## Abstract:

Four methods of endmember detection and spectral unmixing are described. The methods determine endmembers and perform spectral unmixing while simultaneously determining the number of endmembers, representing endmembers as distributions, partitioning the input data set into several convex regions, or performing hyperspectral band selection. Few endmember detection algorithms estimate the number of endmembers in addition to determining their spectral shape. Also, methods which treat endmembers as distributions or treat hyperspectral images as piece-wise convex data sets have not been previously developed.

A hyperspectral image is a three-dimensional data cube containing radiance values collected over an area (or scene) in a range of wavelengths. Endmember detection and spectral unmixing attempt to decompose a hyperspectral image into the pure – separate and individual – spectral signatures of the materials in a scene, and the proportions of each material at every pixel location. Each spectral pixel in the image can then be approximated by a convex combination of proportions and endmember spectra.

The first method, the Sparsity Promoting Iterated Constrained Endmembers (SPICE) algorithm, incorporates sparsity-promoting priors to estimate the number of endmembers. The algorithm is initialized with a large number of endmembers. The sparsity promotion process drives all proportions of some endmembers to zero. These endmembers can be removed by SPICE with no effect on the error incurred by representing the image with endmembers. The second method, the Endmember Distribution detection (ED) algorithm, models each endmember as a distribution rather than a single spectrum incorporating an endmember’s inherent spectral variation or the variation due to differing environmental conditions. The third method, the Piece-wise Convex Endmember (PCE) detection algorithm, partitions the input hyperspectral data set into convex regions while simultaneously estimating endmember distributions for each partition and proportion values for each pixel in the image. The number of convex regions are determined autonomously using the Dirichlet process. The fourth method is known as the Band Selecting Sparsity Promoting Iterated Constrained Endmember (B-SPICE) algorithm and is an extension of SPICE that performs hyperspectral band selection in addition to all of SPICE’s endmember detection and spectral unmixing features. This method applies sparsity promoting priors to discard those hyperspectral bands which do not aid in distinguishing between endmembers in a data set. All of the presented algorithms are effective at handling highly-mixed hyperspectral images where all of the pixels in the scene contain mixtures of multiple endmembers. These methods are capable of extracting endmember spectra from a scene that does not contain pure pixels composed of only a single endmember’s material. Furthermore, the methods conform to the Convex Geometry Model for hyperspectral imagery. This model requires that the proportions associated with an image pixel be non-negative and sum to one.

Results indicate that SPICE and B-SPICE consistently produce the correct number of endmembers and the correct spectral shape for each endmember. The B-SPICE algorithm is shown to significantly decrease the number of hyperspectral bands while maintaining competitive classification accuracy for a data set. The ED algorithm results indicate that the algorithm produces accurate endmembers and can incorporate spectral variation into the endmember representation. The PCE algorithm results on hyperspectral data indicate that PCE produces endmember distributions which represent the true ground truth classes of the input data set.

## Links:

## Citation:

`A. Zare, “Hyperspectral endmember detection and band selection using bayesian methods,” PhD Thesis, Gainesville, FL, 2008. `

```
@PhdThesis{zare2008hyperspectraldissertation,
Title = {Hyperspectral endmember detection and band selection using bayesian methods},
Author = {Alina Zare},
School = {Univ. of Florida},
Year = {2008},
Address = {Gainesville, FL},
Month = {Dec.},
}
```