Image analysis by bidimensional empirical mode decomposition

J.C. Nunes*, Y. Bouaoune, E. Delechelle, O. Niang, Ph. Bunel

Laboratoire d’Etude et de Recherche en Instrumentation, Signaux et Systemes (LERISS), Universite Paris XII-Val de Marne, Bat P2-Piece 230 61 Avenue du

General de Gaulle, 94010 Creteil Cedex, France

Received 30 September 2002; received in revised form 18 April 2003; accepted 9 May 2003

Abstract

Recent developments in analysis methods on the non-linear and non-stationary data have received large attention by the image

analysts. In 1998, Huang introduced the empirical mode decomposition (EMD) in signal processing. The EMD approach, fully

unsupervised, proved reliable monodimensional (seismic and biomedical) signals. The main contribution of our approach is to apply the

EMD to texture extraction and image filtering, which are widely recognized as a difficult and challenging computer vision problem. We

developed an algorithm based on bidimensional empirical mode decomposition (BEMD) to extract features at multiple scales or spatial

frequencies. These features, called intrinsic mode functions, are extracted by a sifting process. The bidimensional sifting process is

realized using morphological operators to detect regional maxima and thanks to radial basis function for surface interpolation. The

performance of the texture extraction algorithms, using BEMD method, is demonstrated in the experiment with both synthetic and

natural images.

q 2003 Elsevier B.V. All rights reserved.

Keywords: Bidimensional empirical mode decomposition; Texture analysis; Unsupervised texture decomposition; Radial basis function; Surface interpolation

1. Introduction

The joint space-spatial frequency representations have

received special attention in the fields of image processing,

vision and pattern recognition. Huang [15] introduces a

multiresolution decomposition technique: the empirical

mode decomposition (EMD), which is adaptive and appears

to be suitable for non-linear, non-stationary data analysis.

We propose a new analysis method of texture images based

on bidimensional empirical mode decomposition (BEMD),

firstly presented in Ref. [24].

This paper is organized as follow. Section 2 presents

state of the art of texture analysis methods. Section 3

presents an introduction to the EMD and its extension on

bidimensional data. It describes too the implementation

details of sifting process, including the extrema detection by

morphological reconstruction and the radial basis function

(RBF) for surface interpolation. In Section 4, experimental

results on different texture images indicate the interest of

this new multiresolution decomposition with statistical

descriptors. Finally, a conclusion is presented in Section 5.

2. Texture analysis methods

Texture features are determined by the spatial relations

between neighbouring pixels. The difficulty of texture

analysis is demonstrated by the number of different texture

definitions attempted by vision researchers [12,17,28].

Approaches to texture feature extraction and recog-

nition span a wide range of methods. Several books and

articles give overviews of the available methodology [9,

33]. There are four major issues in texture analysis [22]:

feature extraction (local texture properties), texture

discrimination (image partition corresponding to different

textures), texture classification (classes definition in finite

number, normal and pathological) and shape from texture

(reconstruction of 3D surface geometry from texture

information).

Four major method categories may be identified [20]:

statistical with cooccurrence and autocorrelation features

[12], geometrical with Voronoi tessellation [33] and

structural features, model based with Markov random field

[5,11] and fractal parameters [6,26,32], and multiscale

features with AM– FM analysis [13], morphological

operators [27], Wigner-Ville Distributions [14,31], Gabor

filters [8,16], and wavelet transforms [21,23].

0262-8856/03/$ - see front matter q 2003 Elsevier B.V. All rights reserved.

doi:10.1016/S0262-8856(03)00094-5

Image and Vision Computing 21 (2003) 1019–1026

www.elsevier.com/locate/imavis

* Corresponding author. Tel.: þ33-1-45-171-492.

E-mail address: nunes@univ-paris12.fr (J.C. Nunes).

A few comparisons between texture feature extraction

schemes have been presented in Refs. [22,29,30]. Com-

parative studies showed different conclusions. Different

setups, test images, and filtering methods may be the

reasons for the contradicting results. No single approach did

perform best or very close to the best for all images. The

multiresolution techniques, which based on human visual

perception, intend to transform images into a representation

in which both spatial and frequency information are present.

The most commonly multiscale features used are Wigner

distributions [14,31], Gabor functions [2,7,16] and wavelets

transforms [18,19,21,34,36]. These multiscale features try

to characterize textures by filter responses directly.

However, one difficulty of the multiresolution analysis is

its non-adaptive nature since it uses filtering schemes.

We have applied the EMD technique to texture images

for two reasons. The first, the advantage of this technique,

the EMD is a fully data driven method [25], does not use any

pre-determined filter [8] or wavelet functions [19]. The

second, we can easily implement a bidimensional extension

of EMD. This paper, which is based on multiscale

decomposition, examines the issue of designing texture

decomposition by BEMD.

