RELEVANCE OF TRANSFORMATION TECHNIQUES IN RAPID ENDMEMBER

IDENTIFICATION AND SPECTRAL UNMIXING: A HYPESPECTRAL REMOTE

SENSING PERSPECTIVE

Keshav Dev Singh1, Desikan Ramakrishnan1 and Lalu Mansinha2 1Department of Earth Sciences, Indian Institute of Technology Bombay, Powai, Mumbai, India

2Department of Earth Sciences, University of Western Ontario, London, Ontario, Canada

Abstract— One of the tedious and time consuming tasks

related to hyperspectral data analysis is the identification

of library candidates for spectral unmixing. In this study,

we evaluated the relevance of different transformation

procedures such as First Derivative (FD), Fast Fourier

Transform (FFT), Discrete Wavelet Transform (DWT),

Hilbert–Huang Transform (HHT) and S-transform (ST) in

automated retrieval of library endmembers for linear

spectral unmixing. The spectral similarity between the

target and library candidates were estimated using

Pearson’s Correlation Coefficient (PCC) and student t-test

based approach. Subsequently, these endmembers are used

to estimate the fractional abundances by Fully Constrained

Least Square Estimation (FCLSE) based Quadratic

Programming (QP) optimization approach. The match

between the target and modeled spectrum was calculated

based on Root Mean Squared Error (RMSE) and spectral

similarity scores estimated using Spectral Angle Mapper

(SAM). In addition to RMSE and SAM scores, the

simulation processing time and appropriateness of

identified endmembers are considered to estimate the

effectiveness of each transformation procedure. It is

observed that DWT, HHT and ST based approaches are

more efficient in identifying correct library endmembers

than the FD and FFT based approaches.

Index Terms— Hyperspectral imaging, Spectral libraries,

Transformation techniques, Pearson correlation

coefficient, Quadratic programming.

1. THE LINEAR MIXING MODEL Spectral unmixing is the process by which a measured

spectrum is decomposed into a collection of constituent

spectra or “end-members” and corresponding fractions or

“abundances”. In case of hyperspectral remote sensing,

unmixing allows us to detect and classify sub-pixel objects

[1]–[2]. The endmember extraction and abundance

estimation are the most challenging and critical aspects of

hyperspectral image processing. Usually, the candidate

spectra for unmixing are visually identified and drawn

from huge repository of spectral libraries based on the

characteristic absorption features at specific wavelength

regions [3]. Mathematically this process can be

represented as resolving a target spectra (r), a w×1 column

vector in terms of M= [m1, m2...ml], a w×l signature

matrix drawn from Johns Hopkins University (JHU) press

spectral library database L [4] of size w×n. If, w×l be the

size of l-likely endmembers selected automatically such as

l<<n and x =[x1, x2,...,xl]T be a l×1 abundance column

vector associated with r, then r can be modeled as per the

equation (1):

Where, represents the additive spectral noise.

This work is aimed at automatically selecting the potential

endmembers (M) and estimating their abundance fractions

(x) in such a way that x =1 and 0 x 1 with the least

possible root mean squared error (RMSE) between the

target and modeled spectra. An error (O-3) of order 3 is

acceptable.

2. METHODOLOGY USED IN

HYPERSPECTRAL DATA UNMIXING

In this study, we attempted to identify the constituent

minerals and its abundances for spectra representing rocks

such as Basalt, Diorite, Gabbro, Granite, Nepheline

Syenite and Norite collected in the field using Fourier

Transform Infra Red (FTIR) spectrometer [5] and JHU

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spectral library [6]. The procedures adopted include (i)

transformation of target and library spectra using FD, FFT,

DWT, HHT and ST techniques, (ii) estimating the PCC for

end-member identification [7], (iii) linear unmixing of the

target spectra with the identified endmembers using

FCLSE (QP) [8]–[10], and (iv) error estimation based on

RMSE and SAM Score [11].

