Jérémy Levallois

Multigrid convergent principal curvature estimators in digital geometry

We propose discrete principal curvature estimators based on integral invariants.We prove the multigrid convergence of these estimators.We provide an experimental evaluation on synthetic and real data. In many geometry processing applications, the estimation of differential geometric quantities such as curvature or normal vector field is an essential step. In this paper, we investigate a new class of estimators on digital shape boundaries based on integral invariants (Pottmann et al., 2007) 39. More precisely, we provide both proofs of multigrid convergence of principal curvature estimators and a complete experimental evaluation of their performances.

Integral based curvature estimators in digital geometry

In many geometry processing applications, the estimation of differential geometric quantities such as curvature or normal vector field is an essential step. In this paper, we investigate a new class of estimators on digital shape boundaries based on Integral Invariants. More precisely, we provide both proofs of multigrid convergence of curvature estimators and a complete experimental evaluation of their performances.

Parameter-Free and Multigrid Convergent Digital Curvature Estimators

In many geometry processing applications, the estimation of differential geometric quantities such as curvature or normal vector field is an essential step. Focusing on multigrid convergent estimators, most of them require a user specified parameter to define the scale at which the analysis is performed (size of a convolution kernel, size of local patches for polynomial fitting, etc). In a previous work, we have proposed a new class of estimators on digital shape boundaries based on Integral Invariants. In this paper, we propose new variants of these estimators which are parameter-free and ensure multigrid convergence in 2D. As far as we know, these are the first parameter-free multigrid convergent curvature estimators.

Scale-space feature extraction on digital surfaces

A classical problem in many computer graphics applications consists in extracting significant zones or points on an object surface, like loci of tangent discontinuity (edges), maxima or minima of curvatures, inflection points, etc. These places have specific local geometrical properties and often called generically features. An important problem is related to the scale, or range of scales, for which a feature is relevant. We propose a new robust method to detect features on digital data (surface of objects in Z 3 , which exploits asymptotic properties of recent digital curvature estimators. In Coeurjolly et al. 1 and Levallois et al. 1,2, authors have proposed curvature estimators (mean, principal and Gaussian) on 2D and 3D digitized shapes and have demonstrated their multigrid convergence (for C3-smooth surfaces). Since such approaches integrate local information within a ball around points of interest, the radius is a crucial parameter. In this paper, we consider the radius as a scale-space parameter. By analyzing the behavior of such curvature estimators as the ball radius tends to zero, we propose a tool to efficiently characterize and extract several relevant features (edges, smooth and flat parts) on digital surfaces. Graphical abstractDisplay Omitted HighlightsWe propose a robust discrete feature estimators based on Integral Invariants.We propose a classification method to detect edges, smooth and flat regions.We provide an experimental evaluation on synthetic and real data.

Robust and Convergent Curvature and Normal Estimators with Digital Integral Invariants

We present, in details, a generic tool to estimate differential geometric quantities on digital shapes, which are subsets of Z^d. This tool, called digital integral invariant, simply places a ball at the point of interest, and then examines the intersection of this ball with input data to infer local geometric information. Just counting the number of input points within the intersection provides curvature estimation in 2D and mean curvature estimation in 3D. The covariance matrix of the same point set allows to recover principal curvatures, principal directions and normal direction estimates in 3D. We show the multigrid convergence of all these estimators, which means that their estimations tend toward the exact geometric quantities on — smooth enough— Euclidean shapes digitized with finer and finer gridsteps. During the course of the chapter, we establish several multigrid convergence results of digital volume and moments estimators in arbitrary dimensions. Afterwards, we show how to set automatically the radius parameter while keeping multigrid convergence properties. Our estimators are then demonstrated to be accurate in practice, with extensive comparisons with state-of-the-art methods. To conclude the exposition, we discuss their robustness to perturbations and noise in input data and we show how such estimators can detect features using scale-space arguments.

Dgtal-Team/Dgtaltools: Release 0.9.4.1

Small fix including: Bugfix parsing options in regularization tool (volSurfaceRegularization) Min DGtal version updated

Interactive Curvature Tensor Visualization on Digital Surfaces

Interactive visualization is a very convenient tool to explore complex scientific data or to try different parameter settings for a given processing algorithm. In this article, we present a tool to efficiently analyze the curvature tensor on the boundary of potentially large and dynamic digital objects mean and Gaussian curvatures, principal curvatures, principal directions and normal vector field. More precisely, we combine a fully parallel pipeline on GPU to extract an adaptive triangulated isosurface of the digital object, with a curvature tensor estimation at each surface point based on integral invariants. Integral invariants being parametrized by a given ball radius, our proposal allows to explore interactively different radii and thus select the appropriate scale at which the computation is performed and visualized.