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Hyperspectral unmixing with spectral variability using a perturbed linear mixing model

Pierre-Antoine Thouvenin, Nicolas Dobigeon, Jean-Yves Tourneret

arXiv:1502.01260v2stat.MEphysics.data-an

TL;DR

Hyperspectral unmixing can be distorted when endmember signatures vary across an image. This paper introduces a perturbed linear mixing model estimated with ADMM, and reports favorable comparisons with state-of-the-art methods while estimating variability relevantly.

  • Problem

    Endmember variability within hyperspectral images is not explicitly represented by classical linear unmixing, motivating estimation of reference signatures and their spatial-spectral deviations.

  • Method

    The method extends the linear mixing model with pixel-dependent additive perturbations and estimates its parameters through alternating minimizations using ADMM.

  • Results

    On synthetic and real data, the method compares favorably with state-of-the-art approaches while providing relevant endmember variability estimates.

  • Takeaways & Limitations

    The approach can estimate variability for each endmember in each pixel and remains robust when pure pixels are absent.

Abstract

from arXiv · show

Given a mixed hyperspectral data set, linear unmixing aims at estimating the reference spectral signatures composing the data - referred to as endmembers - their abundance fractions and their number. In practice, the identified endmembers can vary spectrally within a given image and can thus be construed as variable instances of reference endmembers. Ignoring this variability induces estimation errors that are propagated into the unmixing procedure. To address this issue, endmember variability estimation consists of estimating the reference spectral signatures from which the estimated endmembers have been derived as well as their variability with respect to these references. This paper introduces a new linear mixing model that explicitly accounts for spatial and spectral endmember variabilities. The parameters of this model can be estimated using an optimization algorithm based on the alternating direction method of multipliers. The performance of the proposed unmixing method is evaluated on synthetic and real data. A comparison with state-of-the-art algorithms designed to model and estimate endmember variability allows the interest of the proposed unmixing solution to be appreciated.

I. INTRODUCTION … B. Problem formulation

The paper addresses hyperspectral unmixing under spatial and spectral endmember variability by introducing a perturbed linear mixing model and a constrained optimization formulation. The approach estimates reference endmembers, abundances, and spatially varying additive perturbations while assuming the number of endmembers is known or externally estimated.

  • I. INTRODUCTION: Hyperspectral images provide hundreds of contiguous spectral bands but have lower spatial resolution, and endmembers may vary across images because of acquisition conditions.This variability can cause significant estimation errors to propagate through the unmixing process.
  • II. PROBLEM STATEMENT: The proposed contribution is a linear unmixing model that explicitly accounts for spatial and spectral endmember variabilities.The method treats identified endmembers as variable instances of reference endmembers.
  • I. INTRODUCTION: The number of endmembers K is assumed known or estimated beforehand because jointly determining K and endmember variability is challenging.The choice of K strongly affects scene representation, endmember identification, and abundance estimation.
  • A. Perturbed linear mixing model (PLMM): Under the PLMM, each pixel combines endmembers with abundances, spatially varying additive perturbations, and zero-mean white Gaussian noise.The model reduces to the classical LMM when variability is absent and uses physically motivated non-negativity and sum-to-one constraints.
  • A. Perturbed linear mixing model (PLMM): The PLMM explicitly represents variability as additive perturbations affecting each endmember and thereby captures deviations from the linear mixing model.This formulation is presented as simple and flexible when no specific prior knowledge about variability is available.
  • B. Problem formulation: Parameter estimation is formulated as a constrained optimization problem minimizing a Frobenius-norm data-fitting term between acquisitions and reconstructed data.Because the problem is ill-posed, penalization terms are added for abundances, endmembers, and perturbations, with α, β, and γ controlling tradeoffs.
  • B. Problem formulation: Separable non-negative differentiable convex penalties decompose the optimization into simpler subproblems.The abundance penalty promotes spatially smooth abundances, and the decomposition supports the algorithmic treatment described later.

1) Abundance penalization: … III. AN ADMM-BASED ALGORITHM

The method regularizes abundance differences, endmember geometry and reference proximity, and spectral variability through differentiable penalties. It then alternately minimizes the nonconvex objective over abundances, endmembers and variability using ADMM-solvable subproblems.

