Order-preserving moves for graph-cut-based optimization

In the last decade, graph-cut optimization has been popular for a variety of labeling problems. Typically, graph-cut methods are used to incorporate smoothness constraints on a labeling, encouraging most nearby pixels to have equal or similar labels. In addition to smoothness, ordering constraints on labels are also useful. For example, in object segmentation, a pixel with a "car wheel" label may be prohibited above a pixel with a "car roof" label. We observe that the commonly used graph-cut \alpha-expansion move algorithm is more likely to get stuck in a local minimum when ordering constraints are used. For a certain model with ordering constraints, we develop new graph-cut moves which we call order-preserving. The advantage of order-preserving moves is that they act on all labels simultaneously, unlike \alpha-expansion. More importantly, for most labels \alpha, the set of \alpha-expansion moves is strictly smaller than the set of order-preserving moves. This helps to explain why in practice optimization with order-preserving moves performs significantly better than \alpha-expansion in the presence of ordering constraints. We evaluate order-preserving moves for the geometric class scene labeling (introduced by Hoiem et al.) where the goal is to assign each pixel a label such as "sky," "ground," etc., so ordering constraints arise naturally. In addition, we use order-preserving moves for certain simple shape priors in graph-cut segmentation, which is a novel contribution in itself.