A Kalman filtering approach to stochastic global and region-of-interest tomography

We define two forms of stochastic tomography. In global tomography, the goal is to reconstruct an object from noisy observations of all of its projections. In region-of-interest (ROI) tomography, the goal is to reconstruct a small portion of an object (an ROI) from noisy observations of its projections densely sampled in and near the ROI and sparsely sampled away from the ROI. We solve both problems by expanding the object and its projections in a circular harmonic (Fourier) series in the angular variable so that the Radon transform becomes Abel transforms of integer orders applied to the harmonics. The algorithm has three major components. First, we fit state-space models to each order of Abel transform and thus represent the Radon transform operation as a parallel bank of systems, each of which computes the appropriate Abel transform of a circular harmonic. A variable transformation here allows either the global or ROI problem to be solved. Second, the object harmonics are modeled as a Brownian branch. This is a two-point boundary value system, which is Markovianized into a form suitable for the Kalman filter. Finally, a parallel bank of Kalman smoothing filters independently estimates each circular harmonic from the noisy projection data. Numerical examples illustrate the proposed procedure.