IEEE Geoscience and Remote Sensing Magazine - December 2020 - 68

where D 2 represents 2D wavelet bases, X is the observed
HSI, V contains the orthogonal subspace bases, and N is
the noise and model error. WSRRR simultaneously estimates the low-rank projection matrix and the wavelet coefficients W, which minimizes

	

t ) = argmin 1 X - V T QD 2
( Vt , Q
2
Q, V

2
F

d

+

/ mj

q (Tj)

j=1

1

T

s.t. V V = I.


(5)

t 2.
Note that the extracted features are given by Ft = QD
To capture the spatial (neighboring) information, orthogonal total variation (TV) component analysis (OTVCA)
was proposed in [42], where the HSI is modeled as
X = V T F + N, (6)

	

with matrix F containing the unknown features. OTVCA
assumes that the hyperspectral features are spatially piecewise smooth and, therefore, exploits the TV penalty and
simultaneously estimates F and V using
1
Ft = argmin 2 X - V T F

	

F

2
F

+ m / T V ^f (Tj)h, (7)
d

j=1

where
T V (x) =

(D h (x))2 + (D v (x)) 2

1

and D v and D h are the matrix operators, to calculate the
first-order vertical and horizontal differences, respectively, of a vectorized image. Recently, sparse and smooth
low-rank analysis (SSLRA) was proposed in [43], which
models the HSI based on a combination of sparse and
smooth features:

X = V T(F + S) + N, (8)

	

where F and S contain smooth and sparse features, respectively. SSLRA simultaneously extracts the sparse, S, and
smooth features, F, by taking into account both sparsity
and TV penalties:

	

1
(Ft , St , Vt ) = 2 X - V T(F + S)

2
F

+ m1 F

TV

+ m2 S

1

T

s.t. V V = I.



(9)

GRAPH-EMBEDDING AND/OR MANIFOLDLEARNING TECHNIQUES
Considering the nonlinear characteristic of HSIs, this
group of FE techniques aims to capture the data manifold through the local geometric structure of neighboring
pixels in the feature space. Figure 6(d) demonstrates the
concept of manifold-learning FE techniques applied on the
Swiss roll data set. The pink line in the left image shows the
Euclidean distance between two data points in 3D space. It
is clear that this line is not an effective metric to measure
the similarity of the two points selected in the Swiss roll
data set.
On the other hand, after unfolding of the data set, which
is represented in 2D space in the right image of Figure 6(d),
the Euclidean distance between two data points shown by
the pink line is a better representation of the similarity of
the two points in the data set. The FE techniques categorized in this group are designed to capture such a manifold while representing the data in a lower-dimensional
feature space.
Graph-embedding or manifold-learning FE techniques
often include three main steps: 1) neighborhood pixel selection, 2) weight selection, and 3) embedding construction. Isometric mapping (ISOMAP) [44], [45] is a global

PC1
PCA

PC2

PC2

PC1
Component Space

Original Data Space

Input

Band
Clustering

(a)

Band
Merging

Band
Output
Filtering

(b)

Graph
Embedding
Input

Features

Subspace
Basis
(c)

Noise

Original Data Space

Embedding Space
(d)

FIGURE 6. The four major categories of UFE methods: (a) transform, (b) band-clustering and -merging, (c) low-rank reconstruction, and (d) manifold-

learning based.
68

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

DECEMBER 2020
IEEE Geoscience and Remote Sensing Magazine - December 2020

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - December 2020
