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An Augmented Linear Mixing Model to Address Spectral Variability for Hyperspectral Unmixing

Danfeng Hong, Naoto Yokoya, Jocelyn Chanussot, Xiao Xiang Zhu

arXiv:1810.12000v1cs.CV

TL;DR

Spectral variability limits accurate abundance estimation under the classical LMM. The paper proposes ALMM, which separately models scaling and other variability using endmember and learned spectral variability dictionaries; experiments on synthetic and real datasets demonstrate superiority over previous state-of-the-art methods.

  • Problem

    Spectral variability makes it difficult for the LMM to accurately estimate abundance maps in hyperspectral unmixing.

  • Method

    ALMM models scaling factors with the endmember dictionary and other spectral variability with a learned low-coherent spectral variability dictionary.

  • Results

    Experiments on synthetic and real datasets demonstrate the superiority and effectiveness of ALMM compared with previous state-of-the-art methods.

  • Takeaways & Limitations

    ALMM simultaneously handles intra-class and inter-class spectral variability while estimating abundance maps.

Abstract

from arXiv · show

Hyperspectral imagery collected from airborne or satellite sources inevitably suffers from spectral variability, making it difficult for spectral unmixing to accurately estimate abundance maps. The classical unmixing model, the linear mixing model (LMM), generally fails to handle this sticky issue effectively. To this end, we propose a novel spectral mixture model, called the augmented linear mixing model (ALMM), to address spectral variability by applying a data-driven learning strategy in inverse problems of hyperspectral unmixing. The proposed approach models the main spectral variability (i.e., scaling factors) generated by variations in illumination or typography separately by means of the endmember dictionary. It then models other spectral variabilities caused by environmental conditions (e.g., local temperature and humidity, atmospheric effects) and instrumental configurations (e.g., sensor noise), as well as material nonlinear mixing effects, by introducing a spectral variability dictionary. To effectively run the data-driven learning strategy, we also propose a reasonable prior knowledge for the spectral variability dictionary, whose atoms are assumed to be low-coherent with spectral signatures of endmembers, which leads to a well-known low coherence dictionary learning problem. Thus, a dictionary learning technique is embedded in the framework of spectral unmixing so that the algorithm can learn the spectral variability dictionary and estimate the abundance maps simultaneously. Extensive experiments on synthetic and real datasets are performed to demonstrate the superiority and effectiveness of the proposed method in comparison with previous state-of-the-art methods.

I. INTRODUCTION

Hyperspectral unmixing is hindered by spectral variability that the LMM and its extensions model incompletely. The paper proposes ALMM, which separates scaling variability from other variability and learns the latter through a low-coherent spectral variability dictionary.

  • Spectral variability from illumination, topography, atmospheric effects, and material properties causes significant errors in hyperspectral abundance estimation.
  • ALMM models scaling factors through the endmember dictionary and represents remaining spectral variability with an additional dictionary.
  • The method imposes a low-coherence prior between spectral variability atoms and endmember signatures, enabling data-driven dictionary learning within spectral unmixing.
  • The proposed optimization uses alternating direction method of multipliers to jointly learn the spectral variability dictionary and estimate abundances.
  • The LMM represents observations as Y = AX + R, where A contains endmembers, X contains abundance vectors, and R contains additive noise and other errors.
  • The LMM models mixed pixels linearly but is limited when spectral signatures vary, while ELMM mainly addresses scaling factors and PLMM uses overly general perturbations.

C. Perturbed Linear Mixing Model

The ALMM extends perturbed and scaled mixing models by separately representing pixel-wise scaling and residual spectral variability. It introduces the spectral variability dictionary E and estimates the model variables under regularized constraints.

  • PLMM models spectral variability through additive endmember perturbations, while ADMM is used to estimate abundances.
  • Existing models either cannot effectively handle spectral variability or address only scaling factors, while generalized perturbation models impose restrictive assumptions.
  • ALMM combines shared scaling factors with an additional spectral variability term to model effects unexplained by scaled endmembers.
  • The spectral variability dictionary E contains basis vectors, with coefficient matrix B specifying their contributions across pixels.
  • The ALMM optimization estimates X, S, E, and B while treating the endmember dictionary A as given.

1) Abundance Regularization Φ(X):

The abundance regularization promotes sparse material representations, while the broader ALMM formulation regularizes spectral variability using prior assumptions and physical constraints. These design choices support separation of abundance effects from variability effects.

  • Abundance Regularization Φ(X):: The abundance penalty uses the ℓ1,1 norm to promote sparsity because each hyperspectral pixel usually contains a limited number of materials.
  • Spectral Variability Dictionary Regularization Υ(E):: The spectral variability dictionary E is regularized with a Frobenius norm to support generalization across diverse variability sources.
  • Spectral Variability Dictionary Regularization Υ(E):: E is constrained to be low-coherent with the endmember dictionary A, separating variability components from endmember signatures.
  • Spectral Variability Dictionary Regularization Υ(E):: An additional orthogonality prior encourages basis vectors in E to represent diverse potential spectral variabilities.
  • Physical Meaning of E: Without E, residual intra-class variability may be absorbed by A and produce inaccurate abundance maps; ALMM instead calibrates class-specific variability into a generalized representation.

