The curvature regularization method is well-known for its good geometric interpretability and strong priors in the continuity of edges, which has been applied to various image processing tasks. However, due to the non-convex, non-smooth, and highly non-linear intrinsic limitations, most existing algorithms lack a convergence guarantee. This paper proposes an efficient yet accurate scalar auxiliary variable (SAV) scheme for solving both mean curvature and Gaussian curvature minimization problems. The SAV-based algorithms are shown unconditionally energy diminishing, fast convergent, and very easy to be implemented for different image applications. Numerical experiments on noise removal, image deblurring, and single image super-resolution are presented on both gray and color image datasets to demonstrate the robustness and efficiency of our method. Source codes are made publicly available at https://github.com/Duanlab123/SAV-curvature.