The liver lesion segmentation is the second stage in the framework. The lesion is de- tected from the segmented liver, as presented in Section4.1. The clustering method is widely used in lesion segmentation by dividing the input image into different clusters, based on changes in the intensity value. The different tissues in the medical images of- ten comprise overlapping intensity on the gray level. Accordingly, the fuzzy clustering algorithm is particularly suitable for lesion detection from the medical image such as CT scan.
Fuzzy clustering is a popular unsupervised approach in medical image processing field, especially for liver lesion segmentation (Moghbel et al., 2016). This is because of its fuzzy nature, which allows pixels to belong to multiple clusters with varying de- grees of membership. The advantages of Fuzzy clustering algorithm include a straight- forward implementation and applicability to multichannel data make. Especially, the ability of Fuzzy clustering approach to model uncertainty within the data give a so- lution to deal with the fuzziness of the CT images. The CT images mainly suffer from inhomogeneity and uncertainty due to differences in noise and scanning geome- tries (e.g. starting angles of the helical scan). Furthermore, the substantial variations exist among different manufacturers when measuring CT images in phantoms, poten- tially owing to differences in x-ray spectra, CT detectors, and reconstruction kernels (Birnbaum et al.,2007;Fletcher et al.,2016). Because of our dataset has varied charac- teristics (e.g. resolution, spaces, etc.) and has produced from different manufacturers, the Fuzzy clustering approach has adopted in this work.
The fuzzy c-means (FCM) clustering algorithm assigns pixels to each group by using fuzzy memberships. Let X = {x1, x2, ..., xn} denotes an image with n pixels to
be grouped into c clusters where c is a positive integer greater than one. The c clusters is a partition of X into mutually disjoint sets {x1, x2, ..., xc} where X = x1∪x2∪...∪xc
or equivalently based on indicator function {µ1, µ2, ..., µc} where µi(x) = 1 if x ∈ Xi
and µi(x) = 0 if x /∈ Xi for all i = 1, 2, ..., c. The set {µ1, µ2, ..., µc} of indicator
function is called a hard c-partition of clustering X into c classes (Wu and Yang,2005). The following Equation4.8defined the hard c-means.
H(µ, a) = c X i=1 n X j=1 µijkxj − aik2 (4.8)
Where {µ1, µ2, ..., µc} with µij = µi(xj) is a fuzzy c-partition and {a1, a2, ..., ac}
The hard c-means extended to allow µi(x) to be membership function of fuzzy sets
µion X assuming values in the interval [0, 1], where c
P
i=1
µi(x) = 1 for all x ∈ X. This
extension of {µ1, µ2, ..., µc} is called a fuzzy c-partition of X, as defined in Equation
4.9. Hm(µ, a) = c X i=1 n X j=1 µmijkxj − aik 2 , m > 1 (4.9)
where m (weighting exponent) is the degree of fuzziness, which could be used to enhance the clustering performance of FCM (Yu et al., 2004). The FCM clustering method minimises the objective function Hm in an iterative way, as depicted in Equa-
tion4.10and Equation4.11.
µij = [1/kxj − aik2] 1/(m−1) c P k=1 [1/kxj − akk2] 1/(m−1) (4.10) ai = n P j=1 (µij) m xj n P j=1 (µij)m (4.11)
The fuzzy c-means (FCM) clustering method has been employed widely in le- sion detection. However, the FCM is sensitive to noise and outliers. Therefore, the alternative fuzzy c-means (AFCM) is used for their ability to tolerate noise and out- liers that often happen in medical images (Wu and Yang, 2002). The euclidean norm E2(x, a) = kx − ak2 used in FCM has been replaced in AFCM with new one, as
presented in Equation4.12.
E2(x, x) = 1 − exp(−βkx − ak2) (4.12)
The AFCM clustering method groups the similar pixels iteratively and the centres of the grouped pixels are adjusted for all iterations. The AFCM objective function measures the overall dissimilarity within the grouped pixels and the dissimilarity re- quires to be minimised to get the optimal partition. The objective function of AFCM is shown in Equation4.13. HAF CM = c X i=1 n X j=1 (µij) m1 − exp(−βkx j− aik 2 (4.13)
Chapter 4. Liver Image Analysis in CT
Where β is a constant which can be defined in Equation4.14.
β = n P j=1 kxj − ¯xk2 n −1 with ¯x = n X j=1 xj n (4.14)
Through the objective function, the pixels are assigned to higher membership value when their intensities are close to the centroid of corresponding groups, or the lower membership values are being assigned to them when the intensities are far from the centroid. The membership value of the intensities of pixels and centre of the grouped pixels are solely dependent on the distance between each other to assign these pixels to a specific group. The Equation4.15and Equation4.16are been used to update the membership functions and the group centres.