  • 1) Abundance penalization:: Abundance regularization uses a matrix H that computes differences between each pixel’s abundances and those of its four nearest neighbors.The resulting abundance penalty expression is given in Appendix A.
  • 2) Endmember penalization:: Endmember penalization can constrain simplex size through occupied volume or approximate it using mutual endmember distances.These criteria penalize the volume of the (K −1)-simplex formed by the endmembers.
  • 2) Endmember penalization:: A separate endmember penalty measures distances between estimated endmembers and reference spectral signatures M0.The distance-based formulation is introduced for the reference signatures.
  • 2) Endmember penalization:: The simplex-volume formulation is based on projecting M onto the space spanned by its K −1 principal components.Under pure-pixel and linear-mixture assumptions, the data lie inside a simplex whose vertices are the endmembers.
  • 3) Variability penalization:: Variability penalization limits spectral perturbation energy by penalizing the norms of the perturbation matrices dMn.The proposed penalty is differentiable with respect to dMn.
  • 3) Variability penalization:: No additional spatial regularization is imposed on dM because specific information about variability’s spatial distribution is unavailable.A spatial penalty such as group-Lasso ℓ2,1 can be added when needed to promote spatial sparsity.
  • III. AN ADMM-BASED ALGORITHM: The nonconvex objective is minimized alternately over A, M and dM until a stopping criterion is met.The penalty assumptions split the problem into strictly convex differentiable subproblems, which ADMM solves through separable closed-form steps.

A. ADMM: general principle

ADMM solves constrained optimization by alternating minimization over the variables of an augmented Lagrangian, with iterations monitored through primal and dual residuals. Its penalty parameter ρ may be adjusted finitely many times without altering convergence.

  • General principle: ADMM combines augmented Lagrangian and dual decomposition methods to solve constrained optimization problems through successive minimization with respect to each variable.The general formulation introduces functions f and g, matrices A and B, and a linear constraint.
  • General principle: The algorithm alternates minimization of the scaled augmented Lagrangian with respect to x, z, and u.The scaled augmented Lagrangian uses a penalty parameter ρ > 0, and the dual update is u(k+1) = u(k) + Ax(k+1) + Bz(k+1) −c.
  • Stopping criterion: ADMM iterations stop when the primal and dual residuals satisfy a classical stopping criterion.The residuals are evaluated at each iteration to determine whether the procedure should continue.
  • Parameter adjustment: The penalty parameter ρ can be adjusted finitely many times without altering ADMM convergence.The adjustment follows a rule described in the cited reference and preserves convergence under the finite-adjustment condition.

B. Optimization with respect to A · C. Optimization with respect to M · D. Optimization with respect to dM

The optimization alternates updates for abundances A, reference signatures M, and variability dM using splitting variables and scaled augmented Lagrangians. These procedures are performed sequentially until convergence, after which the unmixing strategy is evaluated on synthetic and real data.

  • B. Optimization with respect to A: The A-update reformulates optimization of J under constraint (3) as subproblems using splitting variables w(A).The resulting scaled augmented Lagrangian defines step a of Algorithm 1, detailed in Algorithm 2.
  • B. Optimization with respect to A: Solutions to the A-update subproblems are provided in Appendix B.
  • C. Optimization with respect to M: The M-update similarly reformulates optimization of J under constraint (3), introducing splitting variables W(M) and a scaled augmented Lagrangian.The formulation uses the ℓth row of M in its subproblem definition.
  • C. Optimization with respect to M: The M-update constitutes step b of Algorithm 1 and is similar to Algorithm 2.Its optimization solutions depend on the selected endmember penalizing function Ψ.
  • D. Optimization with respect to dM: The dM-update reformulates optimization under constraint (3) with splitting variables W(dM) and the relation M_n + dM_n.This formulation yields the scaled augmented Lagrangian for the dM subproblems.
  • D. Optimization with respect to dM: The dM procedure is step c of Algorithm 1 and is similar to Algorithm 2, with solutions given in Appendix B.The A, M, and dM optimization procedures run sequentially until the stopping criterion is satisfied.
  • D. Optimization with respect to dM: The resulting unmixing strategy is evaluated in subsequent experiments on synthetic and real data.

E. Convergence and computational cost · IV. EXPERIMENT WITH SYNTHETIC DATA · A. State-of-the-art methods

The proposed BCD/ADMM scheme converges under exact subproblem minimization, with computational cost dominated by matrix products. Synthetic experiments test endmember, abundance, perturbation, and reconstruction performance against classical and variability-aware unmixing methods.

