Robust Segmentation Using Kernel and Spatial ... - Semantic Scholar

Robust Segmentation Using Kernel and Spatial Based Fuzzy C-means Methods on Breast X-ray Images Xuejun Sun, Dmitry Goldgof, Walker Land* Department of Computer Science and Engineering, University of South Florida *Department of Bioengineering, Binghamton University ABSTRACT Robust methods for precise segmentation of breast region or volume from breast X-ray images, including mammogram and tomosynthetic image, is crucial for applications of these medical images. However, this task is challenging because the acquired images not only are inherent noisy and inhomogeneous, but there are also connected or overlapped artifacts, or noises on the images as well, due to local volume effect of tissues, parametric resolutions and other physical limitations of the imaging device. This paper proposes and develops robust fuzzy c-means (FCM) segmentation methods for segmentation of breast region on breast x-ray images, including mammography and tomosynthesis, respectively. We develop spatial information- and kernel function- based FCM methods to differentiate breast area or breast volume. Spatial information based FCM method incorporates neighborhood pixels’ intensities into segmentation because neighbored pixels on an image are highly correlated. Kernel based FCM algorithm is developed by transforming pixel intensity using kernel functions to better improve segmentation performance. The proposed segmentation methods are implemented on mammograms and tomosynthetic images and compared with conventional FCM results. Experiment results demonstrate the proposed segmentation methods are much better compared with traditional FCM method, and are more robust to noises. The developed kernel and spatial based FCM method will be applied for differentiation of breast density and abnormal regions within the breast region to examine its performance in reducing false positive segmentations.

1. INTRODUCTION Image segmentation plays vital role in biomedical imaging applications, including quantification, biometrics, diagnosis, and computer-aided surgery. Robust methods for precise segmentation of breast region or volume from breast X-ray images, including mammogram and tomosynthetic image, is crucial for applications of these medical images. However, this task is challenging because the acquired images not only are inherent noisy and inhomogeneous, but they have connected or overlapped artifacts, due to local volume effect of tissues, parametric resolutions and other physical limitations of the imaging device. Lots of segmentation methods for breast region have been reported [1-16], including histogram thresholding [1, 2], simple thresholding and morphological filtering [3-6], gradient based segmentation techniques [7-11], polynomial modeling based methods [13-15], active contour methods [17-21], and classification based techniques [22-25]. Existing breast region segmentation methods upon mammograms have been reviewed in detail elsewhere [16]. In this paper, we develop fuzzy based clustering methods for segmenting breast region on breast X-ray images, including mammograms and digital tomosynthesis images. Fuzzy c-means (FCM) clustering is one of well-known unsupervised clustering techniques, which can be used for unsupervised image segmentation. First developed by Dunn [26] and improved by Bezdek [27], FCM based image analysis methods have been widespread applied in biomedical imaging [28]. Conventional FCM has limited segmentation performance in medical application because it solely does clustering based on pixels’ intensities. Tissues on image are compact regions and thus the neighbored pixels on an image are highly correlated. Thus the spatial relationship of neighboring pixels can be of great aid in image segmentation and should be taken into consideration. Recently, researchers have been incorporating spatial information into the original FCM method to improve its segmentation performance [29, 30].

Medical Imaging 2008: Image Processing, edited by Joseph M. Reinhardt, Josien P. W. Pluim, Proc. of SPIE Vol. 6914, 69143X, (2008) 1605-7422/08/$18 · doi: 10.1117/12.770711 Proc. of SPIE Vol. 6914 69143X-1 2008 SPIE Digital Library -- Subscriber Archive Copy

In k-means clustering, data are grouped in an exclusive way that a certain datum belongs to a definite class and then it could not be included in another class. On the contrary, as an overlapping clustering technique, FCM uses fuzzy sets to measure each datum’s degrees of membership to different classes. Original FCM algorithm clusters each pixel by measuring its Euclidean distance from each class’s centroid. In contemporary machine learning technology, such as support vector machine (SVM), kernel method has been being used to transform the feature space in order to maximize the classification accuracy. Most recently kernel function has been applied into fuzzy clustering [31]. In this paper, we develop spatial information based FCM (SI-FCM) and kernel based spatial FCM (KS-FCM) algorithms and apply them onto breast X-ray images, including mammograms and digital tomosynthetic images. We also compare segmentation results of the developed segmentation method with those of conventional FCM method, and Canny edge based segmentation algorithm. This paper is organized as follows: Section 2 describes the proposed segmentation algorithms, including SI-FCM and KS-FCM. Section 3 presents experimental results and discussions of the developed segmentation methods upon breast X-ray images, and section 4 presents concluding remarks.

