Nonlinear hyperspectral unmixing with robust nonnegative matrix factorization (1401.5649v2)

Abstract: This paper introduces a robust mixing model to describe hyperspectral data resulting from the mixture of several pure spectral signatures. This new model not only generalizes the commonly used linear mixing model, but also allows for possible nonlinear effects to be easily handled, relying on mild assumptions regarding these nonlinearities. The standard nonnegativity and sum-to-one constraints inherent to spectral unmixing are coupled with a group-sparse constraint imposed on the nonlinearity component. This results in a new form of robust nonnegative matrix factorization. The data fidelity term is expressed as a beta-divergence, a continuous family of dissimilarity measures that takes the squared Euclidean distance and the generalized Kullback-Leibler divergence as special cases. The penalized objective is minimized with a block-coordinate descent that involves majorization-minimization updates. Simulation results obtained on synthetic and real data show that the proposed strategy competes with state-of-the-art linear and nonlinear unmixing methods.

• The paper introduces a robust nonlinear mixing model that incorporates sparse outlier terms within NMF to capture nonlinear spectral interactions.
• It employs group-sparse constraints and block-coordinate descent for efficient, nonnegative estimation in large-scale hyperspectral datasets.
• Experimental results demonstrate improved unmixing performance over conventional linear models, particularly in complex scenes such as vegetation areas.

Nonlinear Hyperspectral Unmixing with Robust Nonnegative Matrix Factorization

Hyperspectral unmixing is a critical technique in remote sensing and other fields that manage large datasets characterized by spectral information. Such datasets require the decomposition of complex spectral mixtures into simpler constituent components or endmembers. Traditional methods frequently rely on linear mixing models (LMM), however, nonlinear interactions between spectral signatures often occur and need more sophisticated approaches for accurate characterization. The paper "Nonlinear hyperspectral unmixing with robust nonnegative matrix factorization" by Cédric Févotte and Nicolas Dobigeon introduces a robust nonlinear mixing model as an alternative to conventional LMM, offering a more flexible framework for hyperspectral data analysis.

The proposed robust linear mixing model (rLMM) incorporates nonlinearity using an additional term treated as outliers within the nonnegative matrix framework. These outliers are assumed to be sparse and possibly nonzero in limited circumstances, providing a mechanism to capture nonlinear effects more accurately without relying on specific predefined models of nonlinearity. This approach is advantageous as it doesn't require an explicit form of nonlinearity and generalizes various bilinear models prevalent in hyperspectral imagery processing.

The researchers utilized a robust nonnegative matrix factorization (rNMF) technique, introducing group-sparse constraints imposed on the nonlinear component. This detail underscores the nonlinearity encapsulation, ensuring it affects a sparse number of pixels within the imagery, thus preserving the efficacy of the linear segments where applicable. The model’s fidelity is evaluated using the $\beta$-divergence, a divergence measure offering a continuum of noise statistics adjustable to specific cases such as Gaussian or Poisson noise, providing flexible adaptation of the model to various practical scenarios.

The paper proceeds to provide an algorithmic framework leveraging block-coordinate descent and majorization-minimization updates for minimizing the penalized objective. This approach ensures nonnegativity in the estimates throughout iterations while prioritizing computational efficiency. Such adaptations are crucial for handling large-scale datasets often encountered in hyperspectral imagery.

Experimental results were detailed in the paper reflecting robust performance against state-of-the-art linear and nonlinear unmixing methods. The model demonstrated efficiency in various simulated data environments and real datasets, illustrating better adaptability over strict linear models especially in complex scene compositions like vegetation areas where photon interactions invoke significant nonlinear effects.

In a broader context, the implications of this research stand out in its contribution to hyperspectral image analysis, offering a robust alternative to LMM. This research paves the way for refined data analysis in remote sensing, supporting applications in environmental monitoring, agriculture, and geosciences where capturing spectral dynamics accurately enhances decision-making processes. The authors surmise potential continuations and developments in refining nonlinear interactions further and improving computational tractability given the ever-increasing size and complexity of spectral datasets. Future work may explore deeper integration with machine learning techniques, offering possibilities for data-driven unmixing algorithms that intrinsically adapt to the nonlinear characteristics of hyperspectral images.