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Unmixing of Hyperspectral Images using Bayesian Non-negative Matrix Factorization with Volume Prior

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Abstract

Hyperspectral imaging can be used in assessing the quality of foods by decomposing the image into constituents such as protein, starch, and water. Observed data can be considered a mixture of underlying characteristic spectra (endmembers), and estimating the constituents and their abundances requires efficient algorithms for spectral unmixing. We present a Bayesian spectral unmixing algorithm employing a volume constraint and propose an inference procedure based on Gibbs sampling. We evaluate the method on synthetic and real hyperspectral data of wheat kernels. Results show that our method perform as good or better than existing volume constrained methods. Further, our method gives credible intervals for the endmembers and abundances, which allows us to asses the confidence of the results.

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Notes

  1. In the literature, the constraints that abundances must sum to one is sometimes refered to as an additivity constraint.

  2. In the notation used in this paper, matrices and vectors are denoted by capital and lower case bold letters respectively. Two subscripts denotes a sub matrix or sub vector with the corresponding rows and columns, where a colon denotes all indices, and \(\tilde m\) denotes all indices except m. For example, w :k denotes the kth column of W and \(\mathbf{w}_{m\tilde k}\) denotes the mth row of W with the kth element removed. A single element of the matrix W is denoted by w mk .

  3. For simplicity, we use a slightly different notation in this section.

  4. In practice, of course, we cannot generate a random number from this improper prior, so we can think of generating w from a uniform distribution between zero and u w , where u w is some very large number.

  5. All scatterplots in this paper are presented as subspace projections onto the first and second principal component (PC) based on the data points.

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Authors and Affiliations

  1. DTU Informatics, Technical University of Denmark, Bldg. 321, Richard Petersens Plads, 2800, Lyngby, Denmark

    Morten Arngren, Mikkel N. Schmidt & Jan Larsen

  2. FOSS Analytical A/S, Slangerupgade 69, 3400, Hillerød, Denmark

    Morten Arngren

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  1. Morten Arngren
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  2. Mikkel N. Schmidt
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  3. Jan Larsen
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Correspondence to Morten Arngren.

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Arngren, M., Schmidt, M.N. & Larsen, J. Unmixing of Hyperspectral Images using Bayesian Non-negative Matrix Factorization with Volume Prior. J Sign Process Syst 65, 479–496 (2011). https://doi.org/10.1007/s11265-010-0533-2

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  • DOI: https://doi.org/10.1007/s11265-010-0533-2

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