Marcos Craizer

Curvature and torsion estimators based on parametric curve fitting

Many applications of geometry processing and computer vision rely on geometric properties of curves, particularly, their curvature. Several methods have already been proposed to estimate the curvature of a planar curve, most of them for curves in digital spaces. This work proposes a new scheme for estimating curvature and torsion of planar and spatial curves, based on weighted least-squares fitting and local arc-length approximation. The method is simple enough to admit a convergence analysis that takes into account the effect of noise in the samples. The implementation of the method is compared to other curvature estimation methods showing a good performance. Applications to prediction in geometry compression are presented both as a practical application and as a validation of this new scheme.

Arc-length based curvature estimator

Many applications of geometry processing and computer vision rely on geometric properties of curves, particularly their curvature. Several methods have been proposed to estimate the curvature of a planar curve, most of them for curves in digital spaces. This work proposes a new method for curvature estimation based on weighted least square fitting and local arc-length approximation. Convergence analysis of this method and noise impact on the estimator accuracy are given. Numerical robustness issues are addressed with practical solutions. The implementation of the method is compared to other curvature estimation methods.

Area Distances of Convex Plane Curves and Improper Affine Spheres

The area distance to a convex plane curve is an important concept in computer vision. In this paper we describe a strong link between area distances and improper affine spheres. Based on this link, we propose an extremely fast algorithm to compute the inner area distance. Moreover, the concepts of the theory of affine spheres lead to a new definition of an area distance on the outer part of a convex plane curve. On the other hand, area distances provide a good geometrical understanding of improper affine spheres.

Entropy of inner functions

In this paper, we show that an inner functionf has finite entropy if and only if its derivativef′ lies in the Nevanlinna class. We prove also that the entropy off is given by the average of the logarithm of |f′|. The proof is based on the fact that, evenf being highly discontinuous on the circle, the action off−n on Borel subsets is smooth.

Combining Points and Tangents into Parabolic Polygons

Abstract Image and geometry processing applications estimate the local geometry of objects using information localized at points. They usually consider information about the tangents as a side product of the points coordinates. This work proposes parabolic polygons as a model for discrete curves, which intrinsically combines points and tangents. This model is naturally affine invariant, which makes it particularly adapted to computer vision applications. As a direct application of this affine invariance, this paper introduces an affine curvature estimator that has a great potential to improve computer vision tasks such as matching and registering. As a proof-of-concept, this work also proposes an affine invariant curve reconstruction from point and tangent data.

GEncode: Geometry-driven compression for general meshes

Performances of actual mesh compression algorithms vary significantly depending on the type of model it encodes. These methods rely on prior assumptions on the mesh to be efficient, such as regular connectivity, simple topology and similarity between its elements. However, these priors are implicit in usual schemes, harming their suitability for specific models. In particular, connectivity‐driven schemes are difficult to generalize to higher dimensions and to handle topological singularities. GEncode is a new single‐rate, geometry‐driven compression scheme where prior knowledge of the mesh is plugged into the coder in an explicit manner. It encodes meshes of arbitrary dimension without topological restrictions, but can incorporate topological properties, such as manifoldness, to improve the compression ratio. Prior knowledge of the geometry is taken as an input of the algorithm, represented by a function of the local geometry. This suits particularly well for scanned and remeshed models, where exact geometric priors are available. Compression results surfaces and volumes are competitive with existing schemes.

GEncode: Geometry-Driven Compression in Arbitrary Dimension and Co-Dimension

Among the mesh compression algorithms, different schemes compress better specific categories of model. In particular, geometry-driven approaches have shown outstanding performances on isosurfaces. It would be expected these algorithm to also encode well meshes reconstructed from the geometry, or optimized by a geometric re-meshing. GEncode is a new single-rate compression scheme that compresses the connectivity of these meshes at almost zero-cost. It improves existing geometry-driven schemes for general meshes on both geometry and connectivity compression. This scheme extends naturally to meshes of arbitrary dimensions in arbitrary ambient space, and deals gracefully with non-manifold meshes. Compression results for surfaces are competitive with existing schemes.

Alpha-expansions: a class of frame decompositions

This article analyzes a scheme for frame decompositions that is called α-expansion. In this scheme, the choice of a parameter α adequate to a given frame is a central point. We develop a theory that helps choosing the parameter α and also suggests algorithms for obtaining the α-expansions. The method is applied to frame expansions coding. In this context we give conditions under which, for high rate coding, α-expansions are better, in a rate × distortion sense, than schemes that find the frame coefficients first and quantize them in a second step.  2002 Elsevier Science (USA). All rights reserved.

Parabolic Polygons and Discrete Affine Geometry

Geometry processing applications estimate the local geometry of objects using information localized at points. They usually consider information about the normal as a side product of the points coordinates. This work proposes parabolic polygons as a model for discrete curves, which intrinsically combines points and normals. This model is naturally affine invariant, which makes it particularly adapted to computer vision applications. This work introduces estimators for affine length and curvature on this discrete model and presents, as a proof-of-concept, an affine invariant curve reconstruction

Even Dimensional Improper Affine Spheres

Abstract There are exactly two different types of bi-dimensional improper affine spheres: the non-convex ones can be modeled by the center-chord transform of a pair of planar curves while the convex ones can be modeled by a holomorphic map. In this paper, we show that both constructions can be generalized to arbitrary even dimensions: the former class corresponds to the center-chord transform of a pair of Lagrangian submanifolds while the latter is related to special Kahler manifolds. Furthermore, we show that the improper affine spheres obtained in this way are solutions of certain exterior differential systems. Finally, we also discuss the problem of realization of simple stable Legendrian singularities as singularities of these improper affine spheres.