Spectral mixture analysis has been shown to be a powerful, multifaceted tool for analysis of multi- and hyper-spectral data. Applications of AVIRIS data have ranged from mapping soils and bedrock to ecosystem studies. During the first phase of the approach, a set of end-members are selected from an image cube (image end-members) that best account for its spectral variance within a constrained, linear least squares mixing model. These image end-members are usually selected using a priori knowledge and successive trial and error solutions to refine the total number and physical location of the end-members. However, in many situations a more objective method of determining these essential components is desired. We approach the problem of image end-member determination objectively by using the inherent variance of the data. Unlike purely statistical methods such as factor analysis, this approach derives solutions that conform to a physically realistic model

Forsyth, Donald W.

Mustard, John F.

Pieters, Carle M.

Tompkins, Stefanie

English

NASA Technical Reports Server

N95- 23888
OBJECTIVE DETERMINATION OF IMAGE END-MEMBERS IN
SPECTRAL MIXTURE ANALYSIS OF AVIRIS DATA
Stefanie Tompkins, John F. Mustard, Carle M. Pieters, Donald W. Forsyth
Dept. Geol. Sci., Box 1846, Brown University, Providence, RI 02912
INTRODUCTION
Spectral mixture analysis has been shown to be a powerful, multifaceted tool for
analysis of multi- and hyper-spectral data (Adams et at., 1986; Smith et al., 1990).
Applications to AVIRIS data have ranged from mapping soils and bedrock to ecosystem
studies (e.g. Gamon et al., 1993; Roberts et al., 1993; Mustard et at., 1993; Kruse et al.,
1993). During the first phase of the approach, a set of end-members are selected from an
image cube (image end-members) that best account for its spectral variance within a
constrained, linear least squares mixing model. These image end-members are usually
selected using a priori knowledge and successive trial and error solutions to refine the total
number and physical location of the eud-members. However, in many situations a more
objective method of determining these essential components is desired. We approach the
problem of image end-member determination objectively by using the inherent variance of
the data. Unlike purely statistical methods such as factor analysis, this approach derives
solutions that conform to a physically realistic model.
DAMPED LEAST SQUARES MODEL
The underlying assumption of specmd mixture analysis is that each pixel on the
surface is a physical mixture of several comlxments, and the spectrum of the mixture is a
linear combination of the end-member reflectance spectra. The spectral variability of a
scene is thus modeled as a line;u" combination of a small number of image end-members.
The fundamental equations of spectr,'d mixture _malysis are:
_Rb = _ _ ribfi + _Eb (1) and _fi = 1.0 (2)
b b i b i
where fi is the fractional abundance of end-member i in the pixel, Rb is the total
reflectance of the pixel in band b, rib is tile reflectance of end-member i in band b, and E b
is the residual error in band b. In a typic_d spectral mixing analysis, the end-members are
prescribed and the fractional abund,'mces and en'ors ,are then calculated for each pixel,
through a simple least squares inversion of the form:
m=_TG)-tG1_ O)
The matrix G consists of tile p,'u'tial derivatives of the model equations given by (1) and
(2). The vectors d and m represent tile data (Rb) and the model parameters (fi)
respectively. A two-dimensional graphical representation of mixture analysis is shown in
Figure 1. The image end-members (indicated by squares) should reside as close as
possible to the boundaries of the data cloud. Fractions that are >1.0 or <0.0 occur for
data points that lie outside the region defined by the end-members, and cannot be entirely
eliminated no matter how well chosen file hnage end-members.
Consider, instead, ,'m approach where the end-member spectra are not prescribed,
but are treated as unknowns _doug with the fractional abundances. The model equations
are the same as stated above, but because F i and rib ,are both unknowns, they must be
-'iq
/
,/
177
https://ntrs.nasa.gov/search.jsp?R=19950017468 2020-06-16T08:29:20+00:00Z
Figure 1:
S_t
--Cl
a I
B_d 2
A, B, and C indicate image end-members. A', B', and C' are
the model-derived end-members.
Figure 2:
0.81 [I .... I,,,,t,,,,! .... .'''''t.... I ............. I .... i! .... [i
0.2 !
o i .... , .... , .... , .... , .... , .... , .... , .... 1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 09
wavelength
Solid lines represent image end-member spectra. Dashed lines
are model-derived virtual end-members.
solved non-linearly. The approach we have chosen employs a damped least squares non-
linear inversion technique _L,;presented by "l'_uantola ,and Valette (1982). In short, a
starting model is provided (a suite of possible cud-member spectra, estimated fractional
abundances, and the image data to be modeled). Constraints on the solutions are imposed
as additional equations (i.e. I]actions lnust sum to 1.0) or as allowable deviations from
the starting model (damping of solutions). Both the stating model and the constraints are
based on a priori knowledge. Each successive iteration of the equations results in a
calculated change to the previous model that will reduce the error of the fit. These
changes are incorporated ,'rod the inversion is run successively until a prescribed error
threshold is surpassed, as given by:
mj+ 1 =mj + [GjTCnn-I...G__j + Cram-l] [C___jTCnn-I(J_.- g(mj))- .C,.,nm-l(mj - .u_)] (4)
In this case, the model par;uneters ,'rod G-matrix change with each iteration (.tllj is the
