4.2 - Multiscale Image Decompositions and Wavelets

Abstract

This chapter introduces the basic concepts of multiscale image decompositions and wavelets. It focuses on three main techniques—Gaussian pyramids, Laplacian pyramids, and wavelets. The Gaussian pyramid provides a representation of the same image at multiple scales, using simple low-pass filtering and decimation techniques. The Laplacian pyramid provides a coarse representation of the image as well as a set of detailed images at different scales. Both the Gaussian and the Laplacian representations are over-complete in the sense that the total number of pixels is approximately 33% higher than in the original image. Wavelet decompositions are a more recent addition to the arsenal of multiscale signal processing techniques. Unlike the Gaussian and Laplacian pyramids, they provide a complete image representation and perform a decomposition according to both scale and orientation. They are implemented using cascaded filter banks in which the lowpass and highpass filters satisfy certain specific constraints. Although classical signal processing concepts provide an operational understanding of such systems, there exist remarkable connections with work in applied mathematics and in psychophysics, providing a deeper understanding of wavelet decompositions and their role in vision. From a mathematical standpoint, wavelet decompositions are equivalent to signal expansions in a wavelet basis.