Sparse shift-varying FIR preconditioners for fast volume denoising

Splitting-based CT reconstruction algorithms decompose the reconstruction problem into a iterated sequence of “easier” subproblems. One relatively memory-efficient algorithm decomposes the reconstruction problem into a several subproblems, including a volume denoising problem. While easier to solve in isolation than jointly, these subproblems have highly shiftvarying Hessians that are challenging to effectively precondition with circulant operators. In this work, we present an algorithm to design a positive-definite, Schatten p-norm optimal, finite impulse response (FIR) approximation to a given circulant matrix. With this algorithm, we generate efficient space-varying preconditioners for the volume denoising problem. We demonstrate that PCG with an efficient space-varying preconditioner can converge at least quickly as a split-Bregman-like algorithm while using considerably less memory.