Multiscale Image Decompositions and Wavelets

Wavelet decompositions are 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 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. While classical signal processing concepts provide an operational understanding of such systems, there exist remarkable connections with work in applied mathematics and psychophysics, which provide 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. The regularity and vanishing moment properties of the lowpass filter impact the shape of the basis functions and hence their ability to efficiently represent typical images. From a psychophysical perspective, early stages of human visual information processing apparently involve decomposition of retinal images into a set of bandpass components corresponding to different scales and orientations. This suggests that multiscale/multiorientation decompositions are indeed natural and efficient for visual information processing.