IEEE Geoscience and Remote Sensing Magazine - March 2018 - 16

that much redundant information exists among the spectral bands. Thus, dimensionality reduction is a critical preprocessing step for hyperspectral image analysis. In past
decades, dimensionality-reduction methods have leapt forward, coming to play a vital role in the analysis of highdimensional data sets [6]-[11].
There are two main categories of dimensionality reduction in hyperspectral imagery, i.e., band selection and feature extraction. In band selection, we obtain a subset of the
original bands by minimizing spectral redundancy [12]-
[17]. The process selects an appropriate band subset from the
original bands, while feature extraction preserves important
PROGRESS TO DATE
features through mathematical transformations. Typical
Hyperspectral imagery-capturing reflectance values over
feature extraction methods, such as principal component
a wide range of electromagnetic spectra-provides rich
analysis (PCA) [18]-[22] and Fisher's LDA (FLDA) [23]-[27],
spectral information that can help discriminate among
project the original data into a low-dimensional subspace.
spectrally similar object pixels. However, the curse of diWe can view PCA and LDA, which are subspace learning
mensionality [1]-[5], caused by a great number of dimenmethods, as the most popular dimensionality-reduction
sions with only a few samples, makes hyperspectral image
techniques for hyperspectral image classification. PCA, an
classification challenging. Furthermore, these adjacent
unsupervised approach, aims to find projections by maxibands may be heavily redundant.
mizing the data variance in the projected subspace [28].
Figure 1 illustrates the absolute values of the cross-band
This method is suboptimal for classification tasks because it
correlation coefficients of the popular Airborne Visible/Inmay abandon some distinctive information. Some PCA exfrared Imaging Spectrometer (AVIRIS) Indian Pines data.
tensions have been proposed, such as robust PCA (RPCA)
Table 1 further demonstrates that more than 34% of the
[29], probabilistic PCA [30], sparse PCA [31], structured
correlation coefficients are greater than 0.9, which indicates
sparse PCA [32], and mixture of Gaussians-RPCA [33].
Comparatively, supervised LDA [34] seeks to find a
transform that maximizes Fisher's ratio as a Rayleigh quo1
tient in the projected subspace to enhance class separabil0.9
ity. However, under small-sample-size (SSS) situations,
LDA may potentially fail because of ill-conditioned statisti0.8
cal estimates. Thus, researchers have presented numerous
0.7
LDA extensions, such as subspace LDA (SLDA) [35], modified FLDA (MFLDA) [24], regularized LDA (RLDA) [25],
0.6
normalized discriminant analysis (NDA) [36], and noise0.5
adjusted SLDA (NA-SLDA) [37]. This review focuses on su0.4
pervised LDA-related research.
Traditional subspace learning methods assume that
0.3
the class-conditional distributions are Gaussian [38], [39].
0.2
However, hyperspectral data can be affected by illumina0.1
tion conditions [40], atmospheric effects, and geometric
distortion [41]. Thus, real class-conditional distributions
usually possess a complicated multimodal structure. As
a consequence, traditional LDA is most likely unable to
FIGURE 1. The absolute spectral correlation coefficients of the
capture the data manifold structure reflecting the geometIndian Pines data.
ric information (e.g., multiple clusters and
subspace structure). Manifold learning techniques [42]-[47], which depend significantly
TaBLE 1. ThE PrOPOrTION OF SPEcTraL-BaND cOrrELaTION FOr ThE
INDIaN PINES DaTa.
on the data characteristics, are supposed to
remedy this defect. Locality-preserving projecLEVEL
1
2
3
4
5
correlation coefficient
0 ~ 0.1
0.1 ~ 0.2
0.2 ~ 0.3
0.3 ~ 0.4
0.4 ~ 0.5
tions (LPPs) and supervised LPP [48], origiProportion (%)
4.65
5.17
7.03
8.97
10.51
nally developed for facial recognition, can be
LEVEL
6
7
8
9
10
counted as linear manifold learning methods
correlation coefficient
0.5 ~ 0.6
0.6 ~ 0.7
0.7 ~ 0.8
0.8 ~ 0.9
0.9 ~ 1.0
to maintain the local geometric structure
Proportion (%)
9.00
6.63
3.14
10.11
34.81
of neighboring samples in the original data
space. The work in [49]-[53] also developed
3) graph-embedding discriminant analysis, which preserves similarities of pixel pairs and characterizes data
geometry properties
4) semisupervised discriminant analysis (SDA)
5) discriminant analysis in a kernel space.
Experimental results using real hyperspectral data show
the comparative performance of each approach. We also
discuss some open issues and ongoing investigation in
this field. We hope our survey will offer insights for further research.

16

ieee Geoscience and remote sensing magazine

march 2018



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