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/************************************************************************/
/* */
/* Copyright 1998-2002 by Ullrich Koethe */
/* Cognitive Systems Group, University of Hamburg, Germany */
/* */
/* This file is part of the VIGRA computer vision library. */
/* ( Version 1.4.0, Dec 21 2005 ) */
/* The VIGRA Website is */
/* http://kogs-www.informatik.uni-hamburg.de/~koethe/vigra/ */
/* Please direct questions, bug reports, and contributions to */
/* koethe@informatik.uni-hamburg.de or */
/* vigra@kogs1.informatik.uni-hamburg.de */
/* */
/* Permission is hereby granted, free of charge, to any person */
/* obtaining a copy of this software and associated documentation */
/* files (the "Software"), to deal in the Software without */
/* restriction, including without limitation the rights to use, */
/* copy, modify, merge, publish, distribute, sublicense, and/or */
/* sell copies of the Software, and to permit persons to whom the */
/* Software is furnished to do so, subject to the following */
/* conditions: */
/* */
/* The above copyright notice and this permission notice shall be */
/* included in all copies or substantial portions of the */
/* Software. */
/* */
/* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND */
/* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES */
/* OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND */
/* NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT */
/* HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, */
/* WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING */
/* FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR */
/* OTHER DEALINGS IN THE SOFTWARE. */
/* */
/************************************************************************/
#ifndef VIGRA_NONLINEARDIFFUSION_HXX
#define VIGRA_NONLINEARDIFFUSION_HXX
#include <vector>
#include "vigra/stdimage.hxx"
#include "vigra/stdimagefunctions.hxx"
#include "vigra/imageiteratoradapter.hxx"
#include "vigra/functortraits.hxx"
namespace vigra {
template <class SrcIterator, class SrcAccessor,
class CoeffIterator, class DestIterator>
void internalNonlinearDiffusionDiagonalSolver(
SrcIterator sbegin, SrcIterator send, SrcAccessor sa,
CoeffIterator diag, CoeffIterator upper, CoeffIterator lower,
DestIterator dbegin)
{
int w = send - sbegin - 1;
int i;
for(i=0; i<w; ++i)
{
lower[i] = lower[i] / diag[i];
diag[i+1] = diag[i+1] - lower[i] * upper[i];
}
dbegin[0] = sa(sbegin);
for(i=1; i<=w; ++i)
{
dbegin[i] = sa(sbegin, i) - lower[i-1] * dbegin[i-1];
}
dbegin[w] = dbegin[w] / diag[w];
for(i=w-1; i>=0; --i)
{
dbegin[i] = (dbegin[i] - upper[i] * dbegin[i+1]) / diag[i];
}
}
template <class SrcIterator, class SrcAccessor,
class WeightIterator, class WeightAccessor,
class DestIterator, class DestAccessor>
void internalNonlinearDiffusionAOSStep(
SrcIterator sul, SrcIterator slr, SrcAccessor as,
WeightIterator wul, WeightAccessor aw,
DestIterator dul, DestAccessor ad, double timestep)
{
// use traits to determine SumType as to prevent possible overflow
typedef typename
NumericTraits<typename DestAccessor::value_type>::RealPromote
DestType;
typedef typename
NumericTraits<typename WeightAccessor::value_type>::RealPromote
WeightType;
// calculate width and height of the image
int w = slr.x - sul.x;
int h = slr.y - sul.y;
int d = (w < h) ? h : w;
std::vector<WeightType> lower(d),
diag(d),
upper(d);
std::vector<DestType> res(d);
int x,y;
WeightType one = NumericTraits<WeightType>::one();
// create y iterators
SrcIterator ys = sul;
