DGtal  1.3.beta
geometry/meshes/digpoly-curvature-measures-cnc-XY-3d.cpp

Computation of principal curvatures and directions on a mesh defined by a an implicit shape discretized as a digital surface, using constant or interpolated corrected curvature measures (based on the theory of corrected normal currents). It uses a digital normal vector estimator to improve curvature estimations. Errors with respect to true expected curvatures are also computed.

This first example uses constant per face corrected normal vector field to compute curvatures.

# "Al" vol file
./examples/geometry/meshes/digpoly-curvature-measures-cnc-XY-3d torus 10 0.5 1.0 Const

outputs

Using face-*Constant* Corrected Normal Current
- surface has 2584 surfels.
[SurfaceMesh (OK) #V=2584 #VN=0 #E=5168 #F=2584 #FN=0 E[IF]=4 E[IV]=4 E[IFE]=2]
- CTrivial normal t-ring=3 (discrete)
Expected K1 curvatures: min=-0.25 max=0.125
Computed k1 curvatures: min=-0.29482 max=0.159283
Expected k2 curvatures: min=0.5 max=0.5
Computed k2 curvatures: min=0.408469 max=0.580381
|K1-K1_CNC|_oo = 0.0740888
|K1-K1_CNC|_2  = 0.0236997
|K2-K2_CNC|_oo = 0.0915306
|K2-K2_CNC|_2  = 0.0386669

This second example uses vertex-interpolated corrected normal vector field to compute curvatures.

./examples/geometry/meshes/digpoly-curvature-measures-cnc-XY-3d torus 10 0.5 1.0 Interp

outputs

Using vertex-*Interpolated* Corrected Normal Current
- surface has 2584 surfels.
[SurfaceMesh (OK) #V=2584 #VN=0 #E=5168 #F=2584 #FN=0 E[IF]=4 E[IV]=4 E[IFE]=2]
- CTrivial normal t-ring=3 (discrete)
Expected K1 curvatures: min=-0.25 max=0.125
Computed k1 curvatures: min=-0.267532 max=0.142428
Expected k2 curvatures: min=0.5 max=0.5
Computed k2 curvatures: min=0.422805 max=0.534006
|K1-K1_CNC|_oo = 0.0524167
|K1-K1_CNC|_2  = 0.0188969
|K2-K2_CNC|_oo = 0.0771952
|K2-K2_CNC|_2  = 0.0379048

It also produces several OBJ files to display curvature estimation results, example-cnc-K1.obj, example-cnc-D1.obj, example-cnc-K2.obj, and example-cnc-D2.obj as well as the associated MTL file.

