Local Computerized Tomography Using WaveletsChih-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: re=ri+rm+rr, 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)(ri+2rr)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.

[4]A. 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).