International Machine Vision and Image Processing Conference
Fuzzy Logic Based Segmentation of Microcalcification in Breast Using Digital Mammograms Considering Multiresolution Mahua Bhattacharya1 and Arpita Das2 1
Indian Institute of Information Technology & Management, Gwalior Morena Link Road, Gwalior-474003, India 1 e-mail:
[email protected] 2
Institute of Radio Physics & Electronics, University of Calcutta 92 A.P.C. Road, Kolkata-700009, India 2 e- mail:
[email protected] Abstract automatic segmentation of microcalcification clusters in digital mammography [3],[6],[16],[17]. In case of dense breast, it is almost impossible to identify the cluster of microcalcifications. The information which is 3% of actual microcalcifications are visible in the digital mammogram [1],[7]. A number of digital image processing techniques have been applied to address the mentioned problems [1-3],[7],[9]. Recent discoveries show that a multiresolution approach exists in human vision system, thus leading to an idea of using wavelet based multiresolution analysis for mammographic image processing [2],[5]. A multiresolution method using wavelet filters is proposed in [2] for detection of microcalcification. Tree-structured order statistic filter as a prefiltering stage cascaded with a wavelet based detection algorithm has been suggested [3]. Others [4],[5] used wavelet for background removal or for denoising purposes. In this paper, we have two goals. Firstly enhancement of mammographic images for better visibility of the observed phenomenon to the radiologist, and secondly processing of mammograms to obtain automatic detection of microcalcifications. To achieve both goals, we first apply wavelet transform on mammographic images and secondly, the morphological top-hat transformation [11],[12] has been used for accurate decomposition of each microcalcifications. Finally fuzzy c-means clustering algorithm is used to segment the detected microcalcification [13].
Breast cancer is one of the leading causes of death for women. Small clusters of microcalcifications appearing as collection of white spots on mammograms show an early warning of breast cancer. In present paper a novel approach of segmentation implemented on X-ray mammograms for more accurate detection of microcalcification clusters has been introduced. The method is based on discrete wavelet transform due to its multiresolution properties. Morphological tophat algorithm is applied for contrast enhancement of the calcification clusters. Finally fuzzy c-means clustering (FCM) algorithm has been implemented for intensity-based segmentation. The proposed technique is compared with conventional global thresholding method and experimental results show the good properties of the proposed technique. Keywords: Microcalcifications, Wavelet Transform, Morphological Top-hat transformation, FCM algorithm.
1. Introduction Research on digital mammograms aims to develop expert system for automated diagnosis of breast cancer [1],[6-10],[15-19]. Calcifications are important marker for breast cancer identification. Malignant calcifications may occur with or without the presence of a tumor mass. These are typically clustered, varying in size and of irregular shape. Due to their high attenuation properties, they appear as white (or high intensity) spots on mammograms. A large number of microcalcifications in a cluster increase the likelihood of malignancy. Many authors deal with the problem of
0-7695-2887-2/07 $25.00 © 2007 IEEE DOI 10.1109/IMVIP.2007.33
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2. Proposed Method
pyramid. An image pyramid is a collection of decreasing resolution images arranged in the shape of a pyramid as shown in Fig-2. The base of the pyramid contains the highest resolution representation of the image; the apex contains a low-resolution approximation. With moving up the pyramid, both size J J and resolution decrease. The base level J is size 2 × 2 j j or N × N, intermediate level j is size 2 × 2 , where 0 ≤ j ≤ J. Fully populated pyramids are composed of J + 1 J J 0 0 resolution levels from 2 × 2 to 2 × 2 , but in practice most pyramids are truncated to P + 1 levels, where j = J − P, …., J − 2, J − 1, J and 1 ≤ P ≤ J, since a 1 × 1 pixel image is of little value.
