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CT-Reconstruction Based on Content-Adaptive Mesh Modeling

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									    Tomographic Image
Reconstruction Using Content-
  Adaptive Mesh Modeling

           H. Can Aras
      November 29-30, 2004
       Project Presentation
Problem                        Approach

 Reconstruction     • Content-Adaptive
                        Mesh Generation
        of
    CT-Images
                    •   Estimation of Mesh
                        Nodal Values
       from
                    •   Reconstruction of the
  Projection Data       Image by
                        Interpolation
Phantom & Reference Image
Feature Map Extraction
• Second derivative used, below is the theoretical basis for this.
• M2 : the least upper bound on the second directional derivative of
  f(x) over T
• h : the length of the longest side of T
• The formula tells us two things…
Feature Map Extraction (cont.)
To achieve a low approximation error of the image:
• Small elements in large second derivative regions
• Relatively larger elements in relatively small derivative
  regions
Feature Map Extraction (cont.)
• Not practical to calculate directional
  derivatives
• Use max (| fxx |, | fxy |, | fyy |) or the
  magnitude of the second derivative
• Normalization of feature map
• Segmentation of heart and background
  region
• Modification of feature map
Result
Placement of Mesh Nodes

• Floyd-Steinberg error-diffusion algorithm
• Originally designed for digital halftoning
• The objective is to use the spatial density
  of the ink dots to represent the image
  intensity.
• The classical method used varying ink dot
  sizes.
Placement of Mesh Nodes (cont)
• Distribute errors among pixels
• Uses the perception characteristics of the human eye
• Fast, efficient and produces excellent results (almost)
Result
Connecting Mesh Nodes

• Delaunay triangulation
• Returns set of triangles such that no data
  point is contained in any triangle’s
  circumcircle
• Known to yield a well-structured mesh
• Avoids producing excessively elongated
  elements, reducing the error bound
Result
Estimation of Mesh Nodal Values
Reconstruction of Image
• Pixel value is approximated from the nodal
    values of its enclosing triangle
•   Master element
•   Shape functions
Result
Numerical Comparison
•   PSNR of FBP result : 47.51
•   PSNR of MESH-ML result: 46.77
•   compression rate: 5.36
•   Note: A higher PSNR does not always correlate well with
    the perceived image quality (although it provides a
    measure for relative quality)
•   A slight change on MESH-ML result gives higher PSNR.
•   Subtracting only 0.01 from each value of MESH-ML
    result yields a PSNR of 48.81. Subtracting 0.02 yields
    51.05!
•   The authors may be using another trick for PSNR!
A Comment on Results
              • The mesh nodal values
                  tend to increase slightly
                  on average after MESH-
                  ML.
              •   Until a number of
                  iterations, the results get
                  better. Behind this limit,
                  results tend to go bad,
                  even worse than FBP
                  (reference) image.
Problems Faced

• Radon Transform followed by Inverse
  Radon Transform yielded an image with
  negative values because of incomplete set
  of projections.
• I adjusted this image between [0,1] so
  that the initial values of the mesh nodes
  are not negative in MESH-ML algorithm.
Problems Faced (cont.)

• Delaunay Triangulation is sensitive to the
  position of the nodes.
• Degenerate cases occur frequently when
  using integers for position coordinates.
• I randomly changed the position
  coordinates with very small differences
  and used double instead of integer.
Problems Faced (cont.)
• The analytical form of the response function is not
    known.
•   Hence, I calculated the system matrix by probing the
    input with an impulse function as offered in the paper.
•   Specifically, a unit-impulse was applied at each nodal
    location of the mesh model, and the response at each
    detector was computed.
•   This computation is time and memory consuming.
•   Time problem can be overcome by precalculation.
•   I used sparse matrix since most of the system matrix is
    zeros.
•   The MESH-ML algorithm takes longer than expected.
Plan
• Try to make MESH-ML algorithm faster (not the main
    concern, but can be a bottle-neck for the tests below).
•   Run MESH-ML with multiple iterations.
•   Use better reference image in terms of the number of
    projection angles (5 degrees used between consecutive
    projections in the experiments).
•   Use better reference image in terms of the filter used in
    Fourier domain (Ram-Lak ramp filter used in the
    experiments).
•   Test on medical images, which capture different parts of
    the body.
Thank you for listening…

     Wish me more luck!

								
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