Exploring the Links Between Edge-Preserving Collaborative Filters via Gamma Convergence

Sravan Danda (1) , Aditya S Challa (1) , B S Daya Sagar , Laurent Najman (2)

1 Systems Science and Informatics Unit
2 LIGM - Laboratoire d'Informatique Gaspard-Monge

Edge-aware filtering is an important pre-processing step in many computer vision applications. In literature, there exist collaborative edge-aware filters that work well in practice but are based only on heuristics and/or principles. For instance, Tree Filter (TF) which is proposed recently based on a minimum spanning tree (MST) heuristic yields promising results. However the usage of an arbitrary MST for filtering is theoretically not justified. In this article, we introduce an edge-aware generalization of the TF, termed as UMST filter based on all MSTs. The major contribution of this paper is establishing theoretical links between filters based on MSTs and filters based on geodesics via the notion of Γ-convergence. More precisely, we compute the Γ-limit of Shortest Path Filters (SPFs) and show that it is the same as UMST filter. Consequently, TF can be viewed as an approximate Γ-limit of the SPFs, thereby providing a theoretical basis to it's working. Further, we propose and provide a detailed analysis of two different implementations of the UMST filter based on shortest paths.

Mots clés: Optimization; Image Filtering; Γ-convergence; MST; Shortest Paths; 18

Domaines: Informatique [cs]

Sravan Danda, Aditya S Challa, B S Daya Sagar, Laurent Najman. Exploring the Links Between Edge-Preserving Collaborative Filters via Gamma Convergence. 2017. ⟨hal-01617799v1⟩