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