A direct approach to estimating surfaces in tomographic data

Abstract

Under ideal circumstances, the inverse of the radon transform is computable, and sequences of measured projections are sufficient to obtain accurate estimates of volume densities. In situations where the sinogram data is incomplete, the radon transform is noninvertable, and attempts to reconstruct greyscale density values result in reconstruction artifacts that can undermine the effectiveness of subsequent processing. This paper presents a direct approach to the segmentation of incomplete tomographic data. The strategy is to impose a fairly simple model on the data, and treat segmentation as a problem of estimating the interface between two substances of somewhat homogeneous density. The segmentation is achieved by simultaneously deforming a surface model and updating density parameters in order to achieve a best fit between the projected model and the measured sinograms. The deformation is implemented with level-set surface models, calculated at the resolution of the input data. Relative to previous work, this paper makes several contributions. First is a proper derivation of the deformation of surface models that move according to a gradient descent on a likelihood measure. We also present a series of computational innovations that make this direct surface-fitting approach feasible with state-of-the-art computers. Another contribution is the demonstration of the effectiveness of this approach on under-constrained tomographic problems, using both simulated and real datasets.