Fitting Module: Reference Manual

Table of Contents

• Fitting techniques Overview
• Capabilities of Fitting tools
• Computing Curvatures
• Available filters
• Available Weighting kernel

This page provides a detailed list of what the Fitting module has to offers.

Fitting techniques Overview

We provide various approaches to approximate, analyze or characterize the geometric properties of local neighborhoods. In most situations, all these tools are based on the estimation of a geometric Primitive approximating the neighborhood, on top of which we provide more advanced computations. The table below summarizes the Primitives available in the library, as well as the associated fitting techniques:

Computed primitive	Required Input	Fitting techniques
Line	Points only	CovarianceLineFitImpl (nD)
Plane	Points only	CovariancePlaneFitImpl (nD)
Plane	Oriented points	MeanPlaneFitImpl (nD, co-dimension 1)
MongePatch	Points only	MongePatchQuadraticFitImpl and MongePatchRestrictedQuadraticFitImpl (both 3D)
AlgebraicSphere	Points only	SphereFitImpl (nD) [6]
AlgebraicSphere	Oriented points	OrientedSphereFitImpl (nD) [6]
AlgebraicSphere	Non-oriented points	UnorientedSphereFitImpl (nD) [4]

Ponca also provide methods to compute objects that are not geometric primitives, but provide other informations such as curvatures:

Computed object	Required Input	Fitting techniques
Barycenter	Points only	MeanPosition (nD)
Normal Vector	Oriented points	MeanNormal (nD, co-dimension 1)
Covariance Matrix	Points only	CovarianceFitBase (nD)
Weingarten Map	Points only	FundamentalFormWeingartenEstimator (3D) via MongePatch
Weingarten Map	Oriented points	NormalDerivativeWeingartenEstimator (3D)

Capabilities of Fitting tools

As described in fitting_extensions_deps, The tool classes provides different kinds of compute capabilities. To describe what a class provides, we uses flags that allows us to check for compatibility at compile time.

Here is an in depth description of all the labels of the Fitting tools :

Label	Provided by	Required by	Outputs
PROVIDES_MEAN_POSITION Mean of the input points vectors	MeanPosition	MeanPositionDer MeanPlaneFitImpl OrientedSphereFitImpl CovarianceFitBase UnorientedSphereFitImpl	barycenter() barycenterDistance() barycenterLocal()
PROVIDES_MEAN_POSITION_DERIVATIVE Provides derivative of the mean position	MeanPositionDer	CovarianceFitDer OrientedSphereDerImpl UnorientedSphereDerImpl	barycenterDerivatives()
PROVIDES_MEAN_NORMAL Mean of the normal vectors	MeanNormal	MeanPlaneFitImpl OrientedSphereFitImpl	meanNormalVector()
PROVIDES_MEAN_NORMAL_DERIVATIVE Provides derivative of the mean normal	MeanNormalDer	-	dMeanNormal()
PROVIDES_PRIMITIVE_BASE Provides base API for primitives	PrimitiveBase	PrimitiveDer AlgebraicSphere	getWeightSum() setNeighborFilter() getNeighborFilter() getCurrentState() getNumNeighbors() isReady() isStable()
PROVIDES_PRIMITIVE_DERIVATIVE Provides base API for primitive derivatives	PrimitiveDer	GLSDer CovarianceFitDer MeanPositionDer MeanNormalDer MlsSphereFitDer OrientedSphereDerImpl UnorientedSphereDerImpl	isScaleDer() isSpaceDer() derDimension()
PROVIDES_GLS_PARAMETRIZATION Growing Least Squares reparametrization of the OrientedSphereFit	GLSParam	GLSDer	tau() eta() kappa() tau_normalized() eta_normalized() kappa_normalized() fitness() compareTo()
PROVIDES_GLS_DERIVATIVE Differentiation of GLSParam	GLSDer	-	dtau() deta() dkappa() dtau_normalized() deta_normalized() dkappa_normalized()
PROVIDES_GLS_GEOM_VAR Provides Geometric Variation as a weighted sum of the GLS scale-invariant partial derivatives	GLSDer	-	geomVar()
PROVIDES_ALGEBRAIC_SPHERE	AlgebraicSphere	OrientedSphereFitImpl OrientedSphereDerImpl SphereFitImpl UnorientedSphereFitImpl UnorientedSphereDerImpl	potential() project() primitiveGradient() isPlane() isValid() isApprox() changeBasis() prattNorm() prattNorm2() applyPrattNorm() radius() center() isNormalized()
PROVIDES_ALGEBRAIC_SPHERE_DERIVATIVE Provides derivatives for algebraic sphere	OrientedSphereDerImpl UnorientedSphereDerImpl	GLSDer MlsSphereFitDer	dPotential()
PROVIDES_COVARIANCE_PLANE_DERIVATIVE Provides derivatives for hyper-planes	CovariancePlaneDerImpl	-	dPotential()
PROVIDES_NORMAL_DERIVATIVE Provides the derivatives of the primitive normal	CovariancePlaneDerImpl MlsSphereFitDer OrientedSphereDerImpl UnorientedSphereDerImpl	NormalDerivativesCurvatureEstimator	dNormal()
PROVIDES_POSITION_COVARIANCE Provides the covariance matrix	CovarianceFitBase	CovarianceFitDer CovarianceLineFitImpl CovariancePlaneFitImpl	solver()
PROVIDES_POSITION_COVARIANCE_DERIVATIVE	CovarianceFitDer	CovariancePlaneDerImpl	-
PROVIDES_PLANE	Plane	CovariancePlaneFitImpl CovariancePlaneDerImpl ProjectedNormalCovarianceCurvatureEstimator MeanPlaneFitImpl MongePatch	setPlane() potential() project() primitiveGradient()
PROVIDES_TANGENT_PLANE_BASIS Turns a point in ambient 3D space to the tangent plane.	CovariancePlaneFitImpl	MongePatch	worldToTangentPlane() tangentPlaneToWorld()

