IEEE TRANS. ON SIGNAL PROCESSING, VOL. XX, NO. XX, FEBRUARY 2013 1
Empirical wavelet transform
J
´
er
ˆ
ome Gilles
Abstract—Some recent methods, like the Empirical Mode De-
composition (EMD), propose to decompose a signal accordingly
to its contained information. Even though its adaptability seems
useful for many applications, the main issue with this approach
is its lack of theory. This paper presents a new approach to
build adaptive wavelets. The main idea is to extract the different
modes of a signal by designing an appropriate wavelet filter bank.
This construction leads us to a new wavelet transform, called the
empirical wavelet transform. Many experiments are presented
showing the usefulness of this method compared to the classic
EMD.
Index Terms—Wavelet, Empirical mode decomposition, Adap-
tive filtering
I. INTRODUCTION
A
DAPTIVE methods to analyze a signal is of great in-
terest to find sparse representations in the context of
compressive sensing. “Rigid” methods, like the Fourier or
wavelets transforms, correspond to the use of some basis
(or frame) designed independently of the processed signal.
The aim of adaptive methods is to construct such a basis
directly based on the information contained in the signal.
A well known way to build an adaptive representations is
the basis pursuit approach which is used in the wavelet
packets transform. Even though the wavelet packets have
shown interesting results for practical applications, they still
are based on a prescribed subdivision scheme. A completely
different approach to build an adaptive representation is the
algorithm called “Empirical Mode Decomposition” (EMD)
proposed by Huang et al. [9]. The purpose of this method
is to detect the principal “modes” which represent the signal
(roughly speaking, a mode corresponds to a signal which have
a compactly supported Fourier spectrum). This method has
gained a lot of interest in signal analysis this last decade,
mainly because it is able to separate stationary and non-
stationary components from a signal. However, the main issue
of the EMD approach is its lack of mathematical theory.
Indeed, it is an algorithmic approach and, due to its non-
linearity, is difficult to model. Nevertheless, some experiments
[5]–[7] show that EMD behaves like an adaptive filter bank.
Some recent works attempt to model EMD in a variational
framework. In [4], the authors proposed to model a mode
as an amplitude modulated-frequency modulated (AM-FM)
Manuscript received October, 2012. Revised version received February,
2013. Copyright (c) 2012 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org
J. Gilles is with the Department of Mathematics, University of California,
Los Angeles (UCLA), 520 Portola Plaza, Los Angeles, CA 90024, USA email:
jegilles@math.ucla.edu.
This work was partially founded by the following grants NSF DMS-
0914856, NSF DMS-1118971, ONR N00014-08-1-1119, ONR N0014-09-1-
0360, ONR MURI USC, the UC Lab Fees Research and the Keck Foundation.
signal and then use the properties of such signals to build
a functional to represent the whole signal. Then they are able
to retrieve the different modes by minimizing this functional.
Another proposed variational approach is the work of Hou
et al. [8] where the authors also use the AM-FM formalism.
They propose to minimize a functional which is build on
some regularity assumptions about the different components
and uses higher-order total variation priors.
In this paper, we propose a new approach to build adaptive
wavelets capable of extracting AM-FM components of a
signal. The key idea is that such AM-FM components have
a compact support Fourier spectrum. Separating the different
modes is equivalent to segment the Fourier spectrum and to
apply some filtering corresponding to each detected support.
We will show that it is possible to adapt the wavelet formalism
by considering distinct Fourier supports and then build a set
of functions which form an orthonormal basis. Based on this
construction, we propose an empirical wavelet transform (and
its inverse) to analyze a signal.
The remainder of the paper is organized as follows. Section II
has two distinct subsections: in II-A, we recall the principle
of the EMD algorithm and the AM-FM model; while in II-B,
we recall some wavelet formalism which will be useful in
our own construction and we discuss some of the existing
adaptive wavelet methods. In section III, we build the proposed
empirical wavelets and give some of their properties, then the
empirical wavelet transform and its inverse are introduced.
Section IV show many experiments based on simulated and
real signals. The time-frequency representation based on the
Hilbert transform is introduced in section V. In section VI,
we address the question of the estimation of the number of
modes. An extension to 2D signals (images) is presented in
section VII. Finally, we conclude and give some perspectives
in section VIII.
II. EXISTING APPROACHES
A. Empirical Mode Decomposition
In 1998, Huang et al. [9] proposed an original method
called Empirical Mode Decomposition (EMD) to decompose
a signal into specific modes (we define the meaning of
“mode” hereafter). Its particularity is that it does not use any
prescribed function basis but it is self adapting accordingly
to the analyzed signal f(t). In this paper, as we will use the
Fourier formalism in section III, we adopt the description used
in [4] which is slightly different from the original used in [9].
EMD aims to decompose a signal as a (finite) sum of N + 1
Intrinsic Mode Functions (IMF) f
k
(t) such that
f(t) =
N
X
k=0
f
k
(t). (1)
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