Function-Described Graphs for Structural Pattern Recognition

A fundamental problem in pattern recognition is selecting suitable representations for objects and classes. In the decision-theoretic approach to pattern recognition, a pattern is represented by a set of numerical values, which forms a feature vector. Although, in many tasks, objects can be recognised successfully using only global features such as size and compactness, in some applications it is helpful to describe an object in terms of its basic parts and the relations between them. Nevertheless, there are two major problems that practical applications using structural pattern recognition are confronted with. The first problem is the computational complexity of comparing two structures. The time required by any of the optimal algorithms may in the worst case become exponential in the size of the graphs. The approximate algorithms, on the other hand, have only polynomial time complexity, but do not guarantee to find the optimal solution. For some of the applications, this may not be acceptable. The second problem is the fact that there is more than one model graph that must be matched with an input graph, then the conventional graph matching algorithms must be applied to each model-input pair sequentially. As a consequence, the performance is linearly dependent on the size of the database of model graphs. For applications dealing with large database, this may be prohibitive. Function-described graphs (FDGs) are a compact representation of a set of attributed graphs. They have borrowed from “random graphs” proposed by Wong et al. the ability to probabilistically model structural attribute information, while improving the capacity to record structural relationships that consistently appear throughout the data. They do this by incorporating qualitative knowledge of the second-order probabilities of the elements that are expressed as relations (Boolean functions) between pairs of vertices and pairs of arcs in the FDGs. Four approaches and algorithms for building FDGs from an ensemble of attributed graphs are presented. The first synthesises an FDG in a supervised manner. The other three use the supervised clustering algorithms: dynamic, complete and single clustering. The problem of matching attributed graphs (AGs) to FDGs for recognition or classification purposes is studied from a Bayesian perspective. A distance measure between AGs and FDGs is derived from the principle of maximum likelihood, but robustness is enforced by considering only locally the effects of extraneous and missing elements. A second measure is also given in which the second-order constraints are incorporated as additional costs. A branch-and-bound algorithm is proposed to compute these distance measures together with their corresponding optimal labelling. Because of the exponential cost of this algorithm, three efficient algorithms are also proposed and compared to compute sub-optimal distances between AGs and FDGs. Two of them are based on a probabilistic relaxation approach, and the other does not have an iterative technique. Some experimental tests are presented in random graphs and a 3D-object recognition problem. In the 3D-object recognition application, an FDG model is synthesised (in a supervised and an unsupervised method) for each object from a set of views (AGs). The second-order information in FDGs is shown so that the recognition ration is better than when the first-order probability distributions are only used. Results of efficient algorithms show that there is an important decrease in the run time although there is only a slight decrease in effectiveness.