

ORIGINAL ARTICLE 



Year : 2014  Volume
: 39
 Issue : 3  Page : 169183 

A combined approach for the enhancement and segmentation of mammograms using modified fuzzy Cmeans method in wavelet domain
Subodh Srivastava^{1}, Neeraj Sharma^{1}, SK Singh^{2}, R Srivastava^{2}
^{1} School of BioMedical Engineering, Indian Institute of Technology, Banaras Hindu University, Varanasi, Uttar Pradesh, India ^{2} Department of Computer Science and Engineering, Indian Institute of Technology, Banaras Hindu University, Varanasi, Uttar Pradesh, India
Date of Submission  28Dec2013 
Date of Decision  16May2014 
Date of Acceptance  17May2014 
Date of Web Publication  17Aug2014 
Correspondence Address: Subodh Srivastava Ph.D., School of BioMedical Engineering, Indian Institute of Technology, Banaras Hindu University, Varanasi  221 005, Uttar Pradesh India
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/09716203.139007
Abstract   
In this paper, a combined approach for enhancement and segmentation of mammograms is proposed. In preprocessing stage, a contrast limited adaptive histogram equalization (CLAHE) method is applied to obtain the better contrast mammograms. After this, the proposed combined methods are applied. In the first step of the proposed approach, a two dimensional (2D) discrete wavelet transform (DWT) is applied to all the input images. In the second step, a proposed nonlinear complex diffusion based unsharp masking and crispening method is applied on the approximation coefficients of the wavelet transformed images to further highlight the abnormalities such as microcalcifications, tumours, etc., to reduce the false positives (FPs). Thirdly, a modified fuzzy cmeans (FCM) segmentation method is applied on the output of the second step. In the modified FCM method, the mutual information is proposed as a similarity measure in place of conventional Euclidian distance based dissimilarity measure for FCM segmentation. Finally, the inverse 2DDWT is applied. The efficacy of the proposed unsharp masking and crispening method for image enhancement is evaluated in terms of signaltonoise ratio (SNR) and that of the proposed segmentation method is evaluated in terms of random index (RI), global consistency error (GCE), and variation of information (VoI). The performance of the proposed segmentation approach is compared with the other commonly used segmentation approaches such as Otsu's thresholding, texture based, kmeans, and FCM clustering as well as thresholding. From the obtained results, it is observed that the proposed segmentation approach performs better and takes lesser processing time in comparison to the standard FCM and other segmentation methods in consideration.
Keywords: Mammogram segmentation; mammogram enhancement; modified fuzzy cmeans segmentation; mutual information; performance evaluation; wavelet based segmentation
How to cite this article: Srivastava S, Sharma N, Singh S K, Srivastava R. A combined approach for the enhancement and segmentation of mammograms using modified fuzzy Cmeans method in wavelet domain. J Med Phys 2014;39:16983 
How to cite this URL: Srivastava S, Sharma N, Singh S K, Srivastava R. A combined approach for the enhancement and segmentation of mammograms using modified fuzzy Cmeans method in wavelet domain. J Med Phys [serial online] 2014 [cited 2019 Jun 20];39:16983. Available from: http://www.jmp.org.in/text.asp?2014/39/3/169/139007 
Introduction   
According to American Cancer Society's, the Cancer facts and Figures 2013, ^{[1]} breast cancer is the most common cancer among women, except for skin cancers. About 1 in 8 (12%) women in the US will develop invasive breast cancer during their lifetime. The American Cancer Society for breast cancer in the United States for 2013 estimates that about 232,340 new cases of invasive breast cancer will be diagnosed in women, about 64,640 new cases of carcinoma in situ (CIS) will be diagnosed (CIS is noninvasive and is the earliest form of breast cancer), and about 39,620 women will die from breast cancer. Women in the India have about a 1 in 9 lifetime risk of developing invasive breast cancer. The early detection and diagnosis of breast cancer can increase the survival rate and effective treatment options in time. In screening mammography, radiographic imaging of the breast is currently the most effective and cheap tool for early detection of breast cancer. In screening mammogram program, the digital mammographic images are obtained and collected for the suspicious cases and the radiologists visually examine the mammograms for specific abnormalities. Breast image analysis can be performed using many imaging modalities such as digital mammography, magnetic resonance imaging (MRI), nuclear imaging and ultrasound. But the digital mammography is more popular and commonly used imaging tool for breast cancer detection due to its cost effectiveness as well as its higher ability to detect the disease. Mammography is low dose Xray procedure that allows visualisation of internal structure of the breast. The most common breast abnormalities that may indicate breast cancer include masses, calcifications, architectural distortion, and bilateral symmetry. The breast lesions have a wide range of features that can indicate malignant changes, but can also be part of benign changes. They are sometimes indistinguishable from the surrounding tissue which makes the detection and diagnosis of breast cancer more difficult. Knowing the limitations of human observers and its difficulty for radiologists to provide both accurate and uniform evaluation for the enormous number of mammograms generated in widespread screening, automation of the breast cancer detection and diagnosis through a software CAD tool may help in accurate and uniform detection and diagnosis of breast cancer. Computer aided detection (CADe) and diagnosis (CADx), combined called as CAD, is used to help radiologists in interpretation of mammograms and is usually used as a second opinion by the radiologists. Improving CAD performance increases the treatment options and a cure is more likely. Also, to help the radiologists in screening large number of mammograms, the use of a CAD tool maybe helpful in exact prognosis free from human error analysis.
