Multilevel Image Reconstruction with Natural Pixels

The sampled Radon transform of a two-dimensional (2D) function can be represented as a continuous linear map A:L₂ (Ω ) → RN, where (Au)ⱼ = (u,ψ ⱼ ) and ψ ⱼ is the characteristic function of a strip through Ω approximating the set of line integrals in the sample. The image reconstruction problem is: given a vector b ∈ RN,find an image (or density function) $u(x,y)$ such that Au = b. In general there are infinitely many solutions; we seek the solution with minimal 2-norm, which leads to a matrix equation B w = b, where B is a square dense matrix with several convenient properties. We analyze the use of Gauss–Seidel iteration applied to the problem, observing that while the iteration formally converges, there exists a near null space into which the error vectors migrate, after which the iteration stalls. The null space and near null space of B are characterized in order to develop a multilevel scheme. Based on the principles of the multilevel projection method (PML), this scheme leads to somewhat improved performance. Its primary utility, however, is that it facilitates the development of a PML-based method for spotlight tomography, that is, local grid refinement over a portion of the image in which features of interest can be resolved at finer scale than is possible globally.