New characterizations of minimum spanning trees and of saliency maps based on quasi-flat zones

TL;DR

This paper introduces new characterizations of minimum spanning trees and saliency maps through quasi-flat zones, providing a unified framework for hierarchical data representations applicable beyond image processing.

Contribution

It establishes a novel bijection between saliency maps and hierarchies, and characterizes these structures as solutions to constrained minimization problems based on quasi-flat zones.

Findings

Provides a new theoretical framework for hierarchical representations

Characterizes saliency maps and MSTs via constrained minimization

Enables processing of non-image data with morphological hierarchies

Abstract

We study three representations of hierarchies of partitions: dendrograms (direct representations), saliency maps, and minimum spanning trees. We provide a new bijection between saliency maps and hierarchies based on quasi-flat zones as used in image processing and characterize saliency maps and minimum spanning trees as solutions to constrained minimization problems where the constraint is quasi-flat zones preservation. In practice, these results form a toolkit for new hierarchical methods where one can choose the most convenient representation. They also invite us to process non-image data with morphological hierarchies.