Multilayer NMF for Blind Unmixing
of Hyperspectral Imagery with
Additional Constraints
Lei Chen, Shengbo Chen, and Xulin Guo
Abstract
Due to the coincidence of hyperspectral reflectance nonnega-
tivity (and its corresponding abundance) with nonnegative
matrix factorization (
NMF
) methods,
NMF
has been widely ap-
plied to unmix hyperspectral images in recent years. However,
many local minima persist because of the nonconvexity of
the objective function. Thus, the nonnegativity constraint is
not sufficient and additional auxiliary constraints should be
applied to objective functions. In this paper, a new approach
we call constrained multilayer
NMF
(
CMLNMF
), is proposed for
hyperspectral data. In this approach, the mixed spectra are re-
garded as endmember signatures that has been contaminated
by multiplicative noise. The purpose of
CMLNMF
is to eliminate
noise by hierarchical processing until the endmember spectra
are obtained. Also, the hierarchical processing is self-adaptive
to make the algorithm more effective. Furthermore, in each
layer two constraints are implemented on the objective func-
tion. One is sparseness on the abundance matrix and the
other is minimum volume on the spectral matrix. The hierar-
chical processing separates the abundance matrix into a series
of matrices that make the characteristic of sparseness more
obvious and meaningful. The proposed algorithm is applied
to synthetic data and real hyperspectral data for quantitative
evaluation. According to the comparison with other algo-
rithms,
CMLNMF
has better performance and provides effective
solutions for blind unmixing of hyperspectral image data.
Introduction
Hyperspectral images contain hundreds of continuous bands
including visible, near-infrared, and shortwave infrared
spectral bands that can describe the target spectral feature in
detail (Landgrebe, 2002). However, due to the complexity of
targets in one pixel and low spatial resolution of image data,
more than one distinct object exists in one pixel of a hyper-
spectral image (Heinz
et al
., 2001; Keshava
et al
., 2002). To
get more accurate information about each object, it is neces-
sary to extract the endmember spectra and the corresponding
fraction from the image data. Spectral unmixing is a typical
technique for endmember extraction and abundance fraction
inversion that has been widely discussed in many fields, such
as signal analysis, target recognition, and classification (Wu
and Strahler, 2003; Heinz
et al
., 2001; Keshava
et al
., 2002).
Generally, there are two different types of spectral unmix-
ing models based on an assumption of light process transmis-
sion (Somers
et al.
, 2011). One is the linear spectral mixture
model (
LSMM
) and another is non-linear spectral mixture
model (
NLSMM
).
NLSMM
takes object’s multiple reflection into
account, which requires sufficient, appropriate parameters,
and thus making the model much more difficult to implement
(Li and Strahler, 1985 and 1986; Chen
et al
., 2013). On the
contrary,
LSMM
, which assumes that the different endmembers
and pixels do not interfere with each other (Lillesand
et al
.,
2014), has been widely used for spectral unmixing due to its
simplicity and efficiency (Ichoku and Karnieli, 1996; Miao
et
al
., 2006). Recently, a variety of methods based on
LSMM
have
been proposed for hyperspectral unmixing by analyzing the
spectral and spatial features of image data.
Nonnegative matrix factorization (
NMF
) is one of the
methods based on
LSMM
that is proposed for solving blind
source separation problems (Lee and Seung, 1999 and
2001). The purpose of
NMF
is to decompose one nonnegative
matrix into two nonnegative matrices. Consistent with the
characteristics of nonnegativity of hyperspectral data,
NMF
has been successfully developed to separate hyperspectral
imagery (Liu
et al
., 2011; Ye
et al
., 2015; Jia and Qian, 2009;
Esser
et al
., 2012; Miao and Qi, 2007; Yang
et al
., 2015; Yu
et al
., 2007). Considering objective function differences, the
general
NMF
has three distinct expression forms: maximum
likelihood estimation, divergence derivation, and Euclidean
distance (Lee and Seung, 1999 and 2001; Bakir
et al
., 2006).
The Euclidean distance has been most commonly used due
to the simplicity of calculation and high efficiency. However,
due to the nonconvexity of Euclidean distance, the
NMF
based
on Euclidean distance has many local minima which lead
to non-unique separated matrices (Liu
et al
., 2011; Jia and
Qian, 2009). Therefore, to get accurate unmixing results, the
spectral features and spatial features of image data have been
analyzed to add to
NMF
as a constraint to make the separated
results unique (Esser
et al
., 2012; Jain and Rai, 2012).
Features, including independence, smoothness, and
sparseness, have been largely used as constraint conditions by
many researchers for hyperspectral unmixing (Liu
et al
., 2011;
Yu
et al
., 2007). From the view of statistical theory, the end-
member spectra or abundance fraction of each endmember is
independent from one another (Hyvärinen
et al
., 2004). Thus,
the principle of generating original signal as independently
as possible from the mixed spectra can be used for analysis.
This theory has been widely used by independent compo-
nent analysis (
ICA
) for multichannel audio signal separation,
Lei Chen is with the College of Geo-exploration Science and
Technology, Jilin University, Changchun 130026, China; and
the College of Urban and Environmental Science, Tianjin
Normal University, Tianjin, 300387, China.
Shengbo Chen is with the College of Geo-exploration Science
andTechnology, Jilin University, No.938, Ximinzhu Street,
Changchun, 130026, China
Xulin Guo is with the Department of Geography and
Planning, University of Saskatchewan, 117 Science Place,
Saskatoon S7N 5C8, Canada
Photogrammetric Engineering & Remote Sensing
Vol. 83, No. 4, April 2017, pp. 307–316.
0099-1112/17/307–316
© 2017 American Society for Photogrammetry
and Remote Sensing
doi: 10.14358/PERS.83.4.307
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
April 2017
307
