A finite bidimensional wavelet framework for computer graphics

Abstract

A lot of material has been written about wavelet theory. Most of these texts provide an elegant framework from the functional and real analysis point of view. The complete infinite dimensional space (the set of all functions such that ) is generally used to develop the theory, but this cannot be directly applied to computer software, because the concept of a non denumerable infinite set of vectors or functions is practically useless here. We provide foundations for a finite, linear-algebra based toolkit of wavelets that supply a rich set of tools that can be used to manage image processing, equalization and compression. We test a frequency criterion to design orthonormal wavelet generators and a multirresolution analysis. We show that this criterion can be easily interpreted graphically. Despite our approach only constructs orthonormal wavelet basis; we believe that this approach is general enough to explore possibilities in other computer graphic fields and solution of integer-differential equations on simple domains. We strongly believe that this approach simplifies considerably the wavelet analysis. The frequency criterion expands possibilities of testing two dimensional wavelet bases according to specific graphical needs, and can be applied to different problems that involve regular grid reduction. We propose this criterion as a fundamental basis to design bidimensional ortonormal wavelets for matrix equalization. It gives a wider range of possibilities than restricting only to some well known bases and gives a direct interpretation for computer images.