A Graduated Non-Convexity Technique for Dealing Large Point Spread Functions

This paper focuses on reducing the computational cost of a GNC Algorithm for deblurring images when dealing with full symmetric Toeplitz block matrices composed of Toeplitz blocks. Such a case is widespread in real cases when the PSF has a vast range. The analysis in this paper centers around the class of gamma matrices, which can perform vector multiplications quickly. The paper presents a theoretical and experimental analysis of how $\gamma$-matrices can accurately approximate symmetric Toeplitz matrices. The proposed approach involves adding a minimization step for a new approximation of the energy function to the GNC technique. Specifically, we replace the Toeplitz matrices found in the blocks of the blur operator with $\gamma$-matrices in this approximation. The experimental results demonstrate that the new GNC algorithm proposed in this paper reduces computation time by over $20\%$ compared with its previous version. The image reconstruction quality, however, remains unchanged.