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Spectral Unmixing of Hyperspectral Imagery using Multilayer NMF

Roozbeh Rajabi, Hassan Ghassemian

arXiv:1408.2810v1cs.CV

TL;DR

Hyperspectral mixed pixels require decomposition into endmembers and abundance fractions, but NMF-based unmixing faces difficult optimization. The paper proposes multilayer NMF with sparsity constraints on signatures and abundances, and reports more effective unmixing on synthetic and real datasets than prior methods.

  • Problem

    Low spatial resolution creates mixed pixels, motivating unmixing methods that recover endmembers and abundance fractions; NMF-based methods generally involve NP-hard optimization.

  • Method

    MLNMF iteratively decomposes the observation matrix across layers, modeling the signature matrix as a product of sparse matrices and applying sparsity constraints to signatures and abundance fractions.

  • Results

    The method is evaluated on synthetic USGS-library data and AVIRIS Cuprite data using SAD and AAD, with comparisons against VCA and L1/2-NMF showing more effective unmixing.

  • Takeaways & Limitations

    Multilayer decomposition allows a nonsparse spectral-signature matrix to be represented through sparse factors while supporting hyperspectral unmixing.

Abstract

from arXiv · show

Hyperspectral images contain mixed pixels due to low spatial resolution of hyperspectral sensors. Spectral unmixing problem refers to decomposing mixed pixels into a set of endmembers and abundance fractions. Due to nonnegativity constraint on abundance fractions, nonnegative matrix factorization (NMF) methods have been widely used for solving spectral unmixing problem. In this letter we proposed using multilayer NMF (MLNMF) for the purpose of hyperspectral unmixing. In this approach, spectral signature matrix can be modeled as a product of sparse matrices. In fact MLNMF decomposes the observation matrix iteratively in a number of layers. In each layer, we applied sparseness constraint on spectral signature matrix as well as on abundance fractions matrix. In this way signatures matrix can be sparsely decomposed despite the fact that it is not generally a sparse matrix. The proposed algorithm is applied on synthetic and real datasets. Synthetic data is generated based on endmembers from USGS spectral library. AVIRIS Cuprite dataset has been used as a real dataset for evaluation of proposed method. Results of experiments are quantified based on SAD and AAD measures. Results in comparison with previously proposed methods show that the multilayer approach can unmix data more effectively.

I. INTRODUCTION

Hyperspectral sensors’ low spatial resolution creates mixed pixels, motivating spectral unmixing methods that estimate endmembers and abundance fractions. The letter proposes multilayer NMF and evaluates it on synthetic and AVIRIS Cuprite data against existing methods.

  • Low spatial resolution causes pixels to contain multiple materials, whose signatures and presence percentages must be estimated through spectral unmixing.
  • NMF is widely used for spectral unmixing because the linear mixing model imposes nonnegativity constraints, although NMF-based algorithms generally yield NP-hard optimization problems.
  • The proposed MLNMF method uses a multilayer structure to improve NMF-based hyperspectral data unmixing.
  • Experiments use synthetic data generated from the USGS spectral library and the real AVIRIS Cuprite Nevada dataset.
  • Results are evaluated with spectral angle distance and abundance angle distance and compared with previously proposed methods.

II. METHODOLOGY

The methodology assumes a linear mixing model for spectral unmixing, while recognizing that nonlinear mixing models also exist and that the linear model is not universally valid.

  • Spectral unmixing uses either linear or nonlinear mixing models, and this letter considers only the linear mixing model.
  • The linear mixing model is widely used because it is acceptable in many scenarios, although it is not always true.

A. Linear Mixing Model (LMM)

The linear mixture model represents observations through signatures, abundance fractions, and noise, subject to nonnegativity and sum-to-one constraints on abundances.

  • The linear mixture model is formulated mathematically to represent observed hyperspectral data.
  • The observation matrix X contains spectra across B spectral bands and N pixels, while A contains signatures and S contains abundance fractions.
  • The model includes observation noise E in addition to signatures and abundance fractions.
  • Abundance fractions obey abundance nonnegativity and abundance sum-to-one constraints.

B. Sparse Single Layer Nonnegative Matrix Factorization

The paper formulates NMF with Euclidean distance and adds sparsity and abundance-sum constraints to better reflect hyperspectral unmixing structure.

  • NMF decomposes a nonnegative data matrix into nonnegative signature and abundance matrices by minimizing a cost function.
  • The Euclidean-distance cost function is minimized with respect to nonnegative matrices A and S.
  • A multiplicative update algorithm provides an efficient, fast, and easy-to-implement way to minimize the NMF cost function.
  • Sparsity on abundance fractions reflects that each mixed pixel usually contains far fewer endmembers than the total available.
  • The regularization parameter α controls the impact of the sparsity constraint, whose q-norm is defined for matrix S.
  • FCLS is used to enforce the abundance sum-to-one constraint, with δ controlling its impact.

C. Proposed MLNMF Method

MLNMF iteratively decomposes the observation matrix across multiple NMF layers, applying sparsity constraints to both spectral signatures and abundance fractions. The method represents the signature matrix as a product of sparse matrices while refining the decomposition across layers.