In present paper we discuss two approaches for segmentation of microcalcification centers. Fig-1 demonstrates an overview of our present work. The proposed algorithm uses wavelet-based transformation from the multiresolution point of view. This approach has been used for detection of feature of breast tumor which may not be detected in single resolution and may be easily detectable at another resolution. The morphological top-hat transformation has been applied on the transformed image for enhancing detail in the presence of shading. Finally fuzzy clustering technique used as an efficient intensity based segmentation method (LHS division). The quality of the proposed algorithm is compared with conventional Global Thresholding method [14] for image segmentation (RHS division). Image Acquisition
Multiresolution Processing (wavelet transform)
Morphological Top-hat filtering
FCM based segmentation
Fig-2: A Pyramidal Image Structure
Global Thresholding
Multiresolution Expressions In multiresolution analysis (MRA), scaling function is used to create a series of approximations of an image, each differing by a factor of 2 from its nearest neighboring approximations. Additional functions, called wavelets, are then used to encode the difference in information between adjacent approximations.
Erosion for reduction of false positive Fig-1: Brief overview of our work
2.1 Multiresolution Approach
Haar Transform
In an image, if both small (tiny calcifications) and large (background) objects or low and high contrast objects are present simultaneously, it is advantageous to study them at several resolutions. This is the fundamental motivation for multiresolution processing.
It is a multiresolution analysis of any signal or images. The basis functions of Haar Transform are known as orthonormal wavelets. Haar transform can be expressed in matrix form
Image pyramid
where F is an N × N image matrix, H is an N × N transformation matrix and T is the resulting N × N transform. For the Haar transform, transformation
T = HFH
A powerful but simple structure for representing images at more than one resolution is the image
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---------(1)
f(x) = ∑ αkϕj0, k(x)
matrix H contains the Haar basis functions, hk(z). They are defined over the continuous, closed interval z ∈ [0, n 1] for k = 0, 1, 2, …, N − 1, where N = 2 .
--------(6) k More generally, the subspace is spanned over k for any j as Vj = Span {ϕj, k(x)} -------(7) k
Series Expansions A function f(x) can often be better analyzed as a linear combination of expansion functions f(x) = ∑ αkϕk(x) k
Wavelet Functions
--------(2)
Given a scaling function, wavelet function Ψ(x) can be defined that, together with its integer translates and binary scalings spans the difference between any two adjacent scaling subspaces, Vj and Vj+1. The situation is illustrated graphically in Fig-3.
where k is an integer index of the finite sum, the αk are real-valued expansion coefficients and the ϕk(x) are real-valued expansion functions. If the expansion is unique⎯that is, there is only one set of αk for any given f(x)⎯the ϕk(x) are called basis functions and the expansion set, {ϕk(x)}, is called a basis for a function space that is referred to as the closed span of the expansion set, denoted as V = Span {ϕk(x)} k
--------(3)
To say that f(x) ∈ V means that f(x) is in the closed span of {ϕk(x)}. Scaling Functions
Fig-3: The relationship between scaling and wavelet function spaces.
The set of expansion functions composed of integer translations and binary scalings of the real, squareintegrable function ϕ(x); that is the set {ϕ j,k(x)} where ϕ j, k(x) = 2 j/2 ϕ(2j x −k) --------(4)
The set {Ψj,k(x)} of wavelets can be defined as
Ψj,k(x) = 2j/2 Ψ(2j x −k)
for all k ∈ Z that spans the Wj spaces in the Fig-2.1.2. As with scaling functions, it can be written
2
for all j, k ∈ L (R). Here k determines the position of ϕj,k(x) along the x-axis, j determines ϕj, k(x)’ s width⎯how broad or narrow it is along the xj/2 axis⎯and 2 controls its height or amplitude. Because the shape of ϕj, k(x) changes with j, ϕ(x) is called a scaling function. By choosing, ϕ(x) wisely, 2 {ϕk(x)} can made to span L (R), the set of all measurable, square-integrable functions. If we restrict j in eq. (3) to a specific value, say j = j0, the resulting expansion set, {ϕj0, k(x)} is a subset of {ϕj,k(x)}.It will not span L2(R), but a sub-space within it. Thus we can define that subspace as
Vj0 = Span {ϕj0, k(x)}
---------(8)
Wj = Span {Ψj,k(x)} k
---------(9)
and if f(x) ∈ Wj, --------(10) f(x) = ∑αkψj, k(x) k The scaling and wavelet function subspaces in Fig-3 are related by Vj+1 = Vj ⊕ Wj
--------(11)
where ⊕ denotes the union of spaces. Thus all measurable, square-integrable functions are expressed as
--------(5)
2
k
L (R) = V0 ⊕ W0 ⊕ W1⊕ …
That is, Vj0 is the span of ϕj0, k(x) over k. If f(x)∈Vj0, it can be written
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--------(12)
The Discrete Wavelet Transform Wavelet series expansion maps a function of continuous variable into a sequence of coefficients. If the function being expanded is a sequence of numbers, like samples of a continuous function f(x), the resulting coefficients are called the Discrete Wavelet Transform (DWT) of f(x). In this case Wϕ(j0, k) = (1/√ M) × ∑ f(x) ϕj0, k(x) x
---------(13)
WΨ(j, k) = (1/√ M) × ∑ f(x) Ψj, k(x) x
---------(14)
Fig-4: Calcification Clusters obtained before Top-hat filtering.