WeightIterator yw = wul;
DestIterator yd = dul;
// x-direction
for(y=0; y<h; ++y, ++ys.y, ++yd.y, ++yw.y)
{
typename SrcIterator::row_iterator xs = ys.rowIterator();
typename WeightIterator::row_iterator xw = yw.rowIterator();
typename DestIterator::row_iterator xd = yd.rowIterator();
// fill 3-diag matrix
diag[0] = one + timestep * (aw(xw) + aw(xw, 1));
for(x=1; x<w-1; ++x)
{
diag[x] = one + timestep * (2.0 * aw(xw, x) + aw(xw, x+1) + aw(xw, x-1));
}
diag[w-1] = one + timestep * (aw(xw, w-1) + aw(xw, w-2));
for(x=0; x<w-1; ++x)
{
lower[x] = -timestep * (aw(xw, x) + aw(xw, x+1));
upper[x] = lower[x];
}
internalNonlinearDiffusionDiagonalSolver(xs, xs+w, as,
diag.begin(), upper.begin(), lower.begin(), res.begin());
for(x=0; x<w; ++x, ++xd)
{
ad.set(res[x], xd);
}
}
// y-direction
ys = sul;
yw = wul;
yd = dul;
for(x=0; x<w; ++x, ++ys.x, ++yd.x, ++yw.x)
{
typename SrcIterator::column_iterator xs = ys.columnIterator();
typename WeightIterator::column_iterator xw = yw.columnIterator();
typename DestIterator::column_iterator xd = yd.columnIterator();
// fill 3-diag matrix
diag[0] = one + timestep * (aw(xw) + aw(xw, 1));
for(y=1; y<h-1; ++y)
{
diag[y] = one + timestep * (2.0 * aw(xw, y) + aw(xw, y+1) + aw(xw, y-1));
}
diag[h-1] = one + timestep * (aw(xw, h-1) + aw(xw, h-2));
for(y=0; y<h-1; ++y)
{
lower[y] = -timestep * (aw(xw, y) + aw(xw, y+1));
upper[y] = lower[y];
}
internalNonlinearDiffusionDiagonalSolver(xs, xs+h, as,
diag.begin(), upper.begin(), lower.begin(), res.begin());
for(y=0; y<h; ++y, ++xd)
{
ad.set(0.5 * (ad(xd) + res[y]), xd);
}
}
}
/** \addtogroup NonLinearDiffusion Non-linear Diffusion
Perform edge-preserving smoothing.
*/
//@{
/********************************************************/
/* */
/* nonlinearDiffusion */
/* */
/********************************************************/
/** \brief Perform edge-preserving smoothing at the given scale.
The algorithm solves the non-linear diffusion equation
\f[
\frac{\partial}{\partial t} u =
\frac{\partial}{\partial x}
\left( g(|\nabla u|)
\frac{\partial}{\partial x} u
\right)
\f]
where <em> t</em> is the time, <b> x</b> is the location vector,
<em> u(</em><b> x</b><em> , t)</em> is the smoothed image at time <em> t</em>, and
<em> g(.)</em> is the location dependent diffusivity. At time zero, the image
<em> u(</em><b> x</b><em> , 0)</em> is simply the original image. The time is
propotional to the square of the scale parameter: \f$t = s^2\f$.
The diffusion
equation is solved iteratively according
to the Additive Operator Splitting Scheme (AOS) from
J. Weickert: <em>"Recursive Separable Schemes for Nonlinear Diffusion
Filters"</em>,
in: B. ter Haar Romeny, L. Florack, J. Koenderingk, M. Viergever (eds.):
1st Intl. Conf. on Scale-Space Theory in Computer Vision 1997,
Springer LNCS 1252
<TT>DiffusivityFunctor</TT> implements the gradient dependent local diffusivity.
It is passed
as an argument to \ref gradientBasedTransform(). The return value must be
between 0 and 1 and determines the weight a pixel gets when
its neighbors are smoothed. Weickert recommends the use of the diffusivity
implemented by class \ref DiffusivityFunctor. It's also possible to use
other functors, for example one that always returns 1, in which case
we obtain the solution to the linear diffusion equation, i.e.
Gaussian convolution.
The source value type must be a
linear space with internal addition, scalar multiplication, and
NumericTraits defined. The value_type of the DiffusivityFunctor must be the
scalar field over wich the source value type's linear space is defined.