D. ADMM-based Optimization Algorithm

The optimization alternates between ALMM-based spectral unmixing and spectral variability dictionary learning. ADMM handles the constrained multi-variable subproblems, and convergence is evaluated across three datasets.

  • ADMM-based Optimization Algorithm: The ALMM framework consists of spectral unmixing and spectral variability dictionary learning components.
  • ALMM-based Spectral Unmixing: With A and E fixed, spectral unmixing estimates abundances and scaling variables under non-negativity constraints.
  • ALMM-based Spectral Variability Dictionary Learning: Algorithm 2 jointly updates X, E, B, and S when E is unknown, using ADMM for the resulting multi-variable optimization problem.
  • ALMM-based Spectral Unmixing: The pixel-wise unmixing problem is formulated as a constrained bilinear regression problem and solved using ADMM with auxiliary variables.
  • Convergence Analysis: Convergence is experimentally visualized by recording objective-function values for ALMM-based unmixing and dictionary learning on three datasets.

2) ALMM-based Spectral Variability Dictionary Learning:

When the variability dictionary is unknown, ALMM simultaneously learns E and estimates abundances and scaling variables. The nonconvex dictionary-learning problem is solved by alternating ADMM updates with careful initialization.

  • ALMM-based Spectral Variability Dictionary Learning:: Unknown E requires simultaneous spectral unmixing and dictionary learning, alternating updates of X, E, B, and S.
  • ALMM-based Spectral Variability Dictionary Learning:: ADMM provides the optimization strategy for the multi-variable dictionary-learning problem.
  • ALMM-based Spectral Variability Dictionary Learning:: Auxiliary variables G, H, M, T, and Q replace constrained quantities in the augmented Lagrangian formulation.
  • ALMM-based Spectral Variability Dictionary Learning:: Because dictionary learning is nonconvex, initialization uses SCLSU abundances for X0 and a random orthogonal matrix for E0.

E. Convergence and Computational Cost

The proposed alternating schemes are multi-block ADMM problems with theoretical convergence support. Experiments visualize objective-function convergence, while computational cost is dominated by matrix products.

  • Algorithm 1 and Algorithm 2 use alternating multi-block ADMM schemes with theoretical convergence support.
  • Objective-function values are recorded across iterations to visualize convergence for ALMM-based SU and ALMM-based SVDL on three datasets.
  • The overall computational complexity is O(DLN) for ALMM-based SU and O(DL2N) for ALMM-based SVDL.Matrix products dominate the computational cost.

IV. EXPERIMENTS

Experiments on synthetic and real hyperspectral scenes compare ALMM with conventional and state-of-the-art unmixing methods using quantitative, visual, sensitivity, and robustness analyses. ALMM generally learns spectral variability effectively and improves abundance estimation, while failure cases reveal sensitivity to nonconvex optimization and scaling-factor estimation.

  • Experimental design: ALMM is evaluated on synthetic, urban, and Cuprite scenes against FCLSU, CLSU, SCLSU, SUnSAL, SSUnSAL, PLMM, and ELMM.The study reports quantitative and visual comparisons across three datasets.
  • Evaluation criteria: The experiments assess abundance, reconstruction, and spectral-angle performance using aRMSE, rRMSE, and aSAM.These criteria quantify abundance-map accuracy and reconstruction quality, with or without abundance ground truth.
  • Quantitative and visual results: ALMM outperforms the compared algorithms, with abundance difference values mostly close to zero.The comparison suggests that learning spectral variability improves abundance estimation accuracy.
  • Learned spectral variabilities: Most simulated pixels show good agreement between observed signatures, reconstructions, and learned spectral variabilities.Typical cases demonstrate effective learning of scaling factors and other complex variabilities.
  • Learned spectral variabilities: When endmembers do not share a similar scalar, scaling-factor estimation can fail, yet the Eb_k term can still support accurate abundance estimation.The additional spectral-variability term represents effects not explained by a shared scaling factor.
  • Limitations and sensitivity: ALMM has failure cases in which inaccurate scaling factors or nonconvex optimization lead to inaccurate signatures and abundance maps.Prior assumptions and a VCA-derived dictionary narrow the solution range but do not directly guarantee a globally optimal solution.

4) Robustness Study:

The study evaluates ALMM on synthetic noisy data and the Urban real dataset using quantitative, visual, and classification-based assessments. ALMM is reported as robust to varying noise levels and better at identifying materials under spectral variability.

