高光譜分解(hyperspectral unmixing)是指從高光譜數據(hyperspectral data)中抽取出不同物質之光譜特徵(即端元(endmember))與其相應豐度圖(abundance maps)的過程，豐度圖顯示出每個端元在一表面上的含量比例分佈圖。在本論文中，我們著重在如何從受雜訊(noise)影響的數據中，準確地估測出不同物質之光譜特徵。現存的高光譜分解演算法中的其中一個類別是以Winter所提出的端元抽取想法為基礎，其想法為「在純像素(一個像素中僅含有一種物質的光譜特徵)存在的情況下，端元可藉由從量測到的數據雲(data cloud)內找到最大體積單純型(simplex)的頂點決定。」不過，實際上因為數據中存在雜訊的關係，依照Winter的想法所估測出端元之光譜特徵和真正端元之光譜特徵會有相當的誤差。為此，根據強健的Winter的想法和其公式化後的表示式，我們提出兩個演算法，分別稱為最差狀況強健交替式體積最大化(worst-case robust alternating volume maximization, WCR-AVMAX)和最差狀況強健連續式體積最大化(worst-case robust successive volume maximization, WCR-SVMAX)，其中前者是利用了交替式最佳化(alternating optimization)而後者則是連續式最佳化(successive optimization)。前者需要給定初始值但後者不需要且兩者在計算上都有十分有效率。最後，我們利用電腦模擬和真實高光譜實驗(1997年內華達州Cuprite礦區所採集的高光譜數據)並與現存以純像素為基礎的演算法作比較，驗證了我們提出的演算法的優良效能和實用性。

Hyperspectral unmixing is a process of extracting the
spectral signatures (endmember signatures) and the corresponding
fractions (abundance maps), which represent the proportional
contribution of each endmember over the surface, from the given
hyperspectral data. In this thesis, we focus on the study of how to
accurately estimate the endmember signatures in the presence of
noise in the observed data. A branch of existing hyperspectral
unmixing algorithms is based on Winter's endmember extraction
belief, which indicates that in the presence of pure pixels (the
pixels are contributed by a single endmember only), the endmembers
can be determined by finding the vertices of the maximum-volume
simplex inside the data cloud. Nevertheless, in practice the
endmember estimates yielded by Winter's belief are not in the
proximity of true endmember signatures due to inevitable noise
present in the data. Based on the robust Winter's belief and
formulation \cite{Chan2011}, we propose two algorithms, namely
worst-case robust alternating volume maximization (WCR-AVMAX) and
worst-case robust successive volume maximization (WCR-SVMAX) which
respectively apply alternating optimization and successive
optimization to fulfill the robust Winter's belief. The former needs
initialization while the later does not, and both are
computationally efficient. Finally, we present computer simulations
and real data experiments (AVIRIS hyperspectral data taken over the
Cuprite mining site, Nevada, 1997 \cite{AVIRIS}) to demonstrate the
superior performance and practical applicability of our proposed
algorithms compared to several benchmark existing pure-pixel based
algorithms.

[1] T.-H. Chan, W.-K. Ma, A. Ambikapathi, and C.-Y. Chi, “A simplex volume maximization
framework for hyperspectral endmember extraction,” to appear in IEEE
Trans. Geoscience and Remote Sensing - Special Issue on Spectral Unmixing of Re-
motely Sensed Data, 2011.
[2] AVIRIS Free Standard Data Products, available online: http://aviris.jpl.nasa.
gov/html/aviris.freedata.html.
[3] A. Ambikapathi, T.-H. Chan, W.-K. Ma, and C.-Y. Chi, “Chance constrained robust
minimum volume enclosing simplex algorithm for hyperspectral unmixing,” to appear
in IEEE Trans. Geoscience and Remote Sensing - Special Issue on Spectral Unmixing
of Remotely Sensed Data, 2011.
[4] V. P. Pauca, J. Piper, and R. J. Plemmons, “Nonnegative matrix factorization for
spectral data analysis,” Linear Algebra Appl., vol. 1, no. 416, pp. 29–47, 2006.
[5] M. B. Lopes, J. C. Wolff, J. B. Dias, and M. Figueiredo, “NIR hyperspectral unmixing
based on a minimum volume criterion for fast and accurate chemical characterisation
of counterfeit tablets,” Analyticcal Chemistry, vol. 82, no. 4, pp. 1462–1469,
2010.
