Properties of Gauss digitized sets and digital surface integration

International audienceThis paper presents new topological and geometrical properties of Gauss digitizations of Euclidean shapes, most of them holding in arbitrary dimension $d$. We focus on $r$-regular shapes sampled by Gauss digitization at gridstep $h$. The digitized boundary is shown to be close to the Euclidean boundary in the Hausdorff sense, the minimum distance $\frac{\sqrt{d}}{2}h$ being achieved by the projection map $\xi$ induced by the Euclidean distance. Although it is known that Gauss digitized boundaries may not be manifold when $d \ge 3$, we show that non-manifoldness may only occur in places where the normal vector is almost aligned with some digitization axis, and the limit angle decreases with $h$. We then have a closer look at the projection of the digitized boundary onto the continuous boundary by $\xi$. We show that the size of its non-injective part tends to zero with $h$. This leads us to study the classical digital surface integration scheme, which allocates a measure to each surface element that is proportional to the cosine of the angle between an estimated normal vector and the trivial surface element normal vector. We show that digital integration is convergent whenever the normal estimator is multigrid convergent, and we explicit the convergence speed. Since convergent estimators are now available in the litterature, digital integration provides a convergent measure for digitized objects

Implementation of Integral based Digital Curvature Estimators in DGtal

In many geometry processing applications, differential geometric quantities estimation such as curvature or normal vector field is an essential step. In [1], we have defined curvature estimators on digital shape boundaries based on Integral Invariants. In this paper, we focus on implementation details of these estimators

Integral based Curvature Estimators in Digital Geometry

International audienceIn many geometry processing applications, the estimation of differential geometric quantities such as curvature or normal vector field is an essential step. When designing such estimators, we have to pay attention to both its theoretical properties and practical effectiveness. In this paper, we investigate a new class of estimators on digital shape boundaries based on Integral Invariants. More precisely, we provide proofs of multigrid convergence of curvature estimators which are easy to implement on digital data. Furthermore, we discuss about some algorithmic optimisations and detail a complete experimental evaluation

Integral based Curvature Estimators in Digital Geometry

International audienceIn many geometry processing applications, the estimation of differential geometric quantities such as curvature or normal vector field is an essential step. When designing such estimators, we have to pay attention to both its theoretical properties and practical effectiveness. In this paper, we investigate a new class of estimators on digital shape boundaries based on Integral Invariants. More precisely, we provide proofs of multigrid convergence of curvature estimators which are easy to implement on digital data. Furthermore, we discuss about some algorithmic optimisations and detail a complete experimental evaluation

Interactive Curvature Tensor Visualization on Digital Surfaces

International audienceInteractive 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 triangu-lated 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