EL FADILI, H.

QJIDAA, H.

ZENKOUAR, K.

I-Revues

Skeletonization of Noisy Images via the Method of Moment 
K. Zenkouar(1), H. El Fadili (2), and H. Qjidaa(3)
(1) Faculté des science dhar el mehraz Département de physique, LESSI B.P. 1796 Fes Maroc,
e-mail: khalid_zenkouar@ yahoo.fr. 
Abstract – In this paper, we propose a novel approach to robust skeletonization, that is developed based on a statistical method
using the Zernike moment theory controlled by Maximum Entropy Principal (MEP). This new concept of skeletonization is
articuled into three steps. In the first one, estimation of the underlying probability density function (pdf) using Zernike moment is 
carried out. In the second, the estimation of optimal pdf is selected using MEP criterion. Finally, the subset of local maxima
pixels of the optimal pdf are selected as belonging to the skeleton. This new method is applied on noisy and free noisy binary
images. We have tested the proposed Zernike Moment Skeletonization Method (ZMSM) on a variety of real and simulated noisy
images, it produces excellent and visually appealing results, with comparison to some well known traditional methods. 
1. Introduction
The skeleton is widely recognized as one of the most
important descriptors in image processing and pattern
recognition. Since the first study by Blum [1], the
Skeletonization of shapes has attracted attentions from many
researchers in various fields. Commonly used computational
methods for skeleton extraction include topological thinning
[2-4], approaches based on distance transform [5], [6],
hierarchical methods based on Voronoi diagrams [7], voxel
coding based methods [8], and some approaches based on 
physical simulations [9] or curve evolution [10]. These
approaches present several advantages, however the main
drawback of most of these methods is their high sensitivity to
noise.
In this paper a novel skeletonization approach, is developed
using a statistical method based on the estimation of 
probability density function pdf where the skeleton is defined
as the local maxima of this pdf.
Our proposed approach is based on the expansion of a 
multivariate function pdf in terms of Zernike polynomials by
means of Zernike moment. For this purpose the pdf is
approximated by a truncated series of polynomials. As the
determination of the expansion order is a difficult problem in
the framework of unsupervised classification, we propose the
determination of the optimal order for which the estimated pdf
has maximum entropy.
As the solution to this problem is mathematically too
complexe to be tractable, we introduce an exhaustive search 
for the optimal order. We propose to estimate the pdf for
different orders and to select the optimal one as the one for 
which the entropy reaches a maximum according to the 
Maximum Entropy Principal MEP [11-15]. This latter has 
been used for clustering, see for example [16] and for image
restoration [17]. Having the optimal pdf, the true points of the
skeleton are the local maxima of the pdf. Extraction of the
local maxima of the pdf is carried out using the last phase  of 
the proposed algorithm.
As a summary, our proposed ZMSM skeletonization
method based on the combination of the moment theory and
MEP as a selection criterion, is composed of the three
following steps: 
1- Computation of the pdf using the Zernike moment.
2- Estimation of the optimal pdf using MEP method.
3- Extraction of the local maxima of the optimal pdf taken
as the skeleton points. 
The paper is organized as follows: the next section
describes the basis of our statistical model, using Zernike
moment. The maximum entropy principal is given in section
3. The details of our skeletonization algorithm is presented in
section 4. Section 5 performs main results and performances
of our skeletonization method. Finally section 6 deals with the
summary of important results and conclusions of this work. 
2. Statistical Modelisation using Zernike 
Moment
2.2 Zernike Moments Computation 
The use of Zernike moments in image analysis was 
pioneered by Teague [18]. Since then, the Zernike moments
Colloque GRETSI, 11-14 septembre 2007, Troyes 905
have been frequently utilized for a number of image
processing and computer vision tasks [19-25]. In order to
define the Zernike moments, we need to introduce the concept 
of Zernike functions.
The (p,q) order Zernike function is defined as [22]:
1yx,)jqexp()(R)y,x(V 22q,pq,p dIU  (1) 
where
p: Positive integer or zero.
q: Positive and negative integers subject to constraints
qp  even and pqd .
U : Length of the vector from origin to (x,y) pixel . 
I : Angle between vector U  and x-axis in counterclockwise
direction.
In (1), is a polynomial in  of degree 
containing no power of U  less than
)(R q,p U U 0pt
q , the form of this
polynomial is: 
¦ U




 U

 

