Analysis of MinCut , Average Cut , and Normalized Cut Measures ( Extended

Partitioning of a graph representation, defined over low-level image features based on Gestalt inspired relations, is an effective strategy for forming coherent perceptual groups in an image. The usual practice, mainly motivated by efficiency considerations, is to approximate the general K-way partitioning solution by recursive bi-partitioning, where at each step the graph is broken into two parts based on a partitioning measure. We concentrate on four such measures, namely, the minimum cut [9], average cut [4], Shi-Malik normalized cut [5], and a variation of the Shi-Malik normalized cut. 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 Shi-Malik normalized cut measure is also a normalized measure, but the normalizing factor is the product of the total connectivity (valency) of the nodes in each partition. A natural variation of the Shi-Malik normalized cut measure, which one might suggest, is the total edge weight cut normalized by the total association in each partition. The questions we ask in this work are: Does the nature of the cut measure really matter? Are the quality of the groups significantly different for each cut measure? How do the measures vary in the space of all possible partitions? Other studies that also consider similar questions are those by Perona and Freeman [3], Amir and Lindenbaum [1], Weiss [7], who studied similarities of graph spectral methods for segmentation and Williams and Thornber [8]. Our study complements and extends the previous studies in that it (i) considers bi-partitioning in the presence of K-objects in the scene instead of just two, (ii) relates the measures to the underlying image statistics in a probabilistic manner, and (iii) undertakes a rigorous and extensive empirical evaluation.