A spline-regularized minimal residual algorithm for iterative attenuation correction in SPECT

Abstract

In SPECT, regularization is necessary to avoid divergence of the iterative algorithms used for non-uniform attenuation compensation. In this paper, we propose a spline-based regularization method for the minimal residual algorithm. First, the acquisition noise is filtered using a statistical model involving spline smoothing so that the filtered projections belong to a Sobolev space with specific continuity and derivability properties. Then, during the iterative reconstruction procedure, the continuity of the inverse Radon transform between Sobolev spaces is used to design a spline-regularized filtered backprojection method, by which the known regularity properties of the projections determine those of the corresponding reconstructed slices. This ensures that the activity distributions estimated at each iteration present regularity properties, which avoids computational noise amplification, thus stabilizing the iterative process. Analytical and Monte Carlo simulations are used to show that the proposed spline-regularized minimal residual algorithm converges to a satisfactory stable solution in terms of restored activity and homogeneity, using at most 25 iterations, whereas the non regularized version of the algorithm diverges. Choosing the number of iterations is therefore no longer a critical issue for this reconstruction procedure.