2/qp
0s
s2psq,p
)s
2
qp
()!s
2
qp
(!s
)!sp()1(
)(R  (2) 
These polynomials are orthogonal and satisfy:
'q,q'p,p
D
'q,'pq,p
*
)1p(
dxdy)y,x(V)y,x(V GG
S ³³ (3)
With ^ ba1 oterwise0b,a   G
Zernike moments are the projection of the image function
onto these orthogonal functions, the Zernike moment of order
p with repetition q for a continuous image function  that
vanishes outside the unit circle is:
)y,x(f
dydx),(V)y,x(f
)1p(
A
1yx
q,p
*q,p
22
³³ IUS
 
d
 (4) 
The aforementioned favorable properties of the Zernike
moments are valid as long as one use a true analog image
function. In practice, the Zernike moments have to be 
computed from sampled data, i.e., the rectangular sampling of
the original image function , producing the set of 
samples  with an (M,N) array of pixels, thus we define
the discrete version of  in terms of summation by the
traditional commonly used formula [20].
)y,x(f
)y,x(f ji
q,pA
1yx,),(V)y,x(f
1p
A 2j
2
iq,p
*
x y
jiq,p
~
i j
dIU¦¦S
 (5)
Liao and Pawlak in [21,22], proposed a modified version of 
the Zernike moment given by:
1yx,)y,x(H)y,x(f
1p
A 2j2ijiq,p
x y
jiq,p
i j
dS
 ¦¦ (6)
Where (7)dydx),(V)y,x(H
2/xx
2/xx
2/yy
2/yy
q,p
*jiq,p
i
i
j
j
³ ³ IU 
'
'
'
'
Represents the integration of over  pixel.),(V q,p* IU )y,x( ji
2.2 Estimation of the Probability Density 
Function
By taking the orthogonality principle, of the Zernike 
polynomials, into consideration, the image function  can 
be written as an infinite series expansion in terms of the
Zernike polynomials over the unit disk:
)y,x(f
evenqp,),(VA)y,x(f
0p
q,p
p
pq
q,p IU ¦¦
f
  
 (8) 
Suppose that one knows all moments of  up to a 
given order 
q,pA )y,x(f
T . It is desired to reconstruct a discrete function
 where moments exactly match those of  up to
given order 
)y,x(fT )y,x(f
T :
evenqp,),(VA)y,x(f
0p
p
pq
q,pq,p¦¦
T
  
T IU (9)
Note that since , then q,p*q,p AA   q,pq,p AA  , thus one 
can concentrate on q,pA  with , as far as the defined 
Zernike moments are concerned in the reconstruction process. 
The reconstructed image function can be written as: 
0qt
evenqp,),(VA)y,x(f
0p
p
0q
q,pq,p IU ¦¦
T
  

T

 (10) 
Let  be the estimated probability density function
obtained by normalizing  [13], [15]:

)y,x(p ji
)y,x(f ji

¦
 
:



ji y,x
ji
ji
ji
)y,x(f
)y,x(f
)y,x(p (11)
where (12)1)y,x(p
ji y,x
ji  ¦ :

and ,1)y,x(p0 ji dd

:  is the image plane. 
The estimated pdf depends only on the expansion order T ,
a criterion for choosing this order is explained in the next
paragraph according to the maximum entropy principal. 
3. Optimal Order Moments Selection 
using MEP
We introduce the maximum entropy principle MEP for the
search of this optimal order, this automatic technique can
estimate the optimal number of moments directly from the
available data and does not require any a priori image
information specially for noisy images.
Let  be a set of estimated underlying probability density
function for various Zernike moment orders 
wG
T :
}........1/p{Gw Z T T

(13)
By applying the maximum entropy principle for noisy
images, we deduce that among these estimates of the
probability density function, there is one and only one
probability density function denoted  whose entropy is
maximum [13],[17] and which represents the optimal
probability density function, and then gives the optimal order
of moments.
)y,x(p ji

T

906
The Shannon entropy of  is defined as:)y,x(p ji

T

¦ 
:
T

T

T

ji y,x
jiji ))y,x(plog()y,x(p)p(S (14)
 and the optimal  is such that

T

p
}Gp/)p{S(MAX)p(S W T

T

T

 (15) 
The process of determinating the optimal order T  consists
in estimating the pdf for different orders and selecting the
optimal one as the one for which the entropy reaches 
maximum. The following is basic algorithm which consists in
an exhaustive search to determine the optimal order which
maximises :)p(S

T

1- Initialise T
2- Compute the pdf  and its corresponding Shannon 
entropy
T

p
)p(S T

3- If  is maximum, then  is optimal and = ,)p(S T

T T

p

T

p
     else  and go to 2. 1T T
Then, having , we assign to each point of the optimal pdf 
defined by (11). In this case, the “good data” are the
set of points belonging to the mode of . By extracting the
local maxima of , we can determine the exact points of the
skeleton. In the next section the details of our skeleton
extraction algorithm is presented.

