Hyperspectral imaging was originally developed for defense applications but is now widely used in many other areas, including food safety and inspection, agricultural land use and planning, environmental monitoring, as well as intelligence surveillance and medical imaging. Hyperspectral imaging data sets are generally made up of hundreds of spectral bands with relatively narrow, contiguous bandwidths. Analysis of such imagery involves identifying endmembers (pure signatures) that are used to define distinct spectral classes of interest in the data sets.^{1} Convex geometry is commonly used as a key criterion in several endmember extraction—or finding—algorithms (EEAs).

The two most widely used EEAs are the pixel purity index (PPI) and the N-finder algorithm (N-FINDR).^{2,3} These two algorithms have been used to derive many new approaches for finding endmembers. However, all these algorithms suffer from the same two major issues. First, random initial conditions cause inconsistent results, and second, high computational complexity prevents them from being implemented in real-world applications.^{4,5} We have developed new approaches to address the random initial condition problem.^{1} We use a set of initial conditions that are generated through specific means so that the EEAs are initialized and the final results are reproducible.^{5,6} An alternative method is to consider EEAs as random algorithms.^{7,8}

Depending on the abundance constraints that are imposed on the convex geometry, three design criteria are generally used for finding endmembers (see Figure 1 and Table 1). An orthogonal projection (OP) is used when there are no abundance constraints (e.g., for PPI), a convex cone when there is a partial abundance non-negativity constraint (ANC), and a simplex for full abundance constraints with ANC and an abundance sum-to-one constraint (e.g., for N-FINDR).

**Figure 1. **Relationships between convex cone volume analysis (CCVA), convex cone analysis, and the N-FINDR endmember finding algorithms. ANC: Abundance-unconstrained criterion. ASC: Abundance sum-to-one constraint. N-FINDR: N-finder algorithm. *p*: Number of endmembers.

**Table 1. **Abundance constraint and convex geometry criteria for different endmember finding algorithms.

We have also proposed several approaches to reduce the computational complexity of EEAs. These methods include implementing N-FINDR sequentially (iterative, sequential, or successive N-FINDR) or using convex cone volume analysis in a similar manner (see Figure 1).^{5,9,10} Other approaches involve using N-FINDR in parallel processing or with a graphical processing unit, in real time, or in field programmable gate array hardware.^{11–16}

Although several other metrics can be used as criteria in EEAs (e.g., linear unmixing, non-negative matrix factorization, and blind source separation), the convex geometry concept, specifically simplex volume, remains the most effective criterion.^{5,17–19} However, calculating simplex volumes involves high computational complexity. As such, PPI remains a popular endmember finding method because the OP it uses can be computed quickly and without high costs. If the computational issues involved with the implementation of N-FINDR were to be resolved, it (and its several variants) would overtake PPI to become the most widely used endmember finding algorithm.^{1,5,20}

True endmembers are determined by two factors: endmember variability and endmember discriminability. The former can be addressed by endmember identification, endmember selection, and endmember optimization, and the latter can be accomplished by endmember finding, endmember extraction, and endmember determination. We are currently working on ways to discriminate between all six of these tasks instead of using endmember extraction as a one-size-fits-all approach.^{20} Further work on EEAs will also investigate ways to determine the number of endmembers, such as the virtual dimensionality concept.^{21,22}

Chein-I Chang

University of Maryland, Baltimore County

Baltimore, MD

Chein-I Chang is professor of electrical engineering. He has authored and edited a number of books on the subject of hyperspectral data processing. He is also a fellow of IEEE and SPIE.

References:

1. C.-I Chang, *Hyperspectral Data Processing: Algorithm Design and Analysis* , Wiley, 2013.

2. J. W. Boardman, F. A. Kruse, R. O. Green, Mapping target signatures via partial unmixing of AVIRIS data, * Proc. Summer JPL Airborne Earth Sci. Wrkshp.* , p. 23-26, 1995.

