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Non-linear fully-constrained spectral unmixing.

Rob Heylen, D evdet Burazerović, Paul ScheundersẑIBBT-Visionlab, University of Antwerp, Belgium

IGARSS 2011July 25-29, Vancouver, Canada

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Outline

• Introduction: Spectral unmixing• Non-linear endmember extraction• Distance-based unmixing algorithm• Results• Conclusions and future work

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Spectral unmixing

The linear mixing model

Non-linear mixing models are far more generic

One does not always know the function F:• Model-based unmixing• Data-driven manifold techniques

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Non-linear unmixing

A data driven, fully-constrained non-linear unmixing method:

• Endmember extraction by combining graph-based manifold learning with N-findR.

• Unmixing via a distance-geometry based fully-constrained unmixing algorithm, applied to the geodesic distances.

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Distance geometry

All properties are expressed as (Euclidean) distances between the constituents.

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Endmember extraction

• Rewrite N-findR to work with mutual distances. e.g.:

• Use approximate geodesic distances on the data manifold: Shortest-path distances in nearest-neighbor graph

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Endmember extraction

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Non-linear unmixing

Next step: Unmix the data to find abundances of each pixel.

Aim: Use geodesic distance matrix as input of the unmixing algorithm.

The DSPU algorithm is fit for this task.

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The minimization problem

Linear spectral unmixing can be viewed as a minimization problem

Simplex projection is equivalent problem:

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Simplex projection unmixingRecursive simplex projection unmixing (SPU) algorithm:

1. Project the point orthogonally onto the simplex plane.2. If the point lies inside the simplex, finish.3. Else, find which abundance has to be zero.4. Remove the endmember from the set of endmembers and go to step 1.

Can be expressed in distance geometry: DSPU

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Results: Cuprite data set

• Cuprite data set• Linear unmixing via N-findR and FCLSU/DSPU• Non-linear unmixing via the proposed method

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Cuprite: Linear unmixing

Typical situation: 99.7% of abundances differ by less than 10-7 . E.g. for the alunite endmember:

FCLSU DSPU

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Cuprite: Non-linear EEA

Kaolinite 0.056, Montmorrilonite 0.048, Alunite 0.043

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Cuprite: Non-linear unmixing

Alunite endmember:

N-findR + FCLSU Non-lin. EEA + DSPU

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Conclusions and future work

• A data-driven non-linear unmixing algorithm.• Promising results on artificial data.• Significant deviations from linear unmixing results. Hard to quantify on Cuprite data set.

• Assess method on non-linearly mixed data with ground truth.• Lots of testing