Use of operator theory and sub-band filters in the analysis and encoding of signals and images

The thesis is motivated by recent advances in signal and image processing, a part of electrical and computer engineering.

In the first part, we begin with a new approach to the mathematical signal processing as used in the digital processing of images. We prove such results in the 2D case, and we explain their use. A key point we explore is the interplay between the two cases, continuous and discrete. Our discrete algorithms present fast matrix-operations to be applied to images in pixel form. This part of the thesis in turn is based on tools from wavelet analysis, and more generally from the theory of operators in Hilbert space.

In the second part, we address encoding and quantization of wavelet coefficients obtained after applying the DWT (mentioned in first part) to 1-D signals.This is the last crucial step in A/D conversion, i.e., analog to digital. By quantization we mean the conversion and encoding of processing-output into bits; bits that in turn are transmitted and fed into a decoder. We isolate and make mathematically precise a particular family of quantizers which are efficient in that they produce error terms of exponential fall-off. We do this with a family of discrete algorithms, each one governed by a quantizer. In Theorems 3.2, 3.5, 3.11, we obtain quite precise a priori estimates.

In the last part, we address the compression of a matrix (a 2-D image) obtained by applying the DWT on an image mentioned in the first part. Embedded Zerotree Wavelet algorithm is introduced and implemented.