Characterizing urban subpixel composition using spectral mixture analysis - Urban Remote Sensing: Monitoring, Synthesis and Modeling in the Urban Environment - page 108

orthogonal in spectral space (Adams and Gillespie, 2006). These
may represent conflicting goals; for example, in urban envi-
ronments bright impervious spectra such as concrete may be
spectrally similar to bare soil, even when hyperspectral data are
analyzed (Herold et al ., 2004). Additionally, not all spectral com-
ponents in the scene can be represented in SMA models because
the technique greatly simplifies reality (Song, 2005), even when
MESMA is implemented. The goal in building a spectral library,
therefore, is to capture the dominant spectral components of the
scene that are physically meaningful and specific to the goals of
analysis. Spectral libraries should be constructed on a case-by-
case basis and may utilize a number of strategies simultaneously
(Smith et al ., 1990; Franke et al ., 2009).
One method for endmember selection is to extract spectra
of pixels from homogeneous areas of known materials; often
finer spatial resolution imagery, such as an aerial photograph,
is used to guide this process (e.g., Rashed et al ., 2003; Wu and
Murry, 2003; Myint and Okin, 2009). While this methodology
is straightforward, it may not capture the most representative
endmembers for a given material type. A second, very common,
approach for selecting endmembers is to generate a series of
two-dimensional plots; the bounding envelope of the plotted
pixels can be thought of as the ''mixing space'' for that image.
The vertices of the mixing space are considered the ''purest''
pixels in the scene and serve as candidate endmembers (e.g.,
Adams, et al ., 1986; Roberts, et al ., 1998a; Small, 2005). These
two strategies may also be applied in tandem, as each provides
complementary information (e.g.,Weng, Hu and Lu, 2008;Weng
and Lu, 2009).
One commonly used, semiautomated approach to identify
pixels that represent the extremes of mixing space for a particular
image is the pixel purity index (PPI). The PPI algorithm projects
the spectrum of each pixel onto multiple unit vectors randomly
oriented in N -dimensional space (where N is the number of
image bands) and records the number of times the pixel is found
to land on vertices of the mixing space (Boardman et al ., 1995).
Pixels with high PPI counts are candidate image endmembers,
though they should be visually inspected to verify that they
can be assigned to a specific material class, because spectrally
extreme pixels are not always physically meaningful (Rashed
et al ., 2003; Powell et al ., 2007; Rashed, 2008; Franke et al ., 2009).
For example, the PPI algorithm will often identify as extreme
pixels that are saturated in one or more bands or pixels located
on sharply defined borders, such as between land surface and
water (Powell et al ., 2007).
More recently, several methods have been proposed for
endmember selection, particularly when building a spectral
library that contains multiple spectra per material class. These
approaches include count-based endmember selection (CoB),
endmember average root mean square error (EAR), and min-
imum average spectral angle (MASA). In contrast to PPI and
N -dimensional visualization, which depend on the mixing space
as defined by a particular image for endmember selection, each
of these newer approaches starts with a large library of candidate
endmembers and applies the two-endmember case of MESMA to
identify representative spectra (Roberts et al ., 1998b; Dennison,
et al ., 2004). In other words, every spectrum in the candidate
library (plus shade) is used to unmix every other spectrum in
the library. The goal is to rank endmembers that best represent
their material class (e.g., vegetation, impervious surface, soil)
and/or are distinctly different from spectra belonging to other
classes.

CoB identifies the endmembers within a library that model
the greatest number of endmembers within their class and the
fewest endmembers outside of their class (Roberts et al ., 2003).
Candidate endmembers are used to model all other spectra in the
library, subject to fraction, RMS, and/or residual constraints. For
each potential endmember, a count of the total number of spectra
that are modeled within the same class ( in_CoB )andthenumber
of spectra modeled outside the class ( out_CoB ) are recorded. The
optimal endmember for each class is that which has the highest
in_CoB and lowest out_CoB values, where an ideal endmember
would model all of the spectra within its class and none of the
spectra in other classes (Roberts et al ., 2003; Franke et al ., 2009).
The goal of EAR is to identify themost representative spectrum
for amaterial class in a library based on RMS error (Dennison and
Roberts, 2003). The procedure starts with a library of potential
endmember spectra, grouped by class. In this case, the two-
endmember model is applied without constraints. The average
RMS error (EAR) for a given endmember modeling all other
spectra in its class is calculated and assumed to measure how
representative that endmember is of other spectra in its class. The
optimal endmember for each class is the endmember with the
lowest EAR (Dennison and Roberts, 2003; Franke et al ., 2009).
The goal of MASA is similar to EAR, except that the measure
of fit is based on a spectral angle metric instead of RMS error
(Dennison et al ., 2004). Spectral angle mapping (SAM) measures
similarity between two spectra by calculating the spectral angle
between the two spectral vectors (Kruse et al ., 1993). The angle
itself is analogous to RMS error; if the angle falls below a user-
defined threshold, the spectra are identified as belonging to the
same class. MASA is the mean spectral angle between each poten-
tial endmember modeling the other spectra in its own class. The
endmember with the lowest MASA is considered most represen-
tative of its class (Dennison et al ., 2004; Franke et al ., 2009). For
a given endmember library, the optimal endmembers selected by
MASA and EAR will be similar, but may not be identical. RMS
error calculated from MESMA tends to better identify optimal
endmembers for classes with high-albedo spectra, while the spec-
tral angle calculated by SAM better identifies endmembers for
classes with dark-albedo spectra (Dennison et al ., 2004).
Because each method of selecting endmembers is sensitive
to different criteria, the analyst may choose to simultaneously
implement multiple methods and experiment with howwell each
set of ''optimal'' endmembers models the data being analyzed.
Ultimately, performance of potential endmembers should be
assessed based on how well the spectra model the image being
analyzed (Franke et al ., 2009).

8.2.3 SMAmodels

After a preliminary endmember library has been constructed,
the next step is to determine the rules for selecting spectral
mixture models. MESMA tests multiple models and identifies
the one ''best-fit'' model for each pixel based on selection criteria
and a measure of goodness-of-fit (Franke et al ., 2009). This
process involves three steps: (a) specifying which combinations
of endmembers are allowed, (b) identifying model constraints so
that candidate models more accurately represent reality, and (c)
determining the criteria to select the overall ''best-fit'' model for
each pixel (Fig. 8.3, Step 2).
The maximum number of endmembers that can be modeled
per pixel is equal to the number of spectral bands in the image, if