The major steps involved in the design and analysis of an automated CAD tool for cancer detection from mammograms include: Preprocessing (restoration and enhancement), image segmentation, feature extraction, feature selection and classification. The design and analysis of efficient algorithms for each step play an important role in deciding the efficacy and correctness of the overall CAD tool. Image enhancement and segmentation plays an important role in the design and development for the said CAD tool. In image segmentation the basic aim is to separate the suspicious region, that may contain abnormalities in mammograms such as microcalcifications, tumors etc., from the background tissue. The segmentation process partitions the mammogram into several nonoverlapping regions, extract regions of interests (ROIs), and locate the suspicious areas, such as microcalcifications and tumours which are candidates for ROIs. The suspicious area is an area that is brighter than its surroundings, has almost uniform density, has a regular shape with varying size, and has fuzzy boundaries. ^{[2]} A better image enhancement technique, applied prior to segmentation process, for highlighting and enhancing the abnormalities in mammograms may further reduce the false positives (FPs) during cancer detection. Hence, image segmentation is a very essential and important step that determines the sensitivity of the overall CAD tool. The results for segmentation is supposed to include the regions containing all abnormalities even with some FPs, if left out, which can be removed at a later stage of the algorithms for CAD tool design. An overview of enhancement and segmentation techniques for mammograms is given as below.
Overview of enhancement techniques for mammograms
The various methods ^{[3],[4],[5],[6]} which exists in literature for the enhancement of mammograms may be broadly divided into three categories which include global approaches, ^{[7],[8],[9],[10]} local approaches, ^{[11],[12],[13],[14],[15]} and multiscale processing based approaches. ^{[16],[17],[18],[19],[20]} The global approach based methods reassign the intensity values of pixels to make the new distribution of the intensities uniform to the maximum extent. This method is effective in enhancing the entire image with low contrast. The main disadvantages of global schemes are that they cannot enhance the textual information and working only for the images having one object. The local approaches for image enhancement are featurebased or use nonlinear mapping locally. These methods are effective in local texture enhancement. The main disadvantages of the local schemes are that they cannot enhance the entire image very well. The multiscale processing based enhancement techniques are based on wavelet transformation and they are flexible to select local features to be enhanced and able to suppress the noise. If the mother wavelet and weight modification functions are chosen carefully, the wavelet based method can perform very well. ^{[16]} Some of the commonly used methods available in literature for the enhancement of mammograms include contrast limited histogram equalization (CLAHE) based technique, ^{[11]} densityweighted contrast enhancement (DWCE), ^{[21]} logic filters, ^{[9],[22],[23]} iris filters ^{[24],[25]} and difference of Gaussians (DoG). ^{[26]} The DWCE is used in two stages, at first it is applied globally to isolate the suspected area, thenit is used locally to refine the segmentation. It works in conjunction with Laplacian of Gaussian (LoG) filter. The logic filter is a nonlinear filter, and logic operators AND, OR, and XOR are used. The concrete logic expressions depend on the prior information, and the filter structure influences the results. Iris filter is an adaptive filter and it is applied locally. The Gaussian filter ROIs are highlighted by a DOG filter and it can reduce number of FPs during segmentation process.
Here, in this paper, in addition to image enhancement we propose to incorporate an unsharp masking and crispening operator to further highlight and sharpen the abnormalities using a nonlinear complex diffusion based approach.
Overview of segmentation techniques for mammograms
In literature, ^{[28],[29],[31],[32]} supervised and unsupervised are two types of image segmentation approaches. The supervised segmentation or model based method use the prior knowledge about the object and background regions to be segmented. The prior information is used to determine if specific regions are present within an image or not. The unsupervised segmentation partitions an image into a set of regions which are distinct and uniform with respect to specific properties, such as greylevel, texture or color. The classical approaches for solving unsupervised segmentation are divided in three major groups namely regionbased methods, which divide the image into homogeneous and spatially connected regions; contourbased methods, which depends on the boundaries of regions; and clustering methods, which group together those pixels having the same properties and might result in nonconnected regions. According to their natures, there are four broad categories of image segmentation approaches in literature ^{[3]} for the segmentation of mammograms which include classical techniques, fuzzy techniques, bilateral image subtraction, rough set based approaches ^{[57],[58],[59],60} and multiscale techniques. A brief review of various segmentation approaches may be found in paper. ^{[3]}
In this paper, a modified fuzzy cmeans (FCM) segmentation method based on mutual information in wavelet domain is proposed for segmenting the abnormalities in mammograms. Before applying the proposed segmentation approach, a PDE based unsharp masking and crispening method is proposed and applied on the mammograms to highlight the details of the abnormalities such as microcalcifications etc., to reduce the false positives (FPs) during segmentation process. The proposed segmentation method is compared with the Otsu's optimal thresholding, ^{[34],[35],[37]} texture based segmentation method, ^{[36]} kmeans segmentation, ^{[31]} and FCM based thresholding ^{[3],[37]} based segmentation method.
