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 256x256 image, we require 22% of full exposure data.
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