[6] G. Shaw and D. Manolakis, “Signal processing for hyperspectral image exploitation,”
IEEE Signal Process. Mag., vol. 19, no. 1, pp. 12–16, Jan. 2002.
[7] N. Keshava and J. Mustard, “Spectral unmixing,” IEEE Signal Process. Mag.,
vol. 19, no. 1, pp. 44–57, Jan. 2002.
[8] M. O. Smith, P. E. Johnson, and J. B. Adams, “Quantitative determination of
mineral types and abundances from reflectance spectra using principal component
analysis,” Journal Feophys. Res., vol. 90, no. 2, pp. C797–C804, Feb. 1985.
[9] A. A. Green, M. Berman, P. Switzer, andM. D. Craig, “A transformation for ordering
multispectral data in terms of image quality with implications for noise removal,”
IEEE Trans. Geosci. Remote Sens., vol. 26, no. 1, pp. 65–74, Jan. 1988.
[10] J. W. Boardman, F. A. Kruse, and R. O. Green, “Mapping target signatures via partial
unmixing of AVIRIS data,” in Proc. Summ. JPL Airborne Earth Sci. Workshop,
vol. 1, Pasadena, CA, Dec. 9-14, 1995, pp. 23–26.
[11] M. E. Winter, “N-findr: An algorithm for fast autonomous spectral end-member
determination in hyperspectral data,” in Proc. SPIE Conf. Imaging Spectrometry,
Pasadena, CA, Oct. 1999, pp. 266–275.
[12] ——, “A proof of the N-FINDR algorithm for the automated detection of endmembers
in a hyperspectral image,” in Proc. SPIE Conf. Algorithms and Technologies
for Multispectral, Hyperspectral, and Ultraspectral Imagery, vol. 5425, Aug. 2004, pp.
31–41.
[13] C.-C. Wu, S. Chu, and C.-I. Chang, “Sequential N-FINDR algorithms,” Proc. of
SPIE, vol. 7086, Aug. 2008.
[14] C.-I. Chang, C.-C. Wu, W.-M. Liu, and Y.-C. Quyang, “A new growing method
for simplex-based endmember extraction algorithm,” IEEE Trans. Geosci. Remote
Sens., vol. 44, no. 10, pp. 2804–2819, Oct. 2006.
[15] C.-I. Chang, C.-C. Wu, C.-S. Lo, and M.-L. Chang, “Real-time simplex growing
algorithms for hyperspectral endmember extraction,” IEEE Trans. Geosci. Remote
Sens., vol. 48, no. 4, pp. 1834–1850, April 2010.
[16] J.M. P. Nascimento and J. M. B. Dias, “Vertex component analysis: A fast algorithm
to unmix hyperspectral data,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 4,
pp. 898–910, Apr. 2005.
[17] M. Berman, H. Kiiveri, R. Lagerstrom, A. Ernst, R. Dunne, and J. F. Huntington,
“ICE: A statistical approach to identifying endmembers in hyperspectral images,”
IEEE Trans. Geosci. Remote Sens., vol. 42, no. 10, pp. 2085–2095, Oct. 2004.
[18] A. Zymnis, S.-J. Kim, J. Skaf, M. Parente, and S. Boyd, “Hyperspectral image
unmixing via alternating projected subgradients,” in Proc. 41st Asilomar Conference
on Signals, Systems, and Computers, Pacific Grove, CA, Nov. 4-7, 2007.
[19] L. Miao and H. Qi, “Endmember extraction from highly mixed data using minimum
volume constrained nonnegative matrix factorization,” IEEE Trans. Geosci. Remote
Sens., vol. 45, no. 3, pp. 765–777, Mar. 2007.
[20] A. Huck, M. Guillaume, and J. Blanc-Talon, “Minimum dispersion constrained nonnegative
matrix factorization to unmix hyperspectral data,” IEEE Trans. Geosci.
Remote Sens., vol. 48, no. 6, pp. 2590–2602, June 2010.
[21] J. Li and J. Bioucas-Dias, “Minimum volume simplex analysis: A fast algorithm
to unmix hyperspectral data,” in Proc. IEEE International Geoscience and Remote
Sensing Symposium, vol. 4, Boston, MA, Aug. 8-12, 2008, pp. 2369–2371.
