Parameter-Free and Multigrid Convergent Digital Curvature Estimators

Levallois, Jérémy; Cœurjolly, David; Lachaud, Jacques-Olivier

doi:10.1007/978-3-319-09955-2_14

Cited by 6 publications

(6 citation statements)

References 12 publications

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“…A first issue is that in dimension 3, even if digital planes can be defined, no result similar to Lemma 12 exists. In [LCL14], we have presented a slice based approach: intersecting Z with planes aligned with grid axis defines a set of 2D digital curves on which maximal DSS can be computed and thus average maximal DSS length can be obtained. Similarly, at a given surfel s, two specific 2D curves can be defined and thus local parameter-free estimator can be defined considering average DSS length in the two maximal segment pencils containing the projection of s. As in the local 2D case, some pathological configurations may occur preventing us to have a complete convergence proof of such estimators (see details in [LCL14]).…”

Section: Parameter-free Digital Curvature Estimatorsmentioning

confidence: 99%

Robust and Convergent Curvature and Normal Estimators with Digital Integral Invariants

Lachaud¹,

Cœurjolly²,

Levallois³

2017

Modern Approaches to Discrete Curvature

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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 onsmooth 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.

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Section: Parameter-free Digital Curvature Estimatorsmentioning

confidence: 99%

Robust and Convergent Curvature and Normal Estimators with Digital Integral Invariants

Lachaud¹,

Cœurjolly²,

Levallois³

2017

Modern Approaches to Discrete Curvature

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“…The quantity µ(s) is called the measure of the surfel s: it is the area of the projected surfel s onto the tangent plane induced by the estimated normal. Normal vectors are estimated using the estimator presented in [4,15] which has the multigrid convergence property. Note that summing µ for each surfel of the surface leads to an estimation of the global area of the shape boundary, which itself has a multigrid convergence property [13].…”

Section: New Laplace-beltrami Operator On Digital Surfacesmentioning

confidence: 99%

Heat Kernel Laplace-Beltrami Operator on Digital Surfaces

Caissard

Cœurjolly

Lachaud

et al. 2017

Discrete Geometry for Computer Imagery

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Many problems in image analysis, digital processing and shape optimization can be expressed as variational problems involving the discretization of the Laplace-Beltrami operator. Such discretizations have have been widely studied for meshes or polyhedral surfaces. On digital surfaces, direct applications of classical operators are usually not satisfactory (lack of multigrid convergence, lack of precision. . . ). In this paper, we first evaluate previous alternatives and propose a new digital Laplace-Beltrami operator showing interesting properties. This new operator adapts Belkin et al.[1] to digital surfaces embedded in 3D. The core of the method relies on an accurate estimation of measures associated to digital surface elements. We experimentally evaluate the interest of this operator for digital geometry processing tasks.

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“…In [7], a proposal was made for a parameter-free estimation of the radius of the ball by analyzing the shape w.r.t. the local shape geometry using maximal digital straight segments of the digital boundary.…”

Section: Curvature Tensor Estimationmentioning

confidence: 99%

Interactive Curvature Tensor Visualization on Digital Surfaces

Perrier

Levallois

Cœurjolly

et al. 2016

Discrete Geometry for Computer Imagery

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Abstract. 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.

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Parameter-Free and Multigrid Convergent Digital Curvature Estimators

Cited by 6 publications

References 12 publications

Robust and Convergent Curvature and Normal Estimators with Digital Integral Invariants

Robust and Convergent Curvature and Normal Estimators with Digital Integral Invariants

Heat Kernel Laplace-Beltrami Operator on Digital Surfaces

Interactive Curvature Tensor Visualization on Digital Surfaces

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