Total Curvature Regularization and its_Minimization for Surface and Image Smoothing

TL;DR

This work introduces Total Normal Curvature (TNC) regularization for surface and image smoothing, enabling robust edge and corner preservation by aggregating normal curvatures over all directions. It recasts the nonlinear, high-order problem into a time-evolution system and solves it via a Lie-type operator-splitting that decouples the problem into tractable substeps, many of which admit closed-form or FFT-based solutions. The approach demonstrates competitive denoising and smoothing performance, with favorable convergence and limited parameter sensitivity, outperforming several curvature-based baselines in preserving sharp geometry. The combination of a directionally rich curvature prior, auxiliary-variable reformulations, and efficient numerical schemes provides a practical framework for high-quality geometric regularization in 2D surfaces and images.

Abstract

We introduce a novel formulation for curvature regularization by penalizing normal curvatures from multiple directions. This total normal curvature regularization is capable of producing solutions with sharp edges and precise isotropic properties. To tackle the resulting high-order nonlinear optimization problem, we reformulate it as the task of finding the steady-state solution of a time-dependent partial differential equation (PDE) system. Time discretization is achieved through operator splitting, where each subproblem at the fractional steps either has a closed-form solution or can be efficiently solved using advanced algorithms. Our method circumvents the need for complex parameter tuning and demonstrates robustness to parameter choices. The efficiency and effectiveness of our approach have been rigorously validated in the context of surface and image smoothing problems.

Figures (18)

• Figure 1: Numerical comparison among different curvatures on three typical image patterns, where (I)-(III) are test patterns; (a)-(f) are different curvatures and their zoomed-in regions.
• Figure 1: Testing images. Synthetic image: (a) Square ($60 \times 60$); (b) Rings ($100\times 100$). Natural image: (c) Peppers ($256 \times 256$); (d) House ($256 \times 256$); (e) Cameraman ($256 \times 256$); (f) Zelda ($512 \times 512$); (g) Parrot ($512 \times 512$); (h) Plane ($512 \times 512$).
• Figure 1: The image surfaces of the clean images and reconstructed images by our TNC model. (a) Clean surfaces. (b) Noisy surfaces with $\sigma = 10^{-3}$. (c) Smoothed surfaces by the proposed model with $\alpha = 0.1$, $\beta = 0.4$, $\gamma=8$, $\tau=0.01$.
• Figure 1: The points used to compute the Hessian matrix at the central point $(i,j)$, where the red point is the center itself, the blue axial points compute the second-order partial derivatives, and the green diagonal points compute the mixed partial derivatives.
• Figure 1: Illustrative edge patterns and their discretizations. (I) A continuous vertical line pattern (Its discrete representation is identical to its continuous form). (II) A continuous diagonal line pattern. (III) A binary discrete representation of pattern (II). (IV) A multi-level discrete representation of pattern (II), where pixels along the diagonal transition are assigned a value of $1/2$ (black and white correspond to 0 and 1, respectively).

Theorems & Definitions (2)

• Remark 2.1
• Remark 3.1