Convexity is known as an important cue in human vision. We propose shape convexity as a new high-order regularization constraint for binary image segmentation. In the context of discrete optimization, object convexity is represented as a sum of 3-clique potentials penalizing any 1-0-1 configuration on all straight lines. We show that these non-submodular interactions can be efficiently optimize...

A path P = v1, . . . , vt is a triangle path (respectively, monophonic path) of G if no edges exist joining vertices vi and vj of P such that |j − i| > 2; (respectively, |j − i| > 1). A set of vertices S is convex in the triangle path convexity (respectively, monophonic convexity) of G if the vertices of every triangle path (respectively, monophonic path) joining two vertices of S are in S. The...

The study of nontraditional notions of visibility and convexity is a fruitful branch of geometry, which has found applications in vlsi design, graphics, and motion planning. Researchers have investigated multiple variations of these notions, including orthogonal, finitely oriented, and restricted-orientation visibility and convexity, as well as nesw and link convexity. Rawlins introduced the no...

The contributions of the first half of this thesis are on the computational and algebraic aspects of convexity in polynomial optimization. We show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1...