Efficient Deblurring with Neumann Boundary Conditions: A comparative study of DCT, DST, and Tau Preconditioners

Deblurring is a critical task in image processing, where boundary conditions (BCs) significantly impact computational efficiency and restoration quality. This paper systematically evaluates three approaches: (1) the discrete cosine transform (DCT) under Neumann BCs, (2) the discrete sine transform (DST) under Dirichlet BCs, and (3) Tau preconditioners. We demonstrate that Neumann BCs paired with DCT diagonalization achieve superior performance for symmetric blurring functions due to their Toeplitz-plus-Hankel structure, enabling direct diagonalization via fast cosine transforms (FCTs) with $\boldsymbol{O}(\boldsymbol{log} \boldsymbol{n})$ complexity. In contrast, DST under Dirichlet BCs suffers from boundary artifacts unless the blur is symmetric and the signal aligns with zero boundaries. Tau preconditioners, while flexible for general Toeplitz systems, require iterative solvers with slower convergence.