3. Texture analysis based on the empirical mode

decomposition

3.1. The 1D empirical mode decomposition

In this paper, we use the EMD, first introduced by Huang

et al. [15]. Different applications as medical and seismic

signals analysis have showed the effectiveness of this

method. This method permits to analyse non-linear and non-

stationary data. Its principle is to decompose adaptively a

given signal into frequency components, called intrinsic

mode functions (IMF). These components are obtained from

the signal by means of an algorithm called sifting process.

This algorithm extracts locally for each mode the highest

frequency oscillations out of original signal.

3.1.1. Sifting procedure

The sifting procedure decomposes a sampled signal sðkÞ

by means of the EMD. The sifting procedure is based on two

constraints:

† each IMF has the same number of zero crossings and

extrema;

† each IMF is symmetric with respect to the local mean.

Furthermore, it assumes that s has at least two

extrema.

The EMD represent adaptively non-stationary signals as

sums of zero-mean AM–FM components [10], i.e. an IMF

is an AM–FM component. AM–FM analysis [13] has

been used successfully in a variety of applications including

non-stationary analysis, edge detection, image enhance-

ment, recovery of 3D shapes from texture, computational

stereopsis, texture segmentation and classification.

The sifting algorithm for s [ l2ðZÞ reads as follows:

1. Initialise: r0 ¼ s (the residual) and j ¼ 1 (index number

of IMF),

2. Extract the jth IMF:

3. (a) Initialise h0 ¼ rj21; i ¼ 1;

(b) Extract local minima/maxima of hi21;

(c) Compute upper envelope and lower envelope

functions xi21 and yi21 by interpolating, respect-

ively, local minima and local maxima of hi21;

(d) Compute mi21 ¼ ðxi21 þ yi21Þ=2 (mean envel-

ope),

(e) Update hi :¼ hi21 2 mi21 and i :¼ i þ 1;

(f) Calculate stopping criterion (standard deviation

SDji)

(g) Repeat steps (b) to (f) until SDji # SDMAX and put

then sj ¼ hi (jth IMF)

4. Update residual rj ¼ rj21 2 sj;

5. Repeat steps 2–4 with j :¼ j þ 1 until the number of

extrema in rj is less than 2.

3.1.2. Finding all the IMFs

1. Once the first set of ‘siftings’ results in an IMF, define

c1 ¼ h1i: This first component contains the finest spatial

scale in the signal.

2. Generate the residue, r1; of the image by subtracting out

c1; r1 ¼ I 2 c1: The residue now contains information

about larger scales.

3. Resift to find additional components r2 ¼ r1 2

c2;…; rn ¼ rn21 2 cn

The superposition of all the IMF reconstructs the data:

I ¼Pn

j¼1 ðcjÞ þ rn:

We have to determine a criterion for the sifting process

to stop. This can be accomplished by limiting the size

of the SD, computed from the two consecutive sifting

results as:

SD2ji ¼

XKk¼1

lðhjði21ÞðkÞ2 hjiðkÞÞl2

h2jði21ÞðkÞ

" #

3.2. The bidimensional empirical mode decomposition

Texture analysis is considered as a challenging task. The

ability to effectively classify and segment images based on

textural features is of key importance in scene analysis,

medical image analysis, remote sensing and many other

application areas. Feature extraction is the first stage of

image texture analysis. To extract the 2D IMF during the

sifting process, we have used morphological reconstruction

J.C. Nunes et al. / Image and Vision Computing 21 (2003) 1019–10261020

to detect the image extrema and RBF to compute the surface

interpolation. A 2D IMF is a zero-mean 2D AM–FM

component. The image AM–FM decomposition [13] is

partially unsupervised feature based segmentation algor-

ithm, whereas the EMD is fully unsupervised. In the two

methods, the results of the decomposition are very efficient

and competitive with the best analysis results [13,25].

We define a bidimensional sifting process [24]:

† Identify the extrema (both maxima and minima) of the

image I by morphological reconstruction based on

geodesic operators;

† Generate the 2D ‘envelope’ by connecting maxima

points (respectively, minima points) with a RBF;

† Determine the local mean m1; by averaging the two

envelopes;

† Since IMF should have zero local mean, subtract out the

mean from the image: I 2 m1 ¼ h1;

† repeat as h1 is an IMF.

As described above, the process is indeed like sifting to

separate the finest local mode from the image first based

only on the characteristic multiscale. The sifting process,

however, has two effects to eliminate riding waves and to

smooth uneven amplitudes.

3.2.1. Extrema detection

Morphological reconstruction is a very useful operator

provided by mathematical morphology [1]. Its use in

hierarchical segmentation proves its efficiency in all the

steps of the process, from extrema detection to hierarchical

image construction. The image extrema have been detected

by using morphological reconstruction based on geodesic

operators. We define the geodesic reconstruction as follows.

