Introduction

Many estimators of local quantities, such as curvature or normal vector field, on a digital surface can be defined locally as a function which associates a quantity to a local surface patch centered at the surfel s. Such local estimators can be characterized by:

a digital surface in which the computation is performed;
a functor to call on surfels in the neighborhood;
a metric to specify the shape of the neighborhood to consider;
a radius to define the neighborhood (ball for the specified metric);
as any estimator, a gridstep h.

When evaluating the estimator at a given surfel, neighboring surfels are traversed (using a DistanceBreadthFirstVisitor parametrized by the specified metric) and each surfel is sent to the surfel functor.

Implementation details

In DGtal, we implement this class of estimator using the generic LocalEstimatorsFromSurfel class. Such class is parametrized by the following template parameters:

a model of digital surface concept (concepts::CDigitalSurfaceContainer)
a model of concepts::CMetricSpace to be used in the neighborhood construction.
a functor to define the estimator on surfel set (model of concepts::CLocalEstimatorFromSurfelFunctor)
a convolution function to weight neighboring surfels.

This class provides three main methods:

init(h,radius): to init the estimators for a given gridstep and a given kernel neighborhood.
eval( aSurfelIterator ): the evaluate the functor at aSurfelIterator.
eval( aSurfelItBegin, aSurfelItEnd): evaluate the estimator on a range of surfels.

The core of the estimators are thus specified in the surfel functor (model of concepts::CLocalEstimatorFromSurfelFunctor). In DGtal, we have defined several functors:

functors::ElementaryConvolutionNormalVectorEstimator: Normal vector estimation based on weighted average of surfel elementary normal vectors ([81], [94])
functors::TensorVotingFeatureExtraction implements a functor to detect feature points on a digital surface using tensor voting strategy (see [95]). More precisely, the functor accumulates tensor votes \(I - vv^T/\|vv^T\|\) for each surfel added during the scan ( \(v\) being the vector from the center of the neighborhood to the added surfel). Then, the eval() method returns the ratio \(\frac{\lambda_1+\lambda_2}{\lambda_3}\) of the eigenvalues ( \(\lambda_1\leq\lambda_2\leq\lambda_3\)) of the accumulated tensor votes.
functors::SphereFittingEstimator: this functor requires the WITH_PATATE flag and the Patate Library dependency. The functor performs a sphere fitting on the neighborhood following [89] . The output of the estimator is a structure containing the fitted sphere parameters (e.g. its center and radius). Please refer to [89] for details on the tau, kappa and eta parameters.
functors::MongeJetFittingGaussianCurvatureEstimator, functors::MongeJetFittingMeanCurvatureEstimator, functors::MongeJetFittingNormalVectorEstimator, functors::MongeJetFittingPrincipalCurvaturesEstimator: return differential quantity by a polynomial surface fitting [20], [19] . These estimators require CGAL and Eigen3 dependency (WITH_CGAL, WITH_EIGEN).
functors::LinearLeastSquareFittingNormalVectorEstimator: normal vector estimation from least square fitting of the embedded sufel points. This functor also needs CGAL dependency.
functors::SphericalHoughNormalVectorEstimator: Robsut normal vector estimation using a spherical hough voting ([16]).

Advanced:: Implementing your own surfel patch based estimator is quite simple. Please have a look to concepts::CLocalEstimatorFromSurfelFunctor or the source code of any of its models.

Usage Example

In this section, we give a step by step example (see exampleEstimatorFromSurfelFunctors.cpp) . Let's start by defining a digital surface from a implicit digital ellipse.

  using namespace Z3i;
  typedef ImplicitDigitalEllipse3<Point> ImplicitDigitalEllipse;
  typedef LightImplicitDigitalSurface<KSpace,ImplicitDigitalEllipse> SurfaceContainer;
  typedef DigitalSurface< SurfaceContainer > Surface;
  typedef SurfaceContainer::Surfel Surfel;
 
  Point p1( -10, -10, -10 );
  Point p2( 10, 10, 10 );
  KSpace K;
  K.init( p1, p2, true );
  ImplicitDigitalEllipse ellipse( 6.0, 4.5, 3.4 );
  Surfel bel = Surfaces<KSpace>::findABel( K, ellipse, 10000 );
  SurfaceContainer* surfaceContainer = new SurfaceContainer
    ( K, ellipse, SurfelAdjacency<KSpace::dimension>( true ), bel );
  Surface surface( surfaceContainer ); // acquired

We then define some types (note that we use the WITH_CGAL define to make sure that the user has enabled CGAL). Since local functors based on Monge Jet Fitting and CGAL do not require any weigths, we just consider a constant weight functor returning 1.0 for each surfel. For the functors::ElementaryConvolutionNormalVectorEstimator, we consider a Gaussian kernel with \( \sigma=2.0\). Please aslo note that the distance visitor is based on a Euclidean \( l_2\) metric:

