Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Tractable Two-Step Linear Mixing Model Solved with Second-Order Optimization for Spectral Unmixing under Variability

Xander Haijen, Bikram Koirala, Xuanwen Tao, Paul Scheunders

TL;DR

The paper tackles spectral unmixing under material variability by introducing the Two-Step Linear Mixing Model (2LMM), which adds a global endmember scaling and a per-pixel scaling to balance modeling richness with computational tractability. It leverages a novel second-order optimization approach (L-BFGS) to solve the mildly nonconvex ALS problem efficiently, outperforming several state-of-the-art methods in synthetic and real datasets while remaining robust to hyperparameter settings. Key contributions include the moderate-complexity model, an ALS-L-BFGS solution with analytical gradients, and extensive benchmarking on challenging scenarios such as blind and topography-induced variability. The approach demonstrates strong reconstruction and abundance estimation, scalability, and practical applicability, while also outlining avenues for future improvements such as adaptive EM updates and nonlinear extensions.

Abstract

In this paper, we propose a Two-Step Linear Mixing Model (2LMM) that bridges the gap between model complexity and computational tractability. The model achieves this by introducing two distinct scaling steps: an endmember scaling step across the image, and another for pixel-wise scaling. We show that this model leads to only a mildly non-convex optimization problem, which we solve with an optimization algorithm that incorporates second-order information. To the authors' knowledge, this work represents the first application of second-order optimization techniques to solve a spectral unmixing problem that models endmember variability. Our method is highly robust, as it requires virtually no hyperparameter tuning and can therefore be used easily and quickly in a wide range of unmixing tasks. We show through extensive experiments on both simulated and real data that the new model is competitive and in some cases superior to the state of the art in unmixing. The model also performs very well in challenging scenarios, such as blind unmixing.

A Tractable Two-Step Linear Mixing Model Solved with Second-Order Optimization for Spectral Unmixing under Variability

TL;DR

The paper tackles spectral unmixing under material variability by introducing the Two-Step Linear Mixing Model (2LMM), which adds a global endmember scaling and a per-pixel scaling to balance modeling richness with computational tractability. It leverages a novel second-order optimization approach (L-BFGS) to solve the mildly nonconvex ALS problem efficiently, outperforming several state-of-the-art methods in synthetic and real datasets while remaining robust to hyperparameter settings. Key contributions include the moderate-complexity model, an ALS-L-BFGS solution with analytical gradients, and extensive benchmarking on challenging scenarios such as blind and topography-induced variability. The approach demonstrates strong reconstruction and abundance estimation, scalability, and practical applicability, while also outlining avenues for future improvements such as adaptive EM updates and nonlinear extensions.

Abstract

In this paper, we propose a Two-Step Linear Mixing Model (2LMM) that bridges the gap between model complexity and computational tractability. The model achieves this by introducing two distinct scaling steps: an endmember scaling step across the image, and another for pixel-wise scaling. We show that this model leads to only a mildly non-convex optimization problem, which we solve with an optimization algorithm that incorporates second-order information. To the authors' knowledge, this work represents the first application of second-order optimization techniques to solve a spectral unmixing problem that models endmember variability. Our method is highly robust, as it requires virtually no hyperparameter tuning and can therefore be used easily and quickly in a wide range of unmixing tasks. We show through extensive experiments on both simulated and real data that the new model is competitive and in some cases superior to the state of the art in unmixing. The model also performs very well in challenging scenarios, such as blind unmixing.

Paper Structure

This paper contains 42 sections, 1 theorem, 27 equations, 14 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Unmixing under spectral variability
  3. Methods that use spectral libraries
  4. Parametric physics-based models
  5. Parametric physically motivated models
  6. Models jointly estimating EMs and abundances
  7. Machine Learning methods
  8. Endmember extraction algorithms
  9. Contributions
  10. Outline
  11. Related work
  12. Notation
  13. Unmixing with the LMM
  14. Unmixing with the SLMM
  15. Unmixing with the ELMM
  16. ...and 27 more sections

Key Result

Theorem 1

If both ground truth factors $\Tilde{\mathbf{E}}$ and $\Tilde{\mathbf{A}}$ are sufficiently scattered, then any extracted solution $\widehat{\mathbf{E}}$ must satisfy $\widehat{\mathbf{E}} = \Tilde{\mathbf{E}} \bm{\Pi} \mathbf{D}$, where $\bm{\Pi}$ is a permutation matrix and $\mathbf{D}$ is a full-

Figures (14)

  • Figure 1: An illustration of the separability (left) and sufficiently scattered (right) condition.
  • Figure 2: A graphical representation of the 2LMM model assumption. First scaling step: The reference EMs (blue lines) are scaled independently (red lines). Mixing: The scaled EMs are mixed to form the unscaled pixels in the image. Second scaling step: Each pixel is scaled independently to form the final image.
  • Figure 3: A graphical representation of the variability parameter count of several physics-inspired models. (*) the EM perturbation factors of the ELMM are not included since they contribute very little to model complexity.
  • Figure 4: The EMs used for generating the synthetic data: asphalt (gds367), brick (gds350), and cardboard (gds371).
  • Figure 5: Abundance maps generated with Gaussian Random Fields for a synthetic image with three EMs.
  • ...and 9 more figures

Theorems & Definitions (2)

  • Definition 1
  • Theorem 1