IEEE Geoscience and Remote Sensing Magazine - December 2020 - 61

classification accuracies. Furthermore, to push this vibrant
field of research forward, an impressive amount of code and
libraries are shared on GitHub, which can be found in [131].
A BRIEF INTRODUCTION ON FE
HSI technology provides detailed spectral information by
sampling the reflective portion of the electromagnetic spectrum, covering a wide range, from the visible region (0.4-
0.7 µm) to the short-wave infrared region (almost 2.4 µm).
Hyperspectral sensors can also characterize the emissive
properties of objects by acquiring data in the range of the
midwave and long-wave infrared regions, in hundreds of
narrow, contiguous spectral channels.
Detailed spectral information provided by hyperspectral sensors presents both challenges and opportunities.
For instance, HSIs can be used to differentiate between
different classes of interest with slightly different spectral
characteristics [1]. However, most of the commonly used
methods utilized for the analysis of gray scale, color, or
multispectral images cannot be extended to analyze HSIs
for several reasons, as detailed in the " Unique Properties of
High-Dimensional Data " section.
The limited availability of training samples (a common
issue in remote sensing) dramatically impacts the performances of supervised classification approaches due to the
high dimensionality of HSIs, which poses a problem for designing robust statistical estimations. FE can be used to address this. It can be described as finding a set of vectors that
represent an observation while reducing the dimensionality by transforming the input data linearly or nonlinearly
to another domain, thereby extracting informative features
in the new domain. The use of FE techniques can be advantageous for a number of reasons, which are illustrated in
Figure 1 and described in the following sections.
UNIQUE PROPERTIES OF HIGH-DIMENSIONAL DATA
Several studies (e.g., [2]-[4]) have demonstrated the unique
geometrical, statistical, and asymptotic properties of highdimensional data compared with red, green, blue (RGB) and
multispectral images. These properties, which have been
shown through experimental and theoretical examples,

explain why most analytical approaches developed for RGB
and multispectral images are not applicable to HSIs [5].
Among those experimental examples, we can recall that 1)
as dimensionality increases, the volume of a hypercube concentrates in corners, or 2) as dimensionality increases, the
volume of a hypersphere concentrates in an outside shell.
With respect to these examples, the following conclusions
have been drawn.
◗◗ A high-dimensional feature space is almost empty, which
indicates that multivariate data in R p (p represents the
number of bands, spectral channels, or dimensions) can
usually be represented in a lower-dimensional space (referred to as subspace) without losing considerable information in terms of class separability [5].
◗◗ Since the high-dimensional feature space is almost empty (i.e., Gaussian distributed data have a tendency to
concentrate in the tails, whereas uniformly distributed
data have a tendency to be concentrated in the corners),
the density estimation of hyperspectral data for both
Gaussian and uniform distributions becomes extremely
challenging.
Fukunaga [6] claimed that there is a relation between
the type of classifier, required number of training samples,
and number of input dimensions. As reported in [6], the
required number of training samples is linearly related to
the dimensionality for linear classifiers and to the square of
the dimensionality for quadratic classifiers (e.g., the Gaussian maximum likelihood classifier [6]); for nonparametric
classifiers, the number of required samples exponentially
increases as the dimensionality increases. Landgrebe [7]
showed the groundbreaking fact that too many spectral
bands might have negative impacts in terms of expected
classification performance.
When dimensionality increases, with a constant and
limited number of training samples, more statistics must
be estimated. Thus, the accuracy of the statistical estimation decreases, although higher spectral dimensions increase the separability between the classes. This leads to a
decrease in classification accuracies beyond an unknown
number of bands. These problems are related to the curse
of dimensionality, also known as the Hughes phenomenon

Saving Storage
Space; Faster
Transmission

Reflectance

Pixel
Vector

Input Data

0.5
0.4
0.3
0.2
0.1
0

Accurate
Classification
Saving Time

0.5 0.6 0.7 0.8
Wavelength (µm)

Extracted Features

Possible Advantages

FIGURE 1. The FE technique and its advantages for HSI analysis.
DECEMBER 2020

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

IEEE Geoscience and Remote Sensing Magazine - December 2020

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