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

Antoine Bottenmuller, Florent Magaud, Arnaud Demortière, Etienne Decencière, Petr Dokladal

TL;DR

The paper addresses the challenge of estimating abundance or probability maps in spectral images when the linear unmixing model $Y = M A$ is unsuitable due to few bands or highly correlated spectra. It introduces a density function based on the signed Euclidean distances to polyhedral class frontiers defined by a chosen linear classifier and provides an exact algorithm for computing the minimum-norm point to convex polyhedra, enabling precise distance evaluations. The approach yields superior abundance-map reconstruction on the Samson dataset and delivers strong probability-map performance, while also handling datasets (e.g., Li-ion battery spectral images) incompatible with linear mixing. Together, these contributions enable robust, general-purpose class representation in high-dimensional spectral data, with practical applicability beyond traditional unmixing frameworks.

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.

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

TL;DR

The paper addresses the challenge of estimating abundance or probability maps in spectral images when the linear unmixing model is unsuitable due to few bands or highly correlated spectra. It introduces a density function based on the signed Euclidean distances to polyhedral class frontiers defined by a chosen linear classifier and provides an exact algorithm for computing the minimum-norm point to convex polyhedra, enabling precise distance evaluations. The approach yields superior abundance-map reconstruction on the Samson dataset and delivers strong probability-map performance, while also handling datasets (e.g., Li-ion battery spectral images) incompatible with linear mixing. Together, these contributions enable robust, general-purpose class representation in high-dimensional spectral data, with practical applicability beyond traditional unmixing frameworks.

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.

Paper Structure

This paper contains 19 sections, 6 theorems, 12 equations, 11 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Context
  3. Objectives
  4. Density Functions
  5. Problem with Regular Approaches
  6. Proposed Approach
  7. Distance to Convex Polyhedra: Problem Formulation
  8. Definitions and Mathematical Formulation
  9. Existing Algorithms and their Limits
  10. An Exact Minimum-Norm Point Calculation Process for Convex Polyhedra
  11. Support Hyperplanes and Minimum H-Description
  12. Preliminary Algorithm
  13. Main Algorithm
  14. Comparison with a State-of-the-Art Algorithm
  15. Application to Class Representation
  16. ...and 4 more sections

Key Result

Proposition 1

The minimum $H$-description of $P$ is the family of halfspaces $B_j$ in $\mathcal{B}$ such that $B_j^\complement \cap \left( \bigcap_{i \in I \setminus \{j\}} B_i^{\mathrm{o}} \right) \neq \emptyset$.

Figures (11)

  • Figure 1: The distances to clusters' centers (red in \ref{['fig:centers1']}, white in \ref{['fig:centers2']}) given by the $k$-means algorithm (\ref{['fig:centers1']}) are used to compute density functions (color map in \ref{['fig:centers2']}) using Eq.\ref{['eq:inverse_distance']}: endmembers have lower density values than the center of their corresponding class (\ref{['fig:centers2']}).
  • Figure 2: Grayscale image from the first band of a four-band spectral image of a Lithium-ion battery (\ref{['fig:holes1']}), and probability map computed with Eq.\ref{['eq:inverse_distance']} on three $k$-means centroids (\ref{['fig:holes2']}) to segment three chemical phases (high, medium and low values in \ref{['fig:holes1']}): the density function creates holes in the probability map compared to the expected one (\ref{['fig:holes3']}) ; see green phase.
  • Figure 3: The signed distances to clusters' frontiers given by the $k$-means algorithm (\ref{['fig:frontiers1']}) are used to compute density functions (\ref{['fig:frontiers2']}) using Eq.\ref{['eq:softmax']}: endmembers have higher density values than the center (red in \ref{['fig:centers1']}, white in \ref{['fig:centers2']}) of their corresponding class (\ref{['fig:frontiers2']}).
  • Figure 4: Example of an unbounded polyhedron (\ref{['fig:polyhedron']}) and of a bounded one (\ref{['fig:polytope']}) formed by three closed halfspaces.
  • Figure 5: Example of a polyhedron (\ref{['fig:min_h1']}) as the intersection of five halfspaces, and its minimum $H$-description (\ref{['fig:min_h2']}) where only the three first halfspaces have been preserved.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Definition 1: Polyhedron
  • Definition 2: Minimum-norm point
  • Definition 3: Support Hyperplane
  • Definition 4: Minimum $H$-description
  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Proposition 3
  • Proposition 4
  • Proposition 5