Overview

Focusing on mathematical methods in computer tomography, Image Processing: Tensor Transform and Discrete Tomography with MATLAB(R) 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(R)-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.

Full Product Details

Author:   Artyom M. Grigoryan ,  Merughan M. Grigoryan
Publisher:   Taylor & Francis Inc
Imprint:   CRC Press Inc
ISBN:  

9781466509955

ISBN 10:   1466509953
Pages:   466
Publication Date:   03 October 2012
Audience:   General/trade ,  College/higher education ,  General ,  Tertiary & Higher Education
Format:   Electronic book text
Publisher's Status:   Active
Availability:   Available To Order  
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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 x 4 Property of the directed multiresolution Example: Image 8 x 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 x 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

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Author Information

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(R) (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(R) (2009).

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