for j ≥ j0 and f(x) = (1/√ M) × ∑ Wϕ(j0, k) ϕj0(x) + k ∝ (1/√ M) × ∑ ∑ WΨ(j, k) Ψj, k(x) --------(15) j =j0 k Here, f(x), ϕj0, k(x) and Ψj, k(x) are functions of the discrete variable x = 0, 1, 2, …., M − 1. Normally it is considered that j0 = 0 and M is selected as a power of 2 J (i. e. M = 2 ) so that the summations are performed over x = 0, 1, 2, …, M − 1, j = 0, 1, 2, …, J − 1 and k = j 0, 1, 2, …, 2 − 1. For Haar wavelets, the discretized scaling and wavelet functions employed in the transform (i. e., the basis functions) correspond to the rows of the M × M Haar transformation matrix. The transform itself is composed of M coefficients, the minimum scale is 0 and maximum scale is J − 1. The coefficients defined in eq. (13) and eq. (14) are called approximation and detail coefficients respectively.
Fig-5: Enhanced microcalcifications after top-hat filtering
2.3 Fuzzy c-Means Clustering Algorithm The final classification for detection of malignancy / benignancy of microcalcifications, depends on the superiority of the segmentation process. In the proposed method, fuzzy c-means clustering algorithm used for intensity based segmentation of microcalcification clusters. Total number of fuzzy cluster centers is chosen three as shown in Fig-6. Cluster center A represents the normal breast tissue. Second cluster B represents false presence of microcalcifications and C represents actual calcification points.
2.2 Morphological Top-hat Transformation The morphological top-hat transformation of an image, denoted by h, is defined as h = f − (f ο b)
--------(16)
[ f ο b = (f θ b) ⊕ b ] where, f is the input image and b is the structuring element function. This transformation used a cylindrical structuring element function with flattop and is very useful for enhancing detail in the presence of shading. Top-hat filtering can also be used to correct the illumination and make the result more easily visible when background is dark.
Fig-6: Final Fuzzy Partition Membership Functions.
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The ultimate Fuzzy partition membership functions are shown in Fig-6, which show there is an overlapping between the membership functions A, B & C. If the possibility of a region that belongs to the calcification points is greater than 50% (i.e. the membership value of curve C > 0.5), the decision is: the region belongs to the calcification clusters. Thus according to our decision rule, the shaded region of Fig-6 is under the calcification clusters or ROI .
pseudopartition P that minimizes the performance index Jm (Ρ).
2.4 Basic Global thresholding Method Image thresholding plays an important role in application of image segmentation. The simplest of all thresholding techniques is to partition the image histogram by using a single global threshold T. The following algorithm can be used to obtain T automatically.
