Semi-implicit and Bregman parareal method for TV-nonlocal image enhancement

The total variation nonlocal (TV-nonlocal) model integrates several advanced features, including geometric active contours, anisotropic diffusion, regularization, and nonlocal gradients, demonstrating solid mathematical foundations. It achieves remarkable performance and interpretability when processing blurred edge segmentation or enhancement tasks. Although this model has attracted significant research attention, the development of efficient numerical solutions remains unsatisfactory. To enhance the model’s computational efficiency and applicability in image enhancement, this paper proposes a novel parareal method that combines first-order semi-implicit discretization with the split Bregman method, based on fundamental principles of parallel composite methodologies. The rigorous convergence analysis demonstrates that the proposed method exhibits favorable convergence properties under specific conditions. Furthermore, this paper provides a detailed discussion of key implementation aspects of the parallel algorithm. Numerical experiments show that the parareal scheme significantly accelerates the solving process of the TV-nonlocal model while maintaining accuracy, achieving a remarkable improvement in computational efficiency.