1st Edition

Image Processing Tensor Transform and Discrete Tomography with MATLAB ®

    466 Pages 233 B/W Illustrations
    by CRC Press

    466 Pages 233 B/W Illustrations
    by CRC Press

    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

    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


    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).