2006 IEEE Congress on Evolutionary Computation Sheraton Vancouver Wall Centre Hotel, Vancouver, BC, Canada July 16-21, 2006

Evolutionary Neural Networks Applied To The Classification Of Microcalcification Clusters In Digital Mammograms Rolando R. Hern´andez-Cisneros and Hugo Terashima-Mar´ın

Abstract— Breast cancer is one of the main causes of death in women and early diagnosis is an important means to reduce the mortality rate. The presence of microcalcification clusters are primary indicators of early stages of malignant types of breast cancer and its detection is important to prevent the disease. This paper proposes a procedure for the classification of microcalcification clusters in mammograms using sequential Difference of Gaussian filters (DoG) and three Evolutionary Artificial Neural Networks (EANNs) compared against a feedforward Neural Network (NN) trained with backpropagation. We found that the use of Genetic Algorithms (GAs) for 1) finding the optimal weight set for a NN, 2) finding an adequate initial weight set before starting a backpropagation training algorithm and 3) designing its architecture and tuning its parameters, results mainly in improvements in overall accuracy, sensitivity and specificity of a NN, compared with other networks trained with simple backpropagation.

I. I NTRODUCTION Breast cancer is one of the main causes of death in women and early diagnosis is an important means to reduce the mortality rate. Mammography is one of the most common techniques for breast cancer diagnosis, and microcalcifications are one among several types of objects that can be detected in a mammogram. Microcalcifications are calcium accumulations typically 100 microns to several mm in diameter, and they sometimes are early indicators of the presence of breast cancer. Microcalcification clusters are groups of three or more microcalcifications that usually appear in areas smaller than 1 cm2 , and they have a high probability of becoming a malignant lesion. However, the predictive value of mammograms is relatively low, compared to biopsy. This low sensitivity (correct diagnosis of positive cases) [1] is caused by the low contrast between the cancerous tissue and the normal parenchymal tissue, the small size of microcalcifications and possible deficiencies in the image digitalization process. The sensitivity may be improved having each mammogram checked by two or more radiologists, with the consequence of making the process inefficient by reducing the individual productivity of each specialist. A viable alternative is replacing one of the radiologists by a computer system, giving a second opinion [2], [3]. Several computer-aided diagnosis (CAD) systems for early detection of breast cancer have been proposed in literature and some are offered commercially, like Second Rolando R. Hern´andez-Cisneros is with the Center for Intelligent Systems, Tecnol´ogico de Monterrey, Campus Monterrey, Av. Eugenio Garza Sada 2501, Monterrey, N.L. 65841 Mexico (email: [email protected]). Hugo Terashima-Mar´ın is with the Center for Intelligent Systems, Tecnol´ogico de Monterrey, Campus Monterrey, Av. Eugenio Garza Sada 2501, Monterrey, N.L. 65841 Mexico (email: [email protected]).

0-7803-9487-9/06/$20.00/©2006 IEEE

R R Look by iCAD, Inc. and ImageChecker by R2 Technology, Inc. Nevertheless, research in this field continues, trying to improve the efficiency of CAD systems in terms of improved accuracy and less processing time. A computer system intended for microcalcification detection in mammograms may be based on several methods, like wavelets, fractal models, support vector machines, mathematical morphology, bayesian image analysis models, high order statistic, fuzzy logic, etc. The method we selected for this work is the Difference of Gaussian Filters (DoG). DoG filters are adequate for the noise-invariant and size-specific detection of spots, resulting in a DoG image. This DoG image represents the microcalcifications if a thresholding operation is applied to it. The use of DoG for detection of potential microcalcifications has been addressed successfully by Dengler, Behrens and Desaga [4] and Ochoa [5]. We developed a procedure that applies a sequence of Difference of Gaussian Filters, in order to maximize the amount of detected probable individual microcalcifications (signals) in the mammogram, which are later classified in order to detect if they are real microcalcifications or not. Finally, microcalcification clusters are identified and also classified in order to determine which ones are malignant and which ones are benign. Neural networks (NNs) have been successfully used for classification purposes in medical applications, including the classification of microcalcifications in digital mammograms. Unfortunately, for a NN to be successful in a particular domain, its architecture, training algorithm and the domain variables selected as inputs must be adequately chosen. Designing a NN architecture is a trial-and-error process; several parameters must be tuned according to the training data when a training algorithm is chosen and, finally, a classification problem could involve too many variables (features), most of them not relevant at all for the classification process itself. Genetic algorithms (GAs) may be used to address the problems mentioned above, helping to obtain more accurate NNs with better generalization abilities. GAs have been used for searching the optimal weight set of a NN, for designing its architecture, for finding its most adequate parameter set (number of neurons in the hidden layer(s), learning rate, etc.) among others tasks. Exhaustive reviews about evolutionary artificial neural networks (EANNs) have been presented by Yao [6] and Balakrishnan and Honavar [7]. Fogel et al. proposed an approach based in EANNs for early detection of breast cancer in [8], [9] and [10]. However, the scope of these studies was broad, focusing also in detecting other breast lesions beside microcalcifications, like

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Original images

masses, architectural distortions and asymmetric densities. Each type of lesion was described by few features, and each one of the features was measured using a small scale of discrete values. For instance, microcalcifications were described only by its number, morphology, density and distribution, using a scale of four discrete values for each feature. The training data sets for the NNs were constructed with the help of an expert, by visual inspection of the mammograms. In this paper, we propose an automated procedure for feature extraction and training data set construction for training a NN. We also describe the use of GAs for 1) finding the optimal weight set for a NN, 2) finding an adequate initial weight set for a NN before starting a backpropagation training algorithm and 3) designing the architecture and tuning some parameters of a NN. All of these methods are applied to the classification of microcalcifications and microcalcification clusters in digital mammograms, expecting to improve the accuracy of an ordinary feedforward NN performing this task. The rest of this document is organized as follows. In the second section, the mammography database we selected for this study is described and the proposed procedure along with its theoretical framework is discussed. The third section deals with the experiments and the main results of this work. Finally, in the fourth section, the conclusions are presented, and some comments about future work are also made.

Pre-processing Median filter Binarization Automatic cropping

Detection of potential microcalcifications (signals) DoG filters Global binarization Region labeling Point selection by minimum area, minimum gray level and minimum gradient

Classification of Signals Microcalcification data set Feature extraction and selection from microcalc. Selected microcalc. features Microcalcification data set

Neural Networks Neural Networks Overall accuracy

Classification of Microcalc. Clusters Detection of Microcalcification Clusters

Genetic Algorithm Coded weights sets / architectures Neural Networks Neural Networks

There are several mammography databases publicly available for research purposes, and the most known of them are the mammography database from the Mammographic Image Analysis Society (MIAS) [11], the Digital Database for Screening Mammography (DDSM) from the University of South Florida [12] and the Nijmegen digital mammography database. The mammograms used in this project were provided by The Mammographic Image Analysis Society (MIAS). The MIAS is an organization of UK research groups interested in the understanding of mammograms. The MIAS database contains 322 images, all medio-lateral (MLO) view, digitized with a scanning microdensitometer (Joyce-Loebl, SCANDIG3) with resolutions of 50 microns/pixel and 200 microns/pixel. In this work, the images with a resolution of 200 microns/pixel were used. The data has been reviewed by a consultant radiologist and all the abnormalities have been identified and marked. The truth data consists of the location of the abnormality and the radius of a circle which encloses it. The abnormalities represented in the database include calcifications, circumscribed masses, spiculated masses, illdefined masses, architectural distortions and asymmetry. Several normal cases are also included. From the totality of the

Optimal NN for microcalc. cluster classification; best overall accuracy, sensitivity and specificity

Overall accuracy

Fig. 1.

A. The Mammography Database

Cluster data set Feature extraction and selection from microcalc. clusters Selected cluster features

II. P RELIMINARIES In this section, the mammography database we selected for this work is described. Also, the proposed procedure, along with the theoretical framework is discussed in detail.

Genetic Algorithm Coded weights sets / architectures

Optimal NN for individual microcalc. classification; best overall accuracy, sensitivity and specificity

Diagram of the proposed procedure.

database, only 25 images contain microcalcifications. Among these 25 images, 13 cases are diagnosed as malignant and 12 as benign. Some related works have used this same database [13], [14], [15], [16]. B. Methodology The general procedure receives a digital mammogram as an input, and it is conformed by five stages: pre-processing, detection of potential microcalcifications (signals), classification of signals into real microcalcifications, detection of microcalcification clusters and classification of microcalcification clusters into benign and malignant. The diagram of the proposed procedure is shown in Figure 1. As end-products of this process, we obtain two NNs for classifying microcalcifications and microcalcifications clusters respectively, which in this case, are products of the evolutionary approaches that are proposed. 1) Pre-processing: This stage has the aim of eliminating those elements in the images that could interfere in the process of identifying microcalcifications. A secondary goal is to reduce the work area only to the relevant region that exactly contains the breast. The procedure receives the original images as input. First, a 3x3 median filter is applied in order to eliminate the background noise, keeping the significant features of the images. The output is the filtered image. The size of the

