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- 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

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.

**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

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

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

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

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 G-rays

Reconstruction by field transform

Method of circular convolution

Problems

BP and method of splitting-signals

Method of summation of line-integrals

Models with averaging

General case: Probability model

Problems

Appendix A

Appendix B

Index

**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

**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).