[

Optimized Cross-Correlation Spectral Matching In ”traditional” CCSM, the reference spectrum z to which

y

is compared is taken to be a single (pure) endmember.
Next assume M endmembers

X_{k}

of interest, and linear mixing. Seek weighting factors

{a_{k}; k = 1, \dots M}

at match position

m = 0

, places that maximum at

m = 0

, and minimizes its skew, subject to

0 \leq a_{k} \leq 1

and

\sum_{k = 1}^{M} a_{k} = 1

. Significance of different values of

r_{0}

resulting from different choices of endmember set

{X_{k}; k = 1, \dots M}

may be assessed using eq. (10). Coulter [2] chooses endmember set that maximizes

r_{0}

. As written (12) is the cross-correlation of two unit vectors:

which is independent of the normalization of

a

. Likewise, if

\vec{y}

is spectrum at a particular image pixel, then the unit normalization

\hat{y}

rends the

r_{m}

somewhat independent of shadow and striping.

Equation (14) makes

r_{m}

non-linear in the

a_{k}

; eq (13) may be maximized w.r.t the

a_{k}

by the non-linear constrained optimizer of one’s choice. It does not give much insight into the relative band-to-band errors inherent in

y

. As written, it assumes they are all equal, and we can pretend:

4 Linear and Non-Linear Least-Squares Fitting

Least-Squares Fitting Since

r_{m}

is expressed in terms of unit vectors, maximizing

r_{0} = \hat{y} \cdot \hat{z}

is equivalent to minimizing

Minimizing the squared-difference of unit vectors

\hat{y}

and

\hat{z}

in (16) is still a non-linear problem in the

a_{k}

because of how they appear in the denominator of the normalization of

\hat{z}

(14). However, if we relax the restrictions that z be a unit vector (and that

\sum a_{k} \equiv 1

), we can define the equivalent problem

which is linear in the

a_{k}

, but not yet quite as general as we want. At each band

i

there are (at least) three uncertainties in the sensor measurement: those in the received radiance

y_{i}

, those in the center wavelength

x_{i}

, and those in the sensor’s bandwidth, assumed as a fwhm of a gaussian. Fwhm is typically on order of band-to-band wavelength spacing. Fwhm errors are usually of secondary significance and will be ignored.

If we allow for uncertain band centers

x_{i}

and assume normal independent distributions, we can write

as joint probability sensor records band radiance

y_{i}^{0}

at recorded wavelength

x_{i}^{0}

, when it actually received (unknown) radiance

y_{i}

at actual (and unknown) wavelength

x_{i}

. Joint probability of pixel’s measured spectrum across N bands is

Associate the (unknown) actual spectrum

y (x_{i})

with the modeled mixture

z (x_{i}; a_{1}, \dots a_{M}) = \sum_{k = 1}^{M} a_{k} X_{k} (x_{i})

. Then

represents a constrained optimization problem wherein we wish to minimize

χ^{2}

as a function of

N + M

variables

(a_{1}, \dots a_{M}, x_{1}, \dots x_{N})

subject to

a_{i} \geq 0 \forall i

. The motivation for including the

(x_{1}, \dots x_{N})

is to (hopefully) obtain ”more” non-negative

a_{k}

in the optimal solution.

Eqs (28) and (30) make eq. (27) quadratic in the

a_{k}

, so the optimization including center wavelengths uncertainties is non-linear. The

\frac{\partial X_{j} (x)}{\partial x}

can be computed efficiently enough, but the

X_{k} (x_{i})

will need to be reconvolved against the new center wavelengths for each new set of center wavelengths

{x_{i}; i = 1 \dots N}

required during the optimization. The problem will obviously be much simpler if we assume the provided center wavelength values are ”good enough” and simply ignore eqs (27). Dropping the superscript

0

, equations (26) then become

which are the usual linear least-squares normal equations for the

a_{k}

. For what its worth, for the unconstrained problem the minimal solution vector

{a_{k}; k = 1, \dots M}

will be unique provided the matrix

M_{i j} = X_{i} \cdot X_{j}

is non-singular.

5 Image Classification and Statistics

5.1 Image Mean Vector and Covariance Matrix

Image Mean Vector and Covariance Matrix [11, White 2005] If image has

P

bands and

N

pixels, the mean vector is

M_{PxN}

is the matrix of

N

identical mean vectors (

P

rows by

N

columns):

5.2 Principal Component Transformation

Principal Component Transformation [7, Smith 1985] Karhunen-Loeve Transformation[12, White 2005] GRASS imagery i.pca: Let

Magnitudes of

λ_{i}

impose ordering on transformed component vectors

Z_{i k} = \sum_{j = 1}^{P} a_{i j} X_{j k}

. Those with the largest

λ_{i}

s.t.

λ_{i} ∕ λ_{m a x} > t o l

are the Principal Components.

t o l

should be related to the noise floor.

5.3 Minimum Noise Fraction

Minimum Noise Fraction[10, pg. 38] [3, ] We wish to find a particular coefficient matrix

{a_{i j}; i, j = 1, \dots P}

that in some sense maximizes the image S/N, assuming the image pixel vectors

{{\vec{X}}_{k}, k = 1, \dots N}

are the sum of uncorrelated signal and noise:

where

λ_{i}

is generalized eigenvalue of

C_{PxP}

wrt

C_{PxP}^{N}

, and

a_{i}

are corresponding generalized eigenvectors. Compare with PCA:

Noise Covariance

C_{PxP}^{N}

Green suggests

C_{PxP}

be of unit variance and band-to-band uncorrelated.

