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Shaohua Kevin Zhou Center for Automation Research and

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Presentation on theme: "Shaohua Kevin Zhou Center for Automation Research and"— Presentation transcript:

1 Medical Image Processing and Understanding: Algebraic Reconstruction Algorithms
Shaohua Kevin Zhou Center for Automation Research and Department of Electrical and Computer Engineering University of Maryland, College Park 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

2 An Illustration of Line-Projection Method
02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

3 An Illustration of Algebraic Reconstruction
02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

4 Line-projection v.s. Ray-projection
Method Line-projection Ray-projection Formation Line integral Ray sum Solution Fourier slice Linear algebra Algorithmic complexity Complex Simple Accuracy Accurate Not as accurate Computational Speed Fast Slow Other issues # of projections; sometime impossible Noisy 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

5 Image and Projection Representation
Discretization f(x,y) is constant in each cell fj is the value for the jth cell Each ray is a ‘stripe’ of width t Ray-sum N: total # of cells M: total # of rays 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

6 Sj=1:N wij fj = pi ; i=1,2,…,M ($) wj 1xN • f Nx1= pj ; i=1,2,…,M
Linear System A set of linear equations Sj=1:N wij fj = pi ; i=1,2,…,M ($) wj 1xN • f Nx1= pj ; i=1,2,…,M W MxN f Nx1= p Mx1 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

7 Solution Practical values Direct inverse Least square
M = 256*256 ~= 65000 N ~= 65000 W : x 65000 Direct inverse Least square Kaczmarz’37, Tanabe’71 The solution is the intersection of all the hyperplanes defined by ($) 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

8 Kaczmarz Method: Two-Variable Case
Iterative method Alternate projections on hyperplanes 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

9 Kaczmarz Method: Iteration
Equation ($$) 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

10 Derivation of ($$) 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

11 Tanabe’71 Theorem If there exists a unique solution fs to the system of equations ($), then limkinf f(kM) = fs. Convergence Depends on the angle between the two lines (in two-variable case). 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

12 Select the order of the hyperplanes.
Convergence Orthogonalizaiton Gram-Schmidt procedure Select the order of the hyperplanes. Avoid adjacent hyperplanes Enforce prior information Positive image Zero area 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

13 Other issue: M>N and Noise
No solution Kaczmarz method oscillates 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

14 Infinite many solutions
Other issue: M<N Infinite many solutions Kaczmarz method converges to a solution fs such that | f(0) - fs | is minimized 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

15 Difficulty in calculation, storage, & retrieval
Too many weights! 100 x 100 grid, 100 projections, 150 ray/projections  # of weights: 1.5x108 Difficulty in calculation, storage, & retrieval Weight approximations Three techniques: SRT, SIRT, SART Rewrite ($$) 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

16 ATR (Algebraic Reconstruction Technique)
Replace wij by 1’s and 0’s using center checking: wij = 1 if the center of the jth cell is within the ith ray. ($$) becomes Ni: # of image cells whose centers within the ith ray. Li: the length of the ith ray through the image region 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

17 SIRT (Simultaneous Iterative Reconstructive Technique)
Iteratively compute Dfj(i) Average Dfj Simultaneously update fj Noise resistant 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

18 SART (Simultaneous Algebraic Reconstruction Techniques)
Three features Pixel basis replaced by bilinear basis Simultaneous updating weights Hamming windowing 02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

19 Basis ??? Bilinear basis Pixel basis 02/18/2004, ENE739R/CMSC828R
S. Kevin Zhou, UMD

20 Bilinear Interpolation
02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

21 One More Trick: Equidistance
02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

22 Simultaneous Update Sequential | Simultaneous
02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

23 Hamming Windowing SART, 1 iteration, Hamming
02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD

24 Result Ground Truth SART, 2 iterations, Hamming
02/18/2004, ENE739R/CMSC828R S. Kevin Zhou, UMD


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