An in-depth study of graph partitioning measures for perceptual organization

In recent years, one of the effective engines for perceptual organization of low-level image features is based on the partitioning of a graph representation that captures Gestalt inspired local structures, such as similarity, proximity, continuity, parallelism, and perpendicularity, over the low-level image features. Mainly motivated by computational efficiency considerations, this graph partitioning process is usually implemented as a recursive bipartitioning process, where, at each step, the graph is broken into two parts based on a partitioning measure. We focus on three such measures, namely, the minimum, average, and normalized cuts. The minimum cut partition seeks to minimize the total link weights cut. The average cut measure is proportional to the total link weight cut, normalized by the sizes of the partitions. The normalized cut measure is normalized by the product of the total connectivity (valencies) of the nodes in each partition. We provide theoretical and empirical insight into the nature of the three partitioning measures in terms of the underlying image statistics. In particular, we consider for what kinds of image statistics would optimizing a measure, irrespective of the particular algorithm used, result in correct partitioning. Are the quality of the groups significantly different for each cut measure? Are there classes of images for which grouping by partitioning does not work well? Also, can the recursive bipartitioning strategy separate out groups corresponding to K objects from each other? In the analysis, we draw from probability theory and the rich body of work on stochastic ordering of random variables. Our major conclusion is that optimization of none of the three measures is guaranteed to result in the correct partitioning of K objects, in the strict stochastic order sense, for all image statistics.