In addition to <TT>nonlinearDiffusion()</TT>, there is an algorithm
<TT>nonlinearDiffusionExplicit()</TT> which implements the Explicit Scheme
described in the above article. Both algorithms have the same interface,
but the explicit scheme gives slightly more accurate approximations of
the diffusion process at the cost of much slower processing.
<b> Declarations:</b>
pass arguments explicitly:
\code
namespace vigra {
template <class SrcIterator, class SrcAccessor,
class DestIterator, class DestAccessor,
class DiffusivityFunctor>
void nonlinearDiffusion(SrcIterator sul, SrcIterator slr, SrcAccessor as,
DestIterator dul, DestAccessor ad,
DiffusivityFunctor const & weight, double scale);
}
\endcode
use argument objects in conjunction with \ref ArgumentObjectFactories:
\code
namespace vigra {
template <class SrcIterator, class SrcAccessor,
class DestIterator, class DestAccessor,
class DiffusivityFunctor>
void nonlinearDiffusion(
triple<SrcIterator, SrcIterator, SrcAccessor> src,
pair<DestIterator, DestAccessor> dest,
DiffusivityFunctor const & weight, double scale);
}
\endcode
<b> Usage:</b>
<b>\#include</b> "<a href="nonlineardiffusion_8hxx-source.html">vigra/nonlineardiffusion.hxx</a>"
\code
FImage src(w,h), dest(w,h);
float edge_threshold, scale;
...
nonlinearDiffusion(srcImageRange(src), destImage(dest),
DiffusivityFunctor<float>(edge_threshold), scale);
\endcode
<b> Required Interface:</b>
<ul>
<li> <TT>SrcIterator</TT> and <TT>DestIterator</TT> are models of ImageIterator
<li> <TT>SrcAccessor</TT> and <TT>DestAccessor</TT> are models of StandardAccessor
<li> <TT>SrcAccessor::value_type</TT> is a linear space
<li> <TT>DiffusivityFunctor</TT> conforms to the requirements of
\ref gradientBasedTransform(). Its range is between 0 and 1.
<li> <TT>DiffusivityFunctor::value_type</TT> is an algebraic field
</ul>
<b> Precondition:</b>
<TT>scale > 0</TT>
*/
template <class SrcIterator, class SrcAccessor,
class DestIterator, class DestAccessor,
class DiffusivityFunc>
void nonlinearDiffusion(SrcIterator sul, SrcIterator slr, SrcAccessor as,
DestIterator dul, DestAccessor ad,
DiffusivityFunc const & weight, double scale)
{
vigra_precondition(scale > 0.0, "nonlinearDiffusion(): scale must be > 0");
double total_time = scale*scale/2.0;
static const double time_step = 5.0;
int number_of_steps = (int)(total_time / time_step);
double rest_time = total_time - time_step * number_of_steps;
Size2D size(slr.x - sul.x, slr.y - sul.y);
typedef typename
NumericTraits<typename SrcAccessor::value_type>::RealPromote
TmpType;
typedef typename DiffusivityFunc::value_type WeightType;
BasicImage<TmpType> smooth1(size);
BasicImage<TmpType> smooth2(size);
BasicImage<WeightType> weights(size);
typename BasicImage<TmpType>::Iterator s1 = smooth1.upperLeft(),
s2 = smooth2.upperLeft();
typename BasicImage<TmpType>::Accessor a = smooth1.accessor();
typename BasicImage<WeightType>::Iterator wi = weights.upperLeft();
typename BasicImage<WeightType>::Accessor wa = weights.accessor();
gradientBasedTransform(sul, slr, as, wi, wa, weight);
internalNonlinearDiffusionAOSStep(sul, slr, as, wi, wa, s1, a, rest_time);
for(int i = 0; i < number_of_steps; ++i)