Reasons for using fuzzy technique based image segmentation algorithm ^{[3],[37]} are as follows: Since the contrast in mammograms is very low and the boundary between normal tissue and tumours is unclear, the traditional segmentation methods might not work well. The classical region growing based segmentation technique tries to precisely define ROIs, but to find a criterion for segmentation is difficult as most of the malignant tumors with fuzzy boundaries extend from a dense core region to the surrounding tissues. Similarly, the classical global or local thresholding techniques ^{[3]} try to segment ROIs, but the techniques are only effective for the objects with clear boundaries. The fuzzy logic based approaches are useful for segmenting suspicious regions ^{[3],[32]} and are capable of addressing above issues.
The categorization and summary of various commonly used mammogram segmentation methods are presented in [Table 1].
The organization of the paper is as follows: Section 1 of the paper presents the brief introduction of the problem; Section 2 of the paper presents the methods and models, that is the proposed method in detail along with the justification for the proposed models; Section 3 of the paper presents the results, performance analysis, and discussions; Section 4 of the paper presents the conclusions of the work.
Materials and Methods   
In this paper, an image enhancement and segmentation technique is proposed for the segmentation of mammograms for breast cancer detection. The proposed method consists of following steps as illustrated in [Figure 1].
Algorithm for the proposed method
Step 1: In the first step, CLAHE based enhancement technique is applied on original low contrast mammogram to enhance them in to good contrast images.
Step 2: Two levels of 2D discrete wavelet transform (DWT) is applied on the output obtained in the first step.
This step is applied to perform the segmentation at various scales (multiresolution) and also the wavelets can detect more easily the transient signals or abrupt changes caused by various gray level changes in images due to microcalcifications and other abnormalities.
The other advantage being the faster processing of FCM based clustering method used in segmentation step.
Step 3: In this step, the proposed nonlinear complex diffusion based unsharp masking and crispening method is applied on the enhanced mammogram to further enhance and highlight the abnormalities and fine details present in mammograms such as microcalcifications and tumors. This step help the segmentation process in producing the good results and increase the cancer detection rate by the CAD tool by reducing the false positives.
Step 4: The proposed modified FCM thresholding based image segmentation is applied on mammograms obtained in step 3.
Step 5: Inverse wavelet transform is applied to reconstruct the final segmented image in spatial domain.
In step 2 of the algorithm, it had been practically examined that two levels of DWT based segmentation is providing better results and very close to that of third level of DWT decomposition. Three levels of DWT decomposition may also be used but for large scale processing of mammograms computational complexity may increase.
The proposed steps 3 and 4 of the algorithms are described as follows:
Nonlinear complex diffusion based approach for unsharp masking and crispening of mammograms
The basic procedure ^{[27],[39]} for unsharp masking and crispening the image is as follows: In the first step, a lowpass filter is applied on the original image for smoothening the same. In the second step, the edge description and other desired high frequency components of an image are calculated by subtracting the smoothened image obtained in the first step from the original image. In the third and last step, the edge image obtained in second step is used for sharpening the edges and other high variation components of original image by adding back it to the original signal. The unsharp masking produces an edge image I _{e} (x, y) from an input image I (x, y) via
Another method used in place of discrete Laplacian is Laplacian of Gaussian (LoG). In this case since the kernel peak is positive, the edge image is subtracted, rather than added back to the original image. The disadvantages of these schemes are that gradient images produced by both filters, Laplacian and LoG, produces the side effects of ringing or introduction of additional intensity image structure and this ringing occurs at high contrast edges. Hence, the unsharp filter is a powerful sharpening operator, but it also produces a poor result in the presence of noise.
In Eq. (4), the second term of RHS is Laplacian which is used as unsharp mask to produce the edge image defined as, I _{smooth} (x, y) = I (x, y) is a Heat equation which performs the isotropic diffusion to denoise the image. The smoothing process can be regarded as an evolution process governed by a PDE that performs regularization of the image ^{[40]} as follows.
where the function c = c (I, I _{x}, I _{xx}, …) is a function of local image differential structure that depends on local partial derivatives.
The above anisotropic diffusion based process involves the properties of forward diffusion on real axis which is more useful for analysing real valued grey images but may not be useful for reducing those noises which are near to threshold values and may produce staircase and ringing effects. Hence, to overcome these issues, in this paper, a nonlinear complex diffusion based filter as defined in ^{[41]} is used. In complex diffusion based processes, the imaginary part serve as an edge detector, smoothed second derivative scaled by time, when the complex diffusion coefficient approaches the real axis. The complex diffusion based processes do not produce blocky artefacts or stair casing effects during the evolution process of the image. It also preserves the edges and fine structures within the image and results do not change by changing illumination conditions. These properties are helpful in mammographic image analysis for better diagnosis. The nonlinear complex diffusion based filter reads ^{[41]} :
The last step (4) is used to obtain the sharpened with crisped edges and k is a scaling constant, k > 0. The reasonable values for k varies between 0.20.8, with the larger values providing increasing amount of sharpening.
