A Kalman filtering approach to stochastic tomography

The inverse Radon transform problem is formulated using the Fourier series expansion 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 the present approach is that one fits a state-space model to the Abel transform of each order and uses Kalman filters to estimate the function harmonics from noisy observations of the projection harmonics. In applying the Kalman filter to obtain the linear least-squares estimate of each harmonic, two types of object functions are considered: (1) if the function is a realization of an isotropic random field, a state equation for the nth harmonic for each n is derived using Markovianization of a two-point boundary value model of the function harmonics; (2) for an arbitrary function, the model of the Wiener process with a two-point boundary value is proposed.<>