Tomographic reconstruction of piecewise smooth images

In computed tomography, direct inversion of the Radon transform requires more projections than are practical due to constraints in scan time and image accessibility. Therefore, it is necessary to consider the estimation of reconstructed images when the problem is under-constrained, i.e., when a unique solution does not exist. To resolve ambiguities among solutions, it is necessary to place additional constraints on the reconstructed image. In this paper, we present a surface evolution technique to model the reconstructed image as piecewise smooth. We model the reconstructed image as two regions that are each smoothly varying in intensity and are separated by a smooth surface. We define a cost functional to penalize deviation from piecewise smoothness while ensuring that the projections of the estimated image match the measured projections. From this functional, we derive an evolution for the modeled image intensity and an evolution for the surface, thereby defining a variational tomographic estimation technique. We show example reconstructions to highlight the performance of the proposed method on real medical images.