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mask was chosen empirically, trying to avoid the loss of local detail. Next, binary images are created from each filtered image, where each pixel in the binarized image is determined by a window centered in the corresponding pixel in the original image. If the mean gray level of the window is below a certain threshold (established empirically) a zero value is placed in the binary image; otherwise, a unitary value is placed. After observing the histograms of the mean gray level corresponding to windows of sizes 8x8, 16x16, 32x32 and 64x64, it was observed that the histograms corresponding to windows of sizes 8x8 and 16x16 were bimodal, and the visual selection of a threshold was easier. The selected size for this window was 16x16 and the threshold was set to 15. The manual selection of the binarization threshold is allowed by the small size of the data set used in this work. For larger and diverse mammography sets, an automated procedure for the selection of this threshold is proposed as future work. In this stage, the binarized images are intended solely for helping the automatic cropping procedure to delete the background marks and the isolated regions, so the image will contain only the region of interest. The result of this stage is a smaller image, with less noise 2) Detection of potential microcalcification (signals): The main objective of this stage is to detect the mass centers of the potential microcalcifications in the image (signals). The pre-processed image of the previous stage is the input of this procedure. The optimized difference of two gaussian filters (DoG) is used for enhancing those regions containing bright points. A gaussian filter is obtained from a gaussian distribution, and when it is applied to an image, eliminates high frequency noise, acting like a smoothing filter. A DoG filter is built from two simple gaussian filters. These two smoothing filters must have different variances. When two images, obtained by separately applying each filter, are subtracted, then an image containing only the desired frequency range is obtained. The DoG filter is obtained from the difference of two gaussian functions, as it is shown in equation (1), where x and y are the coordinates of a pixel in the image, k is the height of the function and σ1 and σ2 are the standard deviations of the two gaussian filters that construct the DoG filter. DoG(x, y) = k1 e(x

2

+y 2 )/2σ12

− k2 e(x

2

+y 2 )/2σ22

(1)

The resultant image after applying a DoG filter is globally binarized, using an empirically determined threshold. In Figure 2, an example of the application of a DoG filter is shown. A region-labeling algorithm allows the identification of each one of the points (defined as high-contrast regions detected after the application of the DoG filters, which cannot be considered microcalcifications yet). Then, a segmentation algorithm extracts small 9x9 windows, containing the region of interest whose centroid corresponds to the centroid of each point. The size of the windows is adequate for containing the signals, given that at the current resolution of 200 microns, the potentially malignant microcalcifications (whose diameter is typically 100 microns to several mm) have an area of

5x5 Gaussian Filter

Original Image

DoG

Binarized Image

7x7 Gaussian Filter

Fig. 2.

Example of application of a DoG filter (5x5, 7x7).

5x5 pixels on average [17]. In order to detect the greater possible amount of points, six gaussian filters of sizes 5x5, 7x7, 9x9, 11x11, 13x13 and 15x15 are combined, two at a time, to construct 15 DoG filters that are applied sequentially. Each one of the 15 DoG filters was applied 51 times, varying the binarization threshold. The points obtained by applying each filter are added to the points obtained by the previous one, deleting the repeated points. The same procedure is repeated with the points obtained by the remaining DoG filters. All of these points are passed later to three selection procedures. These three selection methods are applied in order to transform a point into a signal (potential microcalcification). The first method performs selection according to the object area, choosing only the points with an area between a predefined minimum and a maximum. For this work, a minimum area of 1 pixel (0.0314 mm2 ) and a maximum of 77 pixels (3.08 mm2 ) were considered. The second methods performs selection according to the gray level of the points. Studying the mean gray levels of pixels surrounding real identified microcalcifications, it was found they have values in the interval [102, 237] with a mean of 164. For this study, we set the minimum gray level for points to be selected to 100. Finally, the third selection method uses the gray gradient (or absolute contrast, the difference between the mean gray level of the point and the mean gray level of the background). Again, studying the mean gray gradient of point surrounding real identified microcalcifications, it was found they have values in the interval [3, 56] with a mean of 9.66. For this study, we set the minimum gray gradient for points to be selected to 3. The result of these three selection processes is a list of signals (potential microcalcifications) represented by their centroids. 3) Classification of Signals into Real Microcalcifications: The objective of this stage is to identify if an obtained signal corresponds to an individual microcalcification or not. With this in mind, a set of features are extracted from the signal, related to their contrast and shape. From each signal, 47 features are extracted: seven related to contrast, seven related to background contrast, three related to relative contrast, 20 related to shape, six related to the moments of the contour sequence and the first four invariants proposed by Hu in a landmark paper [18]. There is not an a priori criterion to determine what features should be used for classification purposes, so the features pass through two feature selection processes [19]:

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the first one attempts to delete the features that present high correlation with other features, and the second one uses a derivation of the forward sequential search algorithm, which is a sub-optimal search algorithm. The algorithm decides what feature must be added depending of the information gain that it provides, finally resulting in a subset of features that minimize the error of the classifier (which in this case was a conventional feedforward NN). After these processes were applied, only three features were selected and used for classification: absolute contrast (the difference between the mean gray levels of the signal and its background), standard deviation of the gray level of the pixels that form the signal and the third moment of contour sequence. Moments of contour sequence are calculated using the signal centroid and the pixels in its perimeter, and are invariant to translation, rotation and scale transformations [20]. In order to process signals and accurately classify the real microcalcifications, we decided to use NNs as classifiers. In the first section, we mentioned some of the difficulties of working with conventional feedforward NNs (its architecture must be designed by hand, the parameters must be adequately tuned according to the particular problem being solved, and there are inherent drawbacks in backpropagation training). Because of these difficulties, we decided to use GAs for evolving populations of NNs, in three different ways, some of them suggested by Cant´u-Paz and Kamath [21]. The first approach uses GAs for searching the optimal set of weights of the NN. Backpropagation training algorithms use some form of gradient search over a high-dimensional, error-dependent search space, which can contain many local optima where the algorithms may be trapped. In this approach, the GA is used only for searching the weights, the architecture is fixed prior to the experiment. In our case, the number of inputs of the NN is three, equal to the number of selected features. For solving nonlinearly separable problems, it is recommended at least one hidden layer in the network, and according to Kolmogorov’s theorem [22], and considering the number of inputs as n = 3, the hidden layer contains 2n + 1 = 7 neurons. The output layer has only one neuron. The second approach is very similar to the previous one, but instead of evaluating the network immediately after the initial weight set which is represented in each chromosome of the GA, is assigned, a backpropagation training starts from this initial weight set, hoping to reach an optimum quickly [23]. Again, the number of inputs (n) for the NN is three, equal to the number of selected features. The hidden layer has 2n + 1 = 7 neurons, and the output layer contains only one neuron. The last approach is not concerned with evolving weights. Instead, a GA is used to evolve a part of the architecture and other features of the NN. The number of nodes in the hidden layer is very important parameter, because too few or to many nodes can affect the learning and generalization capabilities of the NN. In this case, each chromosome encodes the learning rate, a lower and upper limits for the weights before starting the backpropagation training, and the number of

nodes of the hidden layer. The number of inputs is still three, and again there is only one output. For comparison, a conventional feedforward NN is trained also. The number of inputs for this NN is three, the number of hidden layer is fixed to seven and there is one output. Every other parameter of the NN is fixed too. The transfer function of every neuron in all the NNs mentioned above is the sigmoid hyperbolic tangent function, and the error is measured with the mean square error function. At the end of this stage, we obtain three ready-to-use NNs, each one taken from the last generation of the GAs used in each one of the approaches. These NNs have the best performances in terms of overall accuracy (fraction of well classified objects, including microcalcifications and other elements in the image that are not microcalcifications). Additionally, we can obtain the NN with the best performance in terms of sensitivity (fraction of true positives, or well classified microcalcifications) and the NN with the best performance in terms of specificity (fraction of true negatives or well classified objects that are not microcalcifications), from the same last generations. 4) Detection of Microcalcification Clusters: During this stage, the microcalcification clusters are identified. The detection and posterior consideration of every microcalcification cluster in the images may produce better results in a subsequent classification process, as shown by Oporto-D´ıaz, Hern´andez-Cisneros and Terashima-Mar´ın [24]. Because of this, an algorithm for locating microcalcification cluster regions where the quantity of microcalcifications per cm2 (density) is higher, was developed. This algorithm keeps adding microcalcifications to their closest clusters at a reasonable distance until there are no more microcalcifications left or if the remaining ones are too distant for being considered as part of a cluster. Every detected cluster is then labeled. 5) Classification of Microcalcification Clusters into Benign and Malignant: This stage has the objective of classifying each cluster in one of two classes: benign or malignant. This information is provided by the MIAS database. From every microcalcification cluster detected in the mammograms in the previous stage, a cluster feature set is extracted. The feature set is constituted by 30 features: 14 related to the shape of the cluster, six related to the area of the microcalcifications included in the cluster and ten related to the contrast of the microcalcifications in the cluster. The same two feature selection procedures mentioned earlier are also performed in this stage. The first one attempts to delete the features that present high correlation with other features, and the second one uses a derivation of the forward sequential search algorithm, which is a sub-optimal search algorithm. Only three cluster features were selected for the classification process: minimum diameter, minimum radius and mean radius of the clusters. The minimum diameter is the maximum distance that can exist between two microcalcifications within a cluster in such a way that the line connecting them is perpendicular to the maximum diameter, defined as the maximum distance between two microcalcifications in a cluster. The minimum radius is the shortest of the radii

