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

Medial Axis Approximation and Regularization

Medial axis is a classical shape descriptor. Among many good properties, medial axis is thin, centered in the shape, and topology preserving. Therefore, it is constantly sought after by researchers and practitioners in their respective domains. However, two barriers remain that hinder wide adoption of medial axis. First, exact computation of medial axis is very difficult. Hence, in practice medial axis is approximated discretely. Though abundant approximation methods exist, they are either limited in scalability, insufficient in theoretical soundness, or susceptible to numerical issues. Second, medial axis is easily disturbed by small noises on its defining shape. A majority of current works define a significance measure to prune noises on medial axis. Among them, local measures are widely available due to their efficiency, but can be either too aggressive or conservative. While global measures outperform local ones in differentiating noises from features, they are rarely well-defined or efficient to compute. In this dissertation, we attempt to address these issues with sound, robust and efficient solutions. In Chapter 2, we propose a novel medial axis approximation called voxel core. We show voxel core is topologically and geometrically convergent to the true medial axis. We then describe a straightforward implementation as a result of our simple definition. In a variety of experiments, our method is shown to be efficient and robust in delivering topological promises on a wide range of shapes. In Chapter 3, we present Erosion Thickness (ET) to regularize instability. ET is the first global measure in 3D that is well-defined and efficient to compute. To demonstrate its usefulness, we utilize ET to generate a family of shape revealing and topology preserving skeletons. Finally, we point out future directions, and potential applications of our works in real world problems