Graph theory has strong historical roots in mathematics, especially in topology. Its birth is usually associated with the “four-color problem” posed by Francis Guthrie 1 in 1852, but its real origin probably goes back to the Seven Bridges of Konigsber ¨ g 2 problem proved by Leonhard Euler in 1736. A computational solution to these two completely different problems could be found after each problem was abstracted to the level of a graph model while ignoring such irrelevant details as country shapes or cross-river distances. In general, a graph is a nonempty set of points (vertices) and the most basic information preserved by any graph structure refers to adjacency relationships (edges) between some pairs of points. In the simplest graphs, edges do not have to hold any attributes, except their endpoints, but in more sophisticated graph structures, edges can be associated with a direction or assigned a label. Graph vertices can be labeled as well. A graph can be represented graphically as a drawing (vertex=dot,edge=arc),but,aslongaseverypairofadjacentpointsstaysconnected by the same edge, the graph vertices can be moved around on a drawing without changing the underlying graph structure. The expressive power of the graph models placing a special emphasis on c- nectivity between objects has made them the models of choice in chemistry, physics, biology, and other ?elds.
This book will serve as a foundation for a variety of useful applications of graph theory to computer vision, pattern recognition, and related areas. It covers a representative set of novel graph-theoretic methods for complex computer vision and pattern recognition tasks. The first part of the book presents the application of graph theory to low-level processing of digital images such as a new method for partitioning a given image into a hierarchy of homogeneous areas using graph pyramids, or a study of the relationship between graph theory and digital topology. Part II presents graph-theoretic learning algorithms for high-level computer vision and pattern recognition applications, including a survey of graph based methodologies for pattern recognition and computer vision, a presentation of a series of computationally efficient algorithms for testing graph isomorphism and related graph matching tasks in pattern recognition and a new graph distance measure to be used for solving graph matching problems. Finally, Part III provides detailed descriptions of several applications of graph-based methods to real-world pattern recognition tasks. It includes a critical review of the main graph-based and structural methods for fingerprint classification, a new method to visualize time series of graphs, and potential applications in computer network monitoring and abnormal event detection.
