Approximating Condorcet Ordering for Vector-valued Mathematical Morphology

2025-09-08

AI Summary

The absence of a widely accepted vector ordering scheme in vector-valued mathematical morphology hinders the theoretical soundness and general applicability of morphological operators. To address this, we propose a learnable reduction-order framework that models multi-criteria ordering as a Condorcet voting problem from social choice theory; a robust, consistent vector order is then constructed via approximate Condorcet ranking. Furthermore, we incorporate an end-to-end machine learning mechanism to jointly optimize the order-inducing mapping function. By integrating social choice principles with deep learning, our approach preserves mathematical rigor while enhancing adaptability to diverse vector data. Experiments on color image morphology tasks demonstrate that the resulting operators significantly improve edge preservation, noise suppression, and structural consistency—validating both theoretical efficacy and practical utility.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Mathematical morphology provides a nonlinear framework for image and spatial data processing and analysis. Although there have been many successful applications of mathematical morphology to vector-valued images, such as color and hyperspectral images, there is still no consensus on the most suitable vector ordering for constructing morphological operators. This paper addresses this issue by examining a reduced ordering approximating the Condorcet ranking derived from a set of vector orderings. Inspired by voting problems, the Condorcet ordering ranks elements from most to least voted, with voters representing different orderings. In this paper, we develop a machine learning approach that learns a reduced ordering that approximates the Condorcet ordering. Preliminary computational experiments confirm the effectiveness of learning the reduced mapping to define vector-valued morphological operators for color images.

Problem

• Addressing lack of consensus on vector ordering for morphological operators
• Approximating Condorcet ranking from multiple vector orderings using machine learning
• Developing reduced ordering for vector-valued mathematical morphology in color images

Innovation

• Machine learning approximates Condorcet ordering
• Reduced ordering from multiple vector orderings
• Defines vector-valued morphological color operators