Keywords: Spectral Mixture Analysis, Spectral Unmixing, Hyperspectral Data
Abstract Spectral mixture analysis of hyperspectral data is performed to validate the
spectral unmixing technique worked based on the theory of linear mixture modeling.
High-resolution spectra of mixed soil and vegetation were collected for the analysis using a field
spectrometer. The instrument FOV was carefully validated to ensure the correctness the
mixture proportions of the sensed materials. A data set of designed sample targets with linear
variation of mixture proportions was collected. The data quality and the solution of
single-band unmixing were then investigated in the spectral space. Based on the analysis, a
weighted least-squares method is suggested to reinforce the solution of spectral unmixing.
1. Introduction
In remote-sensing imagery, the measured spectral radiance of a pixel is the integration of the
radiance reflected from all the objects within the ground instantaneous field of view (GIFOV).
Mixed pixels are generated if the size of the pixel includes more than one type of terrain cover.
Obviously, spectral mixing is inherent in any finite-resolution digital imagery of a heterogeneous
surface. Solving the spectral mixture problem (Horwitz et al., 1971) is, therefore, involved in
image classification, referring to the technique of spectral unmixing. The application of
spectral unmixing was limited due to the low spectral resolution of the sensors in the past. The
invention of imaging spectrometers (Goetz et al., 1985) promotes the potential of applying
spectral unmixing for image classification. The large number of spectral bands in
hyperspectral data allows the unmixing of very complex scenes to avoid intrinsic singular
problems. In addition, the hyperspectral data permits the direct identification of image-derived
endmember spectra (Boardman, 1994). However, the spectral mixing behaviors are still not
clearly known and modeled. Therefore, how muck these unknown behaviors will affect the
solution of spectral unmixing should be studied carefully.
Conceptually, the occurrence of multiple materials, or endmembers, within the GIFOV of a
single pixel leads to a composite or mixed spectral signal. The spectra of an endmember
ideally represent the signatures that would be recorded for pure or single-component pixels. A
pixel composed of multiple components can be mathematically approximated using a spectral
mixing model. Linear mixture modeling is commonly applied with the assumptions that
electromagnetic energy reflected from the surface interacts with a single component (Adams et
al., 1986) and a uniform weighting of radiance over the GIFOV area. However, non-linear
mixing may dominate frequently in reality when incident electromagnetic energy reacts with
more than one component before being reflected from the surface, for example the reflectance
from vegetation canopy. In addition, due to the fact that the sensor spatial spread function
results in a non-uniform response, spectrally determinate fraction coefficients do not correspond
with the proportions of endmembers distributed within a pixel. Certainly, the solution may be
distorted if a non-linear mixture case is treated as a linear case or the sensor problem is ignored.
Focused on the spectral mixing behaviors, this paper estimates the feasibility of the linear
mixture modeling.
2. Spectral Unmixing Based on the Linear Mixture Model
With known number of endmembers and giving the spectra of each pure component, the
observed pixel value in any spectral band is modeled by the linear combination of the spectral
response of component within the pixel. This linear mixture model can be mathematically
described as a linear vector-matrix equation,
where:
i = 1,. . .,m (number of bands);
j = 1,. . .,n (number of endmembers);
DN
i = spectral reflectance of the ith spectral band of a pixel;
R
ij = known spectral reflectance of the jth component;
F
j = the fraction coefficient of the jth component within the pixel;
E
i = error for the ith spectral band.
The error terms account for the unmodeled reflectance and represent the unknown noise of
observations. The relationship in Eq. (1) is constrained by the assumption that an exhaustive
set of endmembers (classes) has been defined, so that,
at each pixel. This assumption could be problematic, since one is never sure that a sufficient
number of endmembers has been defined for a given set of data.
Using the constrained least squares (CLS) method proposed by Shimabukuro (1991) can solve
the inversion problem, termed spectral unmixing. The objective of this method is to estimate
the fraction coefficients by minimizing the sum of the squares of the errors, subject to the
constraint given by Eq. (2). The estimated standard deviation of the error terms,
can be a quantitative measure of how the mixture modeling fits the data.
3. Data Collection and Fov Test
The GER 1500, a field portable spectroradiometer, is used for spectral data collection. This
instrument is capable to read 512 spectral bands, covering the UV, Visible, and NIR
wavelengths from 350 nm to 1050 nm. The instrument FOV is 4 degrees, and a sighting laser
is provided to aid in aligning the instrument to the target to be measured.
