Delivers an appropriate mix of theory and applications to help readers understand the process and problems of image and signal analysis

Maintaining a comprehensive and accessible treatment of the concepts, methods, and applications of signal and image data transformation, this Second Edition of Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing features updated and revised coverage throughout with an emphasis on key and recent developments in the field of signal and image processing. Topical coverage includes: vector spaces, signals, and images; the discrete Fourier transform; the discrete cosine transform; convolution and filtering; windowing and localization; spectrograms; frames; filter banks; lifting schemes; and wavelets.

Discrete Fourier Analysis and Wavelets introduces a new chapter on frames--a new technology in which signals, images, and other data are redundantly measured. This redundancy allows for more sophisticated signal analysis. The new coverage also expands upon the discussion on spectrograms using a frames approach. In addition, the book 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.

* Features updated and revised content throughout, continues to emphasize discrete and digital methods, and utilizes MATLAB® to illustrate these concepts

* Contains two new chapters on frames and lifting schemes, which take into account crucial new advances in the field of signal and image processing

* Expands the discussion on spectrograms using a frames approach, which is an ideal method for reconstructing signals after information has been lost or corrupted (packet erasure)

* Maintains a comprehensive treatment of linear signal processing for audio and image signals with a well-balanced and accessible selection of topics that appeal to a diverse audience within mathematics and engineering

* Focuses on the underlying mathematics, especially the concepts of finite-dimensional vector spaces and matrix methods, and provides a rigorous model for signals and images based on vector spaces and linear algebra methods

* Supplemented with a companion website containing solution sets and software exploration support for MATLAB and SciPy (Scientific Python)

Thoroughly class-tested over the past fifteen years, Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing is an appropriately self-contained book ideal for a one-semester course on the subject.

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.

Inhalt

Preface xvii

Acknowledgments xxi

1 Vector Spaces, Signals, and Images 1

1.1 Overview 1

1.2 Some Common Image Processing Problems 1

1.2.1 Applications 2

1.2.1.1 Compression 2

1.2.1.2 Restoration 2

1.2.1.3 Edge Detection 3

1.2.1.4 Registration 3

1.2.2 Transform-Based Methods 3

1.3 Signals and Images 3

1.3.1 Signals 4

1.3.2 Sampling, Quantization Error, and Noise 5

1.3.3 Grayscale Images 6

1.3.4 Sampling Images 8

1.3.5 Color 9

1.3.6 Quantization and Noise for Images 9

1.4 Vector Space Models for Signals and Images 10

1.4.1 ExamplesDiscrete Spaces 11

1.4.2 ExamplesFunction Spaces 14

1.5 Basic WaveformsThe Analog Case 16

1.5.1 The One-Dimensional Waveforms 16

1.5.2 2D Basic Waveforms 19

1.6 Sampling and Aliasing 20

1.6.1 Introduction 20

1.6.2 Aliasing for Complex Exponential Waveforms 22

1.6.3 Aliasing for Sines and Cosines 23

1.6.4 The Nyquist Sampling Rate 24

1.6.5 Aliasing in Images 24

1.7 Basic WaveformsThe Discrete Case 25

1.7.1 Discrete Basic Waveforms for Finite Signals 25

1.7.2 Discrete Basic Waveforms for Images 27

1.8 Inner Product Spaces and Orthogonality 28

1.8.1 Inner Products and Norms 28

1.8.1.1 Inner Products 28

1.8.1.2 Norms 29

1.8.2 Examples 30

1.8.3 Orthogonality 33

1.8.4 The CauchySchwarz Inequality 34

1.8.5 Bases and Orthogonal Decomposition 35

1.8.5.1 Bases 35

1.8.5.2 Orthogonal and Orthonormal Bases 37

1.8.5.3 Parseval's Identity 39

1.9 Signal and Image Digitization 39

1.9.1 Quantization and Dequantization 40

1.9.1.1 The General Quantization Scheme 41

1.9.1.2 Dequantization 42

1.9.1.3 Measuring Error 42

1.9.2 Quantifying Signal and Image Distortion More Generally 43

1.10 Infinite-Dimensional Inner Product Spaces 45

1.10.1 Example: An Infinite-Dimensional Space 45

1.10.2 Orthogonal Bases in Inner Product Spaces 46

1.10.3 The CauchySchwarz Inequality and Orthogonal Expansions 48

1.10.4 The Basic Waveforms and Fourier Series 49

1.10.4.1 Complex Exponential Fourier Series 49

1.10.4.2 Sines and Cosines 52

1.10.4.3 Fourier Series on Rectangles 53

1.10.5 Hilbert Spaces and L2(a, b ) 53

1.10.5.1 Expanding the Space of Functions 53

1.10.5.2 Complications 54

1.10.5.3 A Converse to Parseval 55

1.11 Matlab Project 55

Exercises 60

2 The Discrete Fourier Transform 71

2.1 Overview 71

2.2 The Time Domain and Frequency Domain 71

2.3 A Motivational Example 73

2.3.1 A Simple Signal 73

2.3.2 Decomposition into BasicWaveforms 74

2.3.3 Energy at Each Frequency 74

2.3.4 Graphing the Results 75

2.3.5 Removing Noise 77

2.4 The One-Dimensional DFT 78

2.4.1 Definition of the DFT 78

2.4.2 Sample Signal and DFT Pairs 80

2.4.2.1 An Aliasing Example 80

2.4.2.2 Square Pulses 81

2.4.2.3 Noise 82

2.4.3 Suggestions on Plotting DFTs 84

2.4.4 An Audio Example 84

2.5 Properties of the DFT 85

2.5.1 Matrix Formulation and Linearity 85

2.5.1.1 The DFT as a Matrix 85

2.5.1.2 The Inverse DFT as a Matrix 87

2.5.2 Symmetries for Real Signals 88

2.6 The Fast Fourier transform 90

2.6.1 DFT Operation Count 90

2.6.2 The FFT 91

2.6.3 The Operation Count 92

2.7 The Two-Dimensional DFT 93

2.7.1 Interpretation and Examples of the 2-D DFT 96

2.8 Matlab Project 97

2.8.1 Audio Explorations 97

2.8.2 Images 99

Exercises 1...