λ-Connectedness and Its Applications

This paper attempts to provide a systematic view to the λ-connectedness method. This method is for classification/segmentation, fitting/reconstruction, and inference. A common type of partial connectivity that describes the phenomenal of gradual variation is studied in various domains. The previous research work on λ-connectedness was abstracted and integrated into a unified framework: a networkor graph-based system. This method focuses on a way to solve a set of problems that λ-connectedness can apply to rather than to give a solution to a particular problem. This technique is based on a graph G = (V, E) and an associate function ρ on the vertices of the graph, where ρ is called the potential function. A measure Cρ(x, y) is defined for the λconnectedness on the vertices x, y ∈ G with respect to ρ. For a certain λ ∈ [0, 1], x and y are said to be λ-connected if Cρ(x, y) ≥ λ. So, λ-connectedness is a type of fuzzy relation. If every pair of vertices are λ-connected, then ρ is λ-connected on G. Therefore, λ-connectedness is also a measure of continuity in discrete spaces and systems. Based on this concept, the relationship between graph theory and λ-connectedness is discussed, and the λ-connected segmentation and fitting are introduced. Further more, the fast graph-theoretic algorithms to problems of λ-connectedness are described. At the end, the relationship among λ-connectedness, rough sets, belief networks, and network economics and resource management are also investigated. The λ-connectedness method can be viewed as a special application of graph theory, a fuzzy system method, or a systematic method for problems in uncertainty.