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Many imaging problems, such as total variation reconstruction of X-raycomputed tomography (CT) and positron-emission tomography (PET), are solved viaa convex optimization problem with near-circulant, but not actually circulant,linear systems. The popular methods to solve these problems, alternatingdirection method of multipliers (ADMM) and primal-dual hybrid gradient (PDHG),do not directly utilize this structure. Consequently, ADMM requires a costlymatrix inversion as a subroutine, and PDHG takes too many iterations toconverge. In this paper, we present near-circulant splitting (NCS), a novelsplitting method that leverages the near-circulant structure. We show that NCScan converge with an iteration count close to that of ADMM, while paying acomputational cost per iteration close to that of PDHG. Through experiments ona CUDA GPU, we empirically validate the theory and demonstrate that NCS caneffectively utilize the parallel computing capabilities of CUDA.

Splitting with Near-Circulant Linear Systems: Applications to Total Variation CT and PET