Proposed modified fuzzy cmeans thresholding based image segmentation using mutual information
The working of the FCM clustering approach is given as follows: ^{[37],[38],60}
In fuzzy approach based partitioning, the Gaussian membership matrix (U) = [u _{ij}] is randomly initialized according to Eq. (10), where u _{ij} being the degree of membership function of the data point of i ^{th} cluster x _{i} . The membership matrix U is allowed to have elements with values between 0 and 1 but the summation of degrees of belongingness of a data point to all clusters or partitions is always equal to unity:
The FCM algorithm works iteratively through the above two conditions until there is no more improvement.
The limitations of the standard FCM based segmentation algorithms are as follows:
The main requirement of this algorithm is that the number of clusters should be known apriori. The performance of FCM depends on the initial membership matrix values hence the algorithm is run for several times, each starting with different values of membership grades of data points. Although the original intensitybased FCM algorithm functions well on segmenting most noisefree images, it fails to segment images corrupted by noise, outliers, and other imaging artifacts, such as the intensity in homogeneity induced by the various abnormalities such as microcalcifications in mammograms, and thus leads to its nonrobust results mainly due to the use of (a) Nonrobust Euclidean distance and (b) disregard of spatial contextual information in image. ^{[43]} Hence, FCM lacks enough robustness to noise and outliers and is not suitable for revealing nonEuclidean structure of the input data due to the use of Euclidean distance (L2 norm). To deal with the this problem, some researchers adopted robust distance measures such as L _{p} norms (0 < p ≤ 1) ^{[44],[45],[46]} to replace the L2 norm in the FCM objective function for reducing the effect of outliers on clustering results, and while many other algorithms have also been proposed to deal with the second problem by incorporating spatial information into original FCM objective function. ^{[47],[48],[49],[50]}
Hence, to deal with above issues, in this paper, a mutual information based distance measure available in ^{[30]} is used for FCM. The other advantage of using mutual information as distance measure is that it can capture any correlative behaviour (positive, negative, and nonlinear) between image pixel values whereas the Euclidean distance measure can capture only positive correlations between pixel patterns.
Therefore, the Euclidian distance measure, used in classical FCM segmentation as a distance function for dissimilarity measurement to from the clusters of similar pixels, is replaced by the distance measure defined in terms of mutual information due to the reasons discussed as above. In this paper, the idea for the gene clustering based on cluster wide mutual Information used in paper ^{[30]} is adopted to define the distance measure used by FCM for the segmentation of mammograms.
For discrete variables, the mutual information I of two variables X and Y is defined as measure of information about X (or Y) contained in Y (or X) ^{[30]} :
Where H (X) and H (Y) are entropies of X and Y respectively; H (X/Y) and H (Y/X) are conditional entropies of X and Y respectively; H (X, Y) is joint entropy of X and Y; and Nx, Ny are possible values of X and Y that it can take. The mutual information is always nonnegative, which is I (X; Y) ≥ 0. ^{[51]} Since the mutual information of two data variable X and Y defined as above is not normalized; I (X; Y) can be quite small even if X and Y are highly correlated. Hence, the mutual information must be normalized by the maximal entropy of each of the contributing X and Y. The basic advantage of normalization is that it gives a high value for highly correlated data or pixel values in an image independent of the individual entropy. The normalized mutual information is defined as ^{[30]} :
The working of the clustering approach is as follows: In each iteration, the pixel that has a minimal distance to the target pixel to the cluster is added. The process continues until the distance threshold is not crossed. A second candidate cluster is formed by starting with the second pixel and the same procedure is repeated. The pixels from the first candidate cluster are not removed from consideration and this process continues for all pixels. The largest candidate cluster is selected and retained. The pixels in the largest candidate cluster are removed from the whole image pixel set, and the entire procedure is repeated on the smaller pixel set. When the number of clusters reaches to a predefined cluster number, all the remaining pixels to the last cluster are added. The threshold may be chosen as the mean of the distances of all pixel pairs.
