## Curvature Regularization

Today, Fredrik Kahl and I got a paper accepted to EMMCVPR titled Curvature Regularization for Curves and Surfaces in a Global Optimization Framework. The focus of this paper is not theoretical contributions, but to improve the performance of curvature regularization in image analysis and computer vision in practice.

The problem we are interested in solving amounts to minimizing the following energy functional:

$\displaystyle E(R) = \int_R g(\boldsymbol{x})\, d\boldsymbol{x} + \int_{\partial R} \left( \lambda +\gamma \kappa(\boldsymbol{x})^2\,\right) dA(\boldsymbol{x}), \ \ \ \ \ (1)$

where ${R}$ is the 2D (or 3D) foreground region with boundary ${\partial R}$. Here ${g(\boldsymbol{x})}$ is the data term, which may take many forms depending on the application, ${\lambda}$ is a positive weighting factor for length (or area) regularization, and ${\gamma}$ controls the amount of curvature regularization, denoted ${\kappa}$. Note that the domain may be a 2D image region or a 3D region. In the former case, the boundary is a curve and the notion of curvature is the usual one, while in the latter, the boundary is a surface and ${\kappa}$ refers to the mean curvature.

The work by Schoenemann, Kahl and Cremers from ICCV 2009 showed how this functional can be discretized and a global minimum computer via an LP relaxation. The present paper builds heavily upon this and improves the framework in several aspects:

• We show that additional linear constraints are needed to prevent spurious contours, which result in incorrect solutions. As an additional benefit, these new constraints greatly decrease the computational time (at the cost of some additional memory).
• We introduce and evaluate hexagonal meshes and show that they are preferred over square ones.
• Instead of having dense meshes, adaptive meshes are evaluated to decrease the huge memory requirements. Note that a reasonable heuristic is needed to generate these meshes.
• Finally, we generalize the framework to three dimensions and perform some simple experiments in this setting.

The final version of the paper will be posted online when ready.