Euclidean Distance to Convex Polyhedra and Application to Class Representation in Spectral Images

2025-03-26

Proceedings of the 14th International Conference on Pattern Recognition Applications and Methods

AI Summary

To address the failure of linear unmixing in hyperspectral imaging caused by insufficient spectral bands or high inter-band correlation, this paper proposes a novel abundance map reconstruction paradigm that does not rely on spectral independence or the number of bands. Methodologically, we first establish a rigorous mathematical characterization of the distance from a convex polytope to Euclidean space and derive an analytical solution for the minimum-norm point within the polytope. Subsequently, we embed arbitrary linear classifiers as implicit spatial density functions, unifying nonlinear class representation with geometry-driven abundance modeling. The approach integrates convex optimization, computational geometry, and distance field modeling. Experimental results on the Samson dataset demonstrate superior accuracy over state-of-the-art methods. Furthermore, validation on real-world hyperspectral images of lithium-ion batteries confirms the method’s robustness and generalizability under low-dimensional, highly correlated spectral conditions.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

With the aim of estimating the abundance map from observations only, linear unmixing approaches are not always suitable to spectral images, especially when the number of bands is too small or when the spectra of the observed data are too correlated. To address this issue in the general case, we present a novel approach which provides an adapted spatial density function based on any arbitrary linear classifier. A robust mathematical formulation for computing the Euclidean distance to polyhedral sets is presented, along with an efficient algorithm that provides the exact minimum-norm point in a polyhedron. An empirical evaluation on the widely-used Samson hyperspectral dataset demonstrates that the proposed method surpasses state-of-the-art approaches in reconstructing abundance maps. Furthermore, its application to spectral images of a Lithium-ion battery, incompatible with linear unmixing models, validates the method's generality and effectiveness.

Problem

• Estimating abundance maps from spectral images
• Computing Euclidean distance to convex polyhedra
• Improving class representation in hyperspectral data

Innovation

• Exact minimum-norm point in polyhedron algorithm
• Euclidean distance to polyhedral sets formulation
• Spatial density function from linear classifier