IEEE Geoscience and Remote Sensing Magazine - December 2020 - 67

behind this group of methods is to split the spectral bands
into several groups in which the spectral bands have very
high correlation. Hence, the proposed techniques often
use similarity and dissimilarity criteria to split the spectral bands into several nonoverlapping groups. By selecting or fusing the bands of each group, some representative bands or features of different groups are obtained.
Furthermore, followed by the merging step, some bandfiltering and processing operations can be also used to
further improve the discrimination of the resulting features. This group of techniques is often computationally
cheap and, thus, is often used in real applications. On the
other hand, spectral information is often neglected by the
methods in this category.
For band clustering and merging, two algorithms are
proposed in [31]. The first selects discriminative bases by
considering all of the classes simultaneously; however, the
second selects the best bases for a pair of classes at a time.
In [32], a hierarchical clustering algorithm was introduced
to split and cluster the hyperspectral bands, where the representative band for each cluster is selected based on both a
mutual information (MI) criterion and a divergence-based
criterion. Another band-clustering technique was proposed
in [33], where the splitting is done by minimizing an MI criterion iteratively applied on averaged bands. Iterative algorithms were proposed in [34] for both splitting and merging
the bands. The splitting procedure is done using the Pearson correlation coefficient between adjacent bands; then,
the merging is applied by averaging over the split bands.
Besides splitting/clustering and merging hyperspectral
bands, another operation is to further improve the feature
discrimination by band filtering or processing. For example, a hyperspectral FE using image fusion and recursive filtering was given in [35], where the adjacent bands
are fused by averaging, and, then, recursive filtering was
used to extract spatial information. In [36], intrinsic image decomposition was applied for processing the merged
bands, which can effectively remove information that is
not related to the material of different objects. After that,
multiple improved versions of intrinsic decompositionbased band-processing methods were developed [37],
[38]. In [39], a relative total-variation-based structureextraction method was applied for band processing so as
to construct multiscale structural features that are robust
to image noise.

TABLE 3 THE HOUSTON UNIVERSITY 2018 DATA SET: THE
NUMBER OF TRAINING AND TEST SAMPLES AND THE TOTAL
NUMBER OF SAMPLES PER CLASS.
CLASS
NUMBER

CLASS NAME

TRAINING
SAMPLES

TEST
SAMPLES

SAMPLE

1

Healthy grass

1,458

8,341

9,799

2

Stressed grass

4,316

28,186

32,502

3

Synthetic grass

331

353

684

4

Evergreen trees

2,005

11,583

13,588

5

Deciduous trees

676

4,372

5,048

6

Soil

1,757

2,759

4,516

7

Water

147

119

266

8

Residential

3,809

35,953

39,762

9

Commercial

2,789

220,895

223,684

10

Road

3,188

42,622

45,810

11

Sidewalk

2,699

31,303

34,002

12

Crosswalk

225

1,291

1,516

13

Major thoroughfares

5,193

41,165

46,358

14

Highway

700

9,149

9,849

15

Railway

1,224

5,713

6,937

16

Paved parking lot

1,179

10,296

11,475

17

Gravel parking lot

127

22

149

18

Cars

848

5,730

6,578

19

Trains

493

4,872

5,365

20

Seats

1,313

5,511

6,824

34,477

470,235

504,712

Total

TABLE 4 THE DIFFERENT SYMBOLS USED IN THIS ARTICLE
AND THEIR DEFINITIONS.
SYMBOLS

DEFINITION

xi

the ith entry of the vector x

X ij

the (i, j)th entry of the matrix X

xi

the ith column of the matrix X

x( j)

the jth row of the matrix X

x

0

the l0 -norm of the vector x -i.e., the number of nonzero
entries

x

1

the l1 norm of the vector x, obtained by R i x i .

x

2

the l2 norm of the vector x, obtained by

X

1

the l1 norm of the matrix X, obtained by R i, j X ij

X

F

the Frobenius norm of the matrix X, obtained by

Xt

the estimate of the matrix X

tr (X)

the trace of the matrix X

X

LOW-RANK RECONSTRUCTION-BASED TECHNIQUES
Low-rank reconstruction-based FE techniques proposed by
Rasti et al. [40]-[43] are based on finding an orthogonal
subspace by minimizing a constrained cost function. They
exploit low-rank models and reconstruction-based optimization frameworks to extract features. The optimization
frameworks take into account prior knowledge of the data
using different types of penalties. Due to the noise assumption in the low-rank model used, this group of FE techniques
is robust to noise. They are often computationally expensive
DECEMBER 2020

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

TV

R i x 2i

R i, j X 2ij

the TV norm of the matrix X, obtained by R i TV (x (i ))

compared to groups 1 and 2 due to the iterative algorithms
used to solve the (nonconvex) optimization problem.
Wavelet-based sparse reduced rank regression (WSRRR)
[41] applies the sparsity prior on the wavelet coefficients,
considering that the projected data on wavelet bases are
sparse. WSRRR uses the model
	

X = V T QD 2 + N, (4)
67



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