2. METHODS Conventional FCM method clusters each pixel by measuring its Euclidean distance from each class’s centroid as expressed:

Jm

N

C

¦¦ uijm xi c j

2

(1)

i 1 j 1

Where

uij

and

cj

is membership function of pixel

xi

to class j and the centroid of class j, respectively.

Neighbor pixels are highly correlated in breast structure. So in SI-FCM algorithm a spatial function is introduced incorporating neighbor pixels’ membership function [29]:

hij

¦u

kj kNB ( xi )

(2)

Where NB represents a set of neighbor pixels of

xi

in a window, which is selected as 5×5. The spatial function is

utilized to incorporate with membership function:

u ij'

u ijp hijq C

(3)

¦ u ikp hikq k 1

thus membership of the pixel

xi

and its neighbors are taken into account together through this modified membership

function. Consequently, the SI-FCM can be expressed as:

Proc. of SPIE Vol. 6914 69143X-2

N

C

¦¦ u'ijm xi c j

Jm

2 (4)

i 1 j 1

The kernel based spatial FCM (KS-FCM) method incorporates both kernel function and spatial penalty into the conventional FCM [31]. First, Euclidean distance in the FCM algorithm is transformed by using kernel function:

)( xi ) )(c j )

2

K ( xi , xi ) K (c j , c j ) 2 K ( xi , c j )

(5)

There are lots of kernel functions available. In this paper, Gaussian radial basis kernel function is utilized:

§ x y K ( x, y ) exp¨ ¨ V2 ©

2

· ¸ ¸ ¹

(6)

Substituting equation 6 into equation 5, we get:

)( xi ) )(c j )

2

2(1 K ( xi , c j ))

(7)

And penalty function containing spatial neighborhood information is added acting as a regularizer and biases the solution toward piecewise homogeneous labeling:

D

f sp Where

NR

NR

N

C

m ij

¦¦ u ¦ (1 u i 1 j 1

rj

)m

(8)

rN k

represents number of neighbor pixels in a window around

xi

except for

xi

itself, and parameter

controls the effect of the penalty function and lies between zero and one inclusive. Substituting equation 7 and adding equation 8 into equation 1, we get

Jm

N

C

¦¦ u

m ij

(1 K ( xi , c j ))

i 1 j 1 N

C

D NR

§ D m ¨ u ¦¦ ij ¨1 K ( xi , c j ) NR i 1 j 1 ©

N

C

m ij

¦¦ u ¦ (1 u i 1 j 1

rj

)m

rN k

· m ¸ ( 1 u ) ¦ rj ¸ rN k ¹

Incorporating constraint condition


(9)

D

C

¦u

1

ij

(10)

j 1

into equation 9, then equation 9 is equivalent to:

Jm

N

C

§ D ¨1 K ( xi , c j ) u ¦¦ ¨ NR i 1 j 1 © m ij

In equation 11, taking derivative of

Jm

C · (1 urj ) ¸¸ Z (1 ¦ uij ) ¦ rN k j 1 ¹ m

with respect to membership function

uij

(11)

and cluster centroid

cj

,

Z

by incorporating constraint condition, we can get respectively, and setting the derivatives to zero [27], and removing the expression of membership function and clustering centroid for the KS-FCM method: 1 /( m 1)

uijm

· § ¨1 K ( xi , c j ) D ¦ (1 urj ) m ¸ ¸ ¨ N R rN k ¹ © 1 /( m 1) C § · D m ¨1 K ( xi , c j ) (1 urj ) ¸¸ ¦ ¦ ¨ N j 1© R rN k ¹ N

cj

¦u

m ij

(12)

K ( xi , c j ) xi

i 1 N

(13)

