Extended Linear Mixing Model (ELMM)

• Extended Linear Mixing Model (ELMM) is a hyperspectral unmixing technique that captures illumination-induced spectral variability via per-endmember, per-pixel scaling factors.
• It leverages a simplified Hapke radiative transfer model to provide a physically interpretable correction to the classic Linear Mixing Model while preserving convexity.
• The framework uses regularized least-squares with spatial smoothness constraints, demonstrating improved abundance estimation on both synthetic and real hyperspectral datasets.

The Extended Linear Mixing Model (ELMM) is a hyperspectral unmixing framework that explicitly models endmember variability, particularly that induced by illumination changes. The ELMM extends the classic Linear Mixing Model (LMM) by introducing positive, per-endmember, per-pixel scaling factors that capture variability in observed spectra due to geometric effects, while preserving the convex-geometry properties of the LMM. The formal justification and practical deployment of ELMM arise from a physically motivated derivation based on simplifications of Hapke's radiative transfer model, enabling robust unmixing under realistic remote sensing conditions (Drumetz et al., 2019, Imbiriba et al., 2017).

1. Physical Motivation and Derivation from the Hapke Model

The ELMM is grounded in the physics of light scattering and reflectance. The starting point is the Hapke bidirectional reflectance model, which describes how incident light is scattered by a rough particulate surface. This full model incorporates parameters for the single-scattering albedo, incidence and emergence angles, surface roughness, shadowing, opposition effects, and phase functions. However, such completeness renders it intractable for practical spectral unmixing.

To simplify, several assumptions are introduced:

• Smooth Macroscale Surfaces: Neglects macroscopic roughness, such that $S(\mu,\mu_0,\phi)=1$, $\mu_e=\mu$, and $\mu_{0e}=\mu_0$.
• Lambertian Scattering: Assumes isotropic scattering with $P(g)=1$ and $B(g)\approx 0$.
• First Order Taylor Linearization: For small single-scattering albedo $\omega$, reflectance is approximately linear in $\omega$.

This sequence leads to an approximate reflectance relationship that depends only on geometry (incidence/emergence angles) and albedo. The result is that a reference endmember spectrum under standard geometry, $\mathbf{s}_{0p}$, is related to its version in any pixel $n$ by a strictly positive geometric scaling factor:

$\mathbf{s}_{pn} \approx \psi_{pn}\,\mathbf{s}_{0p}$

where $\mu_e=\mu$0 depends only on the local geometry in pixel $\mu_e=\mu$1. This scaling captures illumination-induced spectral variability and justifies the formulation of the ELMM as a tractable, physically interpretable correction to the LMM (Drumetz et al., 2019).

2. Model Specification and Mathematical Formalism

The standard LMM represents each observed hyperspectral pixel $\mu_e=\mu$2 as a convex combination of $\mu_e=\mu$3 fixed endmember signatures, corrupted by additive noise:

$\mu_e=\mu$4

subject to nonnegativity and sum-to-one constraints on the abundances $\mu_e=\mu$5.

The ELMM generalizes this by introducing a strictly positive, per-endmember, per-pixel scaling factor $\mu_e=\mu$6:

$\mu_e=\mu$7

In matrix form, aggregating all pixels:

$\mu_e=\mu$8

where:

• $\mu_e=\mu$9: observed spectra for $\mu_{0e}=\mu_0$0 pixels,
• $\mu_{0e}=\mu_0$1: matrix of reference endmembers,
• $\mu_{0e}=\mu_0$2: pixel abundance matrix,
• $\mu_{0e}=\mu_0$3: scaling factor matrix,
• $\mu_{0e}=\mu_0$4: Hadamard (elementwise) product,
• $\mu_{0e}=\mu_0$5: noise matrix.

Each $\mu_{0e}=\mu_0$6 captures the effect of geometry and illumination on the appearance of material $\mu_{0e}=\mu_0$7 in pixel $\mu_{0e}=\mu_0$8. This formulation preserves the LMM's convexity and interpretability, but enables robust modeling of illumination-induced variability (Drumetz et al., 2019, Imbiriba et al., 2017).

