IEEE Geoscience and Remote Sensing Magazine - March 2023 - 29

Figure 2 displays the dynamics of tensor decompositions
used for HS data processing in the HS community.
The listed numbers contain both scientific journal and
conference papers published in IEEE Xplore, which regards
" hyperspectral " and " tensor decomposition " as the
main keywords in abstracts. To highlight the increasing
trend of the number of publications, the time period has
been divided into four equal time slots [i.e., 2007-2010,
2011-2014, 2015-2018, and 2019-2022 (29 September)].
In this article, we mainly present a systematic overview
from the perspective of state-of-the-art tensor decomposition
techniques for HS data processing in terms of the
five burgeoning topics previously mentioned:
1) To the best of our knowledge, this is the first comprehensive
survey of the state-of-the-art tensor decomposition
techniques for processing and analyzing HS RS images.
More than 100 publications in this field are reviewed and
discussed, most of which were published during the past
five years.
2) For each HS topic, major representative works are
scrupulously presented in terms of the specific categories
of tensor decomposition. We introduce and discuss
the pure tensor decomposition-based methods
and their variants with other HS priors in sequence.
Experimental examples are given for validating and
evaluating theoretical methods, followed by a discussion
of remaining challenges and further
research
directions.
3) This article makes a connection between tensor decomposition
modeling and HS prior information. Table 1 summarizes
the publication years, with a brief description
and prior information. Either beginners or experienced
practitioners are expected to obtain insight into
the tensor decomposition-based frameworks for HS
RS. Code displayed in Table 1 can be used with real
data from Table 2 for the sake of repeatability and further
studies.
TENSOR THEORY PRELIMINARIES
We present some materials and references for practitioners
to establish readers' awareness of tensor-related theory before
learning tensor decomposition-based HS applications,
including Kolda and Bader's tensor decomposition and applications
[20], Sidiropoulos' research on signal processing
and machine learning [22], and Kilmer's theoretical framework
with applications in imaging [25]. These contribute to
a quick understanding of the characteristics and principles
of tensor decomposition. Beginners can benefit from relevant
resources, such as this section's basic tensor theory,
available HS RS data in Table 2, and open source code links,
with a specific classification provided in Table 1.
In this section, we first introduce some vital preliminaries
that are widely used in HS RS applications, containing
needful tensor operations, definitions, and notations. Definitions
are provided for various types of fundamental and
frequently used tensor decompositions. These preliminaries
MARCH 2023 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
DEFINITION 3 (FOUR SPECIAL TENSORS [121])
The conjugate transpose of a three-order tensor RX hv z
is the tensor conj ( ),R
!
X X vh z
= !)# #
##
which can be obtained
by conjugately transposing each front slice and reversing
the order of transposed frontal two through z.
The identity tensor denoted by RI hv z
!
## is the tensor
whose first frontal slice is an identity matrix, and all other
frontal slices are zero. A three-order tensor Q is orthogonal
100
120
140
160
180
200
20
40
60
80
4
9
8
6
2007-2010 2011-2014 2015-2018 2019-2022
Time Period
FIGURE 2. The number of journal and conference papers published
in IEEE Xplore on the subjects of " hyperspectral " and " tensor
decomposition " within different time periods.
29
are part of the theoretical foundation of tensors and run
through this review.
MATHEMATICAL NOTATIONS
The basic notations listed in Tables 3 and 4 give the main abbreviations
used in this article. Common tensor operations
and definitions are reported as follows, and their corresponding
toolbox can be found at http://www.tensortoolbox.org/.
DEFINITION 1 (T-PRODUCT [120])
The T-product of two three-order tensors A Rnn n123
24 3## is denoted by C Rnn n14 3
n2
!
B Rnnn
!
C ik
!
(, ,:)(,,:) (, ,:)
j=1
= /A ij
*B jk
##
:
(1)
where * represents the circular convolution between
two tubes.
DEFINITION 2 (TENSOR n-MODE PRODUCT [20])
The n-mode product of a tensor A Rrr rN12
!
B RBrn
#
!
is the tensor
!
X A .Bn
= #
## #g
X Rrr rB rr
12 nn N11
## ## #g
and matrix
g -+ , defined by
(2)
The unfolding matrix form of (2) is
XB .A
() = # ()
nn
(3)
## and
Journal
Conference
118
24
58
40
Number of Papers
http://www.tensortoolbox.org/
IEEE Geoscience and Remote Sensing Magazine - March 2023

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