2.1 Image Selection

The images used to train and test this methodology were extracted from the mini-mammographic database provided by the Mammographic Image Analysis Society, MIAS [¹⁷]. Each mammogram from the database is 1024×1024 pixels and with a spatial resolution of 200 micron pixel edge. These mammograms have been reviewed by an expert radiologist and all the abnormalities have been identified and classified. The place where these abnormalities such as Microcalcifications, have been located is known as, Region of Interest (ROI). The ROI size in this work is 256×256 pixels.

2.2 ROI Enhancemen

The difficulty for the detection of Microcalcifications depends on some factors, such as, size, shape and distribution with respect to their morphology. On the other hand the Microcalcifications are often located in a non-homogeneous background and due to their low contrast with the background, its intensity may be similar to noise or other structures [⁹-¹⁰]. In this paper, the goal of image enhancement is to improve the contrast between the Microcalcifications clusters and background, for achieving this task morphological operations based on Coordinate Logic Filters (CLF) are implemented.

Mathematical Morphology is a discipline in the field of image processing which involves an analysis of the structure of images. The geometrical structure of image is determined by locally comparing it with a predefined elementary set called structuring element. Image processing using morphological transformations is a process of information removal based on size and shape, in this process irrelevant image content is eliminated selectively, thus the essential image features can be enhanced. Morphological operations are based on the relationships between the two sets: an input image, G, and a processing operator, the structuring element, SE, which is usually much smaller than the input image. By selecting the shape and size of structuring element, different results may be obtained in the output image. The fundamental morphological operations are: erosion and dilation.

CLF were proposed in [¹¹], and constitute a class of non-linear digital filters that are based on the application of Coordinate Logic Operations (CLO) to a single image as dictated by a SE, where CLO are the basic logic operations (NOT, AND, OR, and XOR, and their combinations). CLF are very efficient in digital signal processing applications, such as noise removal, magnification, skeletonization, coding, edge detection, feature extraction, fractal modelling and can execute the morphological operations (erosion, dilation, opening and closing) and the successive filtering and managing of the residues. The execution of CLO (AND, OR) in CLF is performed on the binary values of the image pixels and are analogous to the execution of (MIN, MAX) in the morphological filters. That makes the CLF have analogous operability with the corresponding morphological ones. Moreover, CLF satisfy all the corresponding morphological properties except the increasing property. The CLF coincide with morphological filters in the case of binary images. They could also coincide with gray-scale morphological filters provided that images are quantized and mapped on a specific set of decimal values [⁵]. Given a gray level image G, the decomposition in a set of binary images S_k,k=0,1,…,n-1, according to the decomposition of the (i,j) pixel, is denoted by:

where Sk(i,j),k=0,1,…,n−1 are binary components of the decimal pixel values g(i,j),i=1,2,…,M,j=1,2,…,N, according to equation 1, the Coordinate Logic Dilation (CLD, ⊕) and Coordinate Logic Erosion (CLE, ⊖) of the image G by the structuring element SE, are defined by following equations:

The contrast can be defined as the difference in intensity between an image structure and its background. By combining morphological operations, several image processing tasks can be performed, but in this work morphological operations are used to achieve contrast enhancement. In [¹²], the contrast enhancement technique using mathematical morphology is called morphological contrast enhancement. Morphological contrast enhancement is based on morphological operations known as top-hat and bottom-hat transforms. A Top-Hat is a residual filter which preserves those features in an image that can fit inside the structuring element and removes those that cannot in other words the Top-Hat transform is used to segment objects that differ in brightness from the surrounding background in images with uneven background intensity. The Top-Hat transform is defined by the following equation:

where G is the input image, G_T is the transformed image, SE is the structuring element, ⊖ represents morphological erosion operation, ⊕ represents morphological dilation operation and - image subtraction operation. [(G⊖SE)⊕SE] is also known as the morphological opening operation.

2.3 Image Segmentation by Clustering Algorithms

Image segmentation is an important task in the field of image processing and computer vision and involves identifying objects or regions with the same features in an image. The purpose of image segmentation is to subdivide an image into non-overlapping, constituent regions which are homogeneous with respect to some features such as gray level intensity or texture. The level to which the subdivision is carried out depends on the problem being solved [¹³]. In the field of medical imaging, segmentation plays an important role because it facilitates the delineation of anatomical structures and other regions of interest. For the specific case as Microcalcifications detection, several works based on image segmentation by means of clustering algorithms have been proposed [¹⁴-¹⁶]. In this work, for the image segmentation stage, two techniques based on partitional clustering algorithms are used. The aim of this stage is to segment the ROI images to make easier the Microcalcifications detection. The algorithms used in this stage are k-means and SOM.

