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

S. Allen Broughton and Kurt Bryan
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
Phil Gustafson
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This book is enjoyable to read and pulls together a variety of important topics in the subject at a level that upper level undergraduate mathematics students can understand. Fourier analysis is a broad field, with theory and applications that are widely used in other disciplines and in industry. Applications of Fourier analysis are within reach of students with the help of the computer and software such as MATLAB. However, there seems to be a shortage of books that deliver an appropriate mix of theory and applications to an undergraduate math major. I believe that Discrete Fourier Analysis and Wavelets, Applications to Signal and Image Processing helps fill this void.

The book approaches the material via inner product spaces, which are introduced in the first chapter. In this context many of the foundational results are established, such as the concept of a complete orthonormal set. The inner product space approach allows for a smooth development of the topic and requires some mathematical maturity. Also early in the first chapter are informative discussions on practical issues, such as sampling, quantization, noise, and color schemes (for images). In this manner, as throughout the book, the reader receives a well-rounded exposure to the material, with a good blend of theory and application. Subsequent chapters develop topics such as the discrete Fourier transform, time and frequency domains, the FFT, the discrete cosine transformation, block transforms, JPEG compression, convolution and filtering, windowing and localization, filter banks, and wavelets. The last two chapters, filters banks and wavelets, serve to develop to ideas behind the wavelet-based JPEG 2000 compression, and the basic development of wavelets such as Haar and Daubechies wavelets. Scaling functions and dilation equations are also discussed in the chapter on wavelets.

The authors are knowledgeable in the field, and have good insights. They emphasize many of the key ideas accessible to undergraduates and beginners in the subject. The examples are helpful, and are often illustrated with MATLAB graphs. The writing style is essentially informal. Concepts are often introduced via example, and then followed up with a more formal description.

Exercises are given at the end of the chapters, as opposed to the end of the sections, and are organized by type. Although this reviewer did not work through the exercises, they appear to be interesting and engaging. Depending on the level of student using this book, it may be desirable to augment the exercises with additional routine problems. MATLAB projects are given as part of each exercise set. The material is interesting enough that suggestions for more projects would be a good service to the students, including ideas for extended projects. The authors have a website companion to the text, which is used to post updates to the text and MATLAB programs for the homework problems and projects.

This book is suitable for advanced mathematics undergraduates as well as graduate students. Students in other disciplines such as physics, chemistry and engineering may also find this book helpful. The material in the book could be used for a one semester course, although two semesters would allow for more in-depth treatment.

Phil Gustafson is Professor of Mathematics at Mesa State College in Grand Junction, CO.


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 wave forms the analog case.

1.6 Sampling and aliasing.

1.7 Basic wave forms the discrete case.

1.8 Inner product spaces and orthogonality.

1.9 Signal and image digitization.

1.10 Infinitedimensional inner product spaces.

1.11 Matlab project.


2. The Discrete Fourier Transform. 

2.1 Overview.

2.2 The time domain and frequency domain.

2.3 A motivational example.

2.4 The onedimensional DFT.

2.5 Properties of the DFT.

2.6 The fast Fourier transform.

2.7 The twodimensional DFT.

2.8 Matlab project.


3. The discrete cosine transform. 

3.1 Motivation for the DCT: compression.

3.2 Initial examples thresholding.

3.3 The discrete cosine transform.

3.4 Properties of the DCT.

3.5 The twodimensional DCT.

3.6 Block transforms.

3.7 JPEG compression.

3.8 Matlab project.


4. Convolution and filtering. 

4.1 Overview.

4.2 Onedimensional convolution.

4.3 Convolution theorem and filtering.

4.4 2D convolution filtering images.

4.5 Infinite and biinfinite signal models.

4.6 Matlab project.


5. Windowing and Localization. 

5.1 Overview: Nonlocality of the DFT.

5.2 Localization via windowing.

5.3 Matlab project.


6. Filter banks. 

6.1 Overview.

6.2 The Haar filter bank.

6.3 The general onstage twochannel filter bank.

6.4 Multistage filter banks.

6.5 Filter banks for finite length signals.

6.6 The 2D discrete wavelet transform and JPEG 2000.

6.7 Filter design.

6.8 Matlab project.

6.9 Alternate Matlab project.


7. Wavelets. 

7.1 Overview.

7.2 The Haar Basis.

7.3 Haar wavelets versus the Haar filter bank.

7.4 Orthogonal wavelets.

7.5 Biorthogonal wavelets.

7.6 Matlab Project.