Convexity Shape Prior for Binary Segmentation

Convexity is a known 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 three-clique potentials penalizing any <inline-formula><tex-math notation="LaTeX">$1$</tex-math><alternatives> <inline-graphic xlink:href="veksler-ieq1-2547399.gif"/></alternatives></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$0$</tex-math><alternatives><inline-graphic xlink:href="veksler-ieq2-2547399.gif"/> </alternatives></inline-formula>-<inline-formula><tex-math notation="LaTeX">$1$</tex-math><alternatives> <inline-graphic xlink:href="veksler-ieq3-2547399.gif"/></alternatives></inline-formula> configuration on all straight lines. We show that these non-submodular potentials can be efficiently optimized using an iterative trust region approach. At each iteration the energy is linearly approximated and globally optimized within a small trust region around the current solution. While the quadratic number of all three-cliques is prohibitively high, we design a dynamic programming technique for evaluating and approximating these cliques in linear time. We also derive a second order approximation model that is more accurate but computationally intensive. We discuss limitations of our local optimization and propose <italic>gradual non-submodularization</italic> scheme that alleviates some limitations. Our experiments demonstrate general usefulness of the proposed convexity shape prior on synthetic and real image segmentation examples. Unlike standard second-order length regularization, our convexity prior does not have shrinking bias, and is robust to changes in scale and parameter selection.

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