Maintaining a comprehensive and accessible treatment of the concepts, methods, and applications of signal and image data transformation, the Second Edition features updated and revised coverage throughout with an emphasis on key and recent developments in the field of signal and image processing. The major motivating application continues to be signal and image compression. The authors introduce a new chapter on frames that is appropriately implemented after the coverage on spectrograms. Frames are a new technology in which signals, images, and other data are redundantly measured, and this redundancy allows for more sophisticated signal analysis. This new coverage also expands upon the discussions on spectrograms using a frames approach. In addition, this Second Edition includes a new chapter on lifting schemes for wavelets and provides a variation on the original low-pass/high-pass filter bank approach to the design and implementation of wavelets. These new chapters also include appropriate exercises and MATLAB® projects for further experimentation and practice. The authors clearly explain terms that may be unfamiliar to readers, and they also detail mathematical concepts that are relevant to image compression and signal processing. Most of the existing related texts focus on the concepts of imaging science and algorithms while this book focuses more on the underlying mathematics, with image compression as a motivation. A companion website features updated supplementary solution sets in addition to computer software support, including additional methods for applying the MATLAB routines as well SciPy (Scientific Python) support. Topical coverage includes: vector spaces, signals, and images; the discrete Fourier transform; the discrete cosine transform; convolution and filtering; windowing and localization; frames; filter banks; lifting schemes; and wavelets.
Table of Contents:
Dedication
Preface
Philosophy of the text
Outline of the text
Acknowledgments
Chapter 1: Vector Spaces, Signals, and Images
1.1 Overview
1.2 Some common image processing problems
1.3 Signals and images
1.4 Vector space models for signals and images
1.5 Basic waveforms|the analog case
1.6 Sampling and aliasing
1.7 Basic waveforms|the discrete case
1.8 Inner product spaces and orthogonality
1.9 Signal and image digitization
1.10 Infinite-dimensional inner product spaces
1.11 Matlab project
Exercises
Chapter 2: The Discrete Fourier Transform
2.1 Overview
2.2 The time domain and frequency domain
2.3 A motivational example
2.4 The one-dimensional DFT
2.5 Properties of the DFT
2.6 The fast Fourier transform
2.7 The two-dimensional DFT
2.8 Matlab Project
Exercises
Chapter 3: The discrete cosine transform
3.1 Motivation for the DCT|compression
3.2 Other compression issues
3.3 Initial examples|thresholding
3.4 The discrete cosine transform
3.5 Properties of the DCT
3.6 The two-dimensional DCT
3.7 Block transforms
3.8 JPEG compression
3.9 Matlab Project
Exercises
Chapter 4: Convolution and filtering
4.1 Overview
4.2 One-dimensional convolution
4.3 Convolution theorem and filtering
4.4 2D convolution|filtering images
4.5 Infinite and bi-infinite signal models
4.6 Matlab project
Exercises
Chapter 5: Windowing and Localization
5.1 Overview: Nonlocality of the DFT
5.2 Localization via windowing
5.3 Matlab project
Exercises
Chapter 6: Frames
6.1 Introduction
6.2 Packet loss
6.3 Frames - using more dot products
6.4 Analysis and synthesis with frames
6.5 Initial examples of frames
6.6 More on the frame operator
6.7 Group based frames
6.8 Frame applications
6.9 Matlab project
Exercises
Chapter 7: Filter banks
7.1 Overview
7.2 The Haar filter bank
7.3 The general one-stage two-channel filter bank
7.4 Multistage filter banks
7.5 Filter banks for finite length signals
7.6 The 2D discrete wavelet transform and JPEG 2000
7.7 Filter design
7.8 Matlab project
7.9 Alternate Matlab project
Exercises
Chapter 8: Lifting For Filter Banks and Wavelets
8.1 Overview
8.2 Lifting for the Haar filter bank
8.3 The lifting theorem
8.4 Polyphase analysis for filter banks
8.5 Lifting
8.6 Matlab project
Exercises
Chapter 9: Wavelets
9.1 Overview
9.2 The Haar basis
9.3 Haar wavelets versus the Haar filter bank
9.4 Orthogonal wavelets
9.5 Biorthogonal wavelets
9.6 Matlab project
Exercises
Bibliography
A Solutions to exercises
About the Author :
S. Allen Broughton, PhD, is Professor Emeritus of Mathematics at Rose-Hulman Institute of Technology. Dr. Broughton is a member of the American Mathematical Society (AMS) and the Society for the Industrial Applications of Mathematics (SIAM), and his research interests include the mathematics of image and signal processing, and wavelets.
Kurt Bryan, PhD, is Professor of Mathematics at Rose-Hulman Institute of Technology. Dr. Bryan is a member of MAA and SIAM and has authored over twenty peer-reviewed journal articles.
