Analytical approach to regularization design for isotropic spatial resolution

In emission tomography, conventional quadratic regularization methods lead to nonuniform and anisotropic spatial resolution in penalized-likelihood (or MAP) reconstructed images, even for idealized shift-invariant imaging systems. Previous methods for designing regularizers that improve resolution uniformity have used matrix manipulations and discrete Fourier transforms. This paper describes a simpler approach for designing data-dependent, shift-variant regularizers. We replace the usual discrete system models used in statistical image reconstruction with locally shift-invariant, continuous-space approximations, and design the regularizer using analytical Fourier transforms. We discretize the final analytical solution to compute the regularizer coefficients. This new approach requires even less computation than previous approaches that used FFTs, and provides additional insight into the problem of designing regularization methods to achieve uniform, isotropic spatial resolution.