Algorithm: Let X={x1, x2,……..,x n} be a set of given data. A fuzzy c-partition of X is a family of fuzzy subsets of X, denotes by P = {A1, A2,.……., Ac}, which satisfies c ----------(17) ∑ Ai (x k) = 1 i=1 for all k ∈ Nn and n 0 < ∑ Ai (x k) < n ----------(18) k=1 for all i ∈ Nc, where c is a positive integer. Given a set of data X = { x1, x2,……..,x n}, where x k, in general is a vector, for all k ∈ N n , the problem of fuzzy clustering is to find a fuzzy pseudopartition and the associated cluster centers by which the structure of the data is represented as best as possible. To solve the problem of fuzzy clustering, we need to formulate a performance index. Usually, the performance index is based upon cluster centers, v1, v2, ……., v c associated with the partition and which are calculated by the formula given below n ∑ [Ai (x k)] m x k k=1 --------(19) vi= n ∑ [Ai (x k)] m k=1
an
initial
estimate
Selection of (heuristically).
2.
Segmentation of the image using T. This will produce two groups of pixels; G1 consisting of all pixels with gray level values > T and G2 consisting of pixels with values ≤ T.
3.
Computation of the average gray level values µ1 and µ2 for the pixels in regions G1 and G2.
4.
Computation of a new threshold value: T = 0.5 × (µ1 + µ2 )
5.
for
T
1.
-------(21)
Repetition of steps 2 through 4 until the difference in T in successive iterations is smaller than a predefined parameter T0.
2.5 Morphological Erosion/Dilation 2
With A and B as sets in Z , the erosion of A by B, denoted A θ B, is defined as A θ B = {z | (B)Z ⊆ A}
--------(22)
Set B is commonly referred to as the structuring element in erosion. 2 Similarly for sets A and B as sets in Z , the dilation of A by B, denoted A θ B, is defined as
for all i ∈ Nc, where m >1 is a real number that governs the influence of membership grades. Observe that the vector vi calculated by above equation is viewed as the cluster center of the fuzzy class Ai, is actually weighted average of data in Ai. The performance index of a fuzzy pseudopartition P, Jm (Ρ), is defined in terms of the cluster centers by the formula n c Jm (Ρ) = ∑ ∑ [Ai (x k)] m ⎢⎢ xk − vi ⎢⎢2 -------(20) k=1 i=1
A ⊕ B = {z | (B)Z ∩ A ≠ ∅}
--------(23)
One of the popular uses of erosion is for eliminating irrelevant detail from the binary image. In this paper dilation has been used followed by erosion operation to get back the original size of microcalcifications, which are distorted during erosion.
3. Experimental Results We have applied the proposed algorithm to a database consisting of 65 images from Mammographic Image Analysis Society (MIAS). We have performed Multiresolution Processing to differentiate the tiny and bright spot of microcalcifications from the normal,
where ⎢⎢ xk − vi ⎢⎢2 represents the distance between xk and vi . Clearly, the smaller the value of Jm (Ρ), the better the fuzzy pseudopartition P. Thus, the goal of fuzzy c-means clustering method is to find a fuzzy
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healthy breast tissue and thus Wavelet transform are taken into account. A morphological top-hat operation with circular kernel of radius 2 pixels for background filtering has been used and finally fuzzy c-means based clustering algorithm has been implemented to detect individual microcalcifications.
To evaluate the superiority of the proposed algorithm, we compared it with conventional global thresholding segmentation method followed by erosion/dilation technique for reduction of false positive.
Set-I Images
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Fig-7: [1]-[4]: Original Mammographic Image from MIAS Database [5]-[8]: Segmentation of Microcalcifications using Proposed Algorithm. [9]-[12]: Segmentation of Microcalcifications using Conventional Method.
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Set-II Images
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Fig-8: [1]-[4]: Original Mammographic Image from MIAS Database [5]-[8]: Segmentation of Microcalcifications using Proposed Algorithm. [9]-[12]: Segmentation of Microcalcifications using Conventional Method.
