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Robust Collaborative Nonnegative Matrix Factorization For Hyperspectral Unmixing (R-CoNMF)

Jun Li, Jose M. Bioucas-Dias, Antonio Plaza, Lin Liu

arXiv:1506.04870v1math.OC

TL;DR

CoNMF jointly addresses several stages of hyperspectral unmixing but lacks a convergence guarantee and can have an adverse interaction between its regularizers. R-CoNMF modifies CoNMF and uses an alternating optimization procedure with convergence to a critical point; experiments report effective estimates when the endmember count is unknown and accurate performance when it is known.

  • Problem

    CoNMF jointly performs unmixing stages, but lacks convergence guarantees and can suffer a negative joint effect from its volume and mixed-norm regularizers.

    Source: §1 · p. 1

  • Method

    R-CoNMF modifies CoNMF and uses an alternating optimization procedure with convergence to a critical point.

    Source: §1 · p. 1, §2.2 · p. 2

  • Results

    R-CoNMF estimates endmember counts correctly in p=6 and p=10 tests and performs accurately when the count is known, including q=p=10.

    Source: §3.1 · p. 3, §3.2 · p. 4

  • Takeaways & Limitations

    R-CoNMF is one of the few algorithms that estimates endmember counts, mixing matrices, and abundances fully automatically without external algorithms.

    Source: §4 · p. 4

Abstract

from arXiv · show

The recently introduced collaborative nonnegative matrix factorization (CoNMF) algorithm was conceived to simultaneously estimate the number of endmembers, the mixing matrix, and the fractional abundances from hyperspectral linear mixtures. This paper introduces R-CoNMF, which is a robust version of CoNMF. The robustness has been added by a) including a volume regularizer which penalizes the distance to a mixing matrix inferred by a pure pixel algorithm; and by b) introducing a new proximal alternating optimization (PAO) algorithm for which convergence to a critical point is guaranteed. Our experimental results indicate that R-CoNMF provides effective estimates both when the number of endmembers are unknown and when they are known.

1. INTRODUCTION

The introduction frames hyperspectral unmixing around the linear mixture model and presents R-CoNMF as a modification of CoNMF designed to address convergence and robustness weaknesses.

1. INTRODUCTION

  • Under the linear mixture model, unmixing methods include geometrical, statistical, and sparse-regression approaches, which respectively use simplex geometry, statistical analysis, or known spectral signatures from a library.The model treats each observed pixel as a linear combination of pure spectra, or endmembers.Source: §1 · p. 1
  • Geometrical unmixing commonly divides the process into estimating endmember count, identifying signatures, and estimating pixel abundances.Research has particularly emphasized signature identification, with and without assuming pure pixels in the input data.Source: §1 · p. 1
  • CoNMF jointly addresses endmember-count, signature, and abundance estimation, using abundance sparsity to select active endmembers and volume regularization to promote minimum volume [12].Its volume regularizer also pushes endmembers with zero abundances toward the dataset mean.Source: §1 · p. 1
  • CoNMF lacks guaranteed convergence and can suffer negative interaction between its volume and mixed-norm regularizers; R-CoNMF adds a PAO solver and modifies volume regularization for noise and degenerate simplexes.The PAO solver computes a critical point of the objective function.Source: §1 · p. 1

2. THE PROPOSED APPROACH

The approach represents a hyperspectral image as a matrix of spectral vectors and introduces the linear mixing model (LMM).

  • The image is represented as Y ∈ R^d×n, with n spectral vectors and d spectral bands, under the LMM.Source: §2 · p. 1

A X Y

R-CoNMF estimates the number of endmembers, mixing matrix, and abundances from hyperspectral mixtures using a row-sparse, volume-regularized criterion and a proximal alternating optimization algorithm. It can be applied when the endmember count is known or unknown.

A X Y

  • Under the linear mixing model, Y is represented using a mixing matrix M of endmember signatures and an abundance matrix S of pixel fractions, with nonnegative abundances and a sum-to-one constraint.The figure illustrates the collaborative-sparsity representation: nonzero rows of X form S, and corresponding columns of A form M.Source: A X Y · p. 2, Figure 1 · p. 2

2.1. Estimation criterion

  • The criterion combines reconstruction fidelity, ℓ2,1 row sparsity on X, and a minimum-volume penalty that draws A toward pure-pixel vectors P and reduces sensitivity to ill-conditioning and facet perturbations.The abundance columns are constrained to the probability simplex, enforcing nonnegativity and sum-to-one.Source: §2.1 · p. 2

2.2. Optimization algorithm and convergence

  • The PAO algorithm uses two block updates and is covered by a convergence result guaranteeing convergence to a critical point; its A update is a small quadratic problem, while its X update is solved with CSUNSAL.The X update is a constrained ℓ2−ℓ2,1 problem.Source: §2.2 · p. 2

2.3. Applying R-CoNMF

  • For q>p, an endmember is active when ζ(i)=∥x_i∥2>ξ; with p=6, n=4000 and SNR=30 dB, ξ from 0.5 to 4 correctly estimates p=6.When the count is known, the method sets q=p and α very small; when unknown, it first estimates the count with q>p, then applies R-CoNMF again with q=p.Source: §2.3 · p. 2

3. EXPERIMENTAL RESULTS

Synthetic experiments report that R-CoNMF estimates the correct endmember counts for p = 6 and 10, and remains useful as q = p increases to 15, where MVC-NMF fails for q = p ≥ 8.

3. EXPERIMENTAL RESULTS

  • The synthetic scenes contain 4000 pixels, use USGS-library signatures generated with the LMM, and exclude pure pixels.Pairwise spectral angle distance exceeds 10 degrees; pixels with any abundance above 0.8 are discarded, and each pixel has at most five endmembers when p ≥ 5.Source: §3 · pp. 2–3

3.1. Experiment 1

  • At SNR = 30 dB, R-CoNMF estimates the correct endmember counts for p = 6 and p = 10 across the tested q values.The experiment evaluates whether the estimated count matches the true count; the reported estimates are 6 and 10, respectively.Source: §3.1 · p. 3, Figure 3 · p. 3

3.2. Experiment 2

  • For q = p ≥ 8, R-CoNMF provides useful solutions while MVC-NMF results are unusable; R-CoNMF signatures and abundance estimates also more closely match the references.R-CoNMF remains useful at q = p = 15, while MVC-NMF fails to provide feasible results in the higher-endmember cases.Source: Figure 4 · p. 3, Figure 5 · p. 3, Table 1 · p. 4
  • For q = p, R-CoNMF is compared with MVC-NMF [8] across p = 4, 6, 8, 10, and 15 using 30 dB data averaged over 30 Monte Carlo runs.The evaluation uses RRE, SAD, endmember Frobenius error, and abundance Frobenius error; for unknown endmember count α = 10^-5 and β = 10^-1, while for known count β = 10^-5 and α is hand-tuned.Source: §3.2 · pp. 3–4, §3 · p. 3

4. CONCLUSIONS AND FUTURE LINES

R-CoNMF robustly estimates the number of endmembers, the mixing matrix, and the corresponding abundances in a fully automatic unmixing chain without external algorithms. Future work will evaluate it on real hyperspectral datasets.

4. CONCLUSIONS AND FUTURE LINES

  • R-CoNMF estimates the number of endmembers, the mixing matrix, and the corresponding abundances fully automatically without external algorithms.The paper presents it as a robust version of CoNMF that fills a gap in hyperspectral unmixing.Source: §4 · p. 4
  • Future work will evaluate R-CoNMF using real hyperspectral datasets.Source: §4 · p. 4

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