Indexing Based on Edit-Distance Matching of Shape Graphs

We are investigating a graph matching approach for indexing into pictorial databases using shock graphs, a symmetry-based representation of shape. Each shape (or a collection of edge elements) is represented by a shock graph. Indexing of a query into a pictorial database is accomplished by comparing the corresponding shock graph to the graphs representing database elements and selecting the best match. This paper introduces a new metric for comparing shock graphs. The premise underlying the use of symmetry as a cue for indexing is that the correlation of pairs of edge elements in the sketch query and in the image is a more robust measure of indexing than simply correlating edge elements. Each pair of edge elements is represented by a symmetry curve, and can be extracted from real images and represented as singularities resulting from the propagation of each such orientation element. Previous work shows that these singularities, or shocks, can be detected, classi ed, grouped, and organized into a shock graph embedded in the plane. In previous work, we proposed the use of a graph comparison heuristic called the graduated assignment algorithm which casts the problem as an integer quadratic program. This heuristic has two drawbacks. First, the quadratic program does not fully capture the topological constraints of graph matching, and so even an optimal solution could be topologically incorrect. Second, since solving the quadratic program is infeasible, the algorithm often nds only locally optimal solutions. We now present a matching approach based on the notion of edit distance. The edit distance between two graphs is de ned as the cost of the \least action" path of elementary edit transformations taking one graph to the other. We have de ned a set of elementary edit transformations appropriate to shock graphs. These operations are motivated by intuitive and natural deformations of shape and mathematically correspond closely to the set of shock transitions. Moreover, we observe that shock graphs have special structure: they are unrooted, planarembedded trees. By exploiting this structure, we have developed an algorithm to compute the edit distance (and corresponding edit transformations) between shock graphs precisely and quickly (in polynomial time). We have implemented this algorithm for a subset of these edit operations. In this paper, we illustrate these results for a few shapes. We plan to report on a complete implementation and results in a later paper.