Medical Image Processing and Understanding: Algebraic Reconstruction Algorithms

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# Medical Image Processing and Understanding: Algebraic Reconstruction Algorithms - PowerPoint PPT Presentation

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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### 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

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

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

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

Kaczmarz Method: Two-Variable Case
• Iterative method
• Alternate projections on hyperplanes

S. Kevin Zhou, UMD

Kaczmarz Method: Iteration

Equation (\$\$)

S. Kevin Zhou, UMD

Derivation of (\$\$)

S. Kevin Zhou, UMD

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

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

S. Kevin Zhou, UMD

Other issue: M>N and Noise
• No solution
• Kaczmarz method oscillates

S. Kevin Zhou, UMD

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

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

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

SIRT (Simultaneous Iterative Reconstructive Technique)
• Iteratively compute Dfj(i)
• Average Dfj
• Simultaneously update fj
• Noise resistant

S. Kevin Zhou, UMD

SART (Simultaneous Algebraic Reconstruction Techniques)
• Three features
• Pixel basis replaced by bilinear basis
• Simultaneous updating weights
• Hamming windowing

S. Kevin Zhou, UMD

Basis

???

Bilinear basis

Pixel basis

S. Kevin Zhou, UMD

Bilinear Interpolation

S. Kevin Zhou, UMD

Simultaneous Update

Sequential | Simultaneous

S. Kevin Zhou, UMD

Hamming Windowing

SART, 1 iteration, Hamming

S. Kevin Zhou, UMD

Result

Ground Truth

SART, 2 iterations, Hamming

S. Kevin Zhou, UMD