Grigoryan A.M., Grigoryan M.M. Image Processing. Tensor Transform and Discrete Tomography with MatLAB

CRC Press, 2013. — 466 p.This book is devoted to one of the most interesting applications of mathematical methods in digital image processing. It is computed tomography (CT) or computerized X-tomography, wherein the projection data of the reconstructing image are obtained by means of the roentgen radiation interaction with tissue. The result of the CT is a two-dimensional (2-D) or three-dimensional (3-D) image, which represents, with some degree of accuracy, the image through which the rays pass. Our task is to find the solution of the problem of image reconstruction by a finite number of projections. The fundamental principles and main methods of image reconstruction, which include the Fourier slice theorem, methods of summation and filtered backprojection, and the method of finite series expansion are well known and can be found in many books written on CT. The problem of reconstruction of the 2-D function from its ray-sums, or line-integrals, which was solved by Radon in 1917, faced its main obstacle in CT. The number of possible projections is finite. All known reconstructions in CT result in approximations of the image. The images in CT are not bounded, and Kotelnikov’s theorem of sampling is not valid for them. We still do not know if the exact solution of the problem exists, and if it exists, then for what kind of images and models? Therefore, we will not follow the beaten path, but instead open slightly the door of the box where the solution of the problem can be found.
This book is a brief collection of some notes and results of our research in digital image reconstruction. We present the case of the 2-D image and the parallel projections. The model describing the process of projection data collection is simple. On both sides of the analyzed object, or image which is assumed to be immovable, the X-ray set and detectors are disposed and revolved around the object at different angles. Then, the measurement data of radiation and detection of X-rays are collected along other directions. This set of measurements, or projections, is used by specific mathematical methods to reconstruct the image of the observed object or tissue. Methods of image reconstruction must be fast and accurate because of the desired high-quality images for diagnostic purposes. More importantly, we need methods that reconstruct the image on the discrete Cartesian lattice with a minimal number of projections, in order to not overradiate the body in the CT.
Many concepts, ideas, and methods described in this book have not been presented or published anywhere else. This is written as a textbook with many examples described in detail and programs that are given in short form, to demonstrate the presented concepts, their properties, and methods of image reconstruction. New concepts include the 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, the statistical model of averaging, and others. Our goal is also to give graduate students and other readers solid material for the presented theory of image reconstruction, to benefit those interested in continuing this research and obtaining new results in image reconstruction.
Discrete 2-D Fourier Transform.
Direction Images.
Image Sampling Along Directions.
Main Program of Image Reconstruction.
Reconstruction for Prime Size Image.
Method of Particles.
Methods of Averaging Projections.