model parameter vector at the jth iteration) _md include both fi and rib from equations (1)
and (2). Cnn and Cmm are covarizmce matrices for the data and the model parameters,
respectively. The term g0!!! i) is the predicted vzdue of the data vector when the forward
178
modelisrun,so(d- g(lnj))simplydescribethe residual error. The starting model
parameters, found in II]o, are included at each iteration to penalize large deviations from
the starting model and thus damp tile final solution.
The model results are therefore intimately guided by the inherent spectral
variability of the data, the starting model, and the constraints imposed on the solutions.
The calculated end-member specua ,'ue driven to better bound the data cloud. This is
shown in Figure 1 (new end-members indicated by circles). The effects in a spectral sense
are illustrated conceptually in Figure 2 (initial end-member spectra are shown by solid
lines and calculated end-members by dashed lines). Although the new end-member spectra
are not necessarily identical to any pixel spectrum (they can lie outside the data cloud),
they provide the best mathematical fit to the whole data set and are determined using a
model which is based on a physically realistic description of the surface. One might
think of these as virtual end-members.
PRELIMINARY RESULTS
Artificial Data Set
This model was developed ,'rod refined on a test data set where the end-members
and their abundances are ka_own perfectly. This data set provides a working template for
establishing the necessary conditions under which the best solution can be derived. The
model successfully reproduces the true end-member spectra in the test data sets for a
variety of initial conditions, provided that the model is minimally constlained. An
additional step in spectral mixture analysis is determining the optimum number of end-
members. Using the test data set, model solutions for fewer than and greater than the
correct number of spectral end-members were ,'malyzed. When too few end-members were
used, the solution failed to meet the original constraints on the model. The residual error
steadily declined as the number of end-members increased until the optimum number was
reached; thereafter, the improvement in error was statistically insignificant. Thus, the
evolution of error and successful satislaction of model constraints as a function of the
number of end-members can be used to threshold the optimum solution. We continue to
use the artificial data sets to develop a better understanding of the model, and establish
limits on the amount of a priori information that is needed to calculate the true end-
member spectra.
Lunar CCD Images
The model has also been applied to 8-band CCD images of the lunar crater
Builialdus (Tompkins et al., 1992). This requh'ed some restructuring of the program due
to the increased number of spectral ch_mnels mid pixels; however, the model is the same.
The starting model is guided by previous investigations (Tompkins et al., 1992; Pieters,
1991). Initial results on stmdl data sets withit_ the image show generally the same results
as were obtained with the traditional mixing approach, but provide a much lower rms
error and only very rare occurrences of superlx_sitive (>100%) or negative abundances. In
mixing models, the best image end-members c:m be considered those that best describe the
data cloud (Figure 1), for all bands simultaneously. As was found with the test data, the
nonlinear inversion defines virtual end-membcrs that are outside the data cloud, while still
providing a physically reasonable solution.
AVIRIS Data
The model is now applied to :m AVIRIS image from the western foothills of the
Sierra Nevada. Materials found within the data include dry, grass-covered hills, irrigated
179
orchards, and a complex substrate with rapidly changing bedrock composition (Mustard,
1993). Because the memory requirements for the model increase multiplicatively with the
number of pixels and spectral ch,'mnels in a data set, the initial application will be to a
subset of the image, using a lhnited number of channels. The fraction images that result
from using virtual end-members will be comp;ued to those derived through the traditional
mixing model.
One of the strengths of spectred mixture ,analysis is the ability to calibrate raw
data to reflectance by linking the image end-members to reference end-members from a
spectral library. This can be especially adv:mULgeous when using data from systems with
poorly defined characteristics or areas lacking well characterized reference end-members.
Though this can lead to non-unique solutions for the reference end-members, common
sense and geologic context help govern the choice of library end-members to which the
image end-members are calibrated. The data inversion approach presented here may confer
a distinct advantage for proceeding to the calibration stage. The virtual end-members are
thought to be more spectrally pure th,'m the image end-members, thereby reducing the
number of non-unique solutions. Future work will involve testing this type of
calibration using the AVIRIS bnages, as well as further constraining the mixture model.
REFERENCES
Adams, J.B., M.O. Smith, and P.E. Johnsou, 1986, "Spectral mixture modeling: a new
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Gamon, J., C.B. Field, D.A. Roberts, S.L. Ustin, and R. Valentini, 1993, "Functional
Patterns in an Annual Gr,'tssland during ,'m AVIRIS Overflight," Rein. Sens. Env., vol.
44, pp. 239-253.
Kruse, F.A., A.B. Lekhoff, _md J.B. Dietz, 1993, "Expert System-Based Mineral
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Objective determination of image end-members in spectral mixture analysis of AVIRIS data

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