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connecting each microcalcification to the centroid of the cluster and the mean radius is the mean of these radii. In order to process microcalcification clusters and accurately classify them into benign or malignant, we decided again to use NNs as classifiers. Because of the difficulties of working with conventional feedforward NNs already mentioned in previous sections, we use GAs for evolving populations of NNs, in the same three different approaches we used before for classifying signals. The first approach uses GAs for searching the optimal set of weights of the NN. The number of inputs of the NN is n = 3, equal to the number of selected features. The hidden layer contains 2n + 1 = 7 neurons and the output layer consists of only one neuron. The second approach uses a GA for defining initial weight sets, from which a backpropagation training algorithm is started, hoping to reach an optimum quickly. Again, the number of inputs (n) for the NN is three, equal to the number of selected features. The hidden layer has 2n+ 1 = 7 neurons, and the output layer has only one neuron. The third approach uses a GA for evolving the architecture and other features of the NN. As in a previous stage, when signals were classified, each chromosome encodes the learning rate, a lower and upper limits for the weights before starting the backpropagation training, and the number of nodes of the hidden layer. The number of inputs is still three, and again there is only one output. For comparison, a conventional feedforward NN is used also. The architecture of this NN consists of three inputs, seven units in the hidden layer and one output. Every other parameter of the NN remains the same. The transfer function of every neuron in all the NNs mentioned in this stage is the sigmoid hyperbolic tangent function, and the error is measured with the mean square error function. At the end of this stage, we obtain three ready-to-use NNs, each one taken from the last generation of the GAs used in each of the approaches. These NNs have the best performances in terms of overall accuracy (fraction of well classified clusters). Additionally, we can obtain the NN with the best performance in terms of sensitivity (fraction of true positives or well classified malignant clusters) and the NN with the best performance in terms of specificity (fraction of true negatives or well classified benign clusters), from the same last generations.

positions of the microcalcifications clusters, marked in the additional data that comes with the database, were outside the boundaries of the breast. So, only 22 images were finally used for this study, and they were passed through the preprocessing stage first (application of a 3x3 median filter, binarization and trimming). In the second phase, six gaussian filters of sizes 5x5, 7x7, 9x9, 11x11, 13x13 and 15x15 were combined, two at a time, to construct 15 DoG filters that were applied sequentially. Each one of the 15 DoG filters was applied 51 times to the pre-processed images, varying the binarization threshold in the interval [0, 5] in increments of 0.1. The points obtained by applying each filter were added to the points obtained by the previous one, deleting the repeated points. The same procedure was repeated with the points obtained by the remaining DoG filters. These points passed through the three selection methods for selecting signals (potential microcalcification), according to region area, gray level and the gray gradient. The result was a list of 1,242,179 signals (potential microcalcifications) represented by their centroids. The additional data included with the MIAS database define, with centroids and radii, the areas in the mammograms where microcalcifications are located. It is supposed that signals within these areas are mainly microcalcifications, but there are many signals that lie outside the marked areas. With these data and the support of expert radiologists, all the signals located in these 22 mammograms were preclassified into microcalcification, and not-microcalcifications. From the 1,242,179 signals, only 4,612 (0.37%) were microcalcifications, and the remaining 1,237,567 (99.63%) were not. Because of this imbalanced distribution of elements in each class, an exploratory sampling was made. Several sampling with different proportions of each class were tested and finally we decided to use a sample of 10,000 signals, including 2,500 real microcalcifications in it (25%). After the 47 microcalcification features were extracted from each signal, the feature selection processes mentioned in the previous section reduced the relevant features to only three: absolute contrast, standard deviation of the gray level and the third moment of contour sequence. Finally, a transactional database was obtained, containing 10,000 signals (2500 of them being real microcalcifications randomly distributed) and three features describing each signal.

III. M AIN R ESULTS

In the third stage, a conventional feedforward NN and three evolutionary NNs were developed for the classification of signals into real microcalcifications. The feedforward NN had an architecture of three inputs, seven neurons in the hidden layer and one output. All the units had the sigmoid hyperbolic tangent function as the transfer function. The data (input and targets) were scaled in the range [-1, 1] and divided into ten non-overlapping splits, each one with 90% of the data for training and the remaining 10% for testing. A ten-fold crossvalidation trial was performed; that is, the NN was trained ten times, each time using a different split on the data and the means and

All the programs were written in MATLAB Version 7.0.0.19920 (R14), and executed on a PC with a single 2.1 GHz Intel Pentium IV processor with 1 Gb of memory. The following subsections explain the experimentation and results of every stage of the study. A. From Pre-processing to Feature Extraction As it was mentioned in the previous section, only 25 images from the MIAS database contain microcalcifications. Among these 25 images, 13 cases are diagnosed as malignant and 12 as benign. Three images were discarded because the

B. Classification of Signals into Microcalcifications

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standard deviations of the overall performance, sensitivity and specificity were reported. These results are shown in Table I on the row “BP”. For the three EANNs used to evolve signal classifiers, all of their GAs used a population of 50 individuals. We used simple GAs, with gray encoding, stochastic universal sampling selection, double-point crossover, fitness based reinsertion and a generational gap of 0.9. For all the GAs, the probability of crossover was 0.7 and the probability of mutation was 1/l, where l is the length of the chromosome. The initial population of each GA was always initialized uniformly at random. All the NNs involved in the EANNs are feedforward networks with one hidden layer. All neurons have biases with a constant input of 1.0. The NNs are fully connected, and the transfer functions of every unit is the sigmoid hyperbolic tangent function. The data (input and targets) were normalized to the interval [-1, 1]. For the targets, a value of “-1” means “not-microcalcification” and a value of “1” means “microcalcification”. When backpropagation was used, the training stopped after 20 epochs. For the first approach, where a GA was used to find the NN’s weights, the population consisted of 50 individuals, each one with a length of l = 720 bits and representing 36 weights (including biases) with a precision of 20 bits. There were two crossover points, and the mutation rate was 0.00139. The GA ran for 50 generations. The results of this approach are shown in Table I on the row “WEIGHTS”. In the second approach, where a backpropagation training algorithm is run using the weights represented by the individuals in the GA to initialize the NN, the population consisted of 50 individual also, each one with a length of l = 720 bits and representing 36 weights (including biases) with a precision of 20 bits. There were two crossover points, and the mutation rate was 0.00139 (1/l). In this case, each NN was briefly trained using 20 epochs of backpropagation, with a learning rate of 0.1. The GA ran for 50 generations. The results of this approach are shown in Table I on the row “WEIGHTS+BP”. Finally, in the third approach, where a GA was used to find the size of the hidden layer, the learning rate for the backpropagation algorithm and the range of initial weights before training, the population consisted of 50 individuals, each one with a length of l = 18 bits. The first four bits of the chromosome coded the learning rate in the range [0,1], the next five bits coded the lower value for the initial weights in the range [-10,0], the next five bits coded the upper value for the initial weights in the range [0,10] and the last four bits coded the number of neurons in the hidden layer, in the range [1,15] (if the value was 0, it was changed to 1). There was only one crossover point, and the mutation rate was 0.055555 (1/l). In this case, each NN was built according to the parameters coded in the chromosome, and trained briefly with 20 epochs of backpropagation, in order to favor the NNs that learned quickly. The results of this approach are shown also in Table I, on the row “PARAMETERS”. We performed several two-tailed Student’s t-tests at a level of significance of 5% in order to compare the mean of

TABLE I M EAN AND STANDARD DEVIATION OF THE SENSITIVITY, SPECIFICITY AND OVERALL ACCURACY OF SIMPLE BACKPROPAGATION AND DIFFERENT EVOLUTIONARY METHODS FOR THE CLASSIFICATION OF SIGNALS INTO REAL MICROCALCIFICATIONS .

Method BP WEIGHTS WEIGHTS+BP PARAMETERS

Sensitivity Std. Mean Dev. 75.68 0.044 72.44 0.027 75.81 0.021 73.19 0.177

Specificity Std. Mean Dev. 81.36 0.010 84.32 0.013 86.76 0.025 84.67 0.035

Overall Std. Mean Dev. 80.51 0.013 82.37 0.011 84.68 0.006 83.12 0.028

each method with the means of the other ones in terms of sensitivity, specificity and overall accuracy. In Table II, we show the results of the statistical comparison between the mean of the simple backpropagation method versus the means of the other evolutionary methods. We found that for specificity and overall accuracy, evolutionary methods are significantly better than the simple backpropagation method for the classification of individual microcalcifications. No difference was found in terms of sensitivity, except that simple backpropagation was significantly better than the method that evolves weights. We can notice too that, among the studied EANNs, the one that evolves a set of initial weights and is complemented with backpropagation training is the one that gives better results. In Table III, we show the results of the statistical comparison between this method and the rest of them. We found that in fact, again in terms of specificity and overall accuracy, the method of weight evolution complemented with backpropagation is significantly the best of the methods we studied. Nevertheless, in terms of sensitivity, this method is only significantly better than the method that evolves weights. TABLE II R ESULTS OF THE STATISTICAL COMPARISON BETWEEN THE MEAN OF THE SIMPLE BACKPROPAGATION METHOD VERSUS THE MEANS OF THE EVOLUTIONARY METHODS FOR THE CLASSIFICATION OF SIGNALS INTO REAL MICROCALCIFICATIONS .

Method WEIGHTS WEIGHTS+BP PARAMETERS

Significative Difference with BP Sensitivity Specificity Overall Yes Yes Yes No Yes Yes No Yes Yes

C. Microcalcification Clusters Detection and Classification The process of cluster detection and the subsequent feature extraction phase generates another transactional database, this time containing the information of every microcalcification cluster detected in the images. A total of 40 clusters were detected in the 22 mammograms from the MIAS database that were used in this study. According to MIAS additional data and the advice of expert radiologists, 10 clusters are benign and 30 are malignant. The number of

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TABLE III R ESULTS OF THE STATISTICAL COMPARISON BETWEEN THE MEAN OF

TABLE V R ESULTS OF THE STATISTICAL COMPARISON BETWEEN THE MEAN OF

THE WEIGHT EVOLUTION WITH BACKPROPAGATION METHOD VERSUS

THE SIMPLE BACKPROPAGATION METHOD VERSUS THE MEANS OF THE

THE MEANS OF THE OTHER METHODS FOR THE CLASSIFICATION OF

EVOLUTIONARY METHODS FOR THE CLASSIFICATION OF

SIGNALS INTO REAL MICROCALCIFICATIONS .

