Wavelet-based multiresolution local tomography

• Published in: IEEE transactions on image processing 1997-10, Vol.6 (10), p.1412-1430
• Main Authors: Rashid-Farrokhi, F., Liu, K.J. Ray, Berenstein, C.A., Walnut, D.
• Format: Article
• Language: English
• Subjects: Algorithms; Continuous wavelet transforms; Error analysis; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Image reconstruction; Image reconstruction; tomography; Imaging and optical processing; Optics; Physics; Pixel; Upper bound; Wavelet coefficients; Wavelet transforms; X ray radiography; X-ray imaging; X-ray tomography
• ISSN: 1057-7149; 1941-0042
• DOI: 10.1109/83.624961

We develop an algorithm to reconstruct the wavelet coefficients of an image from the Radon transform data. The proposed method uses the properties of wavelets to localize the Radon transform and can be used to reconstruct a local region of the cross section of a body, using almost completely local data that significantly reduces the amount of exposure and computations in X-ray tomography. The property that distinguishes our algorithm from the previous algorithms is based on the observation that for some wavelet bases with sufficiently many vanishing moments, the ramp-filtered version of the scaling function as well as the wavelet function has extremely rapid decay. We show that the variance of the elements of the null-space is negligible in the locally reconstructed image. Also, we find an upper bound for the reconstruction error in terms of the amount of data used in the algorithm. To reconstruct a local region 16 pixels in radius in a 256/spl times/256 image, we require 22% of full exposure data.