#ifdef WITH_CGAL
  typedef DGtal::functors::MongeJetFittingGaussianCurvatureEstimator<Surfel, CanonicSCellEmbedder<KSpace> > FunctorGaussian;
  typedef functors::MongeJetFittingMeanCurvatureEstimator<Surfel, CanonicSCellEmbedder<KSpace> > FunctorMean;
  typedef functors::MongeJetFittingNormalVectorEstimator<Surfel, CanonicSCellEmbedder<KSpace> > FunctorNormal;
  typedef functors::LinearLeastSquareFittingNormalVectorEstimator<Surfel, CanonicSCellEmbedder<KSpace> > FunctorNormalLeast;
 
  //constant convolution functor
  typedef functors::ConstValue<double> ConstFunctor;
 
  typedef LocalEstimatorFromSurfelFunctorAdapter<SurfaceContainer, LpMetric<Z3i::Space>, FunctorGaussian, ConstFunctor> ReporterK;
  typedef LocalEstimatorFromSurfelFunctorAdapter<SurfaceContainer, LpMetric<Z3i::Space>, FunctorMean, ConstFunctor> ReporterH;
  typedef LocalEstimatorFromSurfelFunctorAdapter<SurfaceContainer, LpMetric<Z3i::Space>, FunctorNormal, ConstFunctor> ReporterNormal;
  typedef LocalEstimatorFromSurfelFunctorAdapter<SurfaceContainer, LpMetric<Z3i::Space>, FunctorNormalLeast, ConstFunctor> ReporterNormalLeast;
#endif
 
  typedef functors::ElementaryConvolutionNormalVectorEstimator<Surfel, CanonicSCellEmbedder<KSpace> > FunctorNormalElem;
  typedef LocalEstimatorFromSurfelFunctorAdapter<SurfaceContainer, LpMetric<Z3i::Space>,
                                                 FunctorNormalElem, DGtal::functors::GaussianKernel> ReporterNormalElem;

We now create the instances for gridstep \( h=1.0 \) and a kernel radius 5.0:

  CanonicSCellEmbedder<KSpace> embedder(surface.container().space());
#ifdef WITH_CGAL
  FunctorGaussian estimatorK(embedder,1.0);
  FunctorMean estimatorH(embedder, 1.0);
  FunctorNormal estimatorN(embedder,1.0);
  FunctorNormalLeast estimatorL(embedder,1.0);
 
  ConstFunctor constFunctor(1.0);
 
  ReporterK reporterK;
  ReporterH reporterH;
  ReporterNormal reporterN;
  ReporterNormalLeast reporterL;
#endif
 
  LpMetric<Z3i::Space> l2(2.0);
  FunctorNormalElem estimatorNormalElem(embedder,1.0);
  DGtal::functors::GaussianKernel gaussian(2.0);
  ReporterNormalElem reporterElem(surface, l2,
                                  estimatorNormalElem, gaussian);

We can now estimate the quantities at a surfel given by an iterator (here, the iterator at surface.begin()):

#ifdef WITH_CGAL
  reporterK.attach(surface);
  reporterH.attach(surface);
  reporterN.attach(surface);
  reporterL.attach(surface);
 
  reporterK.init(1, surface.begin(), surface.end());
  reporterH.init(1, surface.begin(), surface.end());
  reporterN.init(1, surface.begin(), surface.end());
  reporterL.init(1, surface.begin(), surface.end());
 
  reporterK.setParams(l2, estimatorK, constFunctor, 5.0);
  reporterH.setParams(l2, estimatorH, constFunctor, 5.0);
  reporterN.setParams(l2, estimatorN, constFunctor, 5.0);
  reporterL.setParams(l2, estimatorL, constFunctor, 5.0);
 
  FunctorGaussian::Quantity valK = reporterK.eval( surface.begin());
  FunctorMean::Quantity valH = reporterH.eval( surface.begin());
  FunctorNormal::Quantity valN = reporterN.eval( surface.begin());
  FunctorNormalLeast::Quantity valL = reporterL.eval( surface.begin());
#endif
  
  reporterElem.attach(surface);
  reporterElem.setParams(l2,
                         estimatorNormalElem, gaussian, 5.0);
  reporterElem.init(1.0, surface.begin(), surface.end());
  FunctorNormalElem::Quantity valElem = reporterElem.eval( surface.begin());
 
#ifdef WITH_CGAL
  trace.info() << "Gaussian = "<<valK <<std::endl;
  trace.info() << "Mean = "<<valH<< std::endl;
  trace.info() << "Normal Vector (from Monge form) = "<<valN<< std::endl;
  trace.info() << "Normal Vector (linear least square) = "<<valL<< std::endl;
#endif
  trace.info() << "Normal Vector (Elementary conv) = "<<valElem<< std::endl;

See also: exampleEstimatorFromSurfelFunctors.cpp

Table of Contents

Introduction

Implementation details

Usage Example