MICROCALCIFICATION CLUSTERS .

Method BP WEIGHTS PARAMETERS

Significative Difference with WEIGHTS+BP Sensitivity Specificity Overall No Yes Yes Yes Yes Yes No Yes Yes

Method WEIGHTS WEIGHTS+BP PARAMETERS

Significative Difference with BP Sensitivity Specificity Overall Yes Yes Yes Yes Yes Yes Yes No Yes

TABLE VI

features extracted from them is 30, but after the two feature selection processes already discussed in previous sections, the number of relevant features we considered relevant was three: minimum diameter, minimum radius and mean radius of the clusters. As in the stage of signal classification, a conventional feedforward NN and three evolutionary NNs were developed for the classification of signals into real microcalcifications. The four algorithms we use in this step are basically the same ones we used before, except that they receive as input the transactional database containing features about microcalcifications clusters instead of features about signals. Again, the means of the overall performance, sensitivity and specificity for each one of these four approaches are reported and shown in Table IV. TABLE IV M EAN AND STANDARD DEVIATION OF THE SENSITIVITY, SPECIFICITY AND OVERALL ACCURACY OF SIMPLE BACKPROPAGATION AND DIFFERENT EVOLUTIONARY METHODS FOR THE CLASSIFICATION OF MICROCALCIFICATION CLUSTERS .

Method BP WEIGHTS WEIGHTS+BP PARAMETERS

Sensitivity Std. Mean Dev. 55.97 0.072 72.00 0.059 89.34 0.035 63.90 0.163

Specificity Std. Mean Dev. 86.80 0.032 92.09 0.038 95.86 0.025 85.74 0.067

Overall Std. Mean Dev. 76.75 0.032 86.35 0.031 93.88 0.027 80.50 0.043

We also performed several two-tailed Student’s t-tests at a level of significance of 5% in order to compare the mean of each method for cluster classification with the means of the other ones in terms of sensitivity, specificity and overall accuracy. We can observe in Table V that the performance of evolutionary methods is significantly different and better than the performance of the simple backpropagation method, except in one case. Again, the method that evolves initial weights, complemented with backpropagation, is the one that gives the best results. In Table VI, we show the results of the statistical comparison between this method and the rest of them. IV. C ONCLUSIONS This paper presented a comparison of simple backpropagation training and three methods for combining GAs and

R ESULTS OF THE STATISTICAL COMPARISON BETWEEN THE MEAN OF THE WEIGHT EVOLUTION WITH BACKPROPAGATION METHOD VERSUS THE MEANS OF THE OTHER METHODS FOR THE CLASSIFICATION OF MICROCALCIFICATION CLUSTERS .

Method BP WEIGHTS PARAMETERS

Significative Difference with WEIGHTS+BP Sensitivity Specificity Overall Yes Yes Yes Yes Yes Yes Yes Yes Yes

NNs, applied to the classification of signals into real microcalcifications and microcalcification clusters into benign and malignant, on mammograms containing microcalcifications from the MIAS database. Our experimentation suggests that evolutionary methods are significantly better than the simple backpropagation method for the classification of individual microcalcifications, in terms of specificity and overall accuracy. No difference was found in terms of sensitivity, except that simple backpropagation was significantly better than the method that only evolves weights. In the case of the classification of microcalcification clusters, we observed that the performance of evolutionary methods is significantly better than the performance of the simple backpropagation method, except in one case. Again, the method that evolves initial weights, complemented with backpropagation, is the one that gives the best results. As future work, it would be useful to include and process other mammography databases, in order to have more examples and produce transactional feature databases more balanced and complete, and test also how different resolutions could affect system effectiveness. The size of the gaussian filters could be adapted depending on the size of the microcalcifications to be detected and the resolution of images. The correspondence between the spatial frequency of the image and the relation σ1 /σ2 has to be thoroughly studied. Different new features could be extracted from the microcalcifications in the images and tested also. In this study, simple GAs and NNs were used, and more sophisticated versions of these methods could produce better results. The use of real valued chromosomes, chromosomes with indirect representation (metaheuristics, NN construction rules, etc.), use of EANNs for feature selection, etc. are other approaches that could give different results. The inclusion of simple backpropagation training in the EANNs have

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consequences of longer computation times, so alternatives to backpropagation should be tested in order to reduce time costs. ACKNOWLEDGMENT This research was supported in part by the Instituto Tecnol´ogico y de Estudios Superiores de Monterrey (ITESM) under the Research Chair CAT-010 and the National Council of Science and Technology of Mexico (CONACYT) under grant 41515. R EFERENCES [1] M. A. Ganott, K. M. Harris, H. M. Klaman and T. L. Keeling, “Analysis of false-negative cancer cases identified with a mammography audit,” The Breast Journal, 5(3), pp. 166–175, 1999. [2] I. Anttinen, M. Pamilo, M. Soiva and M. Roiha, “Double reading of mammography screening films: one radiologist or two?,” Clin. Radiol., 48, pp. 414–421, 1993. [3] E. L. Thurfjell, K. A. Lernevall and A. A. S. Taube, “Benefit of independent double reading in a population-based mammography screening program,” Radiology, 191, pp. 241–244, 1994. [4] J. Dengler, S. Behrens and J. F. Desaga, “Segmentation of microcalcifications in mammograms,” IEEE Trans. Med. Imaging, 12(4), pp. 634–642, 1993. [5] E. M. Ochoa, Clustered Microcalcification Detection using Optimized Difference of Gaussians, Master Thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, 1996. [6] X. Yao, “Evolving artificial neural networks,” in Proceedings of the IEEE, 87(9), pp. 1423–1447, 1999. [7] K. Balakrishnan, and V. Honavar, “Evolutionary design of neural architectures. A preliminary taxonomy and guide to literature,” Technical Report CS TR 95-01, Department of Computer Sciences, Iowa State University, 1995. [8] D. B. Fogel, E. C. Wasson, E. M. Boughton and V. W. Porto, “A step toward computer-assisted mammography using evolutionary programming and neural networks,” Cancer Letters, Vol. 119, pp. 93–97, 1997. [9] D. B. Fogel, E. C. Wasson, E. M. Boughton, V. W. Porto and P. J. Angeline, “Linear and neural models for classifying breast cancer,” IEEE Trans. Medical Imaging, Vol. 17:3, pp. 485–488, 1998. [10] D. B. Fogel, E. C. Wasson, E. M. Boughton and V. W. Porto, “Evolving artificial neural networks for screening features from mammograms,” Artificial Intelligence in Medicine, Vol. 14, pp. 317–326, 1998. [11] J. Suckling, J. Parker, D. Dance, S. Astley, I. Hutt, C. Boggis, I. Ricketts, E. Stamatakis, N. Cerneaz, S. Kok, P. Taylor, D. Betal and J. Savage, “The Mammographic Images Analysis Society digital mammogram database,” Exerpta Medica International Congress Series, 1069, pp. 375–378, 1994. http://www.wiau.man.ac.uk/services/MIAS/MIASweb.html [12] M. Heath, K. Bowyer, D. Kopans, R. Moore and P. Kegelmeyer Jr., “The digital database for screening mammography,” In Proceedings of the 5th International Workshop on Digital Mammography, Toronto, ON, Canada, 2000. [13] R. Chandrasekhar, and Y. Attikiouzel, “Digitization regime as a cause for variation in algorithm performance across two mammogram databases,” Technical Report 99/05, Centre for Intelligent Information Processing Systems, Department of Electrical and Electronic Engineering, The University of Western Australia, 1999. [14] T. O. Gulsrud, “Analysis of mammographic microcalcifications using a computationally efficient filter bank,” Technical Report, Department of Electrical and Computer Engineering, Stavanger University College, 2001. [15] B.-W. Hong and M. Brady, “Segmentation of mammograms in topographic approach,” In IEE International Conference on Visual Information Engineering, Guildford, UK, 2003. [16] S. Li, T. Hara, Y. Hatanaka, H. Fujita, T. Endo and T. Iwase, “Performance evaluation of a CAD system for detecting masses on mammograms by using the MIAS database,” Medical Imaging and Information Science, 18(3), pp. 144–153, 2001. [17] S. Oporto-D´ıaz, Automatic Detection of Microcalcification Clusters in Digital Mammograms, Master Thesis, Tecnol´ogico de Monterrey, Campus Monterrey, Monterrey, Mexico, 2004.