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mammograms,” IEEE Trans. Med. Imag., vol. 9, pp. 233– 241, (1990). [7]. S. Yu and L. Guan, “A CAD System for the Automatic Detection of Clustered Microcalcifications in Digital Mammogram Films”, IEEE Trans. on Medical Imaging, vol. 19, no. 2, pp. 115-126, 2000. [8]. H. Chan, S. B. Lo, B. Sahiner, K. L. Lam, and M.A. Helvie, “Computer-aided detection of mammographic microcalcifications: Pattern Recognition with artificial neural network," Med. Physics, vol. 22, no. 10, 1995. [9]. N. R. Mudigonda, R.M. Rangayyan, and J. Desautels, “Detection of breast masses in mammograms by density slicing and texture flow-field analysis,” IEEE Trans. Med. Imaging, vol.20, no.12, pp.1215–1227, (2001). [10]. R.M. Nishikawa, Y. Jiang, M.L. Giger, K. Doi, C.J.Vyborny, R.A. Schmidt, “Computer-aided detection of clustered microcalcifications”, Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, pp. 1375–1378, 1992. [11]. J. Goutsias, L. Vincent, D.S. Bloomberg, “Mathematical Morphology and Its Applications to Image and Signal Processing”, Kluwer Academic Publishers, Boston, Mass, 2000. [12]. H. Park and R.T. Chin, “Decomposition of Arbitrary Shaped Morphological Structuring”, IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 17, no. 1, pp. 2-15, 1995. [13]. H.D. Cheng and H. Huijuan Xu, “A Novel Fuzzy Logic Approach to Contrast Enhancement”, Pattern Recognition, vol. 33, no. 5, pp. 809-819, 2000. [14]. A. Perez and R.C. Gonzalez, “An Iterative Thresholding Algorithm for Image Segmentation”, IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. PAMI-9, no. 6, pp. 742-751,1987. [15]. M.Bhattacharya and D.Dutta Majumder, “Breast Cancer Screening Using Mammographic Image Analysis' Sixteen International CODATA (France) Conference (8-12 Nov , Delhi (1998). [16]. M.Bhattacharya, “Development of Mathematical Model for Radiographic Image Analysis”, International Journal of Computational and Numerical Analysis and Applications, Academic Publication ( published 2007). [17]. Craig K. Abbey, Roger J. Zemp, Jie Liu, Karen K. Lindfors, and Michael F. Insana, “Observer Efficiency in Discrimination Tasks Simulating Malignant and Benign Breast Lesions Imaged With Ultrasound”, IEEE Trans. Medical Imaging, vol. 25, no.2, pp: 198 -209 (2006). [18]. Mahua Bhattacharya, “A Computer-Assisted Diagnostic Procedure For Digital Mammograms Using Adaptive Neuro Fuzzy Soft Computing” accepted IEEE Nuclear Science Symposium, Medical Imaging Conference, San Diego, California USA , 29 th Oct -4 th Nov , 2006. [19]. B. Sahiner, H.P. Chan, N. Petrick, M.A. Helvie, and L.M. Hadjiiski, “Improvement of mammographic mass characterization using speculation measures and morphological features,” Med. Physics, vol.28, pp.1455– 1465, (2001).
4. Conclusion We have presented an algorithm for microcalcification segmentation in mammographic Xray images. The proposed algorithm used fuzzy cmeans clustering algorithm for intensity-based segmentation of microcalcification centers. The presence of false positive calcification points has been reduced by choosing three cluster centers. The second cluster center actually posses with the false presence of calcification points, which is suppressed in the final segmented images. In global thresholding method for image segmentation, morphological erosion technique is used for reduction of false positive, but suitable size of structuring element (kernel) is different for different images. Thus it is concluded that our proposed algorithm is fully automatic, parameter free and independent of local statistics. The minute calcification dots (in original X-ray mammograms) are also very prominent, especially in case of dense breast and the decision made is closer to the opinion of radiologists. We have currently applied our algorithm to a large database and investigated the superiority of it over the existing global thresholding method followed by morphological erosion operation.
Acknowledgment The authors would like to thank to Dr. S. K. Sharma, Director, EKO Imaging and X-Ray Institute, Kolkata.
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