INTRODUCTION
Image segmentation is the process of subdividing an image into its constituent
regions or objects based on attributes such as intensity or continuity is one
of the most difficult and challenging problems in the image processing which
is widely used in a variety of applications such as robot vision, object recognition,
geographical imaging and medical imaging (Chaira and Ray,
2003; WeiYi et al., 2010). In the last decades,
fuzzy segmentation methods, especially the fuzzy cmeans algorithm (FCM), have
been widely used in the image segmentation (Hoppner and
Klawonn, 2003; Liew et al., 2005). It can
retain more information from the original image. However, one disadvantage of
FCM is its sensitivity to noise and other imaging artifacts. One of the most
commonly used methods is to modify the objective functions (Liew
et al., 2005). Mohamed et al. (1999)
used a modified fuzzy FCM algorithm for image segmentation. They considered
the dissimilarity measure as the resistance of pixel to the cluster center and
modified the dissimilarity measure such that the pixel can be dragged closer
to the cluster center if it is in homogeneous regions. Pham
(2001) proposed a robust fuzzy clustering based image segmentation method
for noisy image (RFCM) which is presented to achieve the unimodal property of
its membership functions.
However, besides the influence of noises, the spatial intensity variations
caused by uneven illumination are another obstacle in image segmentation based
on FCM. To compensate for this drawback of FCM, the obvious way is to smooth
the image before segmentation such as homomorphic (Brinkmann
et al., 1998). But it is only on images with relatively lowcontrast.
Some researchers reported undesirable artifacts with approach (Dawant
et al., 1993). A different approach used to segment images with intensity
inhomogeneity is to simultaneously compensate for the uneven illumination effect
while segmenting the image (Pham and Prince, 1999).
In their proposed energy function, a term was introduced to ensure the smoothness
of the computed uneven illumination. Xu et al. (1997)
proposed a new adaptive FCM technique to produce fuzzy segmentation while compensation
for uneven illumination. But it is also computationally intensive due to the
introduction of the uneven illumination field smooth term. Mohamed proposed
a modified FCM algorithm for bias field estimation and segmentation (Ahmed
et al., 2002); however, it did not took into the characteristics
of uneven illumination account. In general, the uneven illumination is commonly
assumed to be slowly varying (Li et al., 2009),
theoretically, any function can be approximated by a linear combination of a
set of functions up to arbitrary accuracy, therefore, we propose a novel FCM
image segmentation with simultaneous uneven illumination estimation. The uneven
illumination is modeled as a linear combination of smooth basis functions; furthermore,
introduce it into the objective function in FCM algorithm. Uneven illumination
field and image segmentation ares simultaneously achieved by Lagrange multiplier
method. A salient advantage of our method is that the novel FCM clustering is
robust for uneven illumination and simultaneously estimate the field effect.
BACKGROUND
The observed image signal I (x) is model as a product of the true signal S (x) and uneven illumination field effect B (x):
The application of a logarithmic transformation to the intensities allows the
artifact to be modeled as additive uneven illumination field effect (Wells
et al., 1996).
where, s (x) and i (x) are the true and observed logtransformed intensities at pixel x and the b (x) is the uneven illumination field effect.
ROBUST FCM BASED IMAGE SEGMENTATION WITH SIMULTANEOUS UNEVEN ILLUMINATION ESTIMATION
Image segmentation based on FCM clustering: The standard FCM objective function for partitioning
into c cluster is given by Pham (2001):
where
are the prototypes of the clusters and array
represents a partition matrix, subject to
The parameter m is a weighing exponent on each fuzzy membership and determines the amount of fuzziness of the result classification. I (x)(v)_{i} is the distance of I (x) and the center of class i. the objective function is sensitivity for noise because it omits the structure information. In order to remedy this deficiency, it is a better way that the assignment of a pixel is influenced by its neighboring pixels. This method is known as BCFCM and the modified objective function is:
where, N_{k} stands for the set of neighbors that exist in a window around x_{k} and N_{R} is the cardinality of N_{k} The effect of the neighbors term is controlled by the parameter α.
Although, the improved FCM clustering method is more robust for noises than FCM because the structure information was taken into account, it still sensitivity for the spatial intensity variations caused by uneven illumination field effect.
Robust FCM based image segmentation with simultaneous uneven illumination estimation: In general, the intensity inhomogeneity caused by uneven illumination is assumed to be slowly varying. Moreover, any function can be approximated by a linear combination of a set of basis functions up to arbitrary accuracy theoretically. Let g_{1}, …, g_{M} be a set of basis functions, we estimate the uneven field effect by a linear combination of the basis functions
In our current implementation, we use orthogonal polynomials as the basis functions. So the true image data is denoted as:
Let W = (w_{1}, …, w_{M})^{T}, G (x) = (g_{1}
(x), …, g_{M} (x))^{T} and substitute I (x) in Eq.
6 with 8, then the improved objective function which taken
into the uneven illumination account is defined as:
We can achieve the U, V and W by Lagrange multiplier. Let
Let
As for W, fix the u and v, solve
and let it is 0, we can obtain
where
and
The improved image segmentation algorithm with simultaneous uneven illumination
estimation:
Step 1 
: 
Transform image data to logarithmic field 
Step 2 
: 
Initialization of u, m and g 
Step 3 
: 
Update u, v and w by Eq. 913 
Step 4 
: 
Check convergence criterion. if convergence has been reached, stop
the iteration. otherwise, go to step 3 
RESULTS
We investigate the very applicability of the abovemodified FCM in image segmentation. Present experimental results here demonstrate that when the segmentation for intensity inhomogeneity and noisy images is required, the abovemodified FCM has obvious advantage over FCM.
Figure 1a shows an original rat’s bladder tissue image.
The image is corrupted as its artificial noisy image and intensity inhomogeneity
caused by uneven illumination.

Fig. 1:  (ad)
segmentation results for rat’s bladder tissue image 

Fig. 2:  (ad)
segmentation results for mankind chromosome image 
Figure 2a shows a mankind chromosome image acquired under
the circumstance of uneven illumination. Both in Fig. 1 and
2b and c demonstrate the segmentation results
using BCFCM and the proposed modified FCM, Obviously, the proposed modified
FCM outperforms BCFCM for this test image and get uneven illumination field
effect estimation as Fig. 1d.
CONCLUSION
In this correspondence, we propose a robust fuzzy clustering based segmentation
method for noisy and intensity inhomogeneity caused by uneven illumination.
This method is rooted at the proposed modified FCM which originates from a novel
objective function and uneven illumination presentation. It is robust for noises
and intensity inhomogeneity; meanwhile, it can estimate the uneven illumination
simultaneously.
Future research may be focused on more mathematical analysis about the fitting of uneven illumination field effect and the time efficiency.
ACKNOWLEDGMENT
The study is supported by the Gansu Natural Science Foundation of China under Grant (No. 1010RJZA011).