Wavelet multiresolution representation of curves and

surfaces

L-M Reissell
Department of Computer Science
University of British Columbia
e-mail: [email protected]
UBC Technical Report 93-17

May 1993. Revised February 1995.

Abstract

We develop wavelet methods for the multiresolution representation of parametric curves and surfaces. To support the representation, we construct a new family of compactly supported symmetric biorthogonal wavelets with interpolating scaling functions. The wavelets in these biorthogonal pairs have properties better suited for curves and surfaces than many commonly used filters. We also give examples of the applications of the wavelet approach: these include the derivation of compact hierarchical curve and surface representations using modified wavelet compression, identifying smooth sections of surfaces and a subdivision-like intersection algorithm for discrete plane curves.

1 Introduction

Many applications require the representation of complex multiscale digitized curves and surfaces in a geometric computing environment. Geographic information systems, medical imaging, computer vision, and graphics for scientific visualization are examples of such applications. A hierarchical data representation can be used to overcome the problems of dealing with the large amount of data inherent in such applications: apart from allowing computations at selected accuracy levels, a hierarchical representation has important applications such as rapid data classification, fast display, and surface design.

1.1 Our approach

In this work, we develop multiresolution wavelet methods for the representation of curves and surfaces, and construct a family of wavelets well suited for this purpose. Using these methods, we demonstrate some of the first applications of wavelets to curve and surface representation.

The wavelet coefficients of the surface are found by computing the wavelet transform of each coordinate function separately. The representation can then be compressed; in many cases, such