Joe Qranfal, Charles Byrne
Department of Mathematics
Simon Fraser University
British Columbia, CANADA
Department of Mathematical Sciences
University of Massachusetts at Lowell
Lowellm, USA

Abstract. We present a new filtering algorithm, the SMART filter (simultaneous multiplicative algebraic reconstruction technique) and provide a convergence result. We test it to solve the inverse problem of reconstructing a dynamic medical image where the signal strength changes substantially over the time required for data acquisition. Our test choice is the time-dependent single photon emission computed tomography (SPECT) which is an ill-posed inverse problem. Based on a linear state-space model of the problem, we provide numerical results to corroborate the effectiveness of our reconstruction method. The SMART filter guarantees a nonnegative and temporally regularized solution, filters out errors from modeling the dynamical system as well as the noise from the data, and outputs an optimal recursive estimate. The SMART filter proves itself to be also computationally time efficient which makes it very suitable for large scale systems such as the ones in medical imaging. In addition, it could be used in any discipline which has used, for instance Kalman filter, or in any one that is interested in time-varying variables such as financial risk assesment/evaluation and forecasting, tracking, or control. Tests in both cases, underdetermined and overdetermined, confirm the convergence result. Getting much better results in the latter case supports the fact that the more information we feed the SMART filter the better it behaves.

Received: May 31, 2011

AMS Subject Classification: 93E11, 93E10, 34K29, 49N45, 60G35, 62G05, 62M05, 68U10, 94A08, 90C25

Key Words and Phrases: estimation, stochastic filtering, Kalman filter, optimal filtering, state estimator, convex optimization, medical image, dynamic SPECT, cross-entropy, nonnegative reconstruction, hidden Markov model, algebraic reconstruction, temporal regularization

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