medical image processing and understanding algebraic reconstruction algorithms l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Medical Image Processing and Understanding: Algebraic Reconstruction Algorithms PowerPoint Presentation
Download Presentation
Medical Image Processing and Understanding: Algebraic Reconstruction Algorithms

Loading in 2 Seconds...

play fullscreen
1 / 24

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


  • 285 Views
  • Uploaded on

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/.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Medical Image Processing and Understanding: Algebraic Reconstruction Algorithms' - Faraday


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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/

S. Kevin Zhou, UMD

image and projection representation
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
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
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
Kaczmarz Method: Two-Variable Case
  • Iterative method
  • Alternate projections on hyperplanes

S. Kevin Zhou, UMD

kaczmarz method iteration
Kaczmarz Method: Iteration

Equation ($$)

S. Kevin Zhou, UMD

derivation of
Derivation of ($$)

S. Kevin Zhou, UMD

tanabe 71
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
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

other issue m n and noise
Other issue: M>N and Noise
  • No solution
  • Kaczmarz method oscillates

S. Kevin Zhou, UMD

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

S. Kevin Zhou, UMD

basis
Basis

???

Bilinear basis

Pixel basis

S. Kevin Zhou, UMD

bilinear interpolation
Bilinear Interpolation

S. Kevin Zhou, UMD

simultaneous update
Simultaneous Update

Sequential | Simultaneous

S. Kevin Zhou, UMD

hamming windowing
Hamming Windowing

SART, 1 iteration, Hamming

S. Kevin Zhou, UMD

result
Result

Ground Truth

SART, 2 iterations, Hamming

S. Kevin Zhou, UMD