Nonnegative Definite Quadratic Penalty Design for Penalized-Likelihood Reconstruction

Likelihood-based estimators with conventional regularization methods generally produces images with nonuniform and anisotropic spatial resolution properties. Previous work on penalty design for penalizedlikelihood estimators has led to statistical reconstruction methods that yield approximately uniform “average” resolution. However some asymmetries in the local point-spread functions persist. Such anisotropies result in the elongation of otherwise symmetric features like circular lesions. All previously published penalty functions have used nonnegative values for the weighting coefficients between neighboring voxels. Such nonnegativity provides a sufficient (but not necessary) condition to ensure that the penalty function is convex, which in turn ensures that the objective function has a unique maximizer. This paper describes a novel method for penalty design that allows a subset of the weighting coefficients to take negative values, while still ensuring convexity of the penalty function. We demonstrate that penalties designed under these more flexible constraints yield local pointspread functions that are more isotropic than the previous penalty design methods for 2D PET image reconstruction.