Implicit Component-Graph: A Discussion

Abstract

Component-graphs are defined as the generalization of componenttrees to images taking their values in partially ordered sets. Similarly to component-trees, component-graphs are a lossless image model, and can allow for the development of various image processing approaches (e.g., antiextensive filtering, segmentation by node selection). However, component-graphs are not trees, but directed acyclic graphs. This induces a structural complexity associated to a higher combinatorial cost. These properties make the handling of componentgraphs a non-trivial task. We propose a preliminary discussion on a new way of building and manipulating component-graphs, with the purpose of reaching reasonable space and time costs. Tackling these complexity issues is indeed required for actually involving component-graphs in efficient image processing approaches.

Component-Graph Construction

“…It follows that X ⊆ Z, and since K ′′ w K, we also have Z ⊆ X, and then X = Z. It comes from the definition ofΘ (Equation (17)) that K ′′ = K, and thus u = v. It follows that w {v, w} = u = v. Then we have K s K ′ . Consequently, we have K w K ′ ⇒ K ′ s K, and from Property 1, it comes that K w K ′ ⇔ K ′ w K. The result follows by transitive reduction of w and s .…”

“…By side effect, we also discuss the way to efficiently store the resulting component-graph. This article is an extended and improved version of the conference paper [17].…”

Shape-Based Analysis on Component-Graphs for Multivalued Image Processing

“…In the case ofΘ, we use the component-graph construction algorithm proposed in [47]. (However, an algorithm was recently proposed in [66,67] for building, more generally, the componentgraphs Θ; then, the proposed framework is, of course, also tractable for such "complete" component-graphs.) For the sake of readability, we will now noteΘ as Θ.…”