IEEE Geoscience and Remote Sensing Magazine - March 2023 - 57

conceals useful information and hinders high-level image
processing. SU aims to separate the observed spectrum into
a suite of basic components, also called endmembers, and
their corresponding fractional abundances.
LINEAR MIXING MODEL
With the assumption of a single interaction between the
incident light and material, representative SU methods are
based on the following LMM [203], [204]:
XEAN
=+
where ,, ,XE ARR R
zhvz rr hv
!! !## #
and N Rzhv
#
!
(70)
denote
the observed unfolding HS matrix, endmember matrix,
abundance matrix, and additional noise, respectively.
LMM-based methods have drawn much attention due to
their model simplicity and desirable performance [205],
[206], [207]. Among them, a structured matrix factorization
model, such as nonnegative matrix factorization
(NMF), has physically meaningful interpretation in solving
the inverse ill-posed problem for SU [208]. Over the
past decades, numerous NMF-based methods have been
developed to pursue better unmixing performance by introducing
prior information based on (70), such as spatial
smoothness [209], [210], [211], sparseness [212], [213], and
volume regularization [214], [215], [216]. However, current
LMM-based matrix factorization methods usually convert
the 3D HS cube into a 2D matrix, leading to the loss of spatial
information in the relative positions of pixels. Tensor
factorization-based approaches have been dedicated to SU
to overcome the limitation of the LMM.
CANONICAL DECOMPOSITION/PARALLEL FACTOR
ANALYSIS AND TUCKER DECOMPOSITION MODEL
Zhang et al. [107], [108] first introduced nonnegative
tensor factorization (NTF) into SU via CP decomposition.
However, this NTF-SU method hardly considers the
relationship between the LMM and NTF, giving rise to a
lack of physical interpretation. Imbiriba et al. [109] considered
the underlying variability of spectral signatures
and developed a flexible approach, the unmixing with
LR tensor regularization algorithm, accounting for spectral
variability. The ranks of the abundance tensor and
endmember tensor were estimated with only two easily
adjusted parameters. Sun et al. [116] first introduced
Tucker decomposition for blind unmixing and increased
the sparse characteristic of the abundance tensor.
BLOCK TERM DECOMPOSITION MODEL
In terms of tensor notation, an HS data tensor can be represented
by the sum of the outer products of an endmember
(vector) and its abundance fraction (matrix). This enables a
matrix-vector third-order tensor factorization that consists
of R component tensors:
R
XNr
r
=+
=
$%
=+%
/
/
r
=1
R
1
where ,Er calculated by the product of Ar
the abundance matrix; cr
and
B ,r
T
denotes
is the endmember vector; and
N represents the additional noise. Apparently, this matrix-vector
tensor decomposition has the same form as BT
decomposition, setting up a straightforward link with the
previously mentioned LMM model. Qian et al. [110] proposed
a matrix-vector NTF (MVNTF) unmixing method
by combining the characteristics of CP decomposition and
Tucker decomposition to extract the complete spectral-spatial
structure of HS images. The MVNTF method for SU is
formulated as
R
min X Ecr
Ec,
- / %
r=1
r
F
s.t.AB c $ 0.
r
,,
T
r
r
(72)
MVNTF essentially derived BT decomposition and
established a physical connection with the LMM. Compared
with NMF-based unmixing approaches, MVNTF
can achieve better unmixing performance in most cases.
Nevertheless, the abundance results extracted by MVNTF
may be oversmooth and lose detailed information, due to
2
Ec
AB c
T
r
r
r
r N
(71)
(a)
(b)
(c)
FIGURE 22. HS unmixing based on the linear mixing model and nonlinear mixing model: (a) linear mixing, (b) intimate mixture, and
(c) multilayered mixture.
MARCH 2023 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
57
IEEE Geoscience and Remote Sensing Magazine - March 2023

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