  • Multilayer decomposition: MLNMF decomposes the observation matrix iteratively across a maximum of L layers, producing A1, A2, …, AL and S = SL.The first layer factorizes X, and each subsequent layer factorizes the preceding abundance-related result.
  • Sparsity constraints: Each layer applies sparsity constraints to both the spectral signatures and abundance fractions matrices.The signature matrix itself need not be sparse because it is modeled as a product of sparse matrices.
  • Multilayer refinement: The multilayer process partially decomposes X in the first layer and improves A and S through succeeding layers.The complete decomposition is reached over multiple layers rather than in the initial factorization.
  • Optimization: The cost function uses iteration-dependent sparsity regularization, with αS set to 2αA as an experimental rule of thumb.The regularization schedule is motivated by simulated annealing, while the αS relationship is not presented as optimized.
  • Optimization: The algorithm uses VCA initialization in the first layer, random initialization in later layers, and stops after Tmax iterations or ten successive stopping-criterion matches.The experiments use δ = 25, and VCA initialization is described as more robust than random initialization.

III. EXPERIMENTS AND RESULTS

The experiments evaluate MLNMF using both synthetic and real datasets, with results quantified by spectral and abundance angle measures.

  • Experimental design: The evaluation uses synthetic and real datasets to assess the proposed method.The section includes evaluation metrics and experiments for both dataset types.

A. Evaluation Measures

The study evaluates unmixing accuracy with SAD for spectral signatures and AAD for abundance fractions, using root-mean-square forms for overall performance.

  • Metrics: SAD measures similarity between original and estimated spectral signatures.The metric is defined using the original signatures mi and estimated signatures m̂i.
  • Metrics: AAD measures similarity between original and estimated abundance fractions.The metric compares original abundance vectors ai with estimated vectors âi.
  • Metrics: Root mean squares of SAD and AAD provide overall performance measures independent of scale differences between compared vectors.The scale independence applies to the underlying angle-based measures.

B. Experiment I: (synthetic data) USGS Spectral Library

The synthetic experiment uses USGS spectral signatures and noisy images with known ground truth to evaluate MLNMF across signal-to-noise conditions. Compared with L1/2-NMF and VCA, MLNMF is reported to perform better.

  • Synthetic data: USGS spectral library data provide 224-band signatures spanning 380nm to 2500nm for synthetic-image generation.Six library signatures were randomly selected, and 64 × 64 images without pure pixels were created.
  • Noise evaluation: Zero-mean Gaussian noise at different SNR levels simulates measurement error and sensor noise.The synthetic setting preserves the true signatures and abundance fractions for evaluation.
  • Signature estimates: MLNMF estimated signatures are compared visually with original signatures using blue solid and red dashed curves, respectively.The comparison is presented for the synthetic experiment.
  • Comparative evaluation: rmsSAD and rmsAAD are compared across SNR for VCA, L1/2-NMF, and MLNMF.The comparison evaluates both spectral-signature and abundance-fraction performance.
  • Comparative evaluation: MLNMF is reported to outperform L1/2-NMF and VCA in the synthetic experiments.The comparison includes VCA combined with FCLS for endmember extraction and results from 20 algorithm runs.

C. Experiment II: (real data) AVIRIS Cuprite Nevada

The AVIRIS Cuprite experiment evaluates MLNMF on a selected real-data subscene, using estimated signatures and abundance maps, with results showing improved rmsSAD over VCA and L1/2-NMF.

  • Dataset: The experiment uses a 250 × 191-pixel AVIRIS Cuprite subscene with 188 bands retained after removing low-SNR and water-absorption bands.Band 30 of the subscene is illustrated in Fig. 6.
  • Endmember selection: Twelve endmembers are selected as true signatures after merging similar material pairs and adding Chalcedony based on the mineral map.The selection begins from 14 scene materials identified in prior studies.
  • Evaluation: The methods are compared using mean and standard deviation of rmsSAD and rmsAAD over 20 runs.Figure 5 reports rmsSAD on the left and rmsAAD on the right.
  • MLNMF outputs: MLNMF estimates spectral signatures and abundance-fraction maps for the Cuprite dataset.The experiment uses α0 = 0.1 and τ = 25; estimated signatures and abundance maps are shown in Figs. 7 and 8.
  • Results: MLNMF outperforms VCA and L1/2-NMF on the Cuprite dataset in terms of rmsSAD.The cited result reports this comparison for the real-data experiment.

IV. CONCLUSION

The conclusion presents multilayer NMF with sparsity constraints on both spectral signatures and abundance fractions as an approach for hyperspectral unmixing. Across synthetic and real datasets, the method is reported to unmix hyperspectral data more effectively than the compared methods.

  • Conclusion: The proposed MLNMF method applies sparsity constraints to both spectral signatures and abundance fractions.The method uses a multilayer structure for hyperspectral unmixing.
  • Conclusion: MLNMF decomposes a nonsparse spectral-signature matrix into a few sparse matrices through its multilayer structure.The paper describes this as a physical interpretation of the multilayer formulation.
  • Conclusion: The method is evaluated on synthetic and real datasets using SAD and AAD metrics, with comparisons against VCA and L1/2-NMF.The cited conclusion summarizes the evaluation scope and comparison methods.
  • Conclusion: The comparisons show that MLNMF can unmix hyperspectral data more effectively than the compared methods.This is the paper’s stated overall conclusion.
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