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Medical Image Processing and Understanding: Algebraic Reconstruction Algorithms

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 http://www.cfar.umd.edu/~shaohua/.

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Medical Image Processing and Understanding: Algebraic Reconstruction Algorithms

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  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 http://www.cfar.umd.edu/~shaohua/ S. Kevin Zhou, UMD

  2. An Illustration of Line-Projection Method S. Kevin Zhou, UMD

  3. An Illustration of Algebraic Reconstruction S. Kevin Zhou, UMD

  4. Line-projection v.s. Ray-projection 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 S. Kevin Zhou, UMD

  6. Linear System • A set of linear equations Sj=1:Nwij fj = pi ; i=1,2,…,M ($) wj1xN• fNx1= pj ; i=1,2,…,M WMxNfNx1= p Mx1 S. Kevin Zhou, UMD

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

  8. Kaczmarz Method: Two-Variable Case • Iterative method • Alternate projections on hyperplanes S. Kevin Zhou, UMD

  9. Kaczmarz Method: Iteration Equation ($$) S. Kevin Zhou, UMD

  10. Derivation of ($$) S. Kevin Zhou, UMD

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

  12. Convergence • Orthogonalizaiton • Gram-Schmidt procedure • Select the order of the hyperplanes. • Avoid adjacent hyperplanes • Enforce prior information • Positive image • Zero area S. Kevin Zhou, UMD

  13. Other issue: M>N and Noise • No solution • Kaczmarz method oscillates S. Kevin Zhou, UMD

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

  15. 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 ($$) 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 S. Kevin Zhou, UMD

  17. SIRT (Simultaneous Iterative Reconstructive Technique) • Iteratively compute Dfj(i) • Average Dfj • Simultaneously update fj • Noise resistant S. Kevin Zhou, UMD

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

  19. Basis ??? Bilinear basis Pixel basis S. Kevin Zhou, UMD

  20. Bilinear Interpolation S. Kevin Zhou, UMD

  21. One More Trick: Equidistance S. Kevin Zhou, UMD

  22. Simultaneous Update Sequential | Simultaneous S. Kevin Zhou, UMD

  23. Hamming Windowing SART, 1 iteration, Hamming S. Kevin Zhou, UMD

  24. Result Ground Truth SART, 2 iterations, Hamming S. Kevin Zhou, UMD

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