The grayscale reconstruction IrecIðJÞ of I from J is

obtained by iterating grayscale geodesic dilations ›n1 of J

under I until a stability is reached, i.e. IrecI ¼W

n$1 ›n1ðJÞ:

A well-known use of the morphological reconstruction is

the extraction of the extrema (minima and maxima). If we

take J ¼ I 2 1 (subtract one gray level to every pixel of

original image) and if we perform the reconstruction Irec

(by geodesic dilation) of J by I; the difference I 2 Irec

corresponds to the indicator function of the maxima of I

(Fig. 1).

Conversely, the difference between Irecp (reconstruction

by geodesic erosion) and I (original image) produces

the indicator function of the minima of I: This extrema

detection method is described in Ref. [35].

3.2.2. Surface interpolation by radial basis function

In Ref. [15], Huang proposed to use cubic spline

interpolation on non-equidistant sampled data. We have

choice to use the RBF rather than the bicubic spline for

different reasons developed in Ref. [4]. One of the technical

problems is that the cubic spline fitting creates distortions

near the end points. RBFs are presented as a practical

solution to the problem of interpolating incomplete surfaces

derived from three-dimensional (3D) medical graphics. A

RBF is a function of the form:

sðxÞ ¼ pmðxÞ þXNi¼1

liFðkx 2 xikÞ; x [ Rd; li [ R;

where

† s is the radial basis function,

† pm low degree polynomial, typically linear or quadratic, a

member of mth degree polynomials in d variables,

† k·k denotes the Euclidian norm,

† the li’s are the RBF coefficients,

† -F is a real valued function called the basic function,

† the xi’s are the RBF centres.

The radial basis approximation method offers several

advantages over piecewise polynomial interpolants. The

geometry of the known points is not restricted to a regular

grid. Also, the resulting system of linear equations is

guaranteed to be invertible under very mild conditions.

Finally, polyharmonic RBFs have variational characteriz-

ations, which make them eminently suited to interpolation

of scattered data, even with large data-free regions. These

applications include geodesy, geophysics, signal proces-

sing, and hydrology. RBFs have also been successful

employed for medical imaging and morphing of surfaces

in three dimensions.

Experimental results demonstrate that high-fidelity

reconstruction is possible from a selected set of sparse and

irregular samples. RBF are introduced to compute a

continuous surface through a set of irregularly spaced

points (extrema) during the sifting process.

4. Results and discussion

Real-world texture frame, as well synthetic texture

frames, has been used to test and validate the proposed

approach. The decomposition approach is applied to both

synthetic and natural images, created by composing textures

selected from Brodatz [3]. A study is performed to show the

efficiency and performance of the texture extraction. Then, a

set of texture images is selected to illustrate the application

procedure of the EMD method.Fig. 1. Morphological reconstruction Irec (by geodesic dilation) of I by

I 2 1:

J.C. Nunes et al. / Image and Vision Computing 21 (2003) 1019–1026 1021

4.1. Texture extraction

The 2D decomposition by sifting process of an image

provides a representation that is easy to interpret. Every

mode (IMF) contains information of a specific scale, which

is conveniently separated. Spatial information is retained

within the mode. Our algorithm is able to decompose a

broad range of texture elements. Examples using images

(200 £ 200 in grey scale) from the Brodatz texture album, or

using synthetic and real images are shown in Figs. 3, 6–12.

To stop the sifting process, we used the standard deviation

(SD). We have used SDMAX between 0.05 and 0.75.

In a first step, we applied our algorithm on a synthetic

image (Fig. 2a). It is an image of sinusoidal components,

built with a sum of three horizontal (h1 ¼ 40; h2 ¼ 14;

h3 ¼ 0:2) and three vertical (v1 ¼ 60; v2 ¼ 17; v3 ¼ 0:3)

frequencies with respective amplitudes (ah1 ¼ 40; ah2 ¼ 30;

ah3 ¼ 190; av1 ¼ 40; av2 ¼ 50; av3 ¼ 190). All stages of

results are shown in Figs. 2 and 3.

We observe the images of the minima (Fig. 2b), the

maxima (Fig. 2c) detection, the mean envelope (Fig. 2d) and

the subtraction from mean envelope (Fig. 2e). Fig. 3 shows

the performed decomposition in two modes (Fig. 3b and c)

and the residue image (Fig. 3d). To prove the effectiveness

of our method, we compare the density profiles of the

original image and the various frequencies composing it. In

Fig. 4, we traced the profile of density of sinusoidal images

(corner in top on the left towards the corner in bottom in

right-hand side): the sum of the three frequencies (Fig. 4a),

the high frequency (Fig. 4b), the medium frequency (Fig. 4c)

and the low frequency (Fig. 4d).