3. Statistical Estimation and Regularization

The estimation of abundances and scaling factors within ELMM is typically cast as a joint regularized least-squares optimization. For each pixel $\mu_{0e}=\mu_0$9, denote $P(g)=1$0, and abundances $P(g)=1$1. The estimation problem is:

$P(g)=1$2

where

• $P(g)=1$3: fixed reference endmembers,
• $P(g)=1$4: abundance matrix,
• $P(g)=1$5: collection of per-pixel scaling factors,
• $P(g)=1$6, $P(g)=1$7: operators computing horizontal/vertical spatial gradients (promoting spatial smoothness),
• $P(g)=1$8, $P(g)=1$9: regularization weights,
• $B(g)\approx 0$0: indicator enforcing $B(g)\approx 0$1,
• $B(g)\approx 0$2: Lagrange multipliers for sum-to-one constraint.

This framework enables the simultaneous estimation of abundances and scaling factors, regularized to promote spatial coherence in both variables (Imbiriba et al., 2017).

4. Interpretation, Validity, and Limitations

The ELMM's per-endmember, per-pixel scaling factors $B(g)\approx 0$3 provide a geometric interpretation: the reference endmember acts as a spectral ray in $B(g)\approx 0$4-dimensional space, and the scaling encodes a pixel-specific stretch along that ray. Thus, the observed endmember in every pixel is restricted to lie along the nonnegative scaling of a fixed spectral reference.

The Taylor (linear) approximation from which ELMM is derived is principally valid for:

• Single-scattering albedo $B(g)\approx 0$5
• High emergence/incidence angles, where the reflectance–albedo relationship is near linear

The model neglects:

• Directional (non-Lambertian) scattering effects
• Opposition surge and non-Lambertian lobes
• Macroscopic roughness (shadowing effects are neglected)
• Higher-order multiple scattering nonlinearities

The ELMM performs optimally for materials with moderate albedo and for geometry where specular or strong backscattering is absent. Outside these regimes, band-specific or nonlinear variability may require further model generalization (see Section 6) (Drumetz et al., 2019, Imbiriba et al., 2017).

5. Algorithmic and Implementation Considerations

Efficient implementation of ELMM-based unmixing leverages the convexity of the problem when optimizing subsets of variables, supporting block-coordinate descent and similar frameworks. Key steps include:

• Reference endmember extraction, typically via algorithms such as VCA or using library spectra.
• Initialization using fully constrained least-squares (FCLS) for abundances and uniform scaling factors ($B(g)\approx 0$6).
• Regularization parameter selection, often via cross-validation.
• Use of periodic boundary conditions on abundance and scaling maps to exploit FFT diagonalization in spatial regularization terms.

Subproblems such as abundance update with spatial regularization are typically solved using ADMM, respecting nonnegativity and sum-to-one constraints (Imbiriba et al., 2017).

6. Extensions: Generalized Linear Mixing Model (GLMM) and Beyond

The ELMM restricts endmember variability to uniform stretching of each reference spectrum. To model more general (e.g., band-specific, nonlinear) variability, the Generalized Linear Mixing Model (GLMM) extends the ELMM by replacing each scalar scaling factor with a full, nonnegative, wavelength-dependent scaling matrix $B(g)\approx 0$7 per pixel:

$B(g)\approx 0$8

This formulation allows for arbitrary, band-wise distortions of the endmember spectra within each pixel. Joint estimation of abundances, per-pixel effective endmembers, and variability factors is accomplished via block-coordinate algorithms. Empirically, GLMM achieves lower abundance and reconstruction errors in scenarios exhibiting spectral variability beyond simple intensity scaling, at moderate additional computational cost (Imbiriba et al., 2017).

7. Empirical Performance and Practical Impact

On both synthetic and real hyperspectral datasets, ELMM outperforms traditional LMM in abundance estimation accuracy and produces abundance maps that are more spatially coherent in the presence of illumination variability. The GLMM further improves fit and estimation in cases where endmember variability is band-dependent. Experimental results on datasets such as the DC0 and DC1 synthetic benchmarks and Houston urban imagery demonstrate the practical value of scaling-based variability models, with the GLMM reducing reconstruction errors by up to an order of magnitude relative to the ELMM, and abundance RMSEs by a significant margin (Imbiriba et al., 2017). This suggests that the ELMM and its generalizations offer a rigorous, physically interpretable, and computationally tractable framework for modern hyperspectral unmixing tasks.