DGtal 1.3.0
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dec/exampleHeatLaplace.cpp

Example of the Heat Laplace Operator.

Hands on the Operator in DGtal
#include <DGtal/helpers/StdDefs.h>
#include <DGtal/topology/DigitalSurface.h>
#include <DGtal/topology/DigitalSetBoundary.h>
#include <DGtal/topology/SetOfSurfels.h>
#include <DGtal/topology/LightImplicitDigitalSurface.h>
#include <DGtal/math/linalg/EigenSupport.h>
#include <DGtal/dec/DiscreteExteriorCalculus.h>
#include <DGtal/dec/DiscreteExteriorCalculusFactory.h>
#include <DGtal/dec/DiscreteExteriorCalculusSolver.h>
#include <DGtal/dec/VectorField.h>
#include <DGtal/geometry/surfaces/estimation/IIGeometricFunctors.h>
#include <DGtal/geometry/surfaces/estimation/IntegralInvariantCovarianceEstimator.h>
#include <DGtal/io/colormaps/ColorBrightnessColorMap.h>
#include <DGtal/io/colormaps/TickedColorMap.h>
#include <DGtal/images/IntervalForegroundPredicate.h>
#include <DGtal/images/imagesSetsUtils/SetFromImage.h>
#include <DGtal/shapes/parametric/Ball3D.h>
#include <DGtal/shapes/Shapes.h>
#include <DGtal/shapes/GaussDigitizer.h>
struct Options
{
double h;
};
using namespace DGtal;
using namespace Eigen;
typedef Z3i::Space Space;
typedef Z3i::RealVector RealVector;
typedef Z3i::Point Point;
RealPoint cartesian_to_spherical( const RealPoint& a )
{
return RealPoint( a.norm(), atan2( a[1], a[0] ), acos( a[2] ) );
}
std::function<double(const RealPoint&)> xx_function =
[] ( const RealPoint& p )
{
const RealPoint p_sphere = p / p.norm();
return p_sphere[0] * p_sphere[0];
};
std::function<double(const RealPoint&)> xx_derivative =
[] ( const RealPoint& p )
{
const RealPoint p_s = cartesian_to_spherical( p );
return 2. * cos( p_s[1] ) * cos( p_s[1] ) * ( 2 * cos( p_s[2] ) * cos( p_s[2] ) - sin( p_s[2] ) * sin( p_s[2] ) )
+ 2. * ( sin( p_s[1] ) * sin( p_s[1] ) - cos( p_s[1] ) * cos( p_s[1] ) );
};
template <typename Shape>
void convergence(const Options& options, Shape& shape,
const std::function< double(const RealPoint&) >& input_function,
const std::function< double(const RealPoint&) >& result_function)
{
trace.beginBlock("Laplacian 3D");
trace.beginBlock("Extracting Digital Surface");
Digitizer digitizer;
digitizer.attach(shape);
digitizer.init(shape.getLowerBound() + Z3i::Vector(-1,-1,-1), shape.getUpperBound() + Z3i::Vector(1,1,1), options.h);
Z3i::Domain domain = digitizer.getDomain();
Z3i::KSpace kspace;
bool ok = kspace.init(domain.lowerBound(), domain.upperBound(), true);
if( !ok ) std::cerr << "KSpace init failed" << std::endl;
typedef KSpace::SurfelSet SurfelSet;
typedef SetOfSurfels< KSpace, SurfelSet > MySetOfSurfels;
Surfaces<KSpace>::sMakeBoundary( theSetOfSurfels.surfelSet(),
kspace, digitizer,
domain.lowerBound(),
domain.upperBound() );
MyDigitalSurface digSurf( theSetOfSurfels );
trace.info() << "Digital Surface has " << digSurf.size() << " surfels." << std::endl;
trace.beginBlock("Initializing Normal Functor");
CanonicSCellEmbedder canonicSCellEmbedder(kspace);
typedef functors::IINormalDirectionFunctor<Space> MyIINormalFunctor;
MyIINormalFunctor normalFunctor;
MyIINormalEstimator normalEstimator(normalFunctor);
normalEstimator.attach(kspace, digitizer);
normalEstimator.init(options.h, digSurf.begin(), digSurf.end());
trace.beginBlock("Initializing DEC");
const Calculus calculus = CalculusFactory::createFromNSCells<2>(digSurf.begin(), digSurf.end(), normalEstimator, options.h);
trace.info() << calculus << std::endl;
trace.beginBlock("Computing the input function");
Calculus::DualForm0 input_func(calculus);
for(auto itb = digSurf.begin(), ite = digSurf.end(); itb != ite; itb++)
{
const Calculus::Index i_calc = calculus.getCellIndex( kspace.unsigns( *itb ) );
input_func.myContainer( i_calc ) = input_function( options.h * canonicSCellEmbedder( *itb ) );
}
trace.beginBlock("Computing the Laplace operator");
const double t = options.convolution_radius * pow(options.h, 2. / 3.);
const double K = log( - log1p( t ) ) + 2.;
const Calculus::DualIdentity0 laplace = calculus.heatLaplace<DUAL>(options.h, t, K);
