Wavelets for Multi-resolution Analysis of Triangular Surface Meshes

The application of Wavelets to the Multi-resolution Analysis of surfaces provides an elegant, mathematically rigorous framework for the implementation of subdivision surfaces. We present a method similar to [Lou95] for multiresolution analysis of surfaces with subdivision connectivity. However, due to error incurred during surface remeshing (as with [EDD 95, LSS 98]) we find Wavelets an unsuitable technique for feature preservation during surface compression. 1 Theory As in Lounsbery et. al [Lou95], the technique implemented makes use of a semi-regular multi-resolution framework with biorthogonal surface wavelets. The underlying theory of wavelets and multi-resolution analysis will not be detailed here. For more information regarding terminology and theory the reader is referred to [Mal89, Dau92]. The underlying algorithm is relatively simple: Given a base-mesh / final-mesh pair ( respectively), the algorithm generates hierarchical levels of resolution such that . Note that in order to generate mesh each triangle is split into 4 by introducing new points at the midpoints of each of the edges of (as in Figure 1). Splitting Step Splitting Step Figure 1: Examples of quadrisection. Vertices at consecutive levels introduce vertices at the midpoints of the edges at previous levels. 1.1 Refinable Basis Functions We define "!$# to be the % th scaling function at resolution & , while ! represents a point over the domain. ')( *!$# is hence defined as the matrix consisting of the % functions "!$# . From previous work [Lou95] these functions have been proven to be refinable, and hence "!$# can be written as a linear combination of the functions *!$# . 1 We now write the matrix ' ( !$# as ' ( "!$# ( "!$# ( "! # where ( "! # consists of the scaling functions "!$# associated with the old vertices of , while ( *!$# refers to the scaling functions associated with vertices added to the last mesh. The refinability of the scaling functions allows us to define a matrix ( such that