Image Processing: Tensor Transform and Discrete Tomography with MATLAB ®, 1st Edition (Paperback) book cover

Image Processing

Tensor Transform and Discrete Tomography with MATLAB ®, 1st Edition

By Artyom M. Grigoryan, Merughan M. Grigoryan

CRC Press

466 pages | 233 B/W Illus.

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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


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


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 )


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


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)


Method of Particles

Point-map of projections

Method of G-rays

Reconstruction by field transform

Method of circular convolution


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



Appendix A

Appendix B


About the Authors

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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