In this paper, the three classes of FCM clustering were used. These three classes include small, middle, and large. A switchoff cutposition (SWC) were used to select among the classes. The SWC having value zero and one gives cut between small and middle classes and cut between middle and large classes respectively. The threshold values for segmentation were calculated as follows:
Results and Performance Analysis   
In this section, results and performance analysis of the proposed enhancement and segmentation techniques are presented. For evaluation of the various mammogram segmentation approaches with the proposed one have been performed in terms of random index (RI), variation of information (VoI), and global consistency error (GCE). These performance measures are discussed as follows:
Mammogram segmentation performance measures
Random index
The RI measure ^{[52],[53]} was initially proposed for the evaluation of general clustering algorithms. The RI between test (S) and ground truth (G) is estimated by summing the number of pixel pairs with same label and number of pixel pairs having different labels in both S and G, and then dividing it by total number of pixel pairs. This gives a measure of similarity with value ranging from 0 when the two segmentations have no similarities (when one consists of a single cluster and the other consists only of clusters containing single points) to 1 when the segmentations are identical, that is when a higher value of RI close to 1 is preferred for perfect segmentation.
where and H (Y) are entropies of X and Y; and I (X, Y) is mutual information between X and Y. Vol (X, Y) measures how much the cluster assignment for an item in cluster X reduces the uncertainty about the item's cluster in cluster Y. The value of VoI lies in between 0 and d, where d is the distance between clusters. Since it is a distance measure, hence a lower value of VoI close to zero indicates best segmentation.
Global consistency error
In papers ^{[54],[56]} authors propose two metrics that can be used to evaluate the consistency of a pair of segmentations. These measures are designed in such a way that they are tolerant to refinement, i.e., if subsets of regions in one segmentation consistently merge into some region in the other segmentation the consistency error should be low. To compute the consistency error for a pair of images, at first a measure of the error at each pixel pi is defined as follows:
Since LCE ≤ GCE, hence GCE is a tougher measure than LCE and that's why it is used in this paper. A small value of GCE close to zero represents better segmentation. GCE quantify the amount of error in segmentation i.e., 0 signifies no error and 1 indicates no agreement.
Results and Discussions   
The proposed unsharp masking and crispening techniques were evaluated in terms of improvement in signaltonoise ratio of the sample test mammographic images and its overall effect on the proposed segmentation method is also evaluated. The comparative study of the proposed combined enhancement and segmentation technique is presented with the other popular methods used for segmentation of mammographic images such as Otsu's thresholding, Texture based thresholding, kmeans clustering, and FCM clustering based segmentation method based on Euclidian distance measure. For experimentation purposes, the 256 histogram bins were used in Otsu's gray level thresholding method. For kmeans, fuzzy cmeans, and the proposed segmentation method the initial number of clusters for the proposed FCM based segmentation method was set to three as it was associated with better performance. In texture based segmentation, an entropy based filter was used. For experimentation purposes, 25 test sample digital mammographic images were used. The average performance measures for the 10 sample images are shown in this paper; however the performance trend remained the same for other test images as well. The proposed segmentation approach was also tested on mammographic image analysis society (MIAS) database. [Figure 2] shows the visual results of unsharp masking and crispening procedure in spatial domain. [Table 2] and [Figure 3] present results in terms of signaltonoise ratio (SNR) of original sample mammogram and improvement in SNR (ISNR) after applying proposed unsharp masking and crispening method in wavelet domain. From [Table 1] and [Figure 3], it is observed that the proposed nonlinear complex diffusion (a partial differential equation based approachPDE) based unsharp masking and crispening methods is showing a good improvement over SNR values of the original mammogram which justifies that the proposed method is better capable of enhancing and highlighting the abnormalities in mammogram in details.  Figure 2: Visual results of unsharp masking and crispening procedure in spatial domain
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 Table 2: Results in terms of signaltonoise ratio of original sample mammogram and Improvement in SNR after applying proposed PDE based unsharp masking and crispening method wavelet domain
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 Table 3: Evaluation of segmentation methods in terms of RI, VoI, and GCE for 10 sample mammograms
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 Figure 3: Comparison of SNR values of different mammograms for original image and image obtained after unsharp masking and crispening
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The first top row of [Figure 4] shows visual results for initial steps before segmentation, in initial steps of the proposed method, the visual results of the original mammogram, enhanced image by CLAHE method, unsharp masking and crispening, and results after applying two levels of wavelet decomposition (biorthogonal) used in proposed method. During the 2D discrete wavelet decomposition (DWT) wavelet decomposition, a biorthogonal wavelet was used as a mother wavelet as it is provides complete reconstruction of images.
The bottom row of [Figure 4] shows the visual results for the various segmentation methods such as Otsu's thresholding, texture segmentation, kmeans segmentation, Fuzzy Smeans segmentation and the proposed segmentation method. From visual results, it is observed that the proposed segmentation approach is providing the better segmentation results in comparison to other methods and it is well capable of segmenting the all possible types of abnormalities such as tumours, microcalcifications etc., that may be present in mammogram for breast cancer diagnosis.  Figure 4: Visual results for (i) Initial steps before segmentation (ii) various segmentation methods and proposed one
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[Table 3] presents the evaluation of various segmentation methods and the proposed one in terms of RI, VoI, and GCE for 10 sample mammograms. The averaged values of RI, VoI, and GCE for 10 sample images are also shown which gives the average performance of the methods. For better segmentation results the value RI should be close to one and higher than the values related to other segmentation methods; and the values of VoI and GCE should be lower than that of the other segmentation methods.
