Segmenting an image into multiple components is a central task in computer
vision. In many practical scenarios, prior knowledge about plausible components
is available. Incorporating such prior knowledge into models and algorithms for
image segmentation is highly desirable, yet can be non-trivial. In this work,
we introduce a new approach that allows, for the first time, to constrain some
or all components of a segmentation to have convex shapes. Specifically, we
extend the Minimum Cost Multicut Problem by a class of constraints that enforce
convexity. To solve instances of this APX-hard integer linear program to
optimality, we separate the proposed constraints in the branch-and-cut loop of
a state-of-the-art ILP solver. Results on natural and biological images
demonstrate the effectiveness of the approach as well as its advantage over the
state-of-the-art heuristic.

本文介绍了一种新方法，首次允许将凸形约束应用于图像分割，该方法在最小成本多切问题上通过一类约束来实现，通过在分支 - 切割循环中将所提出的约束分离到最先进的 ILP 求解器的状态，得到了很好的结果。

图像分割的凸形约束

Convexity Shape Constraints for Image Segmentation

The Four Vertex Theorem, one of the earliest results in global differential
geometry, says that a simple closed curve in the plane, other than a circle,
must have at least four "vertices", that is, at least four points where the
curvature has a local maximum or local minimum. In 1909 Syamadas Mukhopadhyaya
proved this for strictly convex curves in the plane, and in 1912 Adolf Kneser
proved it for all simple closed curves in the plane, not just the strictly
convex ones.
The Converse to the Four Vertex Theorem says that any continuous real-valued
function on the circle which has at least two local maxima and two local minima
is the curvature function of a simple closed curve in the plane. In 1971 Herman
Gluck proved this for strictly positive preassigned curvature, and in 1997
Bjorn Dahlberg proved the full converse, without the restriction that the
curvature be strictly positive. Publication was delayed by Dahlberg's untimely
death in January 1998, but his paper was edited afterwards by Vilhelm Adolfsson
and Peter Kumlin, and finally appeared in 2005.
The work of Dahlberg completes the almost hundred-year-long thread of ideas
begun by Mukhopadhyaya, and we take this opportunity to provide a
self-contained exposition.

这篇论文完整地介绍了由 Mukhopadhyaya 开始的关于 Four Vertex Theorem 的研究，探讨了简单封闭曲线的曲率函数和平面上简单封闭曲线之间的关系。