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.

Grigoryan, Artyom M., Grigoryan, Merughan M.

Boca Raton, FL: CRC Press/Taylor & Francis

American Society of Forensic Odontology

CRC Press LLC

United States, United States of America

Boca Raton, FL, Florida, 2012

Boca Raton, FL, ©2013

Boca Raton, Fla, 2013

1, 2012

Includes bibliographical references and index.

Discrete 2-D Fourier TransformSeparable 2-D transformsVector forms of representationPartitioning of 2-D transformsTensor representation of the 2-D DFTDiscrete Fourier transform and its geometryProblemsDirection Images2-D direction images on the latticeThe inverse tensor transform: Case N is prime3-D paired representationComplete system of 2-D paired functionsPaired transform direction imagesL-paired representation of the imageProblemsImage Sampling Along DirectionsImage reconstruction: Model IInverse paired transformExample: Image 4 x 4Property of the directed multiresolutionExample: Image 8 x 8Summary of resultsEquations in the coordinate system (X, 1 - Y)ProblemsMain Program of Image ReconstructionThe main diagram of the reconstructionPart 1: Image modelThe coordinate system and raysPart 2: Projection dataPart 3: Transformation of geometryPart 4: Linear transformation of projectionsPart 5: Calculation the 2-D paired transformFast projection integrals by squaresSelection of projectionsProblemsReconstruction for Prime Size ImageImage reconstruction: Model IIExample with image 7 x 7General algorithm of image reconstructionProgram description and image modelSystem of equationsSolutions of convolution equationsMATLAB R-based code (N prime)ProblemsMethod of ParticlesPoint-map of projectionsMethod of G-raysReconstruction by field transformMethod of circular convolutionProblems Methods of Averaging ProjectionsFiltered backprojectionBP and method of splitting-signalsMethod of summation of line-integralsModels with averagingGeneral case: Probability modelProblemsBibliographyAppendix AAppendix BIndex

xv, 442 p. : 24 cm
"The book is devoted to the problem of image reconstruction from a finite number of projections. It describes in detail 2-D discrete Fourier transform, including properties, fast algorithms, and applications of Fourier transform methods in image processing. It also presents traditional methods of 2D computerized tomography, including Fourier transform-based methods of filtered back projection and algebraic methods. The text shows readers new approaches and new forms of image representation, which can be used effectively in image processing and computerized tomography. All solutions of the image reconstruction problem are accomplished with examples and MATLAB-based codes"--
Includes bibliographical references ( p. 423-426) and index

"The book is devoted to the problem of image reconstruction from a finite number of projections. It describes in detail 2-D discrete Fourier transform, including properties, fast algorithms, and applications of Fourier transform methods in image processing. It also presents traditional methods of 2D computerized tomography, including Fourier transform-based methods of filtered back projection and algebraic methods. The text shows readers new approaches and new forms of image representation, which can be used effectively in image processing and computerized tomography. All solutions of the image reconstruction problem are accomplished with examples and MATLAB-based codes"-- Provided by publisher

Artyom Grigoryan, Merughan M. Grigoryan. Includes Bibliographical References (p. 423-426).

2023-06-28