Multiresolution Curve and Surface Representation: Reversing Subdivision Rules by Least‐Squares Data Fitting

Abstract

This work explores how three techniques for defining and representing curves and surfaces can be related efficiently. The techniques are subdivision, least‐squares data fitting, and wavelets. We show how least‐squares data fitting can be used to “reverse” a subdivision rule, how this reversal is related to wavelets, how this relationship can provide a multilevel representation, and how the decomposition/reconstruction process can be carried out in linear time and space through the use of a matrix factorization. Some insights that this work brings forth are that the inner product used in a multiresolution analysis in uences the support of a wavelet, that wavelets can be constructed by straightforward matrix observations, and that matrix partitioning and factorization can provide alternatives to inverses or duals for building efficient decomposition and reconstruction processes. We illustrate our findings using an example curve, grey‐scale image, and tensor‐product surface.

A ‘subdivision regression’ model for data analysis

“…In this operation, as clarified already, the coarse points f k and the matrix P are given, whereas the fine points f k+1 are the unknown yielded by them. The subdivision (forth) operation may be reversed [19], such that the fine points f k+1 and the matrix P are given, whereas the coarse points f k are the unknown yielded by them. That reverse (back) operation works by finding the solution to the least-squares system in Equation (2b)…”

“…In [19], the traditional reverse subdivision problem is hence formulated by Equation (10a) and solved by Equation (12 In [8,9], the primal and the dual rules (or matrices) trees are shown to extend the standard MRA synthesis (inverse wavelet transform, Equation (11)) and analysis (wavelet transform, Equation (12)) cascades, respectively. The 'generalized perturbed schemes' synthesis model [9] and the 'subdivision regression' analysis model (here, Section 2) are associated with these MRA synthesis and analysis, respectively.…”

“…Other MRA methods, such as wavelet transform and its inverse [18], related subdivision compression methods [11,16] and reverse subdivision [7,12,19], analyse and compress data using specific fixed filters, finally yielding coarser data and displacements. However, our methods analyse and compress the given (fine) data using some sparse (coarse) data, finally yielding the filters themselves, related in this sense to subdivision fitting.…”

“…This in itself is an attribute of the data. Figure 1 describes schematically the 'subdivision regression' model, while comparing it to the existing 'reverse subdivision' method [19]. Note that this figure relates to non-stationary masks P k and to different sets of template masks at each level,…”

Optimal Interpolatory Wavelets Transform for Multiresolution Triangular Meshes

“…By generalizing the uniform subdivision in topology to a new irregular subdivision scheme, Valette and Prost [2,3] extended the work of Lounsbery and proposed a wavelet-based multiresolution analysis, to be applied directly to irregular meshes whose connectivity is unchanged in the wavelet analysis. Samavati et al [4] showed how to use least-squares data fitting to reverse subdivision rules and constructed the wavelets by straightforward matrix observations. Samavati et al [5] constructed multiresolution surfaces of arbitrary topologies by locally reversing the Doo-Sabin subdivision scheme.…”