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E-grāmata: Image Processing: Tensor Transform and Discrete Tomography with MATLAB

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  • Formāts: 466 pages
  • Izdošanas datums: 03-Sep-2018
  • Izdevniecība: CRC Press Inc
  • ISBN-13: 9781466509955
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Image Processing: Tensor Transform and Discrete Tomography with MATLAB Palielināt
  • Formāts: 466 pages
  • Izdošanas datums: 03-Sep-2018
  • Izdevniecība: CRC Press Inc
  • ISBN-13: 9781466509955
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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.
Author Bios xi
Preface xiii
1 Discrete 2-D Fourier Transform
1(40)
1.1 Separable 2-D transforms
2(2)
1.2 Vector forms of representation
4(1)
1.3 Partitioning of 2-D transforms
5(7)
1.3.1 Tensor representation
8(1)
1.3.2 Covering with cyclic groups
9(3)
1.4 Tensor representation of the 2-D DFT
12(20)
1.4.0.1 Code: Splitting-signal calculation
13(1)
1.4.1 Tensor algorithm of the 2-D DFT
13(1)
1.4.2 N is prime
14(6)
1.4.2.1 Code: 2-D DFT by tensor transform
20(1)
1.4.3 N is a power of two
21(6)
1.4.4 N is a power of an odd prime
27(2)
1.4.5 Case N = L1L2 (L1 ≠ L2 > 1)
29(1)
1.4.6 General case
29(1)
1.4.7 Other orders N1 × N2
30(2)
1.5 Discrete Fourier transform and its geometry
32(7)
1.5.1 Inverse DFT
35(4)
Problems
39(2)
2 Direction Images
41(56)
2.1 2-D direction images on the lattice
41(10)
2.1.1 Superposition of directions
44(7)
2.2 The inverse tensor transform: Case N is prime
51(9)
2.2.1 Inverse tensor transform
51(6)
2.2.2 Formula of the inverse tensor transform
57(1)
2.2.2.1 Code for inverse tensor transform
58(2)
2.3 3-D paired representation
60(15)
2.3.1 2D-to-3D paired transform
62(4)
2.3.2 The splitting of the 2-D DFT
66(9)
2.4 Complete system of 2-D paired functions
75(8)
2.4.0.1 Code: System of basic paired functions
80(1)
2.4.1 1-D DFT and paired transform
81(2)
2.5 Paired transform direction images
83(4)
2.6 L-paired representation of the image
87(7)
2.6.1 Principle of superposition: General case
90(4)
Problems
94(3)
3 Image Sampling Along Directions
97(130)
3.1 Image reconstruction: Model I
98(3)
3.1.1 Coordinate systems and rays
100(1)
3.2 Inverse paired transform
101(2)
3.3 Example: Image 4x4
103(17)
3.3.1 Horizontal and vertical projections
103(4)
3.3.2 Diagonal projections
107(2)
3.3.3 Other projections
109(1)
3.3.3.1 Generator (1,3)
109(2)
3.3.3.2 Generator (1,2)
111(4)
3.3.3.3 Generator (2,1)
115(5)
3.4 Property of the directed multiresolution
120(1)
3.5 Example: Image 8x8
121(87)
3.5.1 Horizontal projection
121(3)
3.5.2 Vertical projection
124(1)
3.5.3 Diagonal projection
125(4)
3.5.4 (2,1)- and (1,2)-projections
129(1)
3.5.4.1 (2, 2)-projection
129(8)
3.5.4.2 (1,2)-projection
137(6)
3.5.5 (1,3)-projection
143(15)
3.5.6 (1,4)-and (4,1)-projections
158(14)
3.5.7 (1, 5)-projection
172(17)
3.5.8 (1,6)-projection
189(7)
3.5.9 (6,1)-projection
196(6)
3.5.10 (1,7)-projection
202(6)
3.6 Summary of results
208(6)
3.6.1 Equations of rays
210(3)
3.6.2 Equations for line-integrals
213(1)
3.7 Equations in the coordinate system (X, 1 - Y)
214(10)
3.7.1 Convolution equations
219(5)
Problems
224(3)
4 Main Program of Image Reconstruction
227(44)
4.1 The main diagram of the reconstruction
227(2)
4.2 Part 1: Image model
229(2)
4.3 The coordinate system and rays
231(1)
4.4 Part 2: Projection data
232(5)
4.5 Part 3: Transformation of geometry
237(4)
4.6 Part 4: Linear transformation of projections
241(4)
4.7 Part 5: Calculation the 2-D paired transform
245(9)
4.7.1 Method of incomplete 1-D DPT
246(1)
4.7.2 Fast 1-D paired transform
247(3)
4.7.3 Inverse 2-D DPT
250(2)
4.7.4 Preliminary results
252(2)
4.8 Fast projection integrals by squares
254(11)
4.9 Selection of projections
265(3)
Problems
268(3)
5 Reconstruction for Prime Size Image
271(58)
5.1 Image reconstruction: Model II
271(1)
5.2 Example with image 7x7
272(41)
5.2.1 Horizontal projection
273(1)
5.2.2 Vertical projection
274(1)
5.2.3 Diagonal projection
275(4)
5.2.4 (1,2)-Projection
279(6)
5.2.5 (1,3)-projection
285(6)
5.2.6 (1,4)-projection
291(8)
5.2.7 (1,5)-projection
299(7)
5.2.8 (1,6)-projection
306(5)
5.2.9 Reconstructed image 7x7
311(2)
5.3 General algorithm of image reconstruction
313(2)
5.4 Program description and image model
315(3)
5.5 System of equations
318(1)
5.6 Solutions of convolution equations
319(5)
5.6.1 Splitting-signal composition
321(1)
5.6.2 Inverse 2-D tensor transform
322(2)
5.7 MATLAB®-based code (N prime)
324(3)
Problems
327(2)
6 Method of Particles
329(54)
6.1 Point-map of projections
329(14)
6.1.1 A-particle and the field
332(5)
6.1.2 Representation by field functions
337(6)
6.2 Method of G-rays
343(22)
6.2.1 G-rays for the first set of generators
343(5)
6.2.2 G-rays for the second set of generators
348(3)
6.2.3 G-rays for the third set of generators
351(3)
6.2.4 G-rays for the fourth set of generators
354(1)
6.2.5 Map of projections for one square
355(5)
6.2.5.1 Codes for particles
360(5)
6.3 Reconstruction by field transform
365(9)
6.4 Method of circular convolution
374(6)
6.4.1 Uniform frames
379(1)
Problems
380(3)
7 Methods of Averaging Projections
383(40)
7.1 Filtered backprojection
384(2)
7.2 BP and method of splitting-signals
386(11)
7.2.1 Tensor method of summation of projections
390(7)
7.3 Method of summation of line-integrals
397(1)
7.4 Models with averaging
398(13)
7.4.1 Method of proportion
399(3)
7.4.2 Method with probability model
402(2)
7.4.3 Reconstruction of the shifted image
404(3)
7.4.4 Method of minimization of error
407(2)
7.4.5 Corpuscular approach
409(2)
7.5 General case: Probability model
411(6)
7.5.0.1 Code of the reconstruction
414(3)
Problems
417(6)
Bibliography 423(4)
Appendix A 427(6)
Appendix B 433(8)
Index 441
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).
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