trace.info() << "Matrix has " << ((double)laplace.myContainer.nonZeros() / (double)laplace.myContainer.size() * 100.) << "% of non-zeros elements." << std::endl;
const Eigen::VectorXd laplace_result = (laplace * input_func).myContainer;
Eigen::VectorXd error( digSurf.size() );
Eigen::VectorXd real_laplacian_values( digSurf.size() );
Eigen::VectorXd estimated_laplacian_values( digSurf.size() );
int i = 0;
for(auto itb = digSurf.begin(), ite = digSurf.end(); itb != ite; itb++)
{
const Calculus::Index i_calc = calculus.getCellIndex( kspace.unsigns( *itb ) );
const RealPoint p = options.h * canonicSCellEmbedder( *itb );
const RealPoint p_normalized = p / p.norm();
const RealPoint p_s = cartesian_to_spherical( p );
const double real_laplacian_value = result_function( p_normalized );
const double estimated_laplacian_value = laplace_result( i_calc );
estimated_laplacian_values(i) = estimated_laplacian_value;
real_laplacian_values(i) = real_laplacian_value;
error(i) = estimated_laplacian_value - real_laplacian_value;
++i;
}
trace.info() << "Estimated Laplacian Range : " << estimated_laplacian_values.minCoeff() << " / " << estimated_laplacian_values.maxCoeff() << std::endl;
trace.info() << "Real Laplacian Range : " << real_laplacian_values.minCoeff() << " / " << real_laplacian_values.maxCoeff() << std::endl;
trace.info() << "h = " << options.h << " t = " << t << " K = " << K << std::endl;
trace.info() << "Mean error = " << error.array().abs().mean() << " max error = " << error.array().abs().maxCoeff() << std::endl;
}
int main()
{
Options options;
options.h = 0.1;
Ball ball(Point(0.0,0.0,0.0), 1.0);
std::function<double(const RealPoint&, const RealPoint&)> l2_distance =
[](const RealPoint& a, const RealPoint& b) { return (a - b).norm(); };
convergence<Ball>(options, ball, xx_function, xx_derivative);
return 0;
}
Aim: Model of the concept StarShaped3D represents any Sphere in the space.
Definition: Ball3D.h:61
Aim: Represents a set of n-1-cells in a nD space, together with adjacency relation between these cell...
Aim: This class provides static members to create DEC structures from various other DGtal structures.
Aim: DiscreteExteriorCalculus represents a calculus in the dec package. This is the main structure in...
Aim: A class for computing the Gauss digitization of some Euclidean shape, i.e. its intersection with...
void attach(ConstAlias< EuclideanShape > shape)
Aim: This class implement an Integral Invariant estimator which computes for each surfel the covarian...
Aim: This class is a model of CCellularGridSpaceND. It represents the cubical grid as a cell complex,...
bool init(const Point &lower, const Point &upper, bool isClosed)
Specifies the upper and lower bounds for the maximal cells in this space.
Cell unsigns(const SCell &p) const
Creates an unsigned cell from a signed one.
Aim: Implements basic operations that will be used in Point and Vector classes.
Definition: PointVector.h:593
Aim: A model of CDigitalSurfaceContainer which defines the digital surface as connected surfels....
Definition: SetOfSurfels.h:74
Aim: A utility class for constructing surfaces (i.e. set of (n-1)-cells).
Definition: Surfaces.h:79
Aim: Represent adjacencies between surfel elements, telling if it follows an interior to exterior ord...
void beginBlock(const std::string &keyword="")
std::ostream & info()
double endBlock()
Aim: A functor Matrix -> RealVector that returns the normal direction by diagonalizing the given cova...
DigitalSurface< MyDigitalSurfaceContainer > MyDigitalSurface
MyDigitalSurface::SurfelSet SurfelSet
DGtal is the top-level namespace which contains all DGtal functions and types.
Trace trace
Definition: Common.h:154
@ DUAL
Definition: Duality.h:62
MessageStream error
Aim: A trivial embedder for signed cell, which corresponds to the canonic injection of cell centroids...
Astroid2D< Space > Shape
Z2i::RealPoint RealPoint
int main()
Definition: testBits.cpp:56
MyPointD Point
Definition: testClone2.cpp:383
KSpace K
Domain domain
Ball2D< Space > Ball