  • Synthetic noise robustness: ALMM remains more robust and effective than competing methods as Gaussian noise varies from 5dB to 40dB.The evaluation uses aRMSEs across SNR values sampled at 1dB intervals.
  • Urban dataset: The Urban dataset contains asphalt, grass, trees, and roof as four main endmembers in a 307×307-pixel HYDICE image.The image has 210 spectral bands spanning 400 nm to 2500 nm.
  • Urban experimental setup: VCA and HySime construct the endmember dictionary and determine the number of endmembers for the Urban experiments.Material identification is performed by comparison with reference endmembers.
  • Evaluation protocol: Urban performance is assessed with rRMSE, aSAM, and overall accuracy because abundance-map ground truth is unavailable and reconstruction metrics are insufficient alone.Overall accuracy is calculated using a classification-based strategy based on spectral angle mapper results.
  • Urban results: FCLSU and CLSU struggle with spectral variability, whereas SCLSU improves identification of asphalt, trees, and roof.FCLSU and CLSU fail to detect asphalt, while SCLSU effectively detects it.

3) Results and Analysis:

Experiments on Urban and Cuprite data compare ALMM with established unmixing methods through abundance-map visualization and quantitative evaluation. Across both datasets, ALMM is reported to produce clearer, more distinct, and more accurate material estimates.

  • Urban results: ELMM improves trees and roof identification but leaves asphalt and grass estimation room for improvement, while its non-convexity limits overall performance.The passage reports lower CMMS for ELMM than for CLSU and SCLSU despite lower rRMSE and aSAM.
  • Urban results: PLMM performs relatively poorly on Urban data because it does not address scaling factors, the scene’s main spectral variability.ELMM produces purer trees and roof identification than PLMM in the cited comparison.
  • Urban results: ALMM produces purer asphalt and grass identification and higher-contrast trees and roof abundance maps than the other Urban methods.The authors state that its visual performance is superior and consistent with numerical evaluation in Table II.
  • Urban comparison: Across CLSU, SCLSU, SUnSAL, SSUnSAL, and ALMM, material identification and abundance-map clarity improve progressively, with ALMM giving the clearest maps.CLSU and SUnSAL can produce abundances exceeding 1, while scaling helps SCLSU and the proposed model provides the preferred results.
  • Cuprite evaluation: Cuprite evaluation uses quantitative and visual comparisons, restricting classification-based overall accuracy to alunite, chalcedony, kaolinite, and montmorillonite.The restriction reflects the difficulty of estimating the number of endmembers under highly mixed mineral effects.
  • Cuprite results: On Cuprite, ALMM produces abundance maps with the best visual resemblance to Tetracorder, greater contrast, and more distinct material distributions.Scaled CLSU and SUnSAL keep abundances in an understandable range, while ALMM more closely resembles the reference classification.

APPENDIX A SOLUTION TO ALMM-BASED SPECTRAL UNMIXING

The appendix solves ALMM’s nonconvex optimization by alternating updates over separate variables while holding the others fixed. The procedure includes closed-form, thresholding, NNLS, constraint, and multiplier updates.

  • Alternating optimization: Because the objective is jointly nonconvex but separately convex, the algorithm successively minimizes it with respect to x_k, S_k, b_k, g_k, h_k, λ_k, and ν_k.The alternating strategy fixes the remaining variables during each subproblem.
  • Variable updates: The x_k and S_k subproblem bundles S_kA as D and uses a closed-form solution followed by a sum-to-one update for x_k.The abundance update is further modified to satisfy the sum-to-one constraint.
  • Variable updates: S_k is estimated through a nonnegative least-squares problem after bundling Ax_k as Z.This update enforces the nonnegative structure of the scaling-related variable.
  • Variable updates: The b_k, g_k, and h_k subproblems are solved through their stated update rules, including a well-known soft-threshold solution.The appendix separately identifies the optimization problems and updates for these variables.
  • Multiplier updates: Lagrange multipliers λ_k and ν_k are updated before the next iteration using the residuals g_k−x_k and h_k−x_k.The update uses λ_k ← λ_k + µ(g_k−x_k) and ν_k ← ν_k + µ(h_k−x_k).

DICTIONARY LEARNING

The dictionary-learning appendix alternates optimization over ALMM variables and dictionary-related matrices. It uses analytical, matrix-form, and closed-form updates for the learned components and constraints.

  • Dictionary updates: The appendix formulates optimization subproblems for M and E as part of the dictionary-learning procedure.The E update has an analytical solution.
  • Abundance and scaling updates: B is obtained analytically, while X and S are updated through their respective optimization problems and closed-form solutions.X is rewritten to remove scaling factors while satisfying the sum-to-one constraint, and S is subsequently collected in a separate problem.
  • Notation: The notation uses term-wise Hadamard division in the matrix rewriting of X.The appendix explicitly defines ⊘ as term-wise Hadamard division.
  • Dictionary updates: The Gram matrix E^T E is handled through an auxiliary Q update using the previous-step Q as known.The resulting Q subproblem is specified and then deduced into an update.
  • Auxiliary-variable updates: G and H are summarized in matrix form, while T is updated using the same rule as H.These updates are embedded in the alternating optimization sequence.
  • Multiplier updates: The Lagrange multipliers Λ, V, Ω, Π, and Δ are updated at each iteration following the rule used for the earlier multipliers.The appendix states that these updates can be quickly solved.
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