[18] M.-K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Information Theory, Vol. IT-8, pp. 179–187, 1962. [19] A. Kozlov and D. Koller, “Nonuniform dynamic discretization in hybrid networks,” In Proceedings of the 13th Annual Conference of Uncertainty in AI (UAI), Providence, Rhode Island, USA, pp. 314–325, 2003. [20] L. Gupta and M. D. Srinath, “Contour sequence moments for the classification of closed planar shapes,” Pattern Recognition, 20(3), pp. 267–272, 1987. [21] E. Cant´u-Paz and C. Kamath, “Evolving neural networks for the classification of galaxies,” In Proceedings of the Genetic and Evolutionary Computation Conference, GECCO 2002, San Francisco, CA, USA, pp. 1019-1026, 2002. [22] V. Kurkova, “Kolmogorov’s theorem,” In The handbook of brain theory and neural networks, Edited by M. A. Arbib, MIT Press, Cambridge, Massachusetts, pp. 501-502, 1995. [23] A. Skinner and J. Q. Broughton, “Neural networks in computational material science: traning algorithms,” Modeling and Simulation in Material Science and Engineering, 3, pp. 371–390, 1995. [24] S. Oporto-D´ıaz, R. R. Hern´andez-Cisneros and H. Terashima-Mar´ın, “Detection of microcalcification clusters in mammograms using a difference of optimized gaussian filters”, in Proceedings of the Second International Conference on Image Analysis and Recognition, ICIAR 2005, Toronto, ON, Canada, pp. 998–1005, 2005.

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Evolutionary Neural Networks Applied To The Classification Of Microcalcification Clusters In Digital Mammograms Rolando R. Hern´andez-Cisneros and Hugo Terashima-Mar´ın

Abstract— Breast cancer is one of the main causes of death in women and early diagnosis is an important means to reduce the mortality rate. The presence of microcalcification clusters are primary indicators of early stages of malignant types of breast cancer and its detection is important to prevent the disease. This paper proposes a procedure for the classification of microcalcification clusters in mammograms using sequential Difference of Gaussian filters (DoG) and three Evolutionary Artificial Neural Networks (EANNs) compared against a feedforward Neural Network (NN) trained with backpropagation. We found that the use of Genetic Algorithms (GAs) for 1) finding the optimal weight set for a NN, 2) finding an adequate initial weight set before starting a backpropagation training algorithm and 3) designing its architecture and tuning its parameters, results mainly in improvements in overall accuracy, sensitivity and specificity of a NN, compared with other networks trained with simple backpropagation.

I. I NTRODUCTION Breast cancer is one of the main causes of death in women and early diagnosis is an important means to reduce the mortality rate. Mammography is one of the most common techniques for breast cancer diagnosis, and microcalcifications are one among several types of objects that can be detected in a mammogram. Microcalcifications are calcium accumulations typically 100 microns to several mm in diameter, and they sometimes are early indicators of the presence of breast cancer. Microcalcification clusters are groups of three or more microcalcifications that usually appear in areas smaller than 1 cm2 , and they have a high probability of becoming a malignant lesion. However, the predictive value of mammograms is relatively low, compared to biopsy. This low sensitivity (correct diagnosis of positive cases) [1] is caused by the low contrast between the cancerous tissue and the normal parenchymal tissue, the small size of microcalcifications and possible deficiencies in the image digitalization process. The sensitivity may be improved having each mammogram checked by two or more radiologists, with the consequence of making the process inefficient by reducing the individual productivity of each specialist. A viable alternative is replacing one of the radiologists by a computer system, giving a second opinion [2], [3]. Several computer-aided diagnosis (CAD) systems for early detection of breast cancer have been proposed in literature and some are offered commercially, like Second Rolando R. Hern´andez-Cisneros is with the Center for Intelligent Systems, Tecnol´ogico de Monterrey, Campus Monterrey, Av. Eugenio Garza Sada 2501, Monterrey, N.L. 65841 Mexico (email: [email protected]). Hugo Terashima-Mar´ın is with the Center for Intelligent Systems, Tecnol´ogico de Monterrey, Campus Monterrey, Av. Eugenio Garza Sada 2501, Monterrey, N.L. 65841 Mexico (email: [email protected]).

0-7803-9487-9/06/$20.00/©2006 IEEE

R R Look by iCAD, Inc. and ImageChecker by R2 Technology, Inc. Nevertheless, research in this field continues, trying to improve the efficiency of CAD systems in terms of improved accuracy and less processing time. A computer system intended for microcalcification detection in mammograms may be based on several methods, like wavelets, fractal models, support vector machines, mathematical morphology, bayesian image analysis models, high order statistic, fuzzy logic, etc. The method we selected for this work is the Difference of Gaussian Filters (DoG). DoG filters are adequate for the noise-invariant and size-specific detection of spots, resulting in a DoG image. This DoG image represents the microcalcifications if a thresholding operation is applied to it. The use of DoG for detection of potential microcalcifications has been addressed successfully by Dengler, Behrens and Desaga [4] and Ochoa [5]. We developed a procedure that applies a sequence of Difference of Gaussian Filters, in order to maximize the amount of detected probable individual microcalcifications (signals) in the mammogram, which are later classified in order to detect if they are real microcalcifications or not. Finally, microcalcification clusters are identified and also classified in order to determine which ones are malignant and which ones are benign. Neural networks (NNs) have been successfully used for classification purposes in medical applications, including the classification of microcalcifications in digital mammograms. Unfortunately, for a NN to be successful in a particular domain, its architecture, training algorithm and the domain variables selected as inputs must be adequately chosen. Designing a NN architecture is a trial-and-error process; several parameters must be tuned according to the training data when a training algorithm is chosen and, finally, a classification problem could involve too many variables (features), most of them not relevant at all for the classification process itself. Genetic algorithms (GAs) may be used to address the problems mentioned above, helping to obtain more accurate NNs with better generalization abilities. GAs have been used for searching the optimal weight set of a NN, for designing its architecture, for finding its most adequate parameter set (number of neurons in the hidden layer(s), learning rate, etc.) among others tasks. Exhaustive reviews about evolutionary artificial neural networks (EANNs) have been presented by Yao [6] and Balakrishnan and Honavar [7]. Fogel et al. proposed an approach based in EANNs for early detection of breast cancer in [8], [9] and [10]. However, the scope of these studies was broad, focusing also in detecting other breast lesions beside microcalcifications, like

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Original images

masses, architectural distortions and asymmetric densities. Each type of lesion was described by few features, and each one of the features was measured using a small scale of discrete values. For instance, microcalcifications were described only by its number, morphology, density and distribution, using a scale of four discrete values for each feature. The training data sets for the NNs were constructed with the help of an expert, by visual inspection of the mammograms. In this paper, we propose an automated procedure for feature extraction and training data set construction for training a NN. We also describe the use of GAs for 1) finding the optimal weight set for a NN, 2) finding an adequate initial weight set for a NN before starting a backpropagation training algorithm and 3) designing the architecture and tuning some parameters of a NN. All of these methods are applied to the classification of microcalcifications and microcalcification clusters in digital mammograms, expecting to improve the accuracy of an ordinary feedforward NN performing this task. The rest of this document is organized as follows. In the second section, the mammography database we selected for this study is described and the proposed procedure along with its theoretical framework is discussed. The third section deals with the experiments and the main results of this work. Finally, in the fourth section, the conclusions are presented, and some comments about future work are also made.

Pre-processing Median filter Binarization Automatic cropping

Detection of potential microcalcifications (signals) DoG filters Global binarization Region labeling Point selection by minimum area, minimum gray level and minimum gradient

Classification of Signals Microcalcification data set Feature extraction and selection from microcalc. Selected microcalc. features Microcalcification data set

Neural Networks Neural Networks Overall accuracy

Classification of Microcalc. Clusters Detection of Microcalcification Clusters

Genetic Algorithm Coded weights sets / architectures Neural Networks Neural Networks

There are several mammography databases publicly available for research purposes, and the most known of them are the mammography database from the Mammographic Image Analysis Society (MIAS) [11], the Digital Database for Screening Mammography (DDSM) from the University of South Florida [12] and the Nijmegen digital mammography database. The mammograms used in this project were provided by The Mammographic Image Analysis Society (MIAS). The MIAS is an organization of UK research groups interested in the understanding of mammograms. The MIAS database contains 322 images, all medio-lateral (MLO) view, digitized with a scanning microdensitometer (Joyce-Loebl, SCANDIG3) with resolutions of 50 microns/pixel and 200 microns/pixel. In this work, the images with a resolution of 200 microns/pixel were used. The data has been reviewed by a consultant radiologist and all the abnormalities have been identified and marked. The truth data consists of the location of the abnormality and the radius of a circle which encloses it. The abnormalities represented in the database include calcifications, circumscribed masses, spiculated masses, illdefined masses, architectural distortions and asymmetry. Several normal cases are also included. From the totality of the

Optimal NN for microcalc. cluster classification; best overall accuracy, sensitivity and specificity

Overall accuracy

Fig. 1.

A. The Mammography Database

Cluster data set Feature extraction and selection from microcalc. clusters Selected cluster features

II. P RELIMINARIES In this section, the mammography database we selected for this work is described. Also, the proposed procedure, along with the theoretical framework is discussed in detail.