[22] T.-H. Chan, C.-Y. Chi, Y.-M. Huang, and W.-K. Ma, “A convex analysis based
minimum-volume enclosing simplex algorithm for hyperspectral unmixing,” IEEE
Trans. Signal Processing, vol. 57, no. 11, pp. 4418–4432, Nov. 2009.
[23] M. D. Craig, “Minimum-volume transforms for remotely sensed data,” IEEE Trans.
Geosci. Remote Sens., vol. 32, no. 3, pp. 542–552, May 1994.
[24] N. Dobigeon, S. Moussaoui, M. Coulon, J.-Y. Tourneret, and A. O. Hero, “Joint
Bayesian endmember extraction and linear unmixing for hyperspectral imagery,”
IEEE Trans. Signal Processing, vol. 57, no. 11, pp. 4355–4368, Nov. 2009.
[25] J. M. B. Dias, “A variable splitting augmented Lagrangian approach to linear spectral
unmixing,” in Proc. First IEEE Workshop on Hyperspectral Image and Signal
Processing: Evolution in Remote Sensing, Grenoble, France, Aug. 26-28, 2009.
[26] D. Heinz and C.-I. Chang, “Fully constrained least squares linear mixture analysis
for material quantification in hyperspectral imagery,” IEEE Trans. Geosci. Remote
Sens., vol. 39, no. 3, pp. 529–545, Mar. 2001.
[27] D. Lee and H. S. Seung, “Learning the parts of objects by non-negative matrix
factorization,” Nature, vol. 401, pp. 788–791, Oct. 1999.
[28] J. M. Bioucas-Dias and J. M. P. Nascimento, “Hyperspectral subspace identification,”
IEEE Trans. Geosci. Rem. Sens., vol. 46, no. 8, pp. 2435–2445, 2008.
[29] T.-H. Chan, W.-K.Ma, C.-Y. Chi, and A. Ambikapathi, “Hyperspectral unmixing
from a convex analysis and optimization perspective,” in Proc. First IEEE Work-
shop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing
(WHISPERS), Grenoble, France, August 26-28, 2009, pp. 1–4.
[30] M. T. Eismann and R. C. Hardie, “Application of the stochastic mixing model to
hyperspectral resolution enhancement,” IEEE Trans. Geosci. Remote Sens., vol. 42,
no. 9, pp. 1924–1933, Sept. 2004.
[31] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge Univ. Press, 2004.
[32] T.-H. Chan, W.-K. Ma, C.-Y. Chi, and Y. Wang, “A convex analysis framework for
blind separation of non-negative sources,” IEEE Trans. Signal Processing, vol. 56,
no. 10, pp. 5120–5134, Oct. 2008, available online: http://www.ee.cuhk.edu.hk/
~wkma/.
[33] G. Strang, Linear Algebra and Its Applications, 4th ed. CA: Thomson, 2006.
[34] C.-I. Chang, C.-C. Wu, and C.-T. Tsai, “Random N-finder (N-FINDR) endmember
extraction algorithms for hyperspectral imagery,” IEEE Trans. Image Processing,
vol. 20, no. 3, pp. 641–656, March 2011.
[35] H. W. Kuhn, “The Hungarian method for the assignment method,” Naval Research
Logistics Quarterly, vol. 2, pp. 83–97, 1955.
[36] Tech. Rep., available online: http://speclab.cr.usgs.gov/cuprite.html.
[37] T.-H. Chan, C.-Y. Chi, Y.-M. Huang, and W.-K. Ma, “A convex analysis based
minimum-volume enclosing simplex algorithm for hyperspectral unmixing,” in Proc.
IEEE International Conference on Acoustics, Speech, and Signal Processing, Taipei,
Taiwan, April 19-24, 2009, pp. 1089–1092.
[38] R. N. Clark, G. A. Swayze, A. Gallagher, T. V. King, and W. M. Calvin, “The U.S.
geological survey digital spectral library: Version 1: 0.2 to 3.0 μm,” in U.S. Geol.
Surv., Open File Report 93-592, 1993.
[39] F. A. Kruse, J. W. Boardman, and J. F. Huntington, “Comparison of airborne
hyperspectral data and EO-1 Hyperion for mineral mapping,” IEEE Trans. Geosci.
Remote Sens., vol. 41, no. 6, pp. 1388–1400, June 2003.