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) [9]
AlgebraicSphere	Oriented points	OrientedSphereFitImpl (nD) [9]
AlgebraicSphere	Non-oriented points	UnorientedSphereFitImpl (nD) [6]

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	CovarianceBase (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 requires, we use C++20 concepts. Each module has its own concepts file. For fitting tools and extensions, the requirements are described in this file: Fitting/concepts.h. For each concept, a cast operator is provided (typically first line). All subsequent operation should be valid, with the '->' further describing the return type.

Computing Curvatures

Ponca offers several ways to compute curvatures, most of which are reviewed and compared by Arnal-Anger et al. in [4] and listed in the table below:

Estimator Name	Estimated quantities	Usage
Distance to PCA plane [8]	Mean curvature	Method potential() using FitSmoothNormalCovariance = Basket<Point, WeightSmoothFunc, CovariancePlaneFit>;
Surface Variation [16]	Mean curvature	Method surfaceVariation() using FitSmoothNormalCovariance = Basket<Point, WeightSmoothFunc, CovariancePlaneFit>;
Growing Least Squares [14]	Mean curvature	Basket<P,W,OrientedSphereFit,GLSParam> // method kappa()
Point Set Surfaces (PSS) [3]	Curvature Tensor	API of CurvatureEstimatorBase : using EstimatorSmoothNormalCovariance = BasketDiff<FitSmoothNormalCovariance, FitSpaceDer, CovariancePlaneDer, NormalDerivativeWeingartenEstimator, WeingartenCurvatureEstimatorDer>;
Monge Patch fitting (comparable to jets fitting[5] )	Curvature Tensor	API of CurvatureEstimatorBase using a QuadraticHeightField: using FitMongeSmooth = Basket<Point, WeightSmoothFunc, MongePatchQuadraticFit>;
Monge Patch fitting (comparable to jets fitting[5] )	Curvature Tensor	API of CurvatureEstimatorBase using a RestrictedQuadraticHeightField: using FitMongeRestrictedSmooth = Basket<Point, WeightSmoothFunc, MongePatchRestrictedQuadraticFit>;
Algebraic Point Set Surfaces (APSS) [9]	Curvature Tensor	API of CurvatureEstimatorBase: using FitSmoothOrientedSpatial = BasketDiff<FitSmoothOriented, FitSpaceDer, OrientedSphereDer, NormalDerivativeWeingartenEstimator, WeingartenCurvatureEstimatorDer>;
Algebraic Shape Operator (ASO) [12]	Curvature Tensor	API of CurvatureEstimatorBase: using ASOSmooth = BasketDiff<FitSmoothOriented, FitSpaceDer, OrientedSphereDer, MlsSphereFitDer, 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
DistWeightFilter	Eval location: p, Radius: t	Filter points further than t, and weigh them according to kernel (Available Weighting kernel)
NoWeightFilter	Eval location: p	Constant kernel for every point. Does not check for radius unlike DistWeightFilter with contant kernel. Transform neighbors to local frame.
NoWeightFilterGlobal	Eval location: p	Constant kernel for every point. Does not check for radius unlike DistWeightFilter 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>	\(\phi(x) = \exp\left(-\frac{1}{2}x^2\right)\)