Implement unmixing via blind source separation

[jpolchlo]

We've tried a couple approaches to endmember selection from imagery, notably #146. Efforts thus far have employed "geometric" approaches, which treat spectra as vectors in some high-dimensional space, and the endmembers are selected from the set of available points, such that affine combinations of the chosen spectra have a gamut which has good descriptive power for the remaining spectra. This methodology requires that the spectra we're interested in have "pure pixels" in the image, which is hard to guarantee in many cases.

Another approach is to attempt to infer endmembers that most succinctly represent the content of the image. That is, which M spectra would have affine combinations that can express the observed pixel spectra? We would have no need for pure pixels in this case.

The approach that I'd like to employ is described by Dobigeon, et al.. This employs a hierarchical Bayesian model to represent the joint distribution of both source signals (the endmembers) and the abundances as captured by the weights on a constrained (non-negative, sum-to-one) linear mixture model.

The implementation uses a Gibbs sampler (MCMC method) to generate a chain of plausible spectra and abundance maps, which can be averaged to obtain an estimate of the hidden endmembers.

References:

Dobigeon, N., Moussaoui, S., Tourneret, J. Y., & Carteret, C. (2009). Bayesian separation of spectral sources under non-negativity and full additivity constraints. Signal Processing, 89(12), 2657-2669.

Moussaoui, S., Brie, D., Mohammad-Djafari, A., & Carteret, C. (2006). Separation of non-negative mixture of non-negative sources using a Bayesian approach and MCMC sampling. IEEE transactions on signal processing, 54(11), 4133-4145.

[jpolchlo]

The mechanism of using blind source separation to extract information from a scene is promising at a conceptual level. It should be possible to get some kind of maximum likelihood estimate of a set of signals that decomposes the pixel spectra of a scene. However, that problem description is somewhat fraught.. The key insight is that BSS is a dimension reduction problem combined with a parametrization of the resulting subspace. The dimension reduction is the easy part. Finding a meaningful set of spanning columns for the subspace is the hard part.

(There are any number of ways that one could seek to optimize the columns discovered, and it's not entirely clear how to shoe-horn the Bayesian BSS method into that rubric—or even if BBSS fits the criteria of linear subspace estimation.)

The crux of the issue is is to determine a criteria for how to establish the subspace basis. The pure pixel search methods (RSPA, NFINDR, etc) use a selection criteria that is basically geometric: if abundances are affine weights, then the extreme points should sit on the convex hull of the sampled pixel spectra so that the gamut of the chosen endmembers contains as many of the sampled pixel spectra as possible. Each added endmember should strive to best capture the variation remaining after accounting for the already-discovered endmembers.

In the case of BSS, there should be some scheme for discovering endmembers not present in the sample, but the criteria for which basis to choose is ultimately not clear, as it is contained in a quite complex mathematical framework. But BBSS claims to have a mechanism that discovers underlying structure, picking out meaningful and separated signals. This process was constrained to find non-negative endmembers and non-negative, partition-of-unity abundance weights.

I tested the function of the algorithm by Dobigeon, et al. with the simplest test case we had ready access to: the Deepwater Horizon (DWH) oil spill. By homing in on an area of only oil and water, and searching for 2 endmembers, we should have found an oil signal and a water signal. However, my tests over this area produced a signal for water and a signal for oily water:

That is, we did not discover a signal for oil, which would have been similar to

This concludes our interest in this Bayesian blind source separation algorithm, as it fails to produce the signals we were hoping for. This may not be our last foray into blind source separation, but we will need to find a more innovative approach to the problem if we are to return.