Cone fields and topological sampling in manifolds with bounded curvature

Often noisy point clouds are given as an approximation of a particular compact set of interest. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the Hausdorff distance between two compact sets, for when certain offsets of these two sets are homotopic in terms of the absence of {\mu}-critical points in an annular region. Since an offset of a set deformation retracts to the set itself provided that there are no critical points of the distance function nearby, we can use this theorem to show when the offset of a point cloud is homotopy equivalent to the set it is sampled from. The ambient space can be any Riemannian manifold but we focus on ambient manifolds which have nowhere negative curvature. In the process, we prove stability theorems for {\mu}-critical points when the ambient space is a manifold.Comment: 20 pages, 3 figure

Skeletal representations of orthogonal shapes

In this paper we present two skeletal representations applied to orthogonal shapes of R^n : the cube axis and a family of skeletal representations provided by the scale cube axis. Orthogonal shapes are a subset of polytopes, where the hyperplanes of the bounding facets are restricted to be axis aligned. Both skeletal representations rely on the L∞ metric and are proven to be homotopically equivalent to its shape. The resulting skeleton is composed of n − 1 dimensional facets. We also provide an efficient and robust algorithm to compute the scale cube axis in the plane and compare the resulting skeleton with other skeletal representations.Postprint (published version

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

Multi-Dimensional Medial Geometry: Formulation, Computation, and Applications

Medial axis is a classical shape descriptor. It is a piece of geometry that lies in the middle of the original shape. Compared to the original shape representation, the medial axis is always one dimension lower and it carries many intrinsic shape properties explicitly. Therefore, it is widely used in a large amount of applications in various fields. However, medial axis is unstable to the boundary noise, often referred to as its instability. A small amount of change on the object boundary can cause a dramatic change in the medial axis. To tackle this problem, a significance measure is often associated with the medial axis, so that medial points with small significance are removed and only the stable part remains. In addition to this problem, many applications prefer even lower dimensional medial forms, e.g., shape centers of 2D shapes, and medial curves of 3D shapes. Unfortunately, good significance measures and good definitions of lower dimensional medial forms are still lacking. In this dissertation, we extended Blum\u27s grassfire burning to the medial axis in both 2D and 3D to define a significance measure as a distance function on the medial axis. We show that this distance function is well behaved and it has nice properties. In 2D, we also define a shape center based on this distance function. We then devise an iterative algorithm to compute the distance function and the shape center. We demonstrate usefulness of this distance function and shape center in various applications. Finally we point out the direction for future research based on this dissertation