{
gradientBasedTransform(s1, s1+size, a, wi, wa, weight);
internalNonlinearDiffusionAOSStep(s1, s1+size, a, wi, wa, s2, a, time_step);
std::swap(s1, s2);
}
copyImage(s1, s1+size, a, dul, ad);
}
template <class SrcIterator, class SrcAccessor,
class DestIterator, class DestAccessor,
class DiffusivityFunc>
inline
void nonlinearDiffusion(
triple<SrcIterator, SrcIterator, SrcAccessor> src,
pair<DestIterator, DestAccessor> dest,
DiffusivityFunc const & weight, double scale)
{
nonlinearDiffusion(src.first, src.second, src.third,
dest.first, dest.second,
weight, scale);
}
template <class SrcIterator, class SrcAccessor,
class WeightIterator, class WeightAccessor,
class DestIterator, class DestAccessor>
void internalNonlinearDiffusionExplicitStep(
SrcIterator sul, SrcIterator slr, SrcAccessor as,
WeightIterator wul, WeightAccessor aw,
DestIterator dul, DestAccessor ad,
double time_step)
{
// use traits to determine SumType as to prevent possible overflow
typedef typename
NumericTraits<typename SrcAccessor::value_type>::RealPromote
SumType;
typedef typename
NumericTraits<typename WeightAccessor::value_type>::RealPromote
WeightType;
// calculate width and height of the image
int w = slr.x - sul.x;
int h = slr.y - sul.y;
int x,y;
static const Diff2D left(-1, 0);
static const Diff2D right(1, 0);
static const Diff2D top(0, -1);
static const Diff2D bottom(0, 1);
WeightType gt, gb, gl, gr, g0;
WeightType one = NumericTraits<WeightType>::one();
SumType sum;
time_step /= 2.0;
// create y iterators
SrcIterator ys = sul;
WeightIterator yw = wul;
DestIterator yd = dul;
SrcIterator xs = ys;
WeightIterator xw = yw;
DestIterator xd = yd;
gt = (aw(xw) + aw(xw, bottom)) * time_step;
gb = (aw(xw) + aw(xw, bottom)) * time_step;
gl = (aw(xw) + aw(xw, right)) * time_step;
gr = (aw(xw) + aw(xw, right)) * time_step;
g0 = one - gt - gb - gr - gl;
sum = g0 * as(xs);
sum += gt * as(xs, bottom);
sum += gb * as(xs, bottom);
sum += gl * as(xs, right);
sum += gr * as(xs, right);
ad.set(sum, xd);
for(x=2, ++xs.x, ++xd.x, ++xw.x; x<w; ++x, ++xs.x, ++xd.x, ++xw.x)
{
gt = (aw(xw) + aw(xw, bottom)) * time_step;
gb = (aw(xw) + aw(xw, bottom)) * time_step;
gl = (aw(xw) + aw(xw, left)) * time_step;
gr = (aw(xw) + aw(xw, right)) * time_step;
g0 = one - gt - gb - gr - gl;
sum = g0 * as(xs);
sum += gt * as(xs, bottom);
sum += gb * as(xs, bottom);
sum += gl * as(xs, left);
sum += gr * as(xs, right);
ad.set(sum, xd);
}
gt = (aw(xw) + aw(xw, bottom)) * time_step;
gb = (aw(xw) + aw(xw, bottom)) * time_step;
gl = (aw(xw) + aw(xw, left)) * time_step;
gr = (aw(xw) + aw(xw, left)) * time_step;
g0 = one - gt - gb - gr - gl;
sum = g0 * as(xs);
sum += gt * as(xs, bottom);
sum += gb * as(xs, bottom);
sum += gl * as(xs, left);
sum += gr * as(xs, left);
ad.set(sum, xd);
for(y=2, ++ys.y, ++yd.y, ++yw.y; y<h; ++y, ++ys.y, ++yd.y, ++yw.y)
{
xs = ys;
xd = yd;
xw = yw;
gt = (aw(xw) + aw(xw, top)) * time_step;
gb = (aw(xw) + aw(xw, bottom)) * time_step;
gl = (aw(xw) + aw(xw, right)) * time_step;
gr = (aw(xw) + aw(xw, right)) * time_step;
g0 = one - gt - gb - gr - gl;
sum = g0 * as(xs);
sum += gt * as(xs, top);
sum += gb * as(xs, bottom);
sum += gl * as(xs, right);