2.1 Endmember Extraction from Spectral Library To optimize the end-member identification, we attempted

to calculate PCC with input spectra transformed using

different techniques such as First Derivative Spectra

(FDS), Fast Fourier Transform Spectra (FFTS) [12],

Discrete Wavelet Transformation Spectra (DWTS) [13],

Hilbert–Huang Transform Spectra (HHTS) [14], and S-

Transformation Spectra (STS) [15]. The basic objective of

these approaches is to enhance spectral features in

transform spaces which are otherwise subtle in normal

spectral space. The estimated PCC values range between 0

and 1 corresponding to least and best correlation between

the target and library spectra. In this study, we also

considered optimum PCC values above 0.8 (strong

correlation) to identify the library candidates [16]. With ri

as a target rock-type and mi M L as the i-th library

mineral spectra, then PCC for candidates selection is

expressed as equation (2):

2.2 Abundance Estimation using Fully Constrained

Quadratic programming Approach (FCQPA) Once the library end-members are identified by above

techniques, Quadratic Programming (QP) method was

used to estimate the fractional abundances of each of the

endmember [8]–[9]. The QP method offers a fully

constrained solution for sum-to-unity and non-negativity

simultaneously. Mathematically, a fully constrained

quadratic programming for least square problem can be

expressed as in equation (3).

Where, are matrices and are

vector. The above objective function is convex if and only

if H is positive-semi definite, which is the realm we are

concerned with [10].

2.3 Spectral Similarity Estimation by RMSE and

SAM Score The match between the modeled spectra (Mi) generated

using FCQPA and the target spectra (ri) were evaluated by

estimating the RMSE and Spectral Angle between them

[11], as equation (4) & (5):

3. EXPERIMENTAL RESULTS AND

CONCLUSION Table-I and Figure-I depict the efficiency of various

transformations (FDS, FFTS, DWTS, HHTS, and STS),

PCC and FCQPA in identification of endmembers and its

fractional abundances for the investigated rocks. The

performance of a particular procedure is evaluated in terms

of RMSE, SAM-scores, computation time and

appropriateness of identified end-members. For JHU rock

spectra, the estimated mineral abundances are compared

with measured abundances mentioned in the library. In

case of field spectra, results are compared with

petrography based abundances [17]. It is evident from the

results that DWTS, HHTS and STS based approach yield

satisfactory results over FDS and FFTS based approaches.

STS gives good RMSE (0.001-0.004) and SAM (0.92-

0.97) result, but take more processing time (~16 seconds).

HHTS takes low execution time (0.59-2.0 seconds) and

good in endmember selection, but more RMSE (0.001-

0.006) and lesser SAM Score (0.88-0.96). Results

estimated by DWTS method is moderate in terms of

RMSE (0.001-0.004), SAM score (0.91-0.97) and

processing time (1.2-3.1 seconds).

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Schemes PCC+FCQPA with FDS

PCC+ FCQPA with FFTS

PCC+ FCQPA with DWTS

PCC+ FCQPA with HHTS

PCC+FCQPA with STS

Matching Scores

Rock-

type

RMSE = 0.009-0.032 0.006-0.019 0.001-0.004 0.001-0.006 0.001-0.004

SAM = 0.371-0.815 0.623-0.886 0.919-0.974 0.884-0.964 0.921-0.974

CPU (t) 0.546-2.859 8.140-10.50 1.281-3.156 0.593-2.00 16.07-16.67

JHU-Diorite N N Y Y Y

JHU-Granite N N Y Y Y

JHU-Nepheline Syenite N N Y Y Y

FTIR-Charnockite Y N Y Y Y

FTIR-Norite N N Y Y Y

FTIR-Pink Magmatite Y N Y Y Y

Table-I: Table depicting the comparison of results estimated under different transformation schemes.

(Y-mineral selection appropriate; N- mineral selection inappropriate)

Figure-I: Spectral plots showing similarity between target and modeled spectra for chosen rock-types.

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This study illustrates the relevance of different transformation techniques, PCC and FCLSE (QP) in automated and rapid mineral identification and their abundances estimation. The proposed techniques can also be extended to hyperspectral image cubes for identification of targets and sub-pixel classification. It is concluded that the adopted procedure involving transformation techniques and linear unmixing can be used as an efficient tool for rapid and precise sub-pixel classification.

