Description

This matlab package contains files implementing an approximate subgradient method for solving a minimax nonnegative matrix factorization problem,

   min               max   || X_i - (W)(H_i) ||_F
W in R_{>0}^{b x r}   i 
H in R_{>0}^{r x p}

for some data, {X_i} in R_{>0}^{b x p} for i = 1, ... , n, as described in the paper:

"Hyperspectral Unmixing with Rare Endmembers via Minimax Nonnegative Matrix Factorization." by TIM MARRINAN and NICOLAS GILLIS.

Contents

The matlab package includes the following directories and files:
\

1. README.txt
2. demo.m

\src\
03. main.m
04. minimaxNMF_DataGen.m
05. minimaxNMF.m

(The code in this directory accompanies [3] and was downloaded from https://sites.google.com/site/nicolasgillis/code on 22/10/2019 )

\src\minvolNMF\
06. compareWs.m
07. FGMfcnls.m
08. FGMqpnonneg.m
09. minvolNMF.m
10. munkres.m
11. nnlsHALSupdt.m
12. ReadMe.m
13. sample_dirichlet.m
14. SimplexProj.m
15. SNPA.m
16. synthetic_data.m

\examples\
17. minimaxNMF_Fig1.m
18. minimaxNMF_Fig2.m
19. minimaxNMF_Fig3.m
20. minimaxNMF_ScenarioSpecification.m
21. USGS_Library.mat

Abstract

Hyperspectral images are used for ground-cover classification because many materials can be identified by their spectral signature, even in images with low spatial resolution. Pixels in such an image are often modeled as a convex combination of vectors, called endmembers, that correspond to the reflectance of a material to different wavelengths of light. This is the so-called linear mixing model. Since reflectance is inherently nonnegative, the task of unmixing hyperspectral pixels can be posed as a low-rank nonnegative matrix factorization (NMF) problem, where the data matrix is decomposed into the product of the estimated endmembers and their abundances in the scene. The standard NMF problem then minimizes the residual of the decomposition. Thus, using NMF works well when materials are present in similar amounts, but if some materials are under-represented, they may be missed with this formulation. Alternatively, we propose a novel hyperspectral unmixing model using a collection of NMF subproblems solved for patches of the original image. The endmembers are estimated jointly, such that the the maximum residual across all patches is minimized. In this paper we estimate the solution to the patch-based minimax NMF model, and show that it can estimate rare endmembers with superior accuracy.

File Usage

The file demo.m can be run with no modifications to add the folders to the path and run a synthetic example using endmembers taken from the USGS spectral library [4]. It computes the minimax NMF of the data and the traditional minimum-volume NMF from [3], displays the estimated endmembers, and compares the accuracy of the factorizations.

Alternatively, the file main.m can be run with no modifications to recreate the figures from the associated paper. This will call each of the functions in the 'examples' directory, whose parameters are set by minimaxNMF_ScenarioSpecification.m. The variables 'queued' and 'nRuns' can be modified to control which experiments are run and for how many iterations.

To re-run the experiments and generate the plots from the paper:

1. Run main.m

In the paper the experiments are each run for nRuns = 50 Monte Carlo trials, however this takes numerous hours to complete. Alternatively, the behavior of the methods can be assessed from a much smaller number of independent trials or with less noise variances tested to avoid the long computation time.

On a laptop with a 2.6 GHz Intel Core i7-8850H processor and 16 GB of RAM, the experiments from the paper run for nRuns = 5 take approximately 80 minutes. To run the complete experiment set with nRuns = 50 takes approximately 13 hours.

To run your own experiments:

1. In the file minimaxNMF_ScenarioSpecification.m, modify the parameters of the 'custom' scenario as desired.
2. Depending on your desired outcome, look to the files demo.m, minimaxNMF_Fig1.m, or minimaxNMF_Fig2.mfor examples of how to use the data generation function and the minimax NMF algorithm.

Release Notes

Version 1.0 of the code does not currently provide a standalone function to cut data into patches.

References

[1] Marrinan, Timothy and Nicolas Gillis. "Hyperspectral Unmixing with Rare Endmembers via Minimax Nonnegative Matrix Factorization." In 28th European Signal Processing Conference (EUSIPCO) , pp. 69-78. IEEE, 2020.

[2] Gillis, N., 2014. Successive nonnegative projection algorithm for robust nonnegative blind source separation. SIAM Journal on Imaging Sciences, 7(2), pp.1420-1450.

[3] Leplat, Valentin, Andersen MS Ang, and Nicolas Gillis. "Minimum-volume rank-deficient nonnegative matrix factorizations." In ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3402-3406. IEEE, 2019.

[4] Clark, Roger N., Gregg A. Swayze, A. Gallagher, T. V. V. King, and W. M. Calvin. "The us geological survey digital spectral library." US Geological Survey Open File Report (1993): 93-592.

Citation:

If you find this code useful in your research, please cite:

   Hyperspectral unmixing with rare endmembers via minimax nonnegative matrix factorization.
   T. Marrinan and N. Gillis.
   Proc. IEEE 28th European Signal Processing Conference (EUSIPCO), (2021): 1015-1019.

Contact

In case of questions, suggestions, problems etc. please send an email.

Tim Marrinan:
marrinat@oregonstate.edu

This matlab package is hosted at:
http://www.tmarrinan.com/research-interests-code/
https://sites.google.com/site/nicolasgillis/code