Mathematical morphology based on stochastic permutation orderings

Olivier Lézoray

doi:10.1515/mathm-2021-0101

Abstract

Abstract The extension of mathematical morphology to multivariate data has been an active research topic in recent years. In this paper we propose an approach that relies on the consensus combination of several stochastic permutation orderings. The latter are obtained by searching for a smooth shortest path on a graph representing an image. This path is obtained with a randomized version nearest of neighbors heuristics on a graph. The construction of the graph is of crucial importance and can be based on both spatial and spectral information to enable the obtaining of smoother shortest paths. The starting vertex of a path being taken at random, many different permutation orderings can be obtained and we propose to build a consensus ordering from several permutation orderings. We show the interest of the approach with both quantitative and qualitative results.

Highlights

Mathematical morphology (MM) is a nonlinear image processing framework with many applications in filtering, segmentation and classification [19, 37]
In this paper we propose an approach that relies on the consensus combination of several stochastic permutation orderings
The starting vertex of a path being taken at random, many different permutation orderings can be obtained and we propose to build a consensus ordering from several permutation orderings

Summary

Introduction

Mathematical morphology (MM) is a nonlinear image processing framework with many applications in filtering, segmentation and classification [19, 37]. A similar approach is explored in [11] with a top-down hierarchical clustering, but it is much more computationally demanding and only a subset of the set of vectors T is considered The advantage of these approaches is that the ordering can take into account both spatial and spectral information by means of the induced tree, whatever its construction is top-down or bottom-up. We study the latter and show their interest for processing multivariate images with colors or patches.

Hamiltonian path

Study of the induced orders

Study of the graph influence

Consensus ordering

Instant-Runoff aggregation

Borda count

Weighted Borda count

How many orders to combine ?

Computation complexity

Results

Conclusion