Surface meshes are a prevalent representation of 3D objects consisting of vertices, edges and faces. Features defined on the vertices and faces of surface meshes are at the basis of many works in recent years, guiding algorithms for parametrization, shape partitioning, shape matching and more. However, these features suffer from differing mesh quality, tessellation and feature approximation calculations. As a result these features may be noisy and prone to errors. A method to reduce noise in such complex data is to use a non-parametric clustering algorithm. A versatile and robust method is the mean-shift operator, this method works as a gradient ascend
finding maxima of an estimated probability density function in feature-space, a joint domain of geometry and attributes. In recent years the mean-shift operator was shown to be successful in image, video and volumetric meshes processing. 

We present a method, Geodesic Mean-shift, adapted to analyze and filter features on surface meshes. Geodesic Mean-shift works in feature-space, iteratively seeking the mode of each data point (A vertex and its associated features), the maximal point in the underlying probability density function. For this the algorithm requires a uniform data sampling around each point, and since the data is defined only on the mesh, we must constrain the search to the surface. Using a local parametrization scheme which answers both requirements and hardware assisted rasterization, we average the feature space around the center and find the mean-shift vector with which to iteratively continue. The collected modes of all the vertices are a valuable tool for further algorithms such as segmentation, clustering and filtering. The filtered feature-space retains the salient attributes of the original features. We present examples of mean-shift on scalar and vector features. 

There are many types of features which pertain to surface meshes and are useful in various algorithms. Face and vertex normals, curvature and centricity are just some of the features which can be extracted from the geometry and connectivity of a surface mesh. These features and others have been used extensively in the past and are still in use. However, these features are noisy, highly dependent on the mesh, and reflect only the local shape (not global features) of the mesh. Surface meshes represent volumetric objects, and as such, it is important to find a connection between surface and volume. An effective link is provided by the Medial Axis Transform, however computing the medial axis is expensive and difficult to handle. We define a new surface feature which helps bridge the link between surface and volume. The Volume shape-function expresses the diameter of the volume in the neighborhood of each vertex on the mesh, and is computed using a simple ray shooting scheme. The function is shown to be consistent through pose differences of the same object, and through shape variance of similar but different objects. Its strength lies in the fact that it promotes the distinction among parts based on their
solid shape and functionality rather than their surface attributes. As such, it can be used in many applications such as shape matching, partitioning and more.

As stated above, in order to apply mean-shift on surface meshes we must overcome two major problems, define a convex and uniform domain within which the algorithm can function, and constrain the algorithm to the surface of the mesh. We solve both these problems using a local parametrization scheme. Local Geodesic parametrization creates for each vertex on the mesh, a local map which preserves geodesic distances from that vertex outwards. Much like 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. For each vertex we begin by growing a patch, whose radius is a set geodesic distance. If the patch is not homeomorphic to a disc, we find seams at the farthest points possible from the center (according to geodesic distances) and cut the patch. The patch boundary is then set (either to reflect natural shape using a virtual boundary, or stretched
to the natural convex shape of the patch, as used for mean-shift) and the patch is parametrized. Parametrization is done using Floater’s technique of Mean-value coordinates. The result is a set of overlapping parametrizations, which are used for the mean-shift algorithm.