Computing Curvatures

Ponca offers several ways to compute curvatures, some of which are reviewed and compared by Lejemble et al. in [9] and listed in the table below:

Estimator Name	Estimated quantities	Usage
Distance to PCA plane [5]	Mean curvature	Method potential() using FitSmoothNormalCovariance = Basket<Point, WeightSmoothFunc, CovariancePlaneFit>;
Surface Variation [12]	Mean curvature	Method surfaceVariation() using FitSmoothNormalCovariance = Basket<Point, WeightSmoothFunc, CovariancePlaneFit>;
Growing Least Squares [10]	Mean curvature	Basket<P,W,OrientedSphereFit,GLSParam> // method kappa()
Point Set Surfaces (PSS) [2]	Curvature Tensor	API of CurvatureEstimatorBase : using EstimatorSmoothNormalCovariance = BasketDiff<FitSmoothNormalCovariance, FitSpaceDer, CovariancePlaneDer, CurvatureEstimatorDer, NormalDerivativeWeingartenEstimator>;
Monge Patch fitting (comparable to jets fitting[3] )	Curvature Tensor	API of CurvatureEstimatorBase using a QuadraticHeightField: using FitMongeSmooth = Basket<Point, WeightSmoothFunc, MongePatchQuadraticFit>;
Monge Patch fitting (comparable to jets fitting[3] )	Curvature Tensor	API of CurvatureEstimatorBase using a RestrictedQuadraticHeightField: using FitMongeRestrictedSmooth = Basket<Point, WeightSmoothFunc, MongePatchRestrictedQuadraticFit>;
Algebraic Point Set Surfaces (APSS) [6]	Curvature Tensor	API of CurvatureEstimatorBase: using FitSmoothOrientedSpatial = BasketDiff<FitSmoothOriented, FitSpaceDer, OrientedSphereDer, CurvatureEstimatorDer, NormalDerivativeWeingartenEstimator, WeingartenCurvatureEstimatorDer>;
Algebraic Shape Operator (ASO) [9]	Curvature Tensor	API of CurvatureEstimatorBase: using ASOSmooth = BasketDiff<FitSmoothOriented, FitSpaceDer, OrientedSphereDer, MlsSphereFitDer, CurvatureEstimatorDer, NormalDerivativeWeingartenEstimator, WeingartenCurvatureEstimatorDer>;

Figure 3. Example of mean curvature (GLSParam::kappa) computed at a fine (left) and a coarse (right) scale, and rendered with a simple color map (orange for concavities, blue for convexities).

Available filters

Class name	Parameters	Documentation
DistWeightFunc	Eval location: p, Radius: t	Filter points further than t, and weigh them according to kernel (Available Weighting kernel)
NoWeightFunc	Eval location: p	Constant kernel for every point. Does not check for radius unlike DistWeightFunc with contant kernel. Transform neighbors to local frame.
NoWeightFuncGlobal	Eval location: p	Constant kernel for every point. Does not check for radius unlike DistWeightFunc with contant kernel. Keep neighbors coordinates in global frame
NeighborFilterStoreNormal	Eval location: p, Radius: t, Normal: n	Stores normal. It extends another filter as template param

Available Weighting kernel

During any computation, each neighbor can be weighted by an amount defined by \(w(x) = \phi\left(\frac{||x - p||}{r}\right)\), where \(p\) is the evaluation location and \(r\) is the radius. This sections list the different function \(\phi\) available in Ponca.

Class	Expression
ConstantWeightKernel<_Scalar>	\(\phi(x) = 1\)
SmoothWeightKernel<_Scalar>	\(\phi(x) = (x^2 - 1)^2\)
PolynomialSmoothWeightKernel<_Scalar, m, n>	\(\phi(x) = (x^n - 1)^m\)
WendlandWeightKernel<_Scalar>	\(\phi(x) = (1 - x)^4(4x + 1)\)
SingularWeightKernel<_Scalar>	\(\phi(x) = \frac{1}{x^2}\)
CompactExpWeightKernel<_Scalar>	\(\phi(x) = \exp\left(\frac{-x^2}{1 - x^2}\right)\)
GaussianWeightKernel<_Scalar>	\(\phi(x) = \exp\left(-\frac{1}{2}x^2\right)\)