Geodesics for point based model processing

In this thesis we present a framework for analysis and processing of point set surfaces based on geodesics. The first part deals with computing geodesics on point set surfaces. Two main techniques are presented. The first technique efficiently computes fast approximations of geodesics. It relies on a graph representation of the surface using an extended sphere of influence graph (eSIG). Approximations of geodesics are computed as shortest paths on the eSIG by using Dijkstra-like algorithms. The second technique computes accurate approximations of geodesics and is robust with respect to noise and outliers. It is a novel technique, which computes piecewise linear approximations of geodesics on point set surfaces by minimizing an energy function defined for piecewise linear paths. This thesis presents two surface processing/analysis applications, in which geodesics are used for compression of textured point based surfaces of genus-0, and for point based model retrieval. The compression scheme described in this thesis, is based on a spherical representation of a genus-0 surface represented by a surfel set. It incorporates geometry, texture and other surface attributes in a unified manner. Each surfel, with its attribute vector, is mapped onto a sphere by using a novel and efficient approach for dense surface mapping, which considers rectangular distance matrices. In this approach the geodesic distances between each surfel and a set of key-surfels are optimally preserved in order to improve the resolution and eliminating the need for interpolation that complicates and slows down existing surface unfolding methods. The resulting spherical vector-valued function is then regularly resampled. Its components are decorrelated by the Karhunen-Loève transform, represented by spherical wavelets and encoded using the zerotree algorithm. The compression-induced geometry error is smaller than that of other methods at very low bit-rates. A system for point based model retrieval is also presented. It comprises: a surfelization technique for converting polygonal mesh models into corresponding surfel based representations, and two shape descriptors for comparing efficiently the geometry and the topology of point based models. The presented surfelization technique is designed for converting large databases of triangular mesh models into corresponding surfel based representations. It consists of an enhanced triangle rasterization procedure adapted to the geometric features of a triangulated model, which preserves its main visual and geometric characteristics in the resulting surfel based model. The first shape descriptor focuses on effectively matching the geometry of surfel based models. It is constructed through a pose normalization of surfel based models, and a Fourier analysis of depth-buffer images, which are rendered from different viewpoints. Our depth-buffer image based shape descriptor is able to compare the geometry of several classes of objects. Moreover, it possesses an embedded multi-resolution representation, is invariant with respect to similarity transforms and robust with respect to outliers. The second shape descriptor is based on geodesics, and is designed for geometry and topology matching of point based models. A normalized geodesic distance matrix is computed from a set of reference points, which are selected randomly on a point set surface. The similarity of two point based models is then computed by comparing the associated distance matrices using a histogram based approach. This technique resulted in being an effective method for the comparison of deformed objects of the same class, since it is invariant with respect to similarity and isometric transformations.