Homogeneous Area Method [9, sec 2.9.1] If possible find homogenous area of

N_{h}

pixels in image:

Other methods: ”unsupervised training” derived-endmember classification schemes e.g. LAS’ search ([13, ]) and GRASS’ cluster/maxlik are based upon local covariance minimization.

6 Vertex Component Analysis

Vertex Component Analysis is an “unsupervised training” derived-endmember classification scheme. We use the notation of [5, Nascimento and Dias], which is slightly different than that used in previous sections.

6.1 VCA Algorithm

Our goal is to find a abundance vectors

{α_{i}, i = 1, \dots N}

corresponding to some endmember set

{m_{j}, j = 1, \dots p}

. An appropriate endmember set

{m}

is to be determined as part of the VCA algorithm. Endmember identification – the matching one or more of the VCA generated endmembers to actual specimen spectral samples such as from USGS spectral library or field ground-truth sampling – may be done in a subsequent processing step.

Since the set

{α \in ℜ^{p} : 1^{T} α = 1, α ≽ 0}

is a simplex, the set

S_{x} = {x \in ℜ^{L} : x = M α, 1^{T} α = 1, α ≽ 0; γ \geq 0}

is also a simplex. However, even assuming

n = 0

, th observed vector set belongs to convex cone

C_{p} = {r \in ℜ^{L} : r = M γ α, 1^{T} α = 1, α ≽ 0; γ \geq 0}

owing to different scale factors

γ

But the projective projection of the convec cone

C_{p}

onto a properly chosen hyperplane is a simplex with vertices corresponding to the vertices of simplex

S_{x}

. The simplex

S_{p} = {y \in ℜ^{L} : y = r ∕ (r^{T} u), r \in C_{p}}

is the projective projection of the convex cone

C_{p}

onto the hyperplane

r^{T} u = 1

, where the choice of

u

assures there are no observed vectors

r

orthogonal to the hyperplane:

r_{i}^{T} u ∕ | u | < | r |, 1 = 1, . . . N

6.1.1 SNR

6.1.2 Algorithm 1: Vertex Component Analyis (VCA)

end if
14:

A : = [e_{u}, 0, 0 \dots 0]; e_{u} = {[0, \dots, 0, 1]}^{T}

initialize

A

, a pxp auxiliary matrix, and

A^{+}

by SVD
15: for (

i = 1 to p

) do

22: end for
23: if

(SNR > {SNR}_{th})

then
24:

\hat{M} : = U_{d} {[X]}_{:, [i n d i c e]}; \hat{M}

is a

L \times p

estimated mixing matrix.
25: else
26:

\hat{M} : = U_{d} {[X]}_{:, [i n d i c e]} + \bar{r}; \hat{M}

is a

L \times p

estimated mixing matrix.
27: end if

References

[1] C.-I Chang and Q. Du. Estimation of number of spectrally distinct signal sources in hyperspectral imagery. IEEE Transactions on Geoscience and Remote Sensing, 42(3):608–619, March 2004.

[2] David W. Coulter. Remote Sensing Analysis of Alteration Mineralogy Associated with Natural Acid Drainage in the Grizzly Peak Caldera, Sawatch Range, Colorado. PhD thesis, Colorado School of Mines, Golden, Colorado, 2006.

[3] A.A. Green, M. Berman, P. Switzer, and M.D. Graig. A transformation for ordering multispectral data in terms of image quality with implications for noise removal. Journal of Geophysical Reseach, 90:797 – 804, 1988.

[4] Fred A. Kruse. Comparison of aviris and hyperion for hyperspectral mineral mapping. http://w.hgimaging.com/PDF/Kruse_JPL2002_AVIRIS_Hyperion.pdf, 2002.

[5] José M. P. Nascimento and José M. Bioucas Dias. Vertex component analysis: A fast algorithm to unmix hyperspectral data. IEEE Transactions on Geoscience and Remote Sensing, 43(4), April 2005.

[6] William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes in C. Cambridge University Press, Cambridge, New York, Port Chester, Melbourne, Sidney, 1988.

[7] M.O. Smith, P.E. Johnson, and J.B. Adams. Quantitative determination of mineral types and abundances from reflectance spectra using principal component analysis. IEEE Transactions on Geoscience and Remote Sensing, 36:65 – 74, 1985.

[8] Frank D. van der Meer. Extraction of mineral absorbtion features from high-spectral resolution data using non-parameteric geostatistical techniques. International Journal of Remote Sensing, 15:2193–2214, 1994.

[9] Frank D. van der Meer and Steven M. de Jong. Imaging Spectroscopy. Kluwer Academic Publishers, Dordrecht, Boston, London, 2001.

[10] Frank D. van der Meer, Steven M. de Jong, and W. Bakker. Imaging Spectroscopy: Basic analytical techniques, pages 17–62. Kluwer Academic Publishers, Dordrecht, Boston, London, 2001.

[11] R. A. White. Image mean and covariance: http://dbwww.essc.psu.edu/lasdoc/user/covar.html, 2005.

[12] R. A. White. Karhunen-loeve transformation: http://dbwww.essc.psu.edu/lasdoc/user/karlov.html, 2005.

[13] R. A. White. Search unsupervised training site selection: http://dbwww.essc.psu.edu/lasdoc/user/search.html, 2005.

$C_{PxP}$	the symmetric positive-definite image covariance matrix
${\vec{a}}_{i}$	are its orthonormal eigenvectors with eigenvalue $λ_{i}$ .

[

Contents

1 Introduction

2 Cross-Correlation Spectral Matching

3 Optimized Cross-Correlation Spectral Matching