Face-constant corrected smallest principal curvature and direction, r=1
Face-constant corrected greatest principal curvature and direction, r=1
Vertex-interpolated corrected smallest principal curvature and direction, r=1
Vertex-interpolated corrected greatest principal curvature and direction, r=1
See also
Curvature measures on meshes and digital surfaces
Note
In opposition with Normal Cycle curvature measures, constant or interpolated corrected curvature measures can take into account an external normal vector field to estimate curvatures with better accuracy.
#include <iostream>
#include <fstream>
#include <algorithm>
#include "DGtal/base/Common.h"
#include "DGtal/shapes/SurfaceMesh.h"
#include "DGtal/geometry/meshes/CorrectedNormalCurrentComputer.h"
#include "DGtal/helpers/Shortcuts.h"
#include "DGtal/helpers/ShortcutsGeometry.h"
#include "DGtal/io/writers/SurfaceMeshWriter.h"
#include "DGtal/io/colormaps/GradientColorMap.h"
#include "DGtal/io/colormaps/QuantifiedColorMap.h"
makeColorMap( double min_value, double max_value )
{
DGtal::GradientColorMap< double > gradcmap( min_value, max_value );
gradcmap.addColor( DGtal::Color( 0, 0, 255 ) );
gradcmap.addColor( DGtal::Color( 0, 255, 255 ) );
gradcmap.addColor( DGtal::Color( 255, 255, 255 ) );
gradcmap.addColor( DGtal::Color( 255, 255, 0 ) );
gradcmap.addColor( DGtal::Color( 255, 0, 0 ) );
return gradcmap;
}
void usage( int argc, char* argv[] )
{
using namespace DGtal;
using namespace DGtal::Z3i;
typedef Shortcuts< KSpace > SH;
std::cout << "Usage: " << std::endl
<< "\t" << argv[ 0 ] << " <P> <B> <h> <R> <mode>" << std::endl
<< std::endl
<< "Computation of principal curvatures and directions on" << std::endl
<< "a digitized implicit shape using constant or " << std::endl
<< "interpolated corrected curvature measures (based " << std::endl
<< "on the theory of corrected normal currents)." << std::endl
<< "- builds the surface mesh from polynomial <P>" << std::endl
<< "- <B> defines the digitization space size [-B,B]^3" << std::endl
<< "- <h> is the gridstep digitization" << std::endl
<< "- <R> is the radius of the measuring balls" << std::endl
<< "- <mode> is either Const for constant corrected normal" << std::endl
<< " vector field or Interp for interpolated corrected" << std::endl
<< " normal vector field." << std::endl
<< "It produces several OBJ files to display principal " << std::endl
<< "curvatures and directions estimations: `example-cnc-K1.obj`" << std::endl
<< "`example-cnc-K2.obj`, `example-cnc-D1.obj`, and" << std::endl
<< "`example-cnc-D2.obj` as well as associated MTL files." << std::endl;
std::cout << "You may either write your own polynomial as 3*x^2*y-z^2*x*y+1" << std::endl
<<"or use a predefined polynomial in the following list:" << std::endl;
auto L = SH::getPolynomialList();
for ( const auto& p : L )
std::cout << p.first << " : " << p.second << std::endl;
}
int main( int argc, char* argv[] )
{
if ( argc <= 1 )
{
usage( argc, argv );
return 0;
}
using namespace DGtal;
using namespace DGtal::Z3i;
typedef Shortcuts< KSpace > SH;
std::string poly = argv[ 1 ]; // polynomial
const double B = argc > 2 ? atof( argv[ 2 ] ) : 1.0; // max ||_oo bbox
const double h = argc > 3 ? atof( argv[ 3 ] ) : 1.0; // gridstep
const double R = argc > 4 ? atof( argv[ 4 ] ) : 2.0; // radius of measuring ball
std::string mode = argc > 5 ? argv[ 5 ] : "Const"; // either Const or Interp
bool interpolated = mode == "Interp";
if ( interpolated )
std::cout << "Using vertex-*Interpolated* Corrected Normal Current" << std::endl;
else
std::cout << "Using face-*Constant* Corrected Normal Current" << std::endl;
// Read polynomial and build digital surface
auto params = SH::defaultParameters() | SHG::defaultParameters();
params( "t-ring", 3 )( "surfaceTraversal", "Default" );
params( "polynomial", poly )( "gridstep", h );
params( "minAABB", -B )( "maxAABB", B );
params( "offset", 3.0 );
auto shape = SH::makeImplicitShape3D( params );
auto K = SH::getKSpace( params );
auto dshape = SH::makeDigitizedImplicitShape3D( shape, params );