In Fig. 5, we can observe the density profile of

decomposition, the sum of three frequencies (Fig. 5a), the

first mode (Fig. 5b), the second mode (Fig. 5c) and the

image residue (Fig. 5d). We can compare these different

profiles. We observe that the first mode corresponds to the

sinusoidal image with the highest spatial frequency, the

second to the medium and the residue to the lowest. These

profiles are very similar. However, we can watch in residue

image (Fig. 5d) an irregularity, which is due to interpolation

distortions near the end points. In Fig. 6, the decomposition

is performed in two modes. This fabric texture is quite

regular, mainly diagonal and horizontal structures, making

it well suited to the proposed approach.

Fig. 3. Synthetic texture.

Fig. 4. Synthetic texture (frequency components).

Fig. 2. Photographic texture fabric.

J.C. Nunes et al. / Image and Vision Computing 21 (2003) 1019–10261022

The first mode corresponds to the woven structure, the

second to the pattern and the residue to the horizontal stripes

(black and white). In Fig. 7, the decomposition is performed

in three modes. It is interesting to observe that the first mode

corresponds to the woven structure, the second to the finest

pattern, the third to the medium pattern and the residue to

largest pattern.

In Fig. 8, the decomposition is performed in three modes.

In Fig. 9, the decomposition of MRI is performed in three

modes. We observe the three modes and the residue, which

contain the pattern structures from finest to coarsest.

We can observe that the extraction quality of a mode

depend on the quality of the previous modes. Therefore, the

stop criteria (SD) of sifting process is important.

4.2. Image filtering

Since sifting process extracts firstly the highest fre-

quency, the first modes correspond generally to the noise.

The information of the noise is contained in the very first

modes and in the residual image. Conversely, the image

tendency is contained only in the residual image.

Fig. 5. Synthetic texture (BEMD).

Fig. 6. Photographic fabric texture.

Fig. 7. Photographic texture fabric.

Fig. 8. Photographic texture (D69 Brodatz).

J.C. Nunes et al. / Image and Vision Computing 21 (2003) 1019–1026 1023

The tendency can be represented by a polynom of order

relatively low ð0; 1; 2Þ: The results in Fig. 10 show the

possibility of low level processing (filtering or denoising)

with this technique. After having applied the decompo-

sition, this filtering would be easily carried out by the

subtraction with the original image of one or several modes.

The residue represents the filtered image.

4.3. Extraction of inhomogeneous illumination

Since sifting process extracts firstly the highest fre-

quency, the firsts modes correspond generally to the noise.

Conversely, the image tendency is contained in the latest

mode or more generally in residual image resulting to the

sifting process (Fig. 11).

After the decomposition, we subtract residue image from

original image (Fig. 12c) to perform inhomogeneous

illumination correction.

4.4. Perspectives

In the bidimensional case, the regional extrema are not

always well defined. The saddle points (or more generally,

the pattern of ridges and valleys) should be taken into

account. Thus, we propose the straightforward extension of

the 1D EMD to the 2D case by using morphological

operators. To detect lines peaks (respectively, the line

Fig. 9. Brain MRI.

Fig. 11. Synthetic texture.

Fig. 10. Retinal image.

J.C. Nunes et al. / Image and Vision Computing 21 (2003) 1019–10261024

connecting the lowest points) corresponding to the maxima

(respectively, minima), we use the watershed, a powerful

tool for image segmentation. More details of the BEMD

from these lines (the saddle points) will be presented in

future paper.

We can observe in Fig. 12 the BEMD from these lines

and can compare these results with BEMD from regional

extrema in Fig. 9. For the BEMD from the regional extrema

(Fig. 9), we used SDMAX ¼ 0:75: For the BEMD from the

saddle points (Fig. 12), we used SDMAX ¼ 0:47: By

comparing Figs. 9 and 12, we can observe that the details

corresponding to the modes are finer for the BEMD from the

saddle points. Consequently, BEMD based on saddle points

seem to produced more IMF than the preceding version.

5. Conclusion

In this paper, we present a new modulation domain

feature-based approach for discriminating textured images.

We have applied the EMD to texture image analysis. The

BEMD permits to extract spatial frequency components or

Fig. 12. Brain MRI.

J.C. Nunes et al. / Image and Vision Computing 21 (2003) 1019–1026 1025

different spatial scales, i.e. structures from finest to coarsest

scales. This method, derived from the image data and fully

unsupervised, permits to analyse non-linear and non-

stationary data as texture images. We have shown

experimental results for both natural and synthetic textures.

By having access to these representations of scenes or

objects, we can concentrate on only one or several modes

(one individual or several spatial frequency components)

rather than the image entirety. Clearly the 2D EMD offers a

new and promising way to decompose and extract texture

features without parameter.

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