  The geometric high-order regularization methods such as mean curvature and
Gaussian curvature, have been intensively studied during the last decades due
to their abilities in preserving geometric properties including image edges,
corners, and contrast. However, the dilemma between restoration quality and
computational efficiency is an essential roadblock for high-order methods. In
this paper, we propose fast multi-grid algorithms for minimizing both mean
curvature and Gaussian curvature energy functionals without sacrificing
accuracy for efficiency. Unlike the existing approaches based on operator
splitting and the Augmented Lagrangian method (ALM), no artificial parameters
are introduced in our formulation, which guarantees the robustness of the
proposed algorithm. Meanwhile, we adopt the domain decomposition method to
promote parallel computing and use the fine-to-coarse structure to accelerate
convergence. Numerical experiments are presented on image denoising, CT, and
MRI reconstruction problems to demonstrate the superiority of our method in
preserving geometric structures and fine details. The proposed method is also
shown effective in dealing with large-scale image processing problems by
recovering an image of size $1024\times 1024$ within $40$s, while the ALM
method requires around $200$s.
