Local Computerized Tomography Using Wavelets

1
Local Computerized Tomography Using Wavelets
Chih-ting Wu
Wavelet Reconstruction from projection in 2 D
Motivation
Filtered Backprojection

• Problem The nonlocality of Radon trnasform in
  even dimension
• Goal To reduce exposure to radiation
• Methods 3-D tomography
• Local tomography

• Fourier slice theorem
• Fourier Transform of the projections
• Inversion
• Filtered Backprojection
• Filter step
• Hilbert transform
• Backprojection step

Algorithm 3

• Image R, ROI ri, ROE rerirmrr, N evenly
  spaces angles
• ROE of each projection is filtered by scaling and
  wavelet ramp filters at N angles. The complexity
  is 9/2N re(log re) (using FFT)
• . Extrapolate 4 re pixels at N/2angles( Bandwidth
  is reduced by half after step1. ) The complexity
  is 3N (4re)(log 4re) (using FFT)
• 3. Using backprojection to obtain the wavelet
  coefficients at resolution 2-1. The remaining
  points are set to zero. The complexity is
  (7re/2)(ri2rr)2(using linear interpolation)
• 4. Reconstruct image from the wavelet and scaling
  coefficients. The complexity of filtering is
  4(2ri)2(3rr)

Background

• Radon transform
• Region of interest
• Interior Radon Transform

The Nonlocality of Radon Transform
Results

• Hilbert Transform of a compactly supported
  function can never be compactly supported,
  because it composes a discontinuity in the
  derivative of the Fourier transform of any
  function at the origin.
• The imposition of discontinuity at origin in
  frequency domain will spread the supported
  functions in time domain, i.e., local basis will
  not remain local after filtering

3 F. Rashid-Farrokhi, K.J.R. Liu, C. A.
Berenstein and D. Walnut Wavelet-based
Multiresolution Local Tomography, IEEE
Transactions on Image Processing, 6(1997), pp.
1412-1430.
4A. C. Kak and Malcolm Slaney, Principles of
Computerized Tomographic Imaging, IEEE Press,
1988.
Why wavelets?
5 S. Zhao, G. Welland, G. Wang, Wavelet
Sampling and Localization Schemes for the Radon
Transform in Two Dimensions, 1997 Society for
Industrial and Applied Mathematics.

• Compactly supported function
• Many vanishing moments

References
1 T. Olson, J. DeStefano, Wavelet localization
of the Radon Transform, IEEE Tr. Signal
Proc.42(8) 2055-2067 (1994).
2 C.A. Berenstein, D.F. Walnut, Local inversion
Radon transform in even dimensions using
wavelets, 75 years of Radon transform (Vienna,
1992), S, Gindikin, P. Michor (eds.), pp. 45-69,
International Press, (1994).