auto bimage = SH::makeBinaryImage( dshape, params );
if ( bimage == nullptr )
{
trace.error() << "Unable to read polynomial <"
<< poly.c_str() << ">" << std::endl;
return 1;
}
auto sembedder = SH::getSCellEmbedder( K );
auto embedder = SH::getCellEmbedder( K );
auto surface = SH::makeDigitalSurface( bimage, K, params );
auto surfels = SH::getSurfelRange( surface, params );
trace.info() << "- surface has " << surfels.size()<< " surfels." << std::endl;
SM smesh;
std::vector< SM::Vertices > faces;
SH::Cell2Index c2i;
auto pointels = SH::getPointelRange( c2i, surface );
auto vertices = SH::RealPoints( pointels.size() );
std::transform( pointels.cbegin(), pointels.cend(), vertices.begin(),
[&] (const SH::Cell& c) { return h * embedder( c ); } );
for ( auto&& surfel : *surface )
{
const auto primal_surfel_vtcs = SH::getPointelRange( K, surfel );
SM::Vertices face;
for ( auto&& primal_vtx : primal_surfel_vtcs )
face.push_back( c2i[ primal_vtx ] );
faces.push_back( face );
}
smesh.init( vertices.cbegin(), vertices.cend(),
faces.cbegin(), faces.cend() );
trace.info() << smesh << std::endl;
auto exp_K1 = SHG::getFirstPrincipalCurvatures ( shape, K, surfels, params );
auto exp_K2 = SHG::getSecondPrincipalCurvatures( shape, K, surfels, params );
auto exp_D1 = SHG::getFirstPrincipalDirections ( shape, K, surfels, params );
auto exp_D2 = SHG::getSecondPrincipalDirections( shape, K, surfels, params );
// Builds a CorrectedNormalCurrentComputer object onto the SurfaceMesh object
CNC cnc( smesh );
// Estimates normal vectors using Convolved Trivial Normal estimator
auto face_normals = SHG::getCTrivialNormalVectors( surface, surfels, params );
// Set corrected face normals => Corrected Normal Current with
// constant per face corrected vector field.
smesh.setFaceNormals( face_normals.cbegin(), face_normals.cend() ); // CCNC
// Set corrected vertex normals => Corrected Normal Current with
// smooth linearly interpolated per face corrected vector field.
if ( interpolated ) smesh.computeVertexNormalsFromFaceNormals(); // ICNC
// computes area, anisotropic XY curvature measures
auto mu0 = cnc.computeMu0();
auto muXY = cnc.computeMuXY();
// estimates principal curvatures (K1,K2) and directions (D1,D2) by
// measure normalization.
std::vector< double > K1( smesh.nbFaces() );
std::vector< double > K2( smesh.nbFaces() );
std::vector< RealVector > D1( smesh.nbFaces() );
std::vector< RealVector > D2( smesh.nbFaces() );
for ( auto f = 0; f < smesh.nbFaces(); ++f )
{
const auto b = smesh.faceCentroid( f );
const auto N = smesh.faceNormals()[ f ];
const auto area = mu0 .measure( b, R, f );
const auto M = muXY.measure( b, R, f );
std::tie( K1[ f ], K2[ f ], D1[ f ], D2[ f ] )
= cnc.principalCurvatures( area, M, N );
}
auto exp_K1_min_max = std::minmax_element( exp_K1.cbegin(), exp_K1.cend() );
auto exp_K2_min_max = std::minmax_element( exp_K2.cbegin(), exp_K2.cend() );
auto K1_min_max = std::minmax_element( K1.cbegin(), K1.cend() );
auto K2_min_max = std::minmax_element( K2.cbegin(), K2.cend() );
std::cout << "Expected K1 curvatures:"
<< " min=" << *exp_K1_min_max.first << " max=" << *exp_K1_min_max.second
<< std::endl;
std::cout << "Computed k1 curvatures:"
<< " min=" << *K1_min_max.first << " max=" << *K1_min_max.second
<< std::endl;
std::cout << "Expected k2 curvatures:"
<< " min=" << *exp_K2_min_max.first << " max=" << *exp_K2_min_max.second
<< std::endl;
std::cout << "Computed k2 curvatures:"
<< " min=" << *K2_min_max.first << " max=" << *K2_min_max.second
<< std::endl;
const auto error_K1 = SHG::getScalarsAbsoluteDifference( K1, exp_K1 );
const auto stat_error_K1 = SHG::getStatistic( error_K1 );
const auto error_K1_l2 = SHG::getScalarsNormL2( K1, exp_K1 );
trace.info() << "|K1-K1_CNC|_oo = " << stat_error_K1.max() << std::endl;
trace.info() << "|K1-K1_CNC|_2 = " << error_K1_l2 << std::endl;
const auto error_K2 = SHG::getScalarsAbsoluteDifference( K2, exp_K2 );
const auto stat_error_K2 = SHG::getStatistic( error_K2 );
const auto error_K2_l2 = SHG::getScalarsNormL2( K2, exp_K2 );