[Figure 5] shows comparison of RI values of various segmentation methods for 10 sample images. From [Figure 5], it is observed that the values of RI for each sample image for the proposed method are higher than that of other methods signifying that the proposed method is performing better in comparison to other methods. [Figure 6] shows comparison of average RI values of various segmentation methods for 10 sample images and the average RI value for proposed method is larger in comparison to other methods.  Figure 5: Comparison of random index values of various segmentation methods for 10 sample images
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 Figure 6: Comparison of average random index values of various segmentation methods for 10 sample images
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[Figure 7] shows comparison of GCE of various segmentation methods for 10 sample images and [Figure 8] shows c omparison of average GCE values of various segmentation methods for 10 sample images. From [Figure 7] and [Figure 8], it is observed that the GCE value of the proposed method is smaller than that of the other methods.  Figure 7: Comparison of global consistency errors of various segmentation methods for 10 sample images
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 Figure 8: Comparison of average GCE values of various segmentation methods for 10 sample images
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[Figure 9] shows comparison of VoI values of various segmentation methods for 10 sample images and [Figure 10] shows comparison of average VoI values of various segmentation methods for 10 sample images. From [Figure 9] and [Figure 10], it is observed that the GCE value of the proposed method is smaller than that of the other methods.  Figure 9: Comparison of variation of information values of various segmentation methods for 10 sample images
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 Figure 10: Comparison of average VoI values of various segmentation methods for 10 sample images
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[Table 4] shows comparison of execution time (in seconds) of various segmentation methods for sample image, image1.jpg of 2770 × 1770. Here, again it is observed that the proposed method is taking 6.561 seconds whereas the traditional FCM method is taking 148.94 seconds.
Therefore, from the results obtained it is observed that the proposed segmentation method is performing better in comparison to all other methods in consideration and it is well capable of segmenting the abnormalities in mammograms.  Table 4: Comparison of execution time (in seconds) of various segmentation methods for 10 sample images
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Conclusions   
In this paper, a nonlinear complex diffusion based unsharp masking and crispening method was proposed for enhancement of abnormalities found in mammograms for the breast cancer detection. Further, a modified FCM segmentation method was proposed in wavelet domain. The distance measure for clustering purposes, in the proposed segmentation method, was based on the mutual information of image pixels. Two levels of (DWT) was used for image decomposition and transformation. The mother wavelet used for the wavelet decomposition was biorthogonal wavelets as it provides full reconstruction of images. For experimentation purposes, the initial number of clusters for the kmeans, fuzzy cmeans, and the proposed segmentation method was set to three as it was associated with better performance. The performance of the proposed enhancement method was evaluated in terms of signaltonoise ratio (SNR). The performance of the proposed segmentation method was evaluated in terms of three measures such as RI (RI), GCE, and VoI. The performance of the proposed method and other segmentation methods in consideration were evaluated both qualitatively and quantitatively. The execution time of the proposed method was also lower in comparison to its best counterpart which was FCM with Euclidian distance. The comparisons of the performances of the proposed method with the other segmentation methods were also presented in the paper. Therefore, from the obtained results, it can be concluded that the proposed enhancement and segmentation framework is computationally cheaper, producing better results in comparison to other methods, alleviate the problems related to the Euclidian distance measure in traditional FCM based segmentation and reduces the false positives and outliers during the segmentation of the mammograms. Hence, the proposed method may be a better choice for segmentation of mammograms for the breast cancer detection.
References   
1.  Available from: http://www.cancer.org/Cancer/BreastCancer/DetailedGuide/breastcancerkeystatistics [Last accessed on 2013 Oct 14]. 
2.  Lai SM, Li X, Biscof WF. On techniques for detecting circumscribed masses in mammograms. IEEE Trans Med Imaging 1989;8:37786. 
3.  Cheng HD, Shi XJ, Min R, Hu LM, Cai XP, Du HN. Approaches for automated detection and classification of masses in mammograms. Pattern Recogn 2006;39:64668. 
4.  Ganesan K, Acharya UR, Chua CK, Min LC, Abraham KT, Ng KH. Computeraided breast cancer detection using mammograms: A review. IEEE Rev Biomed Eng 2013;6:7798. 
5.  Tang, J, Rangayyan RM, Xu J, El Naqa I, Yang Y. Computeraided detection and diagnosis of breast cancer with mammography: Recent advances. IEEE Trans Inf Technol Biomed 2009;13:23651. 
6.  Rangayyan RM, Ayres FJ, Desautels JE. A review of computeraided diagnosis of breast cancer: Toward the detection of subtle signs. J Franklin Inst 2007:31248. 
7.  Nam SH, Choi JY. A method of image enhancement and fractal dimension for detection of microcalcifications in mammogram. In: Proceedings of the 20 ^{th} Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Hong Kong. Vol. 2. 1998. p. 100912. 
8.  Wilson M, Hargrave R, Mitra S, Shieh YY, Roberson GH. Automated detection of microcalcifications in mammograms through application of image pixel remapping and statistical filter. In: Proceedings of the IEEE Symposium on Computerbased Medical Systems, Lubbock, TX, USA; 1998. p. 2704. 