  • E. Convergence and computational cost: Exact minimization of each subproblem guarantees that the alternating BCD scheme converges to a stationary point of objective function J.The subproblems in Sections III-B, III-C, and III-D are strongly convex, so ADMM converges to each subproblem’s unique minimum under a constant augmented-Lagrangian parameter.
  • E. Convergence and computational cost: O(LK^2N) is the overall computational cost, dominated by matrix products in the ADMM updates.
  • IV. EXPERIMENT WITH SYNTHETIC DATA: Four synthetic 128 × 64 images with 413 bands test mixtures of K ∈{3, 6}, with or without pure pixels, at SNR = 30dB.The no-pure-pixel setting is included as a challenging scenario, and additive white Gaussian noise corrupts the mixtures.
  • IV. EXPERIMENT WITH SYNTHETIC DATA: cvar = 0.1 in the upper image half and cvar = 0.25 in the lower half generate spatially varying spectral perturbations.Different affine functions are used for different endmembers and pixels, with corresponding perturbed spectra illustrated in Fig. 2.
  • A. State-of-the-art methods: The comparison includes classical VCA/FCLS and SISAL/FCLS methods without variability, plus AEB and FDNS methods that account for variability.These methods are described as the principal comparison algorithms for the proposed experiments.
  • A. State-of-the-art methods: VCA/FCLS extracts endmembers with VCA and estimates abundances using FCLS, while SISAL/FCLS uses SISAL initialized by VCA and stops at tolerance 10−2.Both classical methods assume no variability; the proposed method also uses VCA/FCLS for initialization, although that assumption can be problematic without pure pixels.
  • A. State-of-the-art methods: Performance is assessed through endmember estimation, abundance and perturbation GMSEs, and reconstruction error RE.The reconstructed-pixel matrix bY is formed from parameters estimated by the algorithm, while variability spatial distribution is visualized through perturbation energy.

B. Results … VI. CONCLUSION AND FUTURE WORK

The proposed ADMM-based unmixing method models spatial and spectral endmember variability, remains effective without pure pixels, and generally improves abundance and reconstruction estimates over state-of-the-art methods. Experiments on synthetic and AVIRIS data demonstrate variability detection, while computational cost and performance degradation with more endmembers remain limitations.

  • B. Results: The proposed method is robust to the absence of pure pixels and provides competitive aSAM while estimating variability for each endmember in each pixel.For most simulation scenarios, abundance MSEs and REs are lower than those of state-of-the-art methods.
  • B. Results: Increasing the number of endmembers reduces estimation performance, as expected from initializing with VCA/FCLS.The proposed method is computationally more expensive than existing algorithms, although smoothness penalization on abundances is particularly appropriate in this experiment.
  • B. Results: The method captures spatially varying perturbation intensity consistently with the different variability coefficients applied across the scene.The reported variability examples cover three endmembers and show differing intensities between the upper and lower scene regions.
  • V. EXPERIMENT WITH REAL DATA: The real-data evaluation uses AVIRIS Moffett and Cuprite scenes, each retaining 189 exploitable spectral bands after removing unsuitable bands.Moffett is a 50 × 50 scene containing a lake and coastal area, while Cuprite is a 190 × 250 image with reference abundance maps from prior work.
  • A. Description of the datasets: Real-data parameters follow the synthetic-data settings, with stopping at a successive-objective relative difference below 10^-2 and α, β, γ selected by cross-validation.The stopping threshold was chosen as a compromise between estimation accuracy and computational cost.
  • B. Results: For Moffett, detected variability is more significant in the coastal area, where linear mixing is inadequate and potential nonlinearities are interpreted as variability.Variability peaks also indicate that several corrupted spectral bands may remain in the data.
  • B. Results: For Cuprite, signatures identified as different VCA endmembers for K = 14 are interpreted as multiple instances of single endmembers with K = 10.The algorithm also captures significant variability in pixels containing many different detected endmembers.
  • VI. CONCLUSION AND FUTURE WORK: The paper introduces a linear mixing model with an additive spatially varying perturbation matrix and solves its regularized unmixing problem through alternating minimizations using ADMM.The approach is proposed as a complementary tool for small hyperspectral images expected to exhibit non-negligible variability, while reducing computational complexity remains future work.

APPENDIX A CONSTRAINTS AND PENALIZATION TERMS … 1) Distance between the endmembers and reference signatures:

Appendix A specifies penalization terms for abundance smoothness and for distances between endmembers and reference signatures. The abundance smoothness uses neighbor-difference matrices, while the endmember term is stated as a distance expression.

  • A. Abundance penalization: spatial smoothness: Abundance smoothness is expressed in matrix form.
  • A. Abundance penalization: spatial smoothness: The matrix H computes differences between each pixel’s abundances and those of its four neighbors.
  • A. Abundance penalization: spatial smoothness: For h = 1, . . . , H, the construction introduces a matrix H_h.
  • A. Abundance penalization: spatial smoothness: The displayed H_h structure contains entries including 1, −1, and 0.
  • A. Abundance penalization: spatial smoothness: The abundance-difference operators are assembled as H← = Diag(H_1, . . . , H_H) and H→ = Diag(˜H_1, . . . , ˜H_H).
  • A. Abundance penalization: spatial smoothness: Additional displayed matrix entries include −1, 0, and 1 in the operator construction.
  • B. Endmember penalization: The appendix notes that only specified terms appear in the displayed expression.
  • 1) Distance between the endmembers and reference signatures:: The distance between the endmembers and the available reference signatures is defined by a separate penalization expression.