Genetic Algorithm Coded weights sets / architectures

Optimal NN for individual microcalc. classification; best overall accuracy, sensitivity and specificity

Diagram of the proposed procedure.

database, only 25 images contain microcalcifications. Among these 25 images, 13 cases are diagnosed as malignant and 12 as benign. Some related works have used this same database [13], [14], [15], [16]. B. Methodology The general procedure receives a digital mammogram as an input, and it is conformed by five stages: pre-processing, detection of potential microcalcifications (signals), classification of signals into real microcalcifications, detection of microcalcification clusters and classification of microcalcification clusters into benign and malignant. The diagram of the proposed procedure is shown in Figure 1. As end-products of this process, we obtain two NNs for classifying microcalcifications and microcalcifications clusters respectively, which in this case, are products of the evolutionary approaches that are proposed. 1) Pre-processing: This stage has the aim of eliminating those elements in the images that could interfere in the process of identifying microcalcifications. A secondary goal is to reduce the work area only to the relevant region that exactly contains the breast. The procedure receives the original images as input. First, a 3x3 median filter is applied in order to eliminate the background noise, keeping the significant features of the images. The output is the filtered image. The size of the

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mask was chosen empirically, trying to avoid the loss of local detail. Next, binary images are created from each filtered image, where each pixel in the binarized image is determined by a window centered in the corresponding pixel in the original image. If the mean gray level of the window is below a certain threshold (established empirically) a zero value is placed in the binary image; otherwise, a unitary value is placed. After observing the histograms of the mean gray level corresponding to windows of sizes 8x8, 16x16, 32x32 and 64x64, it was observed that the histograms corresponding to windows of sizes 8x8 and 16x16 were bimodal, and the visual selection of a threshold was easier. The selected size for this window was 16x16 and the threshold was set to 15. The manual selection of the binarization threshold is allowed by the small size of the data set used in this work. For larger and diverse mammography sets, an automated procedure for the selection of this threshold is proposed as future work. In this stage, the binarized images are intended solely for helping the automatic cropping procedure to delete the background marks and the isolated regions, so the image will contain only the region of interest. The result of this stage is a smaller image, with less noise 2) Detection of potential microcalcification (signals): The main objective of this stage is to detect the mass centers of the potential microcalcifications in the image (signals). The pre-processed image of the previous stage is the input of this procedure. The optimized difference of two gaussian filters (DoG) is used for enhancing those regions containing bright points. A gaussian filter is obtained from a gaussian distribution, and when it is applied to an image, eliminates high frequency noise, acting like a smoothing filter. A DoG filter is built from two simple gaussian filters. These two smoothing filters must have different variances. When two images, obtained by separately applying each filter, are subtracted, then an image containing only the desired frequency range is obtained. The DoG filter is obtained from the difference of two gaussian functions, as it is shown in equation (1), where x and y are the coordinates of a pixel in the image, k is the height of the function and σ1 and σ2 are the standard deviations of the two gaussian filters that construct the DoG filter. DoG(x, y) = k1 e(x

2

+y 2 )/2σ12

− k2 e(x

2

+y 2 )/2σ22

(1)

The resultant image after applying a DoG filter is globally binarized, using an empirically determined threshold. In Figure 2, an example of the application of a DoG filter is shown. A region-labeling algorithm allows the identification of each one of the points (defined as high-contrast regions detected after the application of the DoG filters, which cannot be considered microcalcifications yet). Then, a segmentation algorithm extracts small 9x9 windows, containing the region of interest whose centroid corresponds to the centroid of each point. The size of the windows is adequate for containing the signals, given that at the current resolution of 200 microns, the potentially malignant microcalcifications (whose diameter is typically 100 microns to several mm) have an area of

5x5 Gaussian Filter

Original Image

DoG

Binarized Image

7x7 Gaussian Filter

Fig. 2.

Example of application of a DoG filter (5x5, 7x7).

5x5 pixels on average [17]. In order to detect the greater possible amount of points, six gaussian filters of sizes 5x5, 7x7, 9x9, 11x11, 13x13 and 15x15 are combined, two at a time, to construct 15 DoG filters that are applied sequentially. Each one of the 15 DoG filters was applied 51 times, varying the binarization threshold. The points obtained by applying each filter are added to the points obtained by the previous one, deleting the repeated points. The same procedure is repeated with the points obtained by the remaining DoG filters. All of these points are passed later to three selection procedures. These three selection methods are applied in order to transform a point into a signal (potential microcalcification). The first method performs selection according to the object area, choosing only the points with an area between a predefined minimum and a maximum. For this work, a minimum area of 1 pixel (0.0314 mm2 ) and a maximum of 77 pixels (3.08 mm2 ) were considered. The second methods performs selection according to the gray level of the points. Studying the mean gray levels of pixels surrounding real identified microcalcifications, it was found they have values in the interval [102, 237] with a mean of 164. For this study, we set the minimum gray level for points to be selected to 100. Finally, the third selection method uses the gray gradient (or absolute contrast, the difference between the mean gray level of the point and the mean gray level of the background). Again, studying the mean gray gradient of point surrounding real identified microcalcifications, it was found they have values in the interval [3, 56] with a mean of 9.66. For this study, we set the minimum gray gradient for points to be selected to 3. The result of these three selection processes is a list of signals (potential microcalcifications) represented by their centroids. 3) Classification of Signals into Real Microcalcifications: The objective of this stage is to identify if an obtained signal corresponds to an individual microcalcification or not. With this in mind, a set of features are extracted from the signal, related to their contrast and shape. From each signal, 47 features are extracted: seven related to contrast, seven related to background contrast, three related to relative contrast, 20 related to shape, six related to the moments of the contour sequence and the first four invariants proposed by Hu in a landmark paper [18]. There is not an a priori criterion to determine what features should be used for classification purposes, so the features pass through two feature selection processes [19]:

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the first one attempts to delete the features that present high correlation with other features, and the second one uses a derivation of the forward sequential search algorithm, which is a sub-optimal search algorithm. The algorithm decides what feature must be added depending of the information gain that it provides, finally resulting in a subset of features that minimize the error of the classifier (which in this case was a conventional feedforward NN). After these processes were applied, only three features were selected and used for classification: absolute contrast (the difference between the mean gray levels of the signal and its background), standard deviation of the gray level of the pixels that form the signal and the third moment of contour sequence. Moments of contour sequence are calculated using the signal centroid and the pixels in its perimeter, and are invariant to translation, rotation and scale transformations [20]. In order to process signals and accurately classify the real microcalcifications, we decided to use NNs as classifiers. In the first section, we mentioned some of the difficulties of working with conventional feedforward NNs (its architecture must be designed by hand, the parameters must be adequately tuned according to the particular problem being solved, and there are inherent drawbacks in backpropagation training). Because of these difficulties, we decided to use GAs for evolving populations of NNs, in three different ways, some of them suggested by Cant´u-Paz and Kamath [21]. The first approach uses GAs for searching the optimal set of weights of the NN. Backpropagation training algorithms use some form of gradient search over a high-dimensional, error-dependent search space, which can contain many local optima where the algorithms may be trapped. In this approach, the GA is used only for searching the weights, the architecture is fixed prior to the experiment. In our case, the number of inputs of the NN is three, equal to the number of selected features. For solving nonlinearly separable problems, it is recommended at least one hidden layer in the network, and according to Kolmogorov’s theorem [22], and considering the number of inputs as n = 3, the hidden layer contains 2n + 1 = 7 neurons. The output layer has only one neuron. The second approach is very similar to the previous one, but instead of evaluating the network immediately after the initial weight set which is represented in each chromosome of the GA, is assigned, a backpropagation training starts from this initial weight set, hoping to reach an optimum quickly [23]. Again, the number of inputs (n) for the NN is three, equal to the number of selected features. The hidden layer has 2n + 1 = 7 neurons, and the output layer contains only one neuron. The last approach is not concerned with evolving weights. Instead, a GA is used to evolve a part of the architecture and other features of the NN. The number of nodes in the hidden layer is very important parameter, because too few or to many nodes can affect the learning and generalization capabilities of the NN. In this case, each chromosome encodes the learning rate, a lower and upper limits for the weights before starting the backpropagation training, and the number of

nodes of the hidden layer. The number of inputs is still three, and again there is only one output. For comparison, a conventional feedforward NN is trained also. The number of inputs for this NN is three, the number of hidden layer is fixed to seven and there is one output. Every other parameter of the NN is fixed too. The transfer function of every neuron in all the NNs mentioned above is the sigmoid hyperbolic tangent function, and the error is measured with the mean square error function. At the end of this stage, we obtain three ready-to-use NNs, each one taken from the last generation of the GAs used in each one of the approaches. These NNs have the best performances in terms of overall accuracy (fraction of well classified objects, including microcalcifications and other elements in the image that are not microcalcifications). Additionally, we can obtain the NN with the best performance in terms of sensitivity (fraction of true positives, or well classified microcalcifications) and the NN with the best performance in terms of specificity (fraction of true negatives or well classified objects that are not microcalcifications), from the same last generations. 4) Detection of Microcalcification Clusters: During this stage, the microcalcification clusters are identified. The detection and posterior consideration of every microcalcification cluster in the images may produce better results in a subsequent classification process, as shown by Oporto-D´ıaz, Hern´andez-Cisneros and Terashima-Mar´ın [24]. Because of this, an algorithm for locating microcalcification cluster regions where the quantity of microcalcifications per cm2 (density) is higher, was developed. This algorithm keeps adding microcalcifications to their closest clusters at a reasonable distance until there are no more microcalcifications left or if the remaining ones are too distant for being considered as part of a cluster. Every detected cluster is then labeled. 5) Classification of Microcalcification Clusters into Benign and Malignant: This stage has the objective of classifying each cluster in one of two classes: benign or malignant. This information is provided by the MIAS database. From every microcalcification cluster detected in the mammograms in the previous stage, a cluster feature set is extracted. The feature set is constituted by 30 features: 14 related to the shape of the cluster, six related to the area of the microcalcifications included in the cluster and ten related to the contrast of the microcalcifications in the cluster. The same two feature selection procedures mentioned earlier are also performed in this stage. The first one attempts to delete the features that present high correlation with other features, and the second one uses a derivation of the forward sequential search algorithm, which is a sub-optimal search algorithm. Only three cluster features were selected for the classification process: minimum diameter, minimum radius and mean radius of the clusters. The minimum diameter is the maximum distance that can exist between two microcalcifications within a cluster in such a way that the line connecting them is perpendicular to the maximum diameter, defined as the maximum distance between two microcalcifications in a cluster. The minimum radius is the shortest of the radii