9.  Wongsritong K, Kittayaruasiriwat K, Cheevasuvit F, Dejhan K, Somboonkaew A. Contrast enhancement using multipeak histogram equalization with brightness preserving, in proceedings: IEEE AsiaPacific Conference on Circuits and Systems, Chiangmai. 1998. p. 4558. 
10.  Cheng HD, Shi XJ. A simple and effective histogram equalization approach to image enhancement. Dig Sig Proc 2004;14:15870. 
11.  Pisano ED, Zong S, Hemminger BM, DeLuca M, Johnston RE, Muller K, et al. Contrast limited adaptive histogram equalization image processing to improve the detection of simulated spiculations in dense mammograms. J Digit Imaging 1998;11:193200. 
12.  Xiong Y, Lam CF, Frey GD, Croley MR, Contrast enhancement of mammogram by image processing. In proc.: SPIE 1898, Medical Imaging 1993: Image Processing, 8, Newport Beach, CA; 1993. p. 8528. 
13.  Dhawan AP, Le Royer E. Mammographic feature enhancement by computerized image processing. Comput Methods Programs Biomed 1988;27:2335. 
14.  Urias JF. Feature enhancement of images using maximal contrast pixel to pixel. Appl Opt 1991;30:45989. 
15.  Qian W, Clarke LP, Kallergi M, Clark RA. Treestructured nonlinear filters in digital mammography. IEEE Trans Med Imaging 1994;13:2536. 
16.  Chang CM, Laine A. Enhancement of mammograms from oriented information. In: IEEE International Conference on Image Processing., Santa Barbara, CA 1997. vol. 3, p. 5247. 
17.  Mallat S, Zhong S. Characterization of signals from multiscale edges. IEEE Trans Pattern Anal Mach Intell 1992;14:71032. 
18.  Lu J, Healy DM, Weaver JB. Contrast enhancement of medical images using multiscale edge representation. Opt Eng 1994;33:215161. 
19.  Laine A, Fan J, Yang W. Wavelets for contrast enhancement of digital mammography. IEEE Eng Med Biol 1995;14:53650. 
20.  Laine AF, Schuler S, Fan J, Huda W. Mammographic feature enhancement by multiscale analysis. IEEE Trans Med Imaging 1994;13:72540. 
21.  Petrick N, Chan HP, Sahiner B, Wei D. An adaptive density weighted contrast enhancement filter for mammographic breast mass detection. IEEE Trans Med Imaging 1996;15:5967. 
22.  Tsirikolias K, Mertzios BG. Edge extraction and enhancement using coordinate logic filters, Image Process. Theory Appl 1993;2514. 
23.  Rocha A, Tong F, Yan ZZ. Logic filter for tumor detection on mammograms. J Comput Sci Technol 2000;15:62932. 
24.  Kobatake H, Yoshinaga Y. Detection of spicules on mammogram based on skeleton analysis. IEEE Trans Med Imaging 1996;15:23545. 
25.  Kobatake H, Murakami M, Takeo H, Nawano S. Computerized detection of malignant tumors on digital mammograms. IEEE Trans Med Imaging 1999;18:36978. 
26.  Polakowski WE, Cournoyer DA, Rogers SK, DeSimio MP, Ruck DW, Hoffmeister JW, et al. Computeraided breast cancer detection and diagnosis of masses using difference of gaussians and derivativebased feature saliency. IEEE Trans Med Imaging 1997;16:8119. 
27.  Gonzalez RC. Digital image processing. India: Prentice Hall; 2 ^{nd} Edition, 2003. 
28.  Oliver A, Freixenet J, Marti J, Perez E, Pont J, Denton ER, et al. A review of automatic mass detection and segmentation in mammographic images. Med Image Anal 2010;14:87110. 
29.  Srivastava S, Sharma N, Singh SK. "Image analysis and understanding techniques for breast cancer detection from digital mammograms in research developments" in: Srivastava, R., Singh, SK, Shukla, KK.(editors). Computer Vision and Image Processing: Methodologies and Applications. IGI Global, USA; 2013. Ch. 8 p. 123 8. 
30.  Zhou X, Wang X, Dougherty ER, Russ D, Suh E. Gene clustering based on clusterwide mutual information. J Comput Biol 2004;11:14761. 
31.  Sridhar S. Digital image processing. India: Oxford University Press First Edition,; 2011. 
32.  Srivastava S, Sharma N, Singh SK, Srivastava R. Analysis of image segmentation techniques in the design of a CAD tool for early breast cancer detection from mammograms. In proc: International Conference on Artificial Intelligence and Soft Computing (AISC2012), 79 December'2012, IITBHU, Varanasi, India, p. 6571. 
33.  Dogra DP, Majumdar AK, Sural S. Evaluation of segmentation techniques using region area and boundary matching information. J Vis Commun Image Represent January 2012;23, Issue 1,:15060. 
34.  Otsu N. A threshold selection method from graylevel histogram. IEEE Trans Sys Man Cyber 1979;9:626. 
35.  Sezgin M, Sankur B. Survey over image thresholding techniques and quantitative performance evaluation. J Electron Imaging 2004;13:14665. 
