Multiresolution methods for the inverse radon transform.

Abstract

We apply time-frequency and multiresolution representations to three problems of image reconstruction from projections, with applications in X-ray tomography. First, we perform spatially-varying filtering of noisy reconstructed images using time-frequency representations. We consider two alternatives: (i) filtering the reconstructed image directly; and (ii) filtering projection data, and then performing image reconstruction. In (i), we constrain higher-resolution wavelet coefficients to zero in certain regions of the image, which effectively amounts to localized low-pass filtering. We then compute the image requiring the smallest (sum of squared differences) perturbation of the projections while satisfying these constraints; the same procedure can be used to compute the minimum mean-squared error image, subject to the constraints. In (ii), we threshold the wavelet transform or the short-time Fourier transform of the projection data to zero in certain regions of the time-frequency plane. In both alternatives, our approach suppresses noise in slowly-varying regions of the image, while maintaining the sharpness of image features. The second problem is image reconstruction from limited-angle projections. We show that when the extent of missing angles is small, two out of three sets of detail images in the wavelet transform are largely unaffected by the missing data. We also show how a good approximation to very low-resolution images can be obtained by using interpolation to fill the missing angles. We then complete the affected detail images by using a priori knowledge of edges which lie almost parallel to the missing projection angles, and we develop an algorithm based on the wavelet transform to restore the image. The third problem is local tomography, the (possibly approximate) reconstruction of a region of interest of the image using only local projections. We propose an exponential sampling technique which enables us to reconstruct the region of interest with very good resolution and a larger region with poorer resolution. Using circular harmonic decomposition, we develop an algorithm for this sampling technique which is an order of magnitude faster than the usual method of tomographic image reconstruction.