PE&RS July 2019 - page 522

Background
SMA
SMA assumes that more than one land cover class exists in a
mixed pixel. The objective of SMA is to estimate the fraction of
each end member within a mixed pixel. Generally, two
constraints—sum-to-one f j
j
n
∑ = 1 and nonnegative (0 ≤fj ≤1)
—should be met in fully constrained SMA, which can be
expressed as
R f Rk j k j
j
n
k = + ∑ , , ε (1)
where Rk is the spectral reflectance of a mixed pixel on band
k, fj is the fraction of end member j within the pixel, Rk,j is the
spectral reflectance of end member j on band k, εk is the error
of band k, and n is the number of end members.
Mean absolute error (MAE) is used to evaluate the perfor-
mance of SMA. It calculates the absolute difference between
the estimated fraction and the reference fraction of the cor-
responding land cover class:
MAE
ABS
=
− ( )
= ∑ f f
m
e i r i
i
m
, ,
, 1 (2)
where fe,i and fr,i are the estimated and reference fractions,
respectively, of sample i and m is the number of samples.
Spectral Transformations
Thirteen linear and 13 nonlinear transformed schemes were
examined in this study, as well as the untransformed scheme.
These schemes were selected because they are the most com-
mon in the literature. Many commercial software applica-
tions, such as ERDAS and ENVI, have embedded these models.
Researchers can easily implement these transformed schemes
in their applications.
Linear Spectral Transformations
Seven linear spectral transformations were examined (see
Table 1): derivative analysis (DA; Tsai and Philpot 1998), PCA
(Richards and Richards 1999), ICA (Hyvarinen 1999; Hyvärin-
en and Oja 2000), MNF (Green et al. 1988; Boardman and Kruse
1994), TC (Kauth and Thomas 1976), band normalization (BN ),
and DWT (Vetterli and Herley 1992; Strang and Nguyen 1996;
Li 2002). Detailed descriptions of these methods follow.
DA is sensitive to curve shape instead of the scale of re-
flectance (Tsai and Philpot 1998). With this advantage, DAs,
especially those of a higher order, are effective for eliminating
background signals and illumination effects caused by cloud
coverage, sun angle, and topography. Detailed calculation of
first, second, and third DAs is given by Tsai and Philpot (1998).
The purpose of PCA (Richards and Richards 1999; Chu-
vieco and Huete 2009) is to create a new set of orthogonal
axes which can maximize data variance. PCA compresses the
original intercorrelated data into several uncorrelated vari-
ables, called principal components. The amount of variance
decreases with increasing component number. Generally, the
first three components in PCA contribute more than 90% of
the variance in the original data set.
ICA is based on a non-Gaussian assumption of indepen-
dent sources and applies higher-order statistics to extract the
characteristics in non-Gaussian data sets (Hyvarinen 1999;
Hyvärinen and Oja 2000). It commonly contains three steps:
centering and whitening sample data with the mean, eigen-
vectors, and eigenvalues; conducting negentropy maximiza-
tion with the whitened samples to estimate the ICA transform
matrix; and transforming the original data using the indepen-
dent components transform matrix.
MNF transformation, like PCA, separates noise and reduces
data dimension (Green et al. 1988; Boardman and Kruse
1994). First, in the noise-whitening step, the noise covariance
matrix of PCA is used to decorrelate and rescale the noise in
the data. Second, a rotation based on the first step’s outcome is
implemented. Specific channel information can be maintained
in MNF, since each component’s weighting is contributed by
all the original bands. Most of the variance can be explained
with the first three components, while the remaining compo-
nents are contributed mainly by noise (Boardman 1993).
Similarly, TC transformation orthogonally transforms the
original data into a three-dimensional space (Kauth and
Thomas 1976) of brightness, greenness, and wetness (Jen-
sen and Lulla 1987). If the data set is Landsat-7 Enhanced
Thematic Plus with six spectral bands, results also contain
three more outputs—the fourth, fifth, and sixth components.
The first TC band relates to the overall brightness of the image,
while the second output band corresponds to the degree of
greenness. The third band indicates the wetness of the land
surface. TC transformation was originally designed to maxi-
mize separation of the different growth statuses of vegetation.
BN reassigns the range of the pixel values in each band
linearly. Contrasting information is well presented in the
stretched output. BN has two steps: searching the maximum
and minimum values in a band and calculating BN through
normalization for each pixel:
BNk k k k k R R R R = −( ) − ( )
,min ,max ,min , (3)
where BNk is the BN value in band k, Rk is original spectral re-
flectance in band k, and Rk,max and Rk,min are the maximum and
minimum values, respectively, of band k. Light materials will
be displayed as lighter colors, while dark regions will appear
in darker colors.
DWT can be implemented through a fast wavelet transform
(Vetterli and Herley 1992; Strang and Nguyen 1996; Li 2002).
The mother wavelet is represented by a set of HP and LP filters
in the filter bank. In the beginning, the original image signal
goes through the filter bank. The result of the HP filter is the
detail coefficients,while the result of the LP filter is called
Table 1. Spectral transformations.
Transformation Linearity Reference
DA1–3 Linear Tsai and Philpot 1998
PCA Linear Pearson 1901; Byrne et al. 1980;
Richards and Richards 1999
ICA Linear Bayliss, Gualtieri and Cromp 1998;
Chen and Zhang 1999; Hyvärinen and
Oja 2000
MNF Linear Green, Berman, Switzer and Craig
1988; Boardman and Kruse 1994
TC Linear Kauth and Thomas 1976
BN Linear
DWT1–5 Linear Vetterli and Herley 1992; Strang and
Nguyen 1996; Li 2002
CR Nonlinear Kruse 1988
GHP, GLP, HP, LP Nonlinear Green, et al. 1988; Schowengerdt
2006; Yu et al. 2006
NSMA Nonlinear Wu 2004
Tie1–7 Nonlinear Asner and Lobell 2000
BN: band normalization; CR: continuum removal; DA: derivative
analysis; DWT: discrete wavelet transform; GHP : Gaussian high-pass;
GLP: Gaussian low-pass; HP: high-pass; ICA: independent component
analysis; LP: low-pass; MNF: minimum noise fraction; NSMA: normal-
ized spectral mixture analysis; PCA: principal components analysis;
TC: tasselled cap; Tie: tie spectral transformation.
522 July 2019 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING