Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing

PREFACE

PHILOSOPHY OF THE TEXT

In this text we provide an introduction to the mathematics that underlies much of modern signal and image processing. In particular, we

• develop the mathematical framework in which signal and image processing takes place, specifically, vector and inner product spaces;
• develop traditional Fourier-based transform techniques, primarily in the discrete case, but also to some extent in the continuous setting;
• provide entry-level material on filtering, convolution, filter banks, and wavelets; and
• make extensive use of computer-based explorations for concept development.

These topics are extremely beautiful and important areas of classical and modern applied mathematics. They also form the foundation for many techniques in science and engineering. However, while they are usually taught (sometimes in bits and pieces) in the engineering and science curricula, they are often overlooked in mathematics courses, or addressed only as brief asides. We hope to change this.

Throughout the text we often use image compression as a concrete and motivational hook for the mathematics, but our goal here is not a technical manual on how to program a JPEG encoder. Rather, we include this topic so that the reader can begin applying the theory to interesting problems relatively quickly. We also touch upon a number of other important applications in addition to compression, for example, progressive transmission of images, image denoising, spectrographic analysis, and edge ...