3. M. E. Winter, N-finder: an algorithm for fast autonomous spectral endmember determination in hyperspectral data,

*Proc. SPIE* 3753, p. 266-277, 1999.

doi:10.1117/12.366289
4. A. Plaza, C.-I Chang, Impact of initialization on design of endmember extraction algorithms, *IEEE Trans. Geosci. Remote Sens.* 44, p. 3397-3407, 2006.

5. W. Xiong, C.-C. Wu, C.-I Chang, K. Kapalkis, H. M. Chen, Fast algorithms to implement N-FINDR for hyperspectral endmember extraction, *IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens.* 4, p. 545-564, 2011.

6. C.-I Chang, A. Plaza, Fast iterative algorithm for implementation of pixel purity index, *IEEE Trans. Geosci. Remote Sens. Lett.* 3, p. 63-67, 2006.

7. C.-I Chang, C.-C. Wu, H. M. Chen, Random pixel purity index algorithm, *IEEE Trans. Geosci. Remote Sens. Lett.* 7, p. 324-328, 2010.

8. C.-I Chang, C.-C. Wu, C.-T. Tsai, Random N-finder algorithm, *IEEE Trans. Image Process.* 20, p. 641-656, 2011.

9. W. Xiong, C. T. Tsai, C. W. Yang, C.-I Chang, Convex cone-based endmember extraction for hyperspectral imagery,

*Proc.* *SPIE* 7812, p. 78120H, 2010.

doi:10.1117/12.861621
10. C.-I Chang, C. C. Wu, W. Liu, Y. C. Ouyang, A growing method for simplex-based endmember extraction algorithms, *IEEE Trans. Geosci. Remote Sens.* 44, p. 2804-2819, 2006.

11. C. Gonzalez, S. Sanchez, A. Paz, J. Resano, D. Mozos, A. Plaza, Use of FPGA or GPU-based architectures for remotely sensed hyperspectral image processing, *Integration VLSI J.* 46, p. 89-103, 2013.

12. C.-C. Wu, H. M. Chen, C.-I Chang, Real time N-finder processing algorithms, *J. Real-Time Image Process.* 7, p. 105-129, 2010.

13. C.-I Chang, C. C. Wu, C.-S. Lo, M.-L. Chang, Real-time simplex growing algorithms for hyperspectral endmember extraction, *IEEE Trans. Geosci. Remote Sens.* 40, p. 1834-1850, 2010.

14. M. Hsueh, C.-I Chang, Field programmable gate arrays for pixel purity index using blocks of skewers for endmember extraction in hyperspectral imagery, *Int'l J. High Perform. Comput. Appl.* 22, p. 408-423, 2008.

15. C. Gonzalez, D. Mozos, J. Resano, A. Plaza, FPGA implementation of the N-FINDR algorithm for remotely sensed hyperspectral image analysis, *IEEE Trans. Geosci. Remote Sens.* 50, p. 374-388, 2012.

16. C.-I Chang, W. Xiong, C. C. Wu, Field programmable gate array design of implementing simplex growing algorithm for hyperspectral endmember extraction, *IEEE Trans. Geosci. Remote Sens.* 51, p. 1693-1700, 2013.

17. J. Wang, C.-I Chang, Applications of independent component analysis in endmember extraction and abundance quantification for hyperspectral imagery, *IEEE Trans. Geosci. Remote Sens.* 44, p. 2601-2616, 2006.

18. L. Miao, H. Qi, Endmember extraction from highly mixed data using minimum volume constrained nonnegative matrix factorization, *IEEE Trans. Geosci. Remote Sens.* 45, p. 765-777, 2007.

19. C. C. Wu, C.-I Chang, Does a simplex formed by endmembers really yield maximal volume?, *Int'l J Comput. Sci. Eng.* , to appear.

20. C.-I Chang, *Real Time Hyperspectral Image Processing* , Springer, 2014.

21. C.-I Chang, *Hyperspectral Imaging: Techniques for Spectral Detection and Classification* , Kluwer/Plenum Academic Publishers, 2003.

22. C.-I Chang, Q. Du, Estimation of number of spectrally distinct signal sources in hyperspectral imagery, *IEEE Trans. Geosci. Remote Sens.* 42, p. 608-619, 2004.