ACKNOWLEDGEMENTS

The authors are thankful to NASA Jet Propulsion Laboratory for providing Johns Hopkins University (JHU) spectral library database. This research was supported by Department of Science and Technology, Government of India wide project number NRDMS/11/1291/2007.

REFERENCES

[1] N. Keshava and J. F. Mustard, "Spectral unmixing," IEEE Signal Process. Mag., vol. 19, no. 1, pp. 44–57, Jan. 2002.

[2] J. B. Adams, M. O. Smith, and P. E. Johnson, "Spectral Mixture Modeling: A New Analysis of Rock and Soil Types at the Viking Lander 1 Site," J. Geophys. Research, vol. 91, no. B8, pp. 8098–8112, 1986.

[3] R. N. Clark, "Spectroscopy of rocks and minerals, and principles of spectroscopy," Manual of Remote Sensing, New York, USGS, vol 3, pp. 3–58, 1999.

[4] J. W. Salisbury, L. S. Walter, N. Vergo, and D.M. D'Aria, "Mid-infrared (2.1–25 μm) spectra of minerals," Baltimore, MA: Johns Hopkins University Press, 1991b.

[5] A. R. Korb, P. Dybwad, W. Wadsworth, and J. W. Salisbury, "Portable FTIR spectrometer for field measurements of radiance and emissivity," Applied Optics, vol. 35, pp. 1679–1692, 1996.

[6] J. W. Salisbury, D. M. D'Aria, and E. Jarosevich, "Midinfrared (2.5-13.5 micrometers) reflectance spectra of powdered stony meteorites," Icarus, vol. 92, pp. 280–297, 1991a.

[7] N. H. Mahle and J. W. Ashley, "Application of a correlation coefficient pattern recognition technique

to low resolution mass spectra," Computers & Chemistry, vol. 3, no. 1, pp. 19–23, 1979.

[8] Z. Yang, J. Farison, and M. Thompson, "Fully constrained least squares estimation of target quantifications in hyperspectral images," in Proc. IPCV, pp. 910– 915, 2009.

[9] S. Rosario, "Iterative Algorithms for Abundance Estimation on Unmixing of Hyperspectral Imagery," Master of Science Thesis, University of Puerto Rico, Mayagüez, May 2004.

[10] S. P. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge Univ. Press, U.K., 2004.

[11] P. E. Dennison, K. Q. Halligan, and D. A. Roberts, "A comparison of error metrics and constraints for multiple endmember spectral mixture analysis and spectral angle mapper," Remote Sensing of Environment, vol. 93, no. 3, pp. 359–367, Nov. 2004.

[12] E. O. Brigham, "The Fast Fourier Transform," New York: Prentice-Hall, 2002.

[13] Li Jiang, "Wavelet-Based Feature Extraction for Improved Endmember Abundance Estimation in Linear Unmixing of Hyperspectral Signals," IEEE Trans. Geoscience Remote Sensing, vol. 42, no. 3, pp. 644–649, 2004.

[14] N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.C. Yen, C. C. Tung, and H. H. Liu, "The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis," Proc. R. Soc. London, Ser. A, 454, pp. 903–995, Jul. 1998.

[15] R. G. Stockwell, L. Mansinha, and R. P. Lowe, "Localization of the Complex Spectrum: The S Transform," IEEE Trans. on Acoust. Speech and Signal Processing, vol. 44, no. 4, pp. 998–1001, 1996.

[16] H. Xiong, S. Shekhar, P. N. Tan, and V. Kumar, "Exploiting A Support-based Upper Bound of Pearson's Correlation Coefficient for Efficiently Identifying Strongly Correlated Pairs," In ACM SIGKDD, Seattle, WA, USA 2004.

[17] M. S. Ramsey, and P. R. Christensen, "Mineral abundance determination: Quantitative deconvolution of thermal emission spectra," J. Geophys. Res., vol. 103, 577–596, 1998.

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