36.  Karahaliou A, Skiadopoulos S, Boniatis I, Saketllaropoulos P, Likaki E, Panayiotakis G, et al. Texture analysis of tissue surrounding microcalcifications on mammograms for breastcancer diagnosis. Br J Radiol 2007;80:64856. 
37.  Cai W, Chen S, Zhang D. Fast and robust fuzzy cmeans clustering algorithm incorporating local information for image segmentation. Pattern Recogn 2007;40:82538. 
38.  Bezdek JC, Ehrlich, R, Full, W., FCM: The Fuzzy cmeans clustering algorithm, Computers and Geosciences, Vol. 10, N0. 23, p. 191203,1984 . 
39.  Srivastava R, Gupta JR, Parthasarthy H, Srivastava S. PDE based unsharp masking, crispening and high boost filtering of digital images. In: Ranka S, Aluru S, Buyya R, Chung Y.C., Dua S, Grama A, et al. editors. Contemporary Computing, PartI, Communications in Computer and Information Science (CCIS). Vol. 40. Germany: Springer Berlin Heidelberg; 2009. p. 813. 
40.  Perona P, Malik J. Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence. Vol. 12, Issue 7 1990. p. 62939. 
41.  Gilboa G, Sochen N, Zeevi YY. Image enhancement and denoising by complex diffusion processes. IEEE Trans Pattern Anal Mach Intell 2004;25:102036. 
42.  Press HW, Teukolsky A, Vetterling ST, Flannery PB. Numerical Recipes in C: The Art of Scientific Computing. 2 ^{nd} ed.. Cambridge University Press, USA;1992.pp. 827844 
43.  Chen S, Zhang D. Robust image segmentation using FCM with spatial constraints based on new kernelinduced distance measure. IEEE Trans Syst Man Cybern B Cybern 2004;34:190716. 
44.  Hathaway RJ, Bezdek JC. Generalized fuzzy cmeans clustering strategies using Lp norm distance. IEEE Trans Fuzzy Syst 2000;8:576582. 
45.  Jajuga K. L1 norm based fuzzy clustering. Fuzzy Sets Syst 1991;39:4350. 
46.  Leski J. An " insensitive approach to fuzzy clustering. Int J Appl Math Comp Sci 2001;11:9931007. 
47.  Pham DL, Prince JL. An adaptive fuzzy Cmeans algorithm for image segmentation in the presence of intensity in homogeneities. Pattern Recogn Lett 1999;20:5768. 
48.  Tolias YA, Panas SM. On applying spatial constraints in fuzzy image clustering using a fuzzy rulebased system. IEEE Signal Proc Lett 1998;5:2457. 
49.  Liew AW, Leung SH, Lau WH. Fuzzy image clustering incorporating spatial continuity. Inst Elec Eng Vis Image Signal Proc 2000;147:18592. 
50.  Pham DL. Fuzzy clustering with spatial constraints. In: IEEE Proc Int Conf Image Processing, New York; 2002. vol. 2, p. II65II. 
51.  Rand W. Objective criteria for the evaluation of clustering methods. J Acoust Soc Am 1971;66:84685. 
52.  Unnikrishnan R, Pantofaru C, Hebert M. Toward objective evaluation of image segmentation algorithms. IEEE Trans Pattern Anal Mach Intell 2007;29:92944. 
53.  Martin D. An empirical approach to grouping and segmentation. PhD dissertation, Univ. of California, Berkeley; Report No. UCB/CSD31268. August 2003. Computer Science Division (EECS). 
54.  Fowlkes C, Martin D, Malik J. Learning affinity functionsfor image segmentation. Proc IEEE Conf Comput Vision Pattern Recogn 2003;2:5461. 
55.  Martin D, Fowlkes C, Tal D, Malik J. A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. Proc International Conference on Computer Vision ICCV 2001;2:41625. 
56.  Mohabey, Ray AK. Fusion of rough set theoretic approximations and FCM for color image segmentation. IEEE Int Conf Syst Man Cybernet 2000;2:152934. 
57.  Boss RS, Thangavel K, Daniel DA. Mammogram image segmentation using rough clustering. Int J Res Engin Technol 2013:6677. 
58.  Hassanien AE, Abraham A, Peters JF, Schaefer G, Henry C. Rough sets and near sets in medical imaging: A review. IEEE Trans Inf Technol Biomed 2009;13:95568. 
59.  Castillejos H, Volodymyr P. Fuzzy image segmentation algorithms in wavelet domain, fuzzy logic  algorithms, techniques and implementations.Chapter 7, pp. 127146 In: Dadios E, editor. 28 March 2012. Available from: Http://www.intechopen.com/books/fuzzylogicalgorithmstechniquesandimplementations/fuzzyimagesegmentationalgorithmsinwaveletdomain [Accessed online on 2013 Nov 06]. 
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9], [Figure 10]
[Table 1], [Table 2], [Table 3], [Table 4]
