Spatial resolution properties of nonquadratically regularized image reconstruction for PET

This paper examines the spatial resolution properties of non-quadratically regularized image reconstruction methods for positron emission tomography (PET). It is well known that quadratic regularization tends to over-smooth edges. Many types of edge-preserving nonquadratic penalties have been proposed to overcome this problem. However, there has been little research on quantitative analysis of nonquadratic regularization due to its nonlinear nature, whereas quadratic regularization is fairly well-understood in terms of its resolution and variance properties. We analyze the spatial resolution property of nonquadratic regularization by the local perturbation response. Our analysis shows that one can design a non-quadratic penalty to yield improved shift-invariant spatial resolution properties by using the certainty-based method based on the assumption of localized perturbations and locally uniform backgrounds. We also derive a useful scaling property of nonquadratic regularization. Simulations of 2D PET scans illustrate the validity of our analysis