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connecting each microcalcification to the centroid of the cluster and the mean radius is the mean of these radii. In order to process microcalcification clusters and accurately classify them into benign or malignant, we decided again to use NNs as classifiers. Because of the difficulties of working with conventional feedforward NNs already mentioned in previous sections, we use GAs for evolving populations of NNs, in the same three different approaches we used before for classifying signals. The first approach uses GAs for searching the optimal set of weights of the NN. The number of inputs of the NN is n = 3, equal to the number of selected features. The hidden layer contains 2n + 1 = 7 neurons and the output layer consists of only one neuron. The second approach uses a GA for defining initial weight sets, from which a backpropagation training algorithm is started, hoping to reach an optimum quickly. Again, the number of inputs (n) for the NN is three, equal to the number of selected features. The hidden layer has 2n+ 1 = 7 neurons, and the output layer has only one neuron. The third approach uses a GA for evolving the architecture and other features of the NN. As in a previous stage, when signals were classified, each chromosome encodes the learning rate, a lower and upper limits for the weights before starting the backpropagation training, and the number of nodes of the hidden layer. The number of inputs is still three, and again there is only one output. For comparison, a conventional feedforward NN is used also. The architecture of this NN consists of three inputs, seven units in the hidden layer and one output. Every other parameter of the NN remains the same. The transfer function of every neuron in all the NNs mentioned in this stage is the sigmoid hyperbolic tangent function, and the error is measured with the mean square error function. At the end of this stage, we obtain three ready-to-use NNs, each one taken from the last generation of the GAs used in each of the approaches. These NNs have the best performances in terms of overall accuracy (fraction of well classified clusters). Additionally, we can obtain the NN with the best performance in terms of sensitivity (fraction of true positives or well classified malignant clusters) and the NN with the best performance in terms of specificity (fraction of true negatives or well classified benign clusters), from the same last generations.

positions of the microcalcifications clusters, marked in the additional data that comes with the database, were outside the boundaries of the breast. So, only 22 images were finally used for this study, and they were passed through the preprocessing stage first (application of a 3x3 median filter, binarization and trimming). In the second phase, six gaussian filters of sizes 5x5, 7x7, 9x9, 11x11, 13x13 and 15x15 were combined, two at a time, to construct 15 DoG filters that were applied sequentially. Each one of the 15 DoG filters was applied 51 times to the pre-processed images, varying the binarization threshold in the interval [0, 5] in increments of 0.1. The points obtained by applying each filter were added to the points obtained by the previous one, deleting the repeated points. The same procedure was repeated with the points obtained by the remaining DoG filters. These points passed through the three selection methods for selecting signals (potential microcalcification), according to region area, gray level and the gray gradient. The result was a list of 1,242,179 signals (potential microcalcifications) represented by their centroids. The additional data included with the MIAS database define, with centroids and radii, the areas in the mammograms where microcalcifications are located. It is supposed that signals within these areas are mainly microcalcifications, but there are many signals that lie outside the marked areas. With these data and the support of expert radiologists, all the signals located in these 22 mammograms were preclassified into microcalcification, and not-microcalcifications. From the 1,242,179 signals, only 4,612 (0.37%) were microcalcifications, and the remaining 1,237,567 (99.63%) were not. Because of this imbalanced distribution of elements in each class, an exploratory sampling was made. Several sampling with different proportions of each class were tested and finally we decided to use a sample of 10,000 signals, including 2,500 real microcalcifications in it (25%). After the 47 microcalcification features were extracted from each signal, the feature selection processes mentioned in the previous section reduced the relevant features to only three: absolute contrast, standard deviation of the gray level and the third moment of contour sequence. Finally, a transactional database was obtained, containing 10,000 signals (2500 of them being real microcalcifications randomly distributed) and three features describing each signal.

III. M AIN R ESULTS

In the third stage, a conventional feedforward NN and three evolutionary NNs were developed for the classification of signals into real microcalcifications. The feedforward NN had an architecture of three inputs, seven neurons in the hidden layer and one output. All the units had the sigmoid hyperbolic tangent function as the transfer function. The data (input and targets) were scaled in the range [-1, 1] and divided into ten non-overlapping splits, each one with 90% of the data for training and the remaining 10% for testing. A ten-fold crossvalidation trial was performed; that is, the NN was trained ten times, each time using a different split on the data and the means and

All the programs were written in MATLAB Version 7.0.0.19920 (R14), and executed on a PC with a single 2.1 GHz Intel Pentium IV processor with 1 Gb of memory. The following subsections explain the experimentation and results of every stage of the study. A. From Pre-processing to Feature Extraction As it was mentioned in the previous section, only 25 images from the MIAS database contain microcalcifications. Among these 25 images, 13 cases are diagnosed as malignant and 12 as benign. Three images were discarded because the

B. Classification of Signals into Microcalcifications

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standard deviations of the overall performance, sensitivity and specificity were reported. These results are shown in Table I on the row “BP”. For the three EANNs used to evolve signal classifiers, all of their GAs used a population of 50 individuals. We used simple GAs, with gray encoding, stochastic universal sampling selection, double-point crossover, fitness based reinsertion and a generational gap of 0.9. For all the GAs, the probability of crossover was 0.7 and the probability of mutation was 1/l, where l is the length of the chromosome. The initial population of each GA was always initialized uniformly at random. All the NNs involved in the EANNs are feedforward networks with one hidden layer. All neurons have biases with a constant input of 1.0. The NNs are fully connected, and the transfer functions of every unit is the sigmoid hyperbolic tangent function. The data (input and targets) were normalized to the interval [-1, 1]. For the targets, a value of “-1” means “not-microcalcification” and a value of “1” means “microcalcification”. When backpropagation was used, the training stopped after 20 epochs. For the first approach, where a GA was used to find the NN’s weights, the population consisted of 50 individuals, each one with a length of l = 720 bits and representing 36 weights (including biases) with a precision of 20 bits. There were two crossover points, and the mutation rate was 0.00139. The GA ran for 50 generations. The results of this approach are shown in Table I on the row “WEIGHTS”. In the second approach, where a backpropagation training algorithm is run using the weights represented by the individuals in the GA to initialize the NN, the population consisted of 50 individual also, each one with a length of l = 720 bits and representing 36 weights (including biases) with a precision of 20 bits. There were two crossover points, and the mutation rate was 0.00139 (1/l). In this case, each NN was briefly trained using 20 epochs of backpropagation, with a learning rate of 0.1. The GA ran for 50 generations. The results of this approach are shown in Table I on the row “WEIGHTS+BP”. Finally, in the third approach, where a GA was used to find the size of the hidden layer, the learning rate for the backpropagation algorithm and the range of initial weights before training, the population consisted of 50 individuals, each one with a length of l = 18 bits. The first four bits of the chromosome coded the learning rate in the range [0,1], the next five bits coded the lower value for the initial weights in the range [-10,0], the next five bits coded the upper value for the initial weights in the range [0,10] and the last four bits coded the number of neurons in the hidden layer, in the range [1,15] (if the value was 0, it was changed to 1). There was only one crossover point, and the mutation rate was 0.055555 (1/l). In this case, each NN was built according to the parameters coded in the chromosome, and trained briefly with 20 epochs of backpropagation, in order to favor the NNs that learned quickly. The results of this approach are shown also in Table I, on the row “PARAMETERS”. We performed several two-tailed Student’s t-tests at a level of significance of 5% in order to compare the mean of

TABLE I M EAN AND STANDARD DEVIATION OF THE SENSITIVITY, SPECIFICITY AND OVERALL ACCURACY OF SIMPLE BACKPROPAGATION AND DIFFERENT EVOLUTIONARY METHODS FOR THE CLASSIFICATION OF SIGNALS INTO REAL MICROCALCIFICATIONS .

Method BP WEIGHTS WEIGHTS+BP PARAMETERS

Sensitivity Std. Mean Dev. 75.68 0.044 72.44 0.027 75.81 0.021 73.19 0.177

Specificity Std. Mean Dev. 81.36 0.010 84.32 0.013 86.76 0.025 84.67 0.035

Overall Std. Mean Dev. 80.51 0.013 82.37 0.011 84.68 0.006 83.12 0.028

each method with the means of the other ones in terms of sensitivity, specificity and overall accuracy. In Table II, we show the results of the statistical comparison between the mean of the simple backpropagation method versus the means of the other evolutionary methods. We found that for specificity and overall accuracy, evolutionary methods are significantly better than the simple backpropagation method for the classification of individual microcalcifications. No difference was found in terms of sensitivity, except that simple backpropagation was significantly better than the method that evolves weights. We can notice too that, among the studied EANNs, the one that evolves a set of initial weights and is complemented with backpropagation training is the one that gives better results. In Table III, we show the results of the statistical comparison between this method and the rest of them. We found that in fact, again in terms of specificity and overall accuracy, the method of weight evolution complemented with backpropagation is significantly the best of the methods we studied. Nevertheless, in terms of sensitivity, this method is only significantly better than the method that evolves weights. TABLE II R ESULTS OF THE STATISTICAL COMPARISON BETWEEN THE MEAN OF THE SIMPLE BACKPROPAGATION METHOD VERSUS THE MEANS OF THE EVOLUTIONARY METHODS FOR THE CLASSIFICATION OF SIGNALS INTO REAL MICROCALCIFICATIONS .

