A PhD thesis @ NALAG

TW 2009_03

Ward Van Aerschot
Multiscale geometric image approximation
September 18, 2009

Advisor(s): Adhemar Bultheel and Maarten Jansen

Abstract

The application of normal meshes originated from the area of computer graphics applications in the search for surface remeshing and compression techniques. For smooth surfaces it turns out that one can generate multilevel regular mesh approximations by adding detail vectors restricted to a predefined normal direction. These normal mesh approximations only need a series of scalar data to represent the x; y; z-coordinates of its vertices.

In this thesis we use the concept of normal approximations in a different setting. We try to approximate `objects' which are piecewise smooth, i.e. smooth except on a smooth boundary. It is well known that for these kind of objects currently used scale-space techniques have a poor approximation performance. Recently, several methods have emerged using anisotropic elements incorporating directionality to epostciently capture geometric content. We propose the use of normal approximations as an alternative to approximate such objects. The proposed normal approximation scheme generates highly anisotropic triangulations together with a piecewise polynomial surface approximation. Recent results for multiresolution approximation of piecewise smooth geometric objects point out that polynomial approximation on triangulations are very promising, although practical and epostcient implementations are still in their infancy. Using our normal approximation method, however, we come up with a concrete method. The vertices of the generated triangulations are attracted naturally towards places of high functional uctuation.

As normal approximations are in fact unlifted wavelet transforms we enhance the scheme with so-called geometric wavelets, i.e., differences between a local optimal approximating polynomial and the normal approximation on triangle segments. We state results for piecewise smooth function approximation and investigate the application of a normal approximation based scheme in image coding.