius(double sigmaMultiple = 3.0) const
{ return VIGRA_CSTD::ceil(sigma_ * (sigmaMultiple + 0.5 * derivativeOrder())); }
private:
void calculateHermitePolynomial();
T horner(T x) const;
T sigma_, sigma2_, norm_;
unsigned int order_;
ArrayVector<T> hermitePolynomial_;
};
template <class T>
typename Gaussian<T>::result_type
Gaussian<T>::operator()(argument_type x) const
{
T x2 = x * x;
T g = norm_ * VIGRA_CSTD::exp(x2 * sigma2_);
switch(order_)
{
case 0:
return detail::RequiresExplicitCast<result_type>::cast(g);
case 1:
return detail::RequiresExplicitCast<result_type>::cast(x * g);
case 2:
return detail::RequiresExplicitCast<result_type>::cast((1.0 - sq(x / sigma_)) * g);
case 3:
return detail::RequiresExplicitCast<result_type>::cast((3.0 - sq(x / sigma_)) * x * g);
default:
return order_ % 2 == 0 ?
detail::RequiresExplicitCast<result_type>::cast(g * horner(x2))
: detail::RequiresExplicitCast<result_type>::cast(x * g * horner(x2));
}
}
template <class T>
T Gaussian<T>::horner(T x) const
{
int i = order_ / 2;
T res = hermitePolynomial_[i];
for(--i; i >= 0; --i)
res = x * res + hermitePolynomial_[i];
return res;
}
template <class T>
void Gaussian<T>::calculateHermitePolynomial()
{
if(order_ == 0)
{
hermitePolynomial_[0] = 1.0;
}
else if(order_ == 1)
{
hermitePolynomial_[0] = T(-1.0 / sigma_ / sigma_);
}
else
{
// calculate Hermite polynomial for requested derivative
// recursively according to
// (0)
// h (x) = 1
//
// (1)
// h (x) = -x / s^2
//
// (n+1) (n) (n-1)
// h (x) = -1 / s^2 * [ x * h (x) + n * h (x) ]
//
T s2 = T(-1.0 / sigma_ / sigma_);
ArrayVector<T> hn(3*order_+3, 0.0);
typename ArrayVector<T>::iterator hn0 = hn.begin(),
hn1 = hn0 + order_+1,
hn2 = hn1 + order_+1,
ht;
hn2[0] = 1.0;
hn1[1] = s2;
for(unsigned int i = 2; i <= order_; ++i)
{
hn0[0] = s2 * (i-1) * hn2[0];
for(unsigned int j = 1; j <= i; ++j)
hn0[j] = s2 * (hn1[j-1] + (i-1) * hn2[j]);
ht = hn2;
hn2 = hn1;
hn1 = hn0;
hn0 = ht;
}
// keep only non-zero coefficients of the polynomial
for(unsigned int i = 0; i < hermitePolynomial_.size(); ++i)
hermitePolynomial_[i] = order_ % 2 == 0 ?
hn1[2*i]
: hn1[2*i+1];
}
}
////@}
} // namespace vigra
#endif /* VIGRA_GAUSSIANS_HXX */
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