IEEE Geoscience and Remote Sensing Magazine - June 2021 - 68

sum-to-one constraint (ASC). The resulting basic blind SU
model can be written as
min 2
1
EA,
YEAE,AEA s.t.
-+UX !C,
2
F
where the constraint C is
C:, ,.100 1$$ " EA A
==<
,
On the basis of the model (19), Yang et al. [96] proposed
sparse NMF for SU with a well-designed S-measure
sparseness. Qian et al. [97] imposed the sparsity constraint
on abundances and used
,
1 -
to -q, norm q01
##
12/ -regularized NMF for blind
, and
12/ -
SU, which has shown to be more effective than -0
, norm terms. In [98], Sigurdsson et al. relaxed , norm
^h for a better estimation of abundances.
Thouvenin et al. [99] developed an improved LMM,
called a perturbed LMM (PLMM), by their attempts to model
spectral variabilities as perturbed information that simply
meets the Gaussian distribution. A similar work is presented
in [100], where the scaling factor, as a major spectral
variability (SV), is modeled into LMM to yield an extended
LMM (ELMM) for a blind SU task.
He et al. [101] have employed TV and weighted
, norm
1 -
terms to further enhance the smoothness and sparseness
of abundances. Yao et al. [102] sought to explain the NMFbased
SU model by simulating human observations on HS
images, e.g., sparsity, nonlocal, and smooth properties, in a
nonconvex modeling fashion. Another type of interesting
SU strategy is to embed the graph or topological structure
of the HS data. The local neighboring relationship was introduced
into the NMF model and demonstrated robust SU
results [103]. Similarly, Lu et al. [104] enforced abundances
to follow the manifold structure of spectral signatures
in the form of Laplacian regularization for HS unmixing
(HU). Wang et al. [105] used a structuralized hypergraph
regularization in sparse NMF to better depict the underlying
manifolds of the HS data.
Very recently, Qin et al. [106] proposed a novel graph TV
regularization to estimate endmembers and abundances
more effectively. There are still other variants that directly
unmix the 3D HS tensor by preserving spatial structure information
as much as possible. For that, Qian et al. [107]
proposed a matrix-vector nonnegative tensor factorization
framework for blind SU. Imbiriba et al. [108] modeled lowrank
properties in the HS tensor to address the SV for robust
SU. A further modified work based on [108] is proposed
via weighted, nonlocal low-rank tensor decomposition for
sparse HU. Broadly, the key nonconvex priors of the aforementioned
models can be briefly summarized as follows:
◗
◗
, NMF [97]: AA 12
X
◗ PLMM [99]: EE E0
-
and () (/ ),12 n
N
WD =
68
R Dn
12
=1
2
F
, NMF [98]: A Aa()
U =() (/ )
12/ -
qX()=
=
q
=R ,
kn
KN
2
F
/ = R ,KN
, =1
,
kn=1a ()k 12
n()k q
/
n
where E ,0
, X =() (/ ) AH ,
H, and Δ deA
12
2
F
note
the reference endmembers, the matrix differences
() ()
+
(19)
◗ ELMM [100]: EE ES
-
UR= N
()
AH A 21
where Hh
21 +
gradients
◗ TV-regularized reweighted sparse NMF [101]: A()X =
dA A ;,TV
9
11 +
◗ Nonlocal hyperspectral (NLH) TV [102]:
◗ Aa
◗ Graph , NMF [104]: AA AAtr()L
◗ Graph TV [106]: AA AA<
12/ -
X =
()
X =+ <
TV tr().L
12/
+
Due to their powerful data-fitting ability, DL-based SU
approaches have recently been paid increasing attention
and have achieved better unmixing results [109]-[112]. Although
these methods still suffer from the effects of black
box, i.e., a lack of model interpretability, initially, their performances
have shown effectiveness and feasibility in unmixing
HS data more accurately.
ENDMEMBER-GUIDED SU
A mass of blind SU methods has been developed, and they
appear to be effective to simultaneously obtain endmembers
and abundance maps. However,
these blind methods
tend to extract physically meaningless endmembers,
e.g., noisy signals and spectral signatures corresponding
to nonexistent materials, due to a lack of certain interpretable
model guidance or prior knowledge. A straightforward
solution is to provide nearly real endmembers extracted
from HS images. This naturally leads to research on endmember-guided
SU. As the name suggests, the SU process
is performed using given reference endmembers or with the
guidance of extracted endmembers from the HS image. That
is, the endmembers E in (19) are known. Accordingly, the
endmember-guided SU can be implemented in the following
three-stage way:
1) The number of endmembers can be estimated using
subspace-estimation algorithms, e.g., HS signal identification
by minimum error (HySime) [93].
2) The endmembers can be extracted based on the geometric
observations of HS data structures. Several wellknown
methods include vertex component analysis
(VCA) [113], pixel purity index [114], and fast autonomous
endmember extraction (N-FINDER) 115].
3) The abundances of materials are estimated using regression-based
methods, which can generally be written as
min 2
1
A
YEAA A ! C .
-+X() subject to
2
F
(20)
Following the three steps, many well-working nonconvex
models have been successfully developed to estimate the
abundance maps of different materials at a more accurate
level. Heinz et al. [116] thoroughly analyzed the spectral mixture
in the SU issue, yielding a fully constrained least-squares
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE JUNE 2021
()=+Rij lo (| |), where Jw and
S1 are defined as the nonlocal Jacobian operator and
Schatten-1 norm, respectively
()
XR Jxg,,wn S1
:
N
n=1
ij +e
,,v() , and SH SH S
and Hv
W =
()
in spatial four nearest neighbors, and the pixelwise
perturbed information, respectively
()
n=1 nn0
2
F
, AHh
+
X =
()
hv() ,
22
()FF
are the horizontal and vertical
IEEE Geoscience and Remote Sensing Magazine - June 2021

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