Understanding nonlinearity in weighted least-squares image reconstruction for nuclear medicine.

This study investigated a linear approximation of weighted least squares (WLS) and its regularized variants using reductio ad absurdum and numerical simulations. Three phantoms were employed in the simulation: point source 1 represented by a two-dimensional (2D) Gaussian with a full width at half maximum of 3 pixels positioned at the center of the background; point source 2, a 2D Gaussian positioned 24 pixels away from point source 1 along the x-axis; and point source 3 is the sum of them. Ideal 2D parallel-beam projections were used. When the absolute difference of area under curve or the absolute difference of recovery between the reconstructed images of point source 3 and sum of point source 1 and 2, falls within a predetermined threshold, it was defined as linear approximation. The variance of WLS was estimated using either the projections of original image or forward projections. Filtered backprojection and least squares were used as a reference of linear reconstruction. We proved that WLS is nonlinear for small number of iterations but is approximated as linear for large number of iterations by reductio ad absurdum. WLS regularized with a quadratic function (WLS-QU) and a relative difference function (WLS-RD) were both proved to be nonlinear. Simulation results showed that WLS was nonlinear, but it becomes linear approximation when the number of iterations approached 50 with 64 subsets. WLS-QU and WLS-RD was nonlinear.　When the likelihood term itself is nonlinear, the reconstruction becomes nonlinear irrespective of the form of the regularization function.