Medical%20Image%20Processing%20and%20Understanding:%20Algebraic%20Reconstruction%20Algorithms

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Medical Image Processing and UnderstandingAlgebr
aic 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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An Illustration of Line-Projection Method
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An Illustration of Algebraic Reconstruction
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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
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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

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Linear System

• A set of linear equations
• Sj1N wij fj pi i1,2,,M ()
• wj 1xN f Nx1 pj i1,2,,M
• W MxN f Nx1 p Mx1

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Solution

• Practical values
• M 256256 65000
• N 65000
• W 65000 x 65000
• Direct inverse
• Least square
• Kaczmarz37, Tanabe71
• The solution is the intersection of all the
  hyperplanes defined by ()

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Kaczmarz Method Two-Variable Case

• Iterative method
• Alternate projections on hyperplanes

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Kaczmarz Method Iteration

• Equation ()

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Derivation of ()
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Tanabe71

• 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).

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Convergence

• Orthogonalizaiton
• Gram-Schmidt procedure
• Select the order of the hyperplanes.
• Avoid adjacent hyperplanes
• Enforce prior information
• Positive image
• Zero area

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Other issue MgtN and Noise

• No solution
• Kaczmarz method oscillates

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Other issue MltN

• Infinite many solutions
• Kaczmarz method converges to a solution fs such
  that f(0) - fs is minimized

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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 ()

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ATR (Algebraic Reconstruction Technique)

• Replace wij by 1s and 0s 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
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SIRT (Simultaneous Iterative Reconstructive
Technique)

• Iteratively compute Dfj(i)
• Average Dfj
• Simultaneously update fj
• Noise resistant

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SART (Simultaneous Algebraic Reconstruction
Techniques)

• Three features
• Pixel basis replaced by bilinear basis
• Simultaneous updating weights
• Hamming windowing

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Basis
??? Bilinear basis
Pixel basis
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Bilinear Interpolation
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One More Trick Equidistance
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Simultaneous Update
Sequential Simultaneous
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Hamming Windowing
SART, 1 iteration, Hamming
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Result
SART, 2 iterations, Hamming
Ground Truth