Iterative inversion of the Radon transform

Abstract

In many problems in the field of medical imaging, one needs to reconstruct a 2D object or image from its projections. This process is equivalent to computing the inverse Radon transform of the projections. This is the basic problem of X-ray tomography, and the imaging step in magnetic resonance imaging can also be so formulated. The authors have recently developed techniques that use wavelets to perform localized low-pass filtering. They have shown and statistically justified how thresholding can be used to determine the regions in the wavelet domain where wavelet coefficients may be set to zero, affecting spatially varying filtering. Alternatively, a priori information about the image can be used to identify such regions. The authors then use these zero wavelet coefficients as constraints, and compute the minimum mean-squared error (MMSE) image that satisfies these constraints. The distinction between this approach and previous use of wavelets to affect spatially varying image filtering is that by using the wavelet constraints on the image reconstruction process, one is able to improve the reconstructed image not only in regions where the wavelet domain constraints are applied, but also in regions where they are not applied! This is possible since the Radon transform is not a unitary transform, so that constraints applied in a given region effect projections outside that region, which in turn affects the reconstructed image outside the constrained region. Here, the authors briefly illustrate these techniques, minimizing the number of equations, and then present for the first time new results of an iterative implementation of these techniques. The idea is that at each iteration one uses the image reconstructed at the previous iteration as the source of wavelet constraints for the next reconstruction. This is similar to iterative Wiener filtering, except that the iterative filtering is spatially varying, while Wiener filtering is necessarily spatially invariant. The authors' technique allows the advantage of spatially varying low-pass filtering to be applied iteratively. The results are that at the third iteration, the root-mean-square (rms) error is reduced by more than twice as much as in the authors' previous results, while high-resolution image features are preserved.