Image Processing

Image Processing: Tensor Transform and Discrete Tomography with MATLAB ®

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Features

  • Introduces new concepts and novel approaches for solving the problem of image reconstruction on the Cartesian lattice
  • Describes new algorithms and MATLAB®-based codes that can be effectively used in practice
  • Presents the idea of transferring the geometry of rays in the general case of rays with non-zero width
  • Discusses the challenges of implementing the proposed ideas for the fan beam projection scheme and 3-D image reconstruction
  • Includes study problems for computer simulation
  • Proposes new directions for research in the field of computed tomography
  • Contains more than 230 black-and-white illustrations and an eight-page color insert

Summary

Focusing on mathematical methods in computer tomography, Image Processing: Tensor Transform and Discrete Tomography with MATLAB® introduces novel approaches to help in solving the problem of image reconstruction on the Cartesian lattice. Specifically, it discusses methods of image processing along parallel rays to more quickly and accurately reconstruct images from a finite number of projections, thereby avoiding overradiation of the body during a computed tomography (CT) scan.

The book presents several new ideas, concepts, and methods, many of which have not been published elsewhere. New concepts include methods of transferring the geometry of rays from the plane to the Cartesian lattice, the point map of projections, the particle and its field function, and the statistical model of averaging. The authors supply numerous examples, MATLAB®-based programs, end-of-chapter problems, and experimental results of implementation.

The main approach for image reconstruction proposed by the authors differs from existing methods of back-projection, iterative reconstruction, and Fourier and Radon filtering. In this book, the authors explain how to process each projection by a system of linear equations, or linear convolutions, to calculate the corresponding part of the 2-D tensor or paired transform of the discrete image. They then describe how to calculate the inverse transform to obtain the reconstruction. The proposed models for image reconstruction from projections are simple and result in more accurate reconstructions.

Introducing a new theory and methods of image reconstruction, this book provides a solid grounding for those interested in further research and in obtaining new results. It encourages readers to develop effective applications of these methods in CT.

Table of Contents

Discrete 2-D Fourier Transform
Separable 2-D transforms
Vector forms of representation
Partitioning of 2-D transforms
Tensor representation of the 2-D DFT
Discrete Fourier transform and its geometry
Problems

Direction Images
2-D direction images on the lattice
The inverse tensor transform: Case N is prime
3-D paired representation
Complete system of 2-D paired functions
Paired transform direction images
L-paired representation of the image
Problems

Image Sampling Along Directions
Image reconstruction: Model I
Inverse paired transform
Example: Image 4 × 4
Property of the directed multiresolution
Example: Image 8 × 8
Summary of results
Equations in the coordinate system (X, 1 − Y )
Problems

Main Program of Image Reconstruction
The main diagram of the reconstruction
Part 1: Image model
The coordinate system and rays
Part 2: Projection data
Part 3: Transformation of geometry
Part 4: Linear transformation of projections
Part 5: Calculation the 2-D paired transform
Fast projection integrals by squares
Selection of projections
Problems

Reconstruction for Prime Size Image
Image reconstruction: Model II
Example with image 7 × 7
General algorithm of image reconstruction
Program description and image model
System of equations
Solutions of convolution equations
MATLAB R-based code (N prime)
Problems

Method of Particles
Point-map of projections
Method of G-rays
Reconstruction by field transform
Method of circular convolution
Problems

Methods of Averaging Projections
Filtered backprojection
BP and method of splitting-signals
Method of summation of line-integrals
Models with averaging
General case: Probability model
Problems

Bibliography
Appendix A
Appendix B
Index

Author Bio(s)

Artyom M. Grigoryan, Ph.D., is currently an associate professor at the Department of Electrical Engineering, University of Texas at San Antonio. He has authored or co-authored three books, including Brief Notes in Advanced DSP: Fourier Analysis with MATLAB® (2009) and Multidimensional Discrete Unitary Transforms: Representation: Partitioning, and Algorithms (2003) as well as two book chapters and many journal papers. He specializes in the theory and application of fast one- and multi-dimensional Fourier transforms, elliptic Fourier transforms, tensor and paired transforms, integer unitary heap transforms, design of robust linear and nonlinear filters, image encryption, computerized 2-D and 3-D tomography, and processing of biomedical images.

Merughan M. Grigoryan is currently conducting research on the theory and application of quantum mechanics in signal processing, differential equations, Fourier analysis, elliptic Fourier transforms, Hadamard matrices, fast integer unitary transformations, the theory and methods of the fast unitary transforms generated by signals, and methods of encoding in cryptography. He is the coauthor of the book Brief Notes in Advanced DSP: Fourier Analysis with MATLAB® (2009).

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