trace.info() << "|K2-K2_CNC|_oo = " << stat_error_K2.max() << std::endl;
trace.info() << "|K2-K2_CNC|_2 = " << error_K2_l2 << std::endl;
// Remove normals for better blocky display.
smesh.vertexNormals() = SH::RealVectors();
smesh.faceNormals() = SH::RealVectors();
const double Kmax = std::max( fabs( *exp_K1_min_max.first ),
fabs( *exp_K2_min_max.second ) );
const auto colormapK1 = makeQuantifiedColorMap( makeColorMap( -Kmax, Kmax ) );
const auto colormapK2 = makeQuantifiedColorMap( makeColorMap( -Kmax, Kmax ) );
auto colorsK1 = SMW::Colors( smesh.nbFaces() );
auto colorsK2 = SMW::Colors( smesh.nbFaces() );
for ( auto i = 0; i < smesh.nbFaces(); i++ )
{
colorsK1[ i ] = colormapK1( K1[ i ] );
colorsK2[ i ] = colormapK2( K2[ i ] );
}
SMW::writeOBJ( "example-cnc-K1", smesh, colorsK1 );
SMW::writeOBJ( "example-cnc-K2", smesh, colorsK2 );
const auto avg_e = smesh.averageEdgeLength();
SH::RealPoints positions( smesh.nbFaces() );
for ( auto f = 0; f < positions.size(); ++f )
{
D1[ f ] *= smesh.localWindow( f );
positions[ f ] = smesh.faceCentroid( f ) - 0.5 * D1[ f ];
}
SH::saveVectorFieldOBJ( positions, D1, 0.05 * avg_e, SH::Colors(),
"example-cnc-D1",
SH::Color::Black, SH::Color( 0, 128, 0 ) );
for ( auto f = 0; f < positions.size(); ++f )
{
D2[ f ] *= smesh.localWindow( f );
positions[ f ] = smesh.faceCentroid( f ) - 0.5 * D2[ f ];
}
SH::saveVectorFieldOBJ( positions, D2, 0.05 * avg_e, SH::Colors(),
"example-cnc-D2",
SH::Color::Black, SH::Color(128, 0,128 ) );
return 0;
}
max
int max(int a, int b)
Definition: testArithmeticalDSS.cpp:1108
DGtal::Trace::error
std::ostream & error()
DGtal::Color
Structure representing an RGB triple with alpha component.
Definition: Color.h:67
DGtal::trace
Trace trace
Definition: Common.h:154
K
KSpace K
Definition: testCubicalComplex.cpp:62
DGtal::SurfaceMesh
Aim: Represents an embedded mesh as faces and a list of vertices. Vertices may be shared among faces ...
Definition: SurfaceMesh.h:91
DGtal::SurfaceMeshWriter
Aim: An helper class for writing mesh file formats (Waverfront OBJ at this point) and creating a Surf...
Definition: SurfaceMeshWriter.h:64
makeColorMap
DGtal::GradientColorMap< double > makeColorMap(double min_value, double max_value)
[curvature-comparator-Includes]
Definition: curvature-comparator-ii-cnc-3d.cpp:89
DGtal::Shortcuts
Aim: This class is used to simplify shape and surface creation. With it, you can create new shapes an...
Definition: Shortcuts.h:104
DGtal::ProbingMode::R
@ R
DGtal::CorrectedNormalCurrentComputer
Aim: Utility class to compute curvature measures induced by (1) a corrected normal current defined by...
Definition: CorrectedNormalCurrentComputer.h:69
boost::vertices
std::pair< typename graph_traits< DGtal::DigitalSurface< TDigitalSurfaceContainer > >::vertex_iterator, typename graph_traits< DGtal::DigitalSurface< TDigitalSurfaceContainer > >::vertex_iterator > vertices(const DGtal::DigitalSurface< TDigitalSurfaceContainer > &digSurf)
DGtal::Trace::info
std::ostream & info()
Vertices
SMesh::Vertices Vertices
Definition: fullConvexitySphereGeodesics.cpp:118
DGtal
DGtal is the top-level namespace which contains all DGtal functions and types.
DGtal::Z3i
Z3i this namespace gathers the standard of types for 3D imagery.
DGtal::ShortcutsGeometry
Aim: This class is used to simplify shape and surface creation. With it, you can create new shapes an...
Definition: ShortcutsGeometry.h:76
main
int main(int argc, char **argv)
Definition: testArithmeticDSS-benchmark.cpp:147
DGtal::GradientColorMap::addColor
void addColor(const Color &color)
usage
void usage(int, char **argv)
Definition: approximation.cpp:64
Cell
KSpace::Cell Cell
Definition: testCubicalComplex.cpp:56
DGtal::Z2i::K2
KSpace K2
Definition: StdDefs.h:78
DGtal::makeQuantifiedColorMap
QuantifiedColorMap< TColorMap > makeQuantifiedColorMap(TColorMap colormap, int nb=50)
Definition: QuantifiedColorMap.h:113
DGtal::GradientColorMap
Aim: This class template may be used to (linearly) convert scalar values in a given range into a colo...
Definition: GradientColorMap.h:119