T

p
)y,x(p ji

T


T

p

T

p
4. Skeleton Extraction
We define the skeleton as the local maxima of the estimated
probability density function selected in the previous section.
The extraction of these local maxima allows us to determine
the skeleton associated to the shape. The general idea of this 
algorithm consists of a successive points extraction presenting
a local maxima of the selected optimal pdf. 
The procedure consists in making a sweep mask of size 3x3
on all the image. The comparison of the pdf estimated for the
central pixel of the mask with its close eight  neighbours
following the eight directions (Fig.1), allows to confirm if this
central pixel is a point of the skeleton or no. 
Fig. 1. The pixel (i,j) and its eight close neighbour
Indeed two types of comparison are undertaken, a 
comparison following lines and columns and a comparison
following the diagonal. A pixel is a point candidate if it
presents a local maximum compared to its four neighbours
following the lines and column direction (Fig.1c ) or if it
present a local maxima compared to its four neighbours
following the diagonal direction (Fig.1d).
Fig. 2. Rectangle shape skeleton obtained by ZMSM approach: a) rectangle 
shape, b) optimal pdf corresponding to order 8, c) the mask w1 following lines
and columns direction, d) the mask w2 following diagonal direction, e)
extracted skeleton following lines and columns, f) extracted skeleton 
following the diagonal, g) resulting skeleton of the rectangle shape 
5. Experimental Results
In this section, a comparison study is carried out on 
simulated and real images. The proposed ZMSM
skeletonization method is compared to Distance Transform [5]
and Parallel Thinning Algorithms [3].
The first example demonstrates the performance of the 
proposed skeletonization method with respect to noise. The 
experiment is performed on a hand-written word “moi”
scanned and binarised on (100x100) image matrice (Fig.3a),
then corrupted by an impulsive noise affected 10% of pixels
(Fig.3b). The comparison of the skeletons generated by our
algorithm (Fig.3e) with skeletons obtained using distance
transform (Fig.3c) and parallel thinning algorithm (Fig.3d)
illustrates clearly the insensitivity of the proposed method to 
noise.
Figure 4 shows the performance and the potential of the
proposed approach and its insensitivity to different noise 
values. The ZMSM approach is applied to hand-written digit
“6” With different noise values. One notices that the greater
the noise level, the greater is the optimal order obtained by our 
ZMSM method, the presented figures show that our method
performs well even with high noise levels.
Another example investigating the behaviour of the ZMSM
method is presented applied to a ‘plane’ scanned and
binarized on 100x100 image matrix (Fig.5a) corrupted by an 
gaussian noise having a (Signal to Noise Ratio) SNR=10db
(Fig.5b). The pdf obtained by MEP corresponding to optimal
moment order is presented in (Fig.5d). The skeleton obtained
in Fig.5c by our approach ZMSM demonstrates the
consistency of our algorithm against high level gaussian noise.
Colloque GRETSI, 11-14 septembre 2007, Troyes 907
 (f) 
Fig. 3. Skeletonization of Hand-written word ‘moi’ by the proposed
approach. a) original image. b) input noisy image with impulsive noise 
affecting 10% of pixels. c) skeleton obtained using the Distance Transform. d)
skeleton obtained using the Parallel Thinning Algorithms. e) skeleton obtained 
using the proposed ZMSM. f) estimated pdf for optimal order 36.
Fig. 4. Skeletonization of Hand-written digit ‘6’ by the proposed approach. 
Input noisy image, with impulsive noise affecting 15% of pixels (a), 25% of 
pixels (b), 40% of pixels (c), corresponding  skeletons obtained with ZMSM
method for orders 21 (d), 22 (e), 24 (f).
(d)
Fig. 5. Skeletonization of ‘plane’ shape by the proposed ZMSM. a) original
image and b) corresponding  gaussian noisy image with SNR=10db. c)
Skeleton for the noisy ‘plane’ shape by the  proposed ZMSM. e) estimated pdf 
for optimal order 9.
6. Conclusion
In this work, we have proposed a statistical technique for 
skeletonization noisy images using the Zernike moment theory
and the Maximum Entropy Principal. The advantages of our
algorithm is that no a priori information about the original
image is required. Consequently, the practical implementation
of the approach does not require any parameter setting.
Through a comparative study with conventional methods, it
performed quite well in experimental tests and the
skeletonization has been greatly improved which demonstrates
the robustness of the proposed approach against different and 
high noise levels.
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Skeletonization of Noisy Images via the Method of Moment

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Skeletonization of Noisy Images via the Method of Moment

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