STUDY OF THE EMPIRICAL MODE
DECOMPOSITION AND
APPLICATION TO PICTURES
EMPIRICAL MODE DECOMPOSITION (EMD)
Pioneered by N.E. Huang in 1998
 Nonlinear and data-driven technique of nonstationary signals decomposition
 Firstly defined for mono-dimensional signals
 Main idea: to represent a signal as a sum of
components, each of them being a zero-mean AMFM function

PURPOSE OF MY INTERNSHIP
Understand the principle of EMD in monodimensional case and compute the algorithm
 Develop an algorithm for bi-dimensional signals
 Try to identify matters in extension of EMD to
tri-dimensional signals

EMD OF MONODIMENSIONAL SIGNALS
PRINCIPLE
Let x(t) be a mono-dimensional signal:
1- Identify all local maxima of x(t):
Do the same thing with the local minima:
2- Interpolate between maxima ending up with some envelope
e max (t):
Likewise for the minimal envelope
e min(t ) :
3 - Compute the mean m(t) 
e min (t)  e max (t)
2
4- Extract the detail d(t) = x(t) – m(t)
5- Iterate on the residual m(t) until the number of extrema in the
signal is less than 2.
We finally obtain the decomposition of the signal:
x(t )   dk (t )  r (t )
k
INTRINSIC MODE FUNCTIONS
The functions dk (t ) we want to obtain by this
decomposition are called Intrinsic Mode
Functions (IMF) and they satisfy two conditions :
 In the whole data set, the number of extrema and
the number of zero crossings must either equal or
differ at most by one.
 At any point, the mean value of the envelope
defined by the local maxima and the envelope
defined by the local minima is zero.
SIFTING PROCESS
In practice, the details dk (t ) obtained are not
necessarily IMFs.
We have to refine by a Sifting Process:
Iterate steps 1 to 4 upon the detail d(t), until this can
be considered as an IMF.
 The corresponding residue is then computed and step
5 applies.

EMD OF BIDIMENSIONAL SIGNALS
The picture can be considered as a depth map in 2D:
PRINCIPLE
1.
2.
Identify local extrema of f(x,y)
Interpolate between minima (resp. maxima)
ending up with some envelope e min( x, y) (resp.
e max( x, y))
3.
4.
5.
e min( x, y )  e max( x, y )
Compute the mean m( x, y ) 
2
Extract the detail d(x,y) = f(x,y) – m(x,y)
Iterate on the residual m(x,y)
RESULTS
GRAYSCALE PICTURE
DECOMPOSITION
Mode 1
Mode 2
DECOMPOSITION
Mode 3
Mode 4
DECOMPOSITION
Mode 5
Mode 6
DECOMPOSITION
Mode 7
Mode 8
DECOMPOSITION
Mode 9
Residue
CHECKING
Original picture
Sum of the IMF
COLORED PICTURE
(a)
(a) Original red picture
(b) Original green picture
(c) Original blue picture
(b)
(c)
DECOMPOSITION
Mode 1
Mode 2
DECOMPOSITION
Mode 3
Mode 4
DECOMPOSITION
Mode 5
Mode 6
DECOMPOSITION
Mode 7
Mode 8
DECOMPOSITION
Mode 9
Residue
CHECKING
Original picture
Sum of the IMF