Method WEIGHTS WEIGHTS+BP PARAMETERS

Significative Difference with BP Sensitivity Specificity Overall Yes Yes Yes No Yes Yes No Yes Yes

C. Microcalcification Clusters Detection and Classification The process of cluster detection and the subsequent feature extraction phase generates another transactional database, this time containing the information of every microcalcification cluster detected in the images. A total of 40 clusters were detected in the 22 mammograms from the MIAS database that were used in this study. According to MIAS additional data and the advice of expert radiologists, 10 clusters are benign and 30 are malignant. The number of

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TABLE III R ESULTS OF THE STATISTICAL COMPARISON BETWEEN THE MEAN OF

TABLE V R ESULTS OF THE STATISTICAL COMPARISON BETWEEN THE MEAN OF

THE WEIGHT EVOLUTION WITH BACKPROPAGATION METHOD VERSUS

THE SIMPLE BACKPROPAGATION METHOD VERSUS THE MEANS OF THE

THE MEANS OF THE OTHER METHODS FOR THE CLASSIFICATION OF

EVOLUTIONARY METHODS FOR THE CLASSIFICATION OF

SIGNALS INTO REAL MICROCALCIFICATIONS .

MICROCALCIFICATION CLUSTERS .

Method BP WEIGHTS PARAMETERS

Significative Difference with WEIGHTS+BP Sensitivity Specificity Overall No Yes Yes Yes Yes Yes No Yes Yes

Method WEIGHTS WEIGHTS+BP PARAMETERS

Significative Difference with BP Sensitivity Specificity Overall Yes Yes Yes Yes Yes Yes Yes No Yes

TABLE VI

features extracted from them is 30, but after the two feature selection processes already discussed in previous sections, the number of relevant features we considered relevant was three: minimum diameter, minimum radius and mean radius of the clusters. As in the stage of signal classification, a conventional feedforward NN and three evolutionary NNs were developed for the classification of signals into real microcalcifications. The four algorithms we use in this step are basically the same ones we used before, except that they receive as input the transactional database containing features about microcalcifications clusters instead of features about signals. Again, the means of the overall performance, sensitivity and specificity for each one of these four approaches are reported and shown in Table IV. TABLE IV M EAN AND STANDARD DEVIATION OF THE SENSITIVITY, SPECIFICITY AND OVERALL ACCURACY OF SIMPLE BACKPROPAGATION AND DIFFERENT EVOLUTIONARY METHODS FOR THE CLASSIFICATION OF MICROCALCIFICATION CLUSTERS .

Method BP WEIGHTS WEIGHTS+BP PARAMETERS

Sensitivity Std. Mean Dev. 55.97 0.072 72.00 0.059 89.34 0.035 63.90 0.163

Specificity Std. Mean Dev. 86.80 0.032 92.09 0.038 95.86 0.025 85.74 0.067

Overall Std. Mean Dev. 76.75 0.032 86.35 0.031 93.88 0.027 80.50 0.043

We also performed several two-tailed Student’s t-tests at a level of significance of 5% in order to compare the mean of each method for cluster classification with the means of the other ones in terms of sensitivity, specificity and overall accuracy. We can observe in Table V that the performance of evolutionary methods is significantly different and better than the performance of the simple backpropagation method, except in one case. Again, the method that evolves initial weights, complemented with backpropagation, is the one that gives the best results. In Table VI, we show the results of the statistical comparison between this method and the rest of them. IV. C ONCLUSIONS This paper presented a comparison of simple backpropagation training and three methods for combining GAs and

R ESULTS OF THE STATISTICAL COMPARISON BETWEEN THE MEAN OF THE WEIGHT EVOLUTION WITH BACKPROPAGATION METHOD VERSUS THE MEANS OF THE OTHER METHODS FOR THE CLASSIFICATION OF MICROCALCIFICATION CLUSTERS .

Method BP WEIGHTS PARAMETERS

Significative Difference with WEIGHTS+BP Sensitivity Specificity Overall Yes Yes Yes Yes Yes Yes Yes Yes Yes

NNs, applied to the classification of signals into real microcalcifications and microcalcification clusters into benign and malignant, on mammograms containing microcalcifications from the MIAS database. Our experimentation suggests that evolutionary methods are significantly better than the simple backpropagation method for the classification of individual microcalcifications, in terms of specificity and overall accuracy. No difference was found in terms of sensitivity, except that simple backpropagation was significantly better than the method that only evolves weights. In the case of the classification of microcalcification clusters, we observed that the performance of evolutionary methods is significantly better than the performance of the simple backpropagation method, except in one case. Again, the method that evolves initial weights, complemented with backpropagation, is the one that gives the best results. As future work, it would be useful to include and process other mammography databases, in order to have more examples and produce transactional feature databases more balanced and complete, and test also how different resolutions could affect system effectiveness. The size of the gaussian filters could be adapted depending on the size of the microcalcifications to be detected and the resolution of images. The correspondence between the spatial frequency of the image and the relation σ1 /σ2 has to be thoroughly studied. Different new features could be extracted from the microcalcifications in the images and tested also. In this study, simple GAs and NNs were used, and more sophisticated versions of these methods could produce better results. The use of real valued chromosomes, chromosomes with indirect representation (metaheuristics, NN construction rules, etc.), use of EANNs for feature selection, etc. are other approaches that could give different results. The inclusion of simple backpropagation training in the EANNs have

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consequences of longer computation times, so alternatives to backpropagation should be tested in order to reduce time costs. ACKNOWLEDGMENT This research was supported in part by the Instituto Tecnol´ogico y de Estudios Superiores de Monterrey (ITESM) under the Research Chair CAT-010 and the National Council of Science and Technology of Mexico (CONACYT) under grant 41515. R EFERENCES [1] M. A. Ganott, K. M. Harris, H. M. Klaman and T. L. Keeling, “Analysis of false-negative cancer cases identified with a mammography audit,” The Breast Journal, 5(3), pp. 166–175, 1999. [2] I. Anttinen, M. Pamilo, M. Soiva and M. Roiha, “Double reading of mammography screening films: one radiologist or two?,” Clin. Radiol., 48, pp. 414–421, 1993. [3] E. L. Thurfjell, K. A. Lernevall and A. A. S. Taube, “Benefit of independent double reading in a population-based mammography screening program,” Radiology, 191, pp. 241–244, 1994. [4] J. Dengler, S. Behrens and J. F. Desaga, “Segmentation of microcalcifications in mammograms,” IEEE Trans. Med. Imaging, 12(4), pp. 634–642, 1993. [5] E. M. Ochoa, Clustered Microcalcification Detection using Optimized Difference of Gaussians, Master Thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, 1996. [6] X. Yao, “Evolving artificial neural networks,” in Proceedings of the IEEE, 87(9), pp. 1423–1447, 1999. [7] K. Balakrishnan, and V. Honavar, “Evolutionary design of neural architectures. A preliminary taxonomy and guide to literature,” Technical Report CS TR 95-01, Department of Computer Sciences, Iowa State University, 1995. [8] D. B. Fogel, E. C. Wasson, E. M. Boughton and V. W. Porto, “A step toward computer-assisted mammography using evolutionary programming and neural networks,” Cancer Letters, Vol. 119, pp. 93–97, 1997. [9] D. B. Fogel, E. C. Wasson, E. M. Boughton, V. W. Porto and P. J. Angeline, “Linear and neural models for classifying breast cancer,” IEEE Trans. Medical Imaging, Vol. 17:3, pp. 485–488, 1998. [10] D. B. Fogel, E. C. Wasson, E. M. Boughton and V. W. Porto, “Evolving artificial neural networks for screening features from mammograms,” Artificial Intelligence in Medicine, Vol. 14, pp. 317–326, 1998. [11] J. Suckling, J. Parker, D. Dance, S. Astley, I. Hutt, C. Boggis, I. Ricketts, E. Stamatakis, N. Cerneaz, S. Kok, P. Taylor, D. Betal and J. Savage, “The Mammographic Images Analysis Society digital mammogram database,” Exerpta Medica International Congress Series, 1069, pp. 375–378, 1994. http://www.wiau.man.ac.uk/services/MIAS/MIASweb.html [12] M. Heath, K. Bowyer, D. Kopans, R. Moore and P. Kegelmeyer Jr., “The digital database for screening mammography,” In Proceedings of the 5th International Workshop on Digital Mammography, Toronto, ON, Canada, 2000. [13] R. Chandrasekhar, and Y. Attikiouzel, “Digitization regime as a cause for variation in algorithm performance across two mammogram databases,” Technical Report 99/05, Centre for Intelligent Information Processing Systems, Department of Electrical and Electronic Engineering, The University of Western Australia, 1999. [14] T. O. Gulsrud, “Analysis of mammographic microcalcifications using a computationally efficient filter bank,” Technical Report, Department of Electrical and Computer Engineering, Stavanger University College, 2001. [15] B.-W. Hong and M. Brady, “Segmentation of mammograms in topographic approach,” In IEE International Conference on Visual Information Engineering, Guildford, UK, 2003. [16] S. Li, T. Hara, Y. Hatanaka, H. Fujita, T. Endo and T. Iwase, “Performance evaluation of a CAD system for detecting masses on mammograms by using the MIAS database,” Medical Imaging and Information Science, 18(3), pp. 144–153, 2001. [17] S. Oporto-D´ıaz, Automatic Detection of Microcalcification Clusters in Digital Mammograms, Master Thesis, Tecnol´ogico de Monterrey, Campus Monterrey, Monterrey, Mexico, 2004.

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