Curvature Regularization for Curves and Surfaces in a Global Optimization Framework

Abstract

Abstract. Length and area regularization are commonplace for inverse problems today. It has however turned out to be much more difficult to incorporate a curvature prior. In this paper we propose several improvements to a recently proposed framework based on global optimization. We identify and solve an issue with extraneous arcs in the original formulation by introducing region consistency constraints. The mesh geometry is analyzed both from a theoretical and experimental viewpoint and hexagonal meshes are shown to be superior. We demonstrate that adaptively generated meshes significantly improve the performance. Our final contribution is that we generalize the framework to handle mean curvature regularization for 3D surface completion and segmentation.

Curvature Regularity for Multi-label Problems - Standard and Customized Linear Programming

“…They express that for every edge e only one line pair in the boundary may contain an orientation of e as its first element. Simultaneous to our work appears the work [27] that proposes an alternative constraint system to prevent touching lines. We leave this for future work.…”

Tighter Relaxations for Higher-Order Models Based on Generalized Roof Duality

“…We will use the pseudo-boolean optimization approach suggested in [5] for an 8-connected grid which requires quartic interactions. The construction can be understood by examining Figure 1.…”

“…While the runtime for GRD-gen was much faster than GRD it was considerably slower than both of the other methods. The GRD-gen method has potential for this problem but the approximation tightness needs to be improved in order to beat state-of-the-art based on LP relaxations [5]. …”

“…For example, to avoid the well known shrinking bias of length or area based approaches, curvature regularization which requires higher-order models is considered in [4,5]. Other examples are [6] where approximate belief propagation is used for inference of a higher-order learned model, [7] where second order smoothness priors are used for stereo and [8] which uses a higher-order model for texture restoration.…”

Shortest Paths with Curvature and Torsion

“…In [10], it is shown that Euler's elastica -the line integral of the squared curvature -conforms better to intuitive completion of boundary curves. Other applications where curvature plays an important role include saliency [17], inpainting [9], stereo [21], region-based image segmentation [15] and surface reconstruction [19].…”