# Download PDF by Artyom M. Grigoryan: Image Processing: Tensor Transform and Discrete Tomography

By Artyom M. Grigoryan

Targeting mathematical equipment in laptop tomography, photograph Processing: Tensor remodel and Discrete Tomography with MATLAB® introduces novel ways to assist in fixing the matter of snapshot reconstruction at the Cartesian lattice. in particular, it discusses tools of snapshot processing alongside parallel rays to extra speedy and safely reconstruct photos from a finite variety of projections, thereby averting overradiation of the physique in the course of a computed tomography (CT) scan.

The e-book provides numerous new rules, thoughts, and techniques, a lot of that have no longer been released in other places. New strategies contain equipment of shifting the geometry of rays from the airplane to the Cartesian lattice, the purpose map of projections, the particle and its box functionality, and the statistical version of averaging. The authors provide a number of examples, MATLAB®-based courses, end-of-chapter difficulties, and experimental result of implementation.

The major method for photograph reconstruction proposed by means of the authors differs from latest equipment of back-projection, iterative reconstruction, and Fourier and Radon filtering. during this booklet, the authors clarify how one can approach every one projection by way of a process of linear equations, or linear convolutions, to calculate the corresponding a part of the 2-D tensor or paired rework of the discrete picture. They then describe the way to calculate the inverse rework to procure the reconstruction. The proposed versions for picture reconstruction from projections are easy and lead to extra exact reconstructions.

Introducing a brand new conception and strategies of picture reconstruction, this booklet presents a great grounding for these attracted to extra learn and in acquiring new effects. It encourages readers to increase potent functions of those tools in CT.

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**Extra info for Image Processing: Tensor Transform and Discrete Tomography with MATLAB**

**Sample text**

38) These binary functions determine the tensor transform χσ and are defined as χp,s,t (n1 , n2) = 1, if n1 p + n2 s = t mod 2r , 0, otherwise, (n1 , n2 ) ∈ X. 39) 22 CHAPTER 1: 2-D DFT All ones in the masks of the functions lie on parallel lines passing the knots of the corresponding sets Vp,s,t . 7 (4 × 4-point DFT) Consider the following image 4 × 4 and its discrete Fourier transform: 1 2 {fn,m } = 1 1 2 1 3 1 1 1 2 3 27 j 1 −j 3 −2 + j 1 −3j 1 + 6j 2 . → {Fp,s } = −3 −2 + j −5 −2 − j 1 −2 − j 1 − 6j 3j 1 2 For (p, s) = (1, 1), values of t in the equations np+ms = t mod N, t = 0 : 3, can be written in the form of the following matrix: ||t = (n · 1 + m · 1) mod 4||n,m=0:3 0 1 = 2 3 1 2 3 0 2 3 0 1 3 0 .

Xp1 + yp2 =t + kN where k ≤ p1 + p2 . We denote this family by Lp1 ,p2 ,t . For different values of t1 = t2 < N, the families of lines Lp1 ,p2 ,t1 and Lp1 ,p2 ,t2 do not intersect. All 41 42 CHAPTER 2: Direction Images together, the sets Vp1 ,p2 ,t , t = 0 : (N − 1), compose a partition of the period X. It is interesting to note, that the direction of parallel lines of Lp1 ,p2 ,t is perpendicular to the direction of frequency-points of the cyclic group Tp1 ,p2 . 1 (Lattice 8 × 8) On the lattice X8,8 , we consider two sets of parallel lines L2,1,1 and L2,1,2.

Namely, each value fp,s,t is placed at all points that are located on the parallel lines of the corresponding family Lp,s,t . The interesting property of the tensor transform is derived. The direction image is composed of N values of the splitting-signal, that are placed at all points of the image N × N along the parallel lines. 7 (a) The image 257 × 257, (b) the splitting-signal {f1,5,t ; t = 0 : 256}, and (c) the 1-D DFT of the splitting-signal and frequency-points of T1,5 , and (d) the corresponding direction image dn1,n2 (the image has been scaled).