Tomographic Image Reconstruction Using 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

2
Problem Approach

• Reconstruction
• of
• CT-Images
• from
• Projection Data

• Content-Adaptive Mesh Generation
• Estimation of Mesh Nodal Values
• Reconstruction of the Image by Interpolation

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Phantom Reference Image
4
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

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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

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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

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Result
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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.

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Placement of Mesh Nodes (cont)

• Distribute errors among pixels
• Uses the perception characteristics of the human
  eye
• Fast, efficient and produces excellent results
  (almost)

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Result
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Connecting Mesh Nodes

• Delaunay triangulation
• Returns set of triangles such that no data point
  is contained in any triangles circumcircle
• Known to yield a well-structured mesh
• Avoids producing excessively elongated elements,
  reducing the error bound

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Result
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Estimation of Mesh Nodal Values
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Reconstruction of Image

• Pixel value is approximated from the nodal values
  of its enclosing triangle
• Master element
• Shape functions

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Result
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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!

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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.

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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.

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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.

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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.

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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.

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Thank you for listening

• Wish me more luck!