sum += gr * as(xs, right);
ad.set(sum, xd);
for(x=2, ++xs.x, ++xd.x, ++xw.x; x<w; ++x, ++xs.x, ++xd.x, ++xw.x)
{
gt = (aw(xw) + aw(xw, top)) * time_step;
gb = (aw(xw) + aw(xw, bottom)) * time_step;
gl = (aw(xw) + aw(xw, left)) * time_step;
gr = (aw(xw) + aw(xw, right)) * time_step;
g0 = one - gt - gb - gr - gl;
sum = g0 * as(xs);
sum += gt * as(xs, top);
sum += gb * as(xs, bottom);
sum += gl * as(xs, left);
sum += gr * as(xs, right);
ad.set(sum, xd);
}
gt = (aw(xw) + aw(xw, top)) * time_step;
gb = (aw(xw) + aw(xw, bottom)) * time_step;
gl = (aw(xw) + aw(xw, left)) * time_step;
gr = (aw(xw) + aw(xw, left)) * time_step;
g0 = one - gt - gb - gr - gl;
sum = g0 * as(xs);
sum += gt * as(xs, top);
sum += gb * as(xs, bottom);
sum += gl * as(xs, left);
sum += gr * as(xs, left);
ad.set(sum, xd);
}
xs = ys;
xd = yd;
xw = yw;
gt = (aw(xw) + aw(xw, top)) * time_step;
gb = (aw(xw) + aw(xw, top)) * time_step;
gl = (aw(xw) + aw(xw, right)) * time_step;
gr = (aw(xw) + aw(xw, right)) * time_step;
g0 = one - gt - gb - gr - gl;
sum = g0 * as(xs);
sum += gt * as(xs, top);
sum += gb * as(xs, top);
sum += gl * as(xs, right);
sum += gr * as(xs, right);
ad.set(sum, xd);
for(x=2, ++xs.x, ++xd.x, ++xw.x; x<w; ++x, ++xs.x, ++xd.x, ++xw.x)
{
gt = (aw(xw) + aw(xw, top)) * time_step;
gb = (aw(xw) + aw(xw, top)) * time_step;
gl = (aw(xw) + aw(xw, left)) * time_step;
gr = (aw(xw) + aw(xw, right)) * time_step;
g0 = one - gt - gb - gr - gl;
sum = g0 * as(xs);
sum += gt * as(xs, top);
sum += gb * as(xs, top);
sum += gl * as(xs, left);
sum += gr * as(xs, right);
ad.set(sum, xd);
}
gt = (aw(xw) + aw(xw, top)) * time_step;
gb = (aw(xw) + aw(xw, top)) * time_step;
gl = (aw(xw) + aw(xw, left)) * time_step;
gr = (aw(xw) + aw(xw, left)) * time_step;
g0 = one - gt - gb - gr - gl;
sum = g0 * as(xs);
sum += gt * as(xs, top);
sum += gb * as(xs, top);
sum += gl * as(xs, left);
sum += gr * as(xs, left);
ad.set(sum, xd);
}
template <class SrcIterator, class SrcAccessor,
class DestIterator, class DestAccessor,
class DiffusivityFunc>
void nonlinearDiffusionExplicit(SrcIterator sul, SrcIterator slr, SrcAccessor as,
DestIterator dul, DestAccessor ad,
DiffusivityFunc const & weight, double scale)
{
vigra_precondition(scale > 0.0, "nonlinearDiffusionExplicit(): scale must be > 0");
double total_time = scale*scale/2.0;
static const double time_step = 0.25;
int number_of_steps = total_time / time_step;
double rest_time = total_time - time_step * number_of_steps;
Size2D size(slr.x - sul.x, slr.y - sul.y);
typedef typename
NumericTraits<typename SrcAccessor::value_type>::RealPromote
TmpType;
typedef typename DiffusivityFunc::value_type WeightType;
BasicImage<TmpType> smooth1(size);
BasicImage<TmpType> smooth2(size);
BasicImage<WeightType> weights(size);
typename BasicImage<TmpType>::Iterator s1 = smooth1.upperLeft(),
s2 = smooth2.upperLeft();
typename BasicImage<TmpType>::Accessor a = smooth1.accessor();
typename BasicImage<WeightType>::Iterator wi = weights.upperLeft();
typename BasicImage<WeightType>::Accessor wa = weights.accessor();
gradientBasedTransform(sul, slr, as, wi, wa, weight);
internalNonlinearDiffusionExplicitStep(sul, slr, as, wi, wa, s1, a, rest_time);