¦ uijm K ( xi , c j ) i 1

3. EXPERIMENT RESULTS AND DISCUSSION The developed SI-FCM and KS-FCM methods are implemented for segmenting breast region upon digitized mammograms and digital tomosynthesis images, respectively, and the segmentation results are compared with segmentation results using Canny edge detection and conventional FCM methods. The digitized mammogram used in this study has spatial resolution of 50 micron with 16-bit gray value distribution. And the digital tomosynthetic images used has spatial resolution of 100 micron with 16-bit gray value distribution. Figure 1 presents segmentation results on a digitized mammogram using Canny edge detection, conventional FCM, SIFCM, and KS-FCM segmentation methods. It can be seen from the original mammogram Figure 1(a), there are not only characters on the mammogram, but there is also a connected scratch with the breast region to be segmented as well. Furthermore, the mammogram is very noisy with gray value standard deviation of 6978.87 in the background of the image. From segmentation results Figure 1(b)-(e) it can be observed that both the SI-FCM and the KS-FCM methods outperform Canny edge detection and standard FCM segmentation methods. Canny edge detection based segmentation method does not detect breast boundary on the image even incorporated with Gaussian smooth filtering, while standard FCM algorithm clusters lots of background pixels as the same class as the breast region due to high noise in the background region.


Segmentation result from the SI-FCM algorithm is substantially improved over the one using standard FCM method although there are still some false clustered pixels in the background region, and the KS-FCM algorithm performs the best segmentation result among the utilized segmentation methods as shown in figure 1(e). The segmented characters in the background region can be removed easily by setting up area criteria because areas of those characters are much smaller than that of breast region, while the connected scratch can be removed by smoothing breast boundary. Figure 1(f) presents final segmentation result after removing characters and scratch on figure 1(e). From the segmentation results, it can be concluded that under high noise mammogram, the developed KS-FCM performs the best segmentation result, while the developed SI-FCM performs better segmentation result, respectively, among the segmentation methods including Canny edge detection based and standard FCM methods. Figure 2 demonstrates segmentation results on a digital tomosynthesis images using various segmentation methods including Canny edge detection, standard FCM, SI-FCM, and KS-FCM algorithms. The variance in background region of the image is 803.71. Breast boundary is detected using Canny edge detection but the breast region cannot be segmented corrected due to gray value variance in breast region as shown in figure 2(b), while segmentation result using standard FCM method is worse because it can only detect pixels with high gray value as displayed on figure 2(c). Both the SI-FCM and KS-FCM methods perform the same segmentation results on this image as indicated in figure 2(d) and (e). In addition, we also have test segmentation performance of the segmentation methods on digital images with regions connected to the breast region as displayed on figure 3(a), which is a projection image from tomosynthesis system. From segmentation results it can be seen that both Canny edge detection and standard FCM methods cannot separate the connected region from the breast region. Figure 2(d) and (e) demonstrate that both the SI-FCM and KS-FCM methods can separate the connected area from breast region. We compared segmentation speed between the SI-FCM and the KS-FCM on the same digital tomosynthetic images. It is observed that the SI-FCM is faster than the KS-FCM (Order of seconds vs. order of minutes). We observed that segmentation result varies with initial centroid. In our further study, we will be investigating methods for improving selection of a prior knowledge based initial centroids. Furthermore, we will also examine capabilities of the proposed methods in extracting breast density and abnormal areas within breast region.

4. CONCLUSION In this paper, spatial information based FCM (SI-FCM) has been developed by incorporating neighbor pixels’ fuzzy membership function, and kernel based spatial FCM (KS-FCM) method has been developed by transforming Euclidean distance using kernel function and adding spatial penalty function. Both SI-FCM and KS-FCM methods have been applied onto breast X-ray images, including digitized mammograms and digital tomosynthetic images, and the segmentation results have been compared with those using Canny edge detection and standard FCM methods. Upon high noisy images, the KS-FCM performs the best segmentation results over other ones, while the SI-FCM is the better method. Upon low noisy images, both KS-FCM and SI-FCM perform the same segmentation results and are much better than standard FCM method, and SI-FCM is faster than KS-FCM.

ACKNOWLEDGEMENTS This work is supported in part by Suan G. Komen Breast Cancer Foundation, grant No. BCTR0600283. The authors would like to thank our industrial collaborator, Hologic, Inc., for providing clinical tomosynthesis images for this research project.


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Figure 1, segmentation results on a digitized mammogram, (a) original image, (b) Canny edge detection, (c) standard FCM, (d) SIFCM, (e) KS-FCM


a

b

d

c

e

Figure 2, segmentation results on a digital tomosynthetic image, (a) original image, (b) Canny edge detection, (c) standard FCM, (d) SI-FCM, (e) KS-FCM


a

b

d

c

e

Figure 3, Three-class segmentation results on a digital tomosynthetic image, (a) original image, (b) Canny edge detection, (c) standard FCM, (d) SI-FCM, (e) KS-FCM

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