Geodesic Mean-Shift

Lior Shapira, Ariel Shamir, Daniel Cohen-Or

Mesh Analysis Using Geodesic Mean-Shift

In this paper we introduce a versatile and robust method for analyzing the feature space associated with a given mesh surface. The method is based on the mean-shift operator which was shown to be successful in image and video processing. Its strength lays in the fact that it works in a single joint space of geometry and attributes called the feature-space. The mean-shift procedure works as a gradient ascend finding maxima of an estimated probability density function in feature-space. Our method for using the meanshift technique on surfaces solves several difficulties. First, meshes as opposed to images do not present a regular and uniform sampling of domain. Second, on surfaces meshes the shifting procedure must be constrained to stay on the surface and preserve geodesic distances. We define a special local geodesic parameterizations scheme, and use it to generalize the mean-shift procedure to unstructured surface meshes. Our method can support piecewise linear attribute definitions as well as piecewise constant attributes.

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Local Geodesic Parametrization: An Ant’s Perspective

Two dimensional parameterizations of meshes is a dynamic field of research. Most works focus on parameterizing complete surfaces, attempting to satisfy various constraints on distances and angles and produce a 2D map with minimal errors. Except for developable surfaces no single map can be devoid of errors, and a parametrization produced for one purpose usually doesn’t suit others. This work presents a different viewpoint. We try and acquire the perspective of an ant living on the surface. The point on which it stands is the center of its world, and importance diminishes from there onward. Distances and angles measured relative to its position have higher importance than those measured elsewhere. Hence, the local parametrization of the geodesic neighborhood should convey this perspective by mostly preserving geodesic distances from the center. We present a method for producing such overlapping localparametrization for each vertex on the mesh. Our method provides an accurate rendition of the local area of each vertex and can be used for several purposes, including clustering algorithms which focus on local areas of the surface within a certain window such as Mean Shift.

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