Image Processing

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).

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 × 4
Property of the directed multiresolution
Example: Image 8 × 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 × 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