Fourier Methods in Imaging

Fourier Methods in Imaging introduces the mathematical tools for modeling linear imaging systems to predict the action of the system or for solving for the input. The chapters are grouped into five sections, the first introduces the imaging “tasks” (direct, inverse, and system analysis), the basic concepts of linear algebra for vectors and functions, including complex-valued vectors, and inner products of vectors and functions. The second section defines “special” functions, mathematical operations, and transformations that are useful for describing imaging systems. Among these are the Fourier transforms of 1-D and 2-D function, and the Hankel and Radon transforms. This section also considers approximations of the Fourier transform. The third and fourth sections examine the discrete Fourier transform and the description of imaging systems as linear “filters”, including the inverse, matched, Wiener and Wiener-Helstrom filters. The final section examines applications of linear system models to optical imaging systems, including holography.

• Provides a unified mathematical description of imaging systems.
• Develops a consistent mathematical formalism for characterizing imaging systems.
• Helps the reader develop an intuitive grasp of the most common mathematical methods, useful for describing the action of general linear systems on signals of one or more spatial dimensions.
• Offers parallel descriptions of continuous and discrete cases.
• Includes many graphical and pictorial examples to illustrate the concepts.

This book helps students develop an understanding of mathematical tools for describing general one- and two-dimensional linear imaging systems, and will also serve as a reference for engineers and scientists

Contents

1 Introduction

2 Operators and Functions

3 Vectors with Real-Valued Components

4 Complex Numbers and Functions

5 Complex-Valued Matrices and Systems

6 1-D Special Functions

7 2-D Special Functions

8 Linear Operators

9 Fourier Transforms of 1-D Functions

10 Multidimensional Fourier Transforms

11 Spectra of Circular Functions

12 The Radon Transform

13 Approximations to Fourier Transforms

14 Discrete Systems, Sampling, and Quantization

15 Discrete Fourier Transforms

16 Magnitude Filtering

17 Allpass (Phase) Filters

18 Magnitude-Phase Filters

19 Applications of Linear Filters

20 Filtering in Discrete Systems

21 Optical Imaging in Monochromatic Light

22 Incoherent Optical Imaging Systems

23 Holography

Review

“Overall, this is an excellent text, appropriate for the graduate student approaching this material for the first time, and for the seasoned professional looking for an up-to-date reference.” (Journal of Electronic Imaging, 1 April 2011)

“This comprehensive textbook represents a practical review of Fourier techniques in imaging methods. It will be very useful for graduate students (in engineering, science, computer science, and applied mathematics) as well as engineers interested in linear imaging systems.” (Zentralblatt Math, 2010)

About the Author

Professor Roger L. Easton, Jr

Chester F. Carlson Center for Imaging Science, Rochester Institute of Technology

Professor Easton teaches undergraduate and graduate courses in linear systems, optical imaging, and digital image processing at Rochester Institute of Technology. He received a B.S. degree in Astronomy from Haverford College, an M.S. in physics from the University of Maryland, and an M.S. and Ph.D. degree in Optical Sciences from the University of Arizona.

His research interests include the application of digital image processing to text documents and manuscripts. He has contributed to work on the Dead Sea Scrolls and is now part of an imaging team helping scolars to read the original Archimiedes Palimpsest. Professor Easton also conducts research into optical signal processing and computer-generated holography, publishing articles on both.