for(int i = 0; i < number_of_steps; ++i)
{
gradientBasedTransform(s1, s1+size, a, wi, wa, weight);
internalNonlinearDiffusionExplicitStep(s1, s1+size, a, wi, wa, s2, a, time_step);
swap(s1, s2);
}
copyImage(s1, s1+size, a, dul, ad);
}
template <class SrcIterator, class SrcAccessor,
class DestIterator, class DestAccessor,
class DiffusivityFunc>
inline
void nonlinearDiffusionExplicit(
triple<SrcIterator, SrcIterator, SrcAccessor> src,
pair<DestIterator, DestAccessor> dest,
DiffusivityFunc const & weight, double scale)
{
nonlinearDiffusionExplicit(src.first, src.second, src.third,
dest.first, dest.second,
weight, scale);
}
/********************************************************/
/* */
/* DiffusivityFunctor */
/* */
/********************************************************/
/** \brief Diffusivity functor for non-linear diffusion.
This functor implements the diffusivity recommended by Weickert:
\f[
g(|\nabla u|) = 1 -
\exp{\left(\frac{-3.315}{(|\nabla u| / thresh)^4}\right)}
\f]
where <TT>thresh</TT> is a threshold that determines whether a specific gradient
magnitude is interpreted as a significant edge (above the threshold)
or as noise. It is meant to be used with \ref nonlinearDiffusion().
*/
template <class Value>
class DiffusivityFunctor
{
public:
/** the functors first argument type (must be an algebraic field with <TT>exp()</TT> defined).
However, the functor also works with RGBValue<first_argument_type> (this hack is
necessary since Microsoft C++ doesn't support partial specialization).
*/
typedef Value first_argument_type;
/** the functors second argument type (same as first).
However, the functor also works with RGBValue<second_argument_type> (this hack is
necessary since Microsoft C++ doesn't support partial specialization).
*/
typedef Value second_argument_type;
/** the functors result type
*/
typedef typename NumericTraits<Value>::RealPromote result_type;
/** \deprecated use first_argument_type, second_argument_type, result_type
*/
typedef Value value_type;
/** init functor with given edge threshold
*/
DiffusivityFunctor(Value const & thresh)
: weight_(thresh*thresh),
one_(NumericTraits<result_type>::one()),
zero_(NumericTraits<result_type>::zero())
{}
/** calculate diffusivity from scalar arguments
*/
result_type operator()(first_argument_type const & gx, second_argument_type const & gy) const
{
Value mag = (gx*gx + gy*gy) / weight_;
return (mag == zero_) ? one_ : one_ - exp(-3.315 / mag / mag);
}
/** calculate diffusivity from RGB arguments
*/
result_type operator()(RGBValue<Value> const & gx, RGBValue<Value> const & gy) const
{
result_type mag = (gx.red()*gx.red() +
gx.green()*gx.green() +
gx.blue()*gx.blue() +
gy.red()*gy.red() +
gy.green()*gy.green() +
gy.blue()*gy.blue()) / weight_;
return (mag == zero_) ? one_ : one_ - exp(-3.315 / mag / mag);
}
result_type weight_;
result_type one_;
result_type zero_;
};
template <class ValueType>
class FunctorTraits<DiffusivityFunctor<ValueType> >
: public FunctorTraitsBase<DiffusivityFunctor<ValueType> >
{
public:
typedef VigraTrueType isBinaryFunctor;
};
//@}
} // namespace vigra
#endif /* VIGRA_NONLINEARDIFFUSION_HXX */