µij = [1/(1 − exp(−βkxj− aik2))] 1/(m−1) c P k=1 [1/(1 − exp(−βkxj − akk2))] 1/(m−1) (4.15) and ai = n P j=1 (µij) m exp(−βkxj − aik 2 )xj n P j=1 (µij)mexp(−βkxj− aik2) (4.16)
The lesion is extracted from the segmented liver using AFCM clustering method (Wu and Yang,2002) to segregates each pixel into one of three defined classes which are liver, lesion and vessels. According to the pre-knowledge, the liver parenchyma appears less brighter intensity compared to the vessels where the lesions appear darker (Sahani and Kalva, 2004a; Oliveira et al., 2011a). The pixels in the lesion with low intensity are assigned to the first cluster, the liver pixels are assigned to the second cluster and third cluster contains the vessels pixels with the high-intensity value. The AFCM clustering algorithm assigns pixels for each class using fuzzy membership.
Figure4.11 depicts the liver lesion segmentation process. The original CT image shown in Figure 4.11.a. The segmented liver is presented in Figure 4.11.b. Figure
4.11.c shows the image pixels clustering into one of three defined class by applying AFCM algorithm where Liver area defined as green colour, Lesion is red and vessels is blue. Figure4.11.d. shows the final result of liver lesion segmentation.
Figure 4.11: Steps of liver lesion segmentation process. (a) The original input CT image. (b) The segmented liver. (c) Operates AFCM to cluster the segmented liver into three areas where the green colour presents liver, lesion is red and vessels are blue . (d) The final result of the liver lesion segmentation.
Table4.1shows the liver lesion detection algorithm that analyses a segmented liver by using AFCM algorithm to classify image pixels into three classes which are liver, lesion and vessels.
ALGORITHM 1: Clustering the segmented liver into three classes (Liver, Lesion and Vessels)
INPUT:
Liver Image X = {x1, x2, ..., xn}
SET number of cluster c=3; SET degree of fuzziness m=2; SET improvement value ε = 0.0001;
SET Random initial centre ao = {a(o)1 , ..., a(o)c };
OPERATION:
Using Equation4.14to estimate parameter (β); WHILE a(k+1)− a(k)
< ε DO
Using Equation4.15to calculate µ(k)with a(k);
Using Equation4.16to calculate a(k+1)with µ(k)and a(k);
IF a(k+1)− a(k)
> ε THEN k=k+1 END WHILE
OUTPUT L = {l1, l2, . . . , ln}, liis a label for point xi;
Table 4.1: Our algorithm to cluster image pixels into three classes (liver, lesion and vessels) based on AFCM.
Chapter 4. Liver Image Analysis in CT
In the above algorithm, the number of cluster is three classes which are liver, lesion and vessels. The degree of fuzziness (m) selected the value two where the updated fuzzy membership value is proportional to the square of the inverse distance from a specific segment location to each cluster’s centroid.
Figure4.12illustrates the graphical evaluation results of liver lesion segmentation based on Dice similarity coefficient (DSC) metrics. The vertical axis (x-axis) repre- sents the number of cases that achieved the DSC values in the horizontal axis (y-axis). The highest DSC value means the better segmentation result.
Figure 4.12: Number of cases and obtained DSC values of liver lesion segmentation for evaluating the method.
Figure 4.13:Samples of liver lesion detection results (our method versus ground truth). Our method is shown in blue and the reference detection in yellow.
Figure4.13 illustrates the liver lesion detection accuracy of the method compared to the manual radiologist detection. The results of lesion detection shows over-segmentation and under-segmentation compared to the ground truth on some of cases especially with low intensity contrast between lesion and liver or with weak lesion boundary. This is due to the our detection method mainly relies on the gray level intensity difference between lesion and normal liver tissue.
Towards a better liver lesion segmentation, the post-processing step is applied to refine the lesion boundaries. In order to reduce the false negative/ false positive seg- mentation due to under/ over-segmentation of the lesion, the fast marching approach is used to enumerate the majority of the lesion voxels. Subsequently, level set approach refines lesion final shape by attaching the segmented contour to edges in the image while maintaining smoothness. The main advantages of these techniques are that arbi- trarily complex shapes can be modeled and topological changes are handled implicitly. The idea behind this process is to refine the delineation of lesion boundaries that rep- resents the abnormality area with respect to the healthy liver, as shown in Figure4.14. Figure4.14.a depicts the segmentation results using our method based on AFCM ap- proach and before the refinement process. Figure4.14. illustrates the results after the refinement process.
Figure 4.14: Liver lesion segmentation versus ground truth (GT) where GT is delin- eated as red contours. (a) The segmentation results before refinement process. (b) The segmentation results after refinement process.
Chapter 4. Liver Image Analysis in CT