2) Mutual distance between the endmembers: · 3) Volume and endmember positivity constraint:

The section formulates mutual distance between endmembers and develops volume penalization through T. It also seeks conditions equivalent to positivity of M, including the variability-adjusted matrix M+dMn.

  • 2) Mutual distance between the endmembers:: The mutual distance between different endmembers is expressed using the endmember vectors.
  • 2) Mutual distance between the endmembers:: The derivation introduces G1 · · · GK and identifies ek as the kth canonical-basis vector of R^K.
  • 3) Volume and endmember positivity constraint:: The volume penalization is expressed using T, motivating a condition equivalent to positivity of M.
  • 3) Volume and endmember positivity constraint:: The analysis first considers the general volume-penalization expression with respect to e_tk.
  • 3) Volume and endmember positivity constraint:: For variable endmembers, the positivity condition is extended from M to M+dM_n.
  • 3) Volume and endmember positivity constraint:: The volume-penalization expression is developed along its ith row using previous developments.
  • 3) Volume and endmember positivity constraint:: The positivity analysis introduces Y = UYproj. + ¯Y1 and M = UT + ¯Y2, with replicated mean-vector matrices ¯Y1 and ¯Y2.
  • 3) Volume and endmember positivity constraint:: The derivation subsequently introduces two sets of integers for the positivity analysis.

4) Positivity constraint on M and dM: … A. Resolution with respect to A

The paper derives positivity conditions that jointly constrain M and its variable instances, then presents optimization subproblem details for variability penalization and resolution with respect to A.

  • 4) Positivity constraint on M and dM:: The positivity constraint must hold simultaneously for M and M_n = M + dM_n.The derivation therefore seeks a condition analogous to equation (38).
  • 4) Positivity constraint on M and dM:: M_n is represented in the PCA subspace as M_n = UT_n + ¯Y_2.
  • 4) Positivity constraint on M and dM:: The variability coordinates dT_n are related to dM_n through the PCA-space transformation involving V, M, and ¯Y_2.
  • 4) Positivity constraint on M and dM:: The positivity constraint is reformulated using functions g_k defined for this purpose.
  • 4) Positivity constraint on M and dM:: The final positivity condition applies to the sum of the endmembers and their variability and is expressed through g_k(e_tk) ⪰ 0 for all k = 1, ..., K − 1.
  • C. Variability penalization: The variability penalization subsection introduces a variability energy penalty.
  • A. Resolution with respect to A: For the A subproblem, substituting equation (32) into the scaled augmented Lagrangian produces terms involving (M + dM_n)^T(M + dM_n) and μ(A).
  • A. Resolution with respect to A: The vector λ(A) contributes through its first K elements, while the maximum operator is applied term-wise.Without penalization, the solution is obtained by setting α = 0 in the preceding equations.

B. Resolution with respect to M … 3) Volume penalization:

The section derives scaled augmented-Lagrangian formulations for endmember-distance penalties and adapts the optimization subproblems for volume penalization. It also introduces splitting variables and associated matrix definitions for the volume-penalized formulation.

  • 1) Distance between the endmembers and reference spectral signatures:: Using (34), the scaled augmented Lagrangian (26) is formulated for the distance between endmembers and reference spectral signatures.
  • 1) Distance between the endmembers and reference spectral signatures:: Without endmember penalization, the corresponding solution is obtained by setting β = 0 in the previous equation.
  • 2) Mutual distance between the endmembers:: The mutual-distance formulation includes ℓ given by (50).
  • 3) Volume penalization:: Because the volume penalty is expressed with respect to T, the endmember-related optimization subproblems must be rewritten accordingly.
  • 3) Volume penalization:: For the volume-penalized formulation, the terms depending on T are isolated and subproblems are specified for k = 1, . . . , K −1.
  • 3) Volume penalization:: Splitting variables W(T) are introduced, and the scaled augmented Lagrangian uses Z = V ¯Y2 for p = 1, . . . , K.

C. Resolution with respect to dM

The section presents abundance results for the Cuprite scene and visualizes spatial endmember variability using an energy measure.

  • C. Resolution with respect to dM: For the Cuprite scene, ssvcaBCD/ADMM abundance results are identified through visual comparison with results obtained in [28].
  • C. Resolution with respect to dM: Spatial variability is shown for endmembers with the most significant variability levels, using energy for visualization.The energy is defined for the kth endmember in the nth pixel as 1 √ L ∥dmn,k∥2.
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