A Kalman filtering approach to stochastic tomography.

Abstract

The problem of reconstructing a function from its projections (i.e., inverting the Radon transform) is formulated using the Fourier series expansions of the function and its projections in the angular variable. It is regularized by assuming band-limitedness of the function in the angular direction. Using the Fourier series expansion, the Radon transform of the function decouples into Abel transforms of various orders of the circular harmonics of the function. The novelty of our approach is that we fit a state-space model to the Abel transform of each order and use Kalman filters to estimate the function harmonics from noisy observations of the projection harmonics. The function harmonics can be modelled in two different ways. If the function is a realization of an isotropic random field, the state-space model of the n-th order Abel transform is augmented with a differential equation describing the n-th harmonic, for each n. The differential equation is derived using Markovianization of a two-point boundary value model of the function harmonics. The Kalman filter then computes the linear least-squares estimate of the function, given noisy observations of its projections. We also propose modelling the function harmonics as a linear combination of two Wiener precesses; this seems to give better results in reconstructing an arbitrary function. Our numerical results show that the Kalman filtering approach works roughly as well as filtered back-projection, with slightly better performance when few projection angles are available and noise levels are high. Mathematically, the Kalman filtering approach has three main advantages over filtered back-projection: (1) the reconstructed function is consistent, in that projection of the reconstructed function agrees with the given projection data; (2) the derivative-Hilbert transform filtering operation is avoided; and (3) the stochastic formulation of the problem allows incorporation of a priori information about the function, noise in the projection data, and control over their relative importance in the problem.