k-means is one of the simplest unsupervised learning algorithms that solve the well known clustering problem. The procedure follows a simple and easy way to classify a given data set Z in a d-dimensional space, through a certain number of clusters (assume k clusters) fixed a priori. The main idea is to define k prototypes, one for each cluster. The next step is to take each point belonging to a given Z and associate it to the nearest prototype. When no point is pending, the first step is completed and an early group is done. At this point is necessary to re-calculate k new prototypes as barycenters of the clusters resulting from the previous step. Then to obtain these k new prototypes, a new binding has to be done between the same data set points and the nearest new prototype. A loop has been generated. As a result of this loop we may notice that the k prototypes change their location step by step until no more changes are done. In other words, prototypes do not move any more. Finally, this algorithm aims at minimizing an objective function (5), in this case a squared error function:

where ∥Zi(j)−vj∥2 is chosen distance measure between a data point zi(j) and the cluster vj is an indicator of the distance of the data points from their cluster prototypes. SOM neural networks, introduced by [¹⁰], are simple analogues to the brain’s way to organize information in a logical manner. The main purpose of this neural information processing is the transformation of a feature vector of arbitrary dimension drawn from the given feature space into simplified generally two-dimensional discrete maps. This type of neural network utilizes an unsupervised learning method, known as competitive learning, and is useful for analyzing data with unknown relationships. The basic SOM Neural Network consists of an input layer, and an output (Kohonen) layer which is fully connected with the input layer by the adjusted weights (prototype vectors). The number of units in the input layer corresponds to the dimension of the data. The number of units in the output layer is the number of reference vectors in the data space. There are several steps in the application of the algorithm. These are competition and learning, to get the winner in the process. In the training (learning) phase, the SOM forms an elastic net that folds onto the “cloud” formed by the input data. Similar input vectors should be mapped close together on nearby neurons, and group them into clusters. If a single neuron in the Kohonen layer is excited by some stimulus, neurons in the surrounding area are also excited. That means for the given task of interpreting multidimensional image data, each feature vector † x, which is presented to the four neurons of the input layer, typically causes a localized region of active neurons against the quiet background in the Kohonen layer. The degree of lateral interaction between a stimulated neuron j (x) and neighbouring neurons is usually described by a Gaussian function:

where d is the lateral neuron distance in the Kohonen layer, σ is the effective width that changes during the learning process, and h is the activity of the neighbouring neurons. Feature vectors occur in the Kohonen layer in the same topological order as they are presented by the metric (similarity) relations in the original feature space, while performing a dimensionality reduction of the original feature space. Before the self-organizing procedure begins, the link values, called weights. w→k connects the n input layer neurons to the l neurons in the Kohonen layer. n is the dimension of the input space (Microcalcifications feature space), and l is the number of all Kohonen neurons with k = 1,2, ..., m, ..., l.

The neurons of the Kohonen layer compete to see which neuron will be stimulated by the feature vector x→. The weights w→k are used to determine only one stimulated neuron in the Kohonen layer after the winner-takes-all principle. This principle can be summarized as follows: for each x→, the Kohonen neurons compute their respective values of a discriminant function (i.e., Euclidean distance ∥x→j−w→k∥). These values are used to define the winner neuron. That means the network determines the index j of that neuron, whose weight w→k is the closest to vector x→i by:

The neighbourhood function or Gaussian function determines how much the neighbouring neurons become modified. Neurons within the winners neighbourhood participate in the learning process. During the self-organizing process, the neighbourhood size decreases until its size σ is zero. In this case, only the winning neuron is modified each time an input vector is presented to the network. The learning rate η − the amount each weight can be modified − decreases during the learning as well. Once the SOM algorithm has converged, two-dimensional feature maps of Kohonen neurons display the following important statistical characteristics of the represented feature space [¹⁰].

2.4 Classification of Microcalcifications by AMMLPs

Artificial Neural Networks (ANNs) are used in a wide variety of data processing applications where real-time data analysis and information extraction are required. One advantage of the ARTIFICIAL NEURAL NETWORKS approach is that most of the intense computation takes place during the training process. Once ANNs are trained for a particular task, their operation is relatively fast and unknown samples can be rapidly identified in the field.

In this work, Artificial Metaplasticity (AMP) is applied to the learning algorithm of a Multi-Layer Perceptron. Metaplasticity is a biological concept related to the way information is stored by synapses in the brain. Artificial metaplasticity was recently proposed in [¹⁷], where it is modelled by giving more importance to less frequent patterns in the training process. One advantage of the use of AMP is that it usually results in a more efficient training because a small set of training patterns is usually required. This approach is in fact equivalent to the application of importance sampling to artificial neural networks training [¹⁸].

In the classic backpropagation algorithm the learning rate is usually fixed for all input patterns meaning that it gives the same importance to all patterns. AMP can be included in the training process by simply introducing a weighting function 1/fX*(x) in the weight update equation of the learning algorithm [¹⁹]. As a matter of fact, the learning rate of the backpropagation algorithm is replaced by a variable learning rate η(x)=η/fX*(x) which is a function of the input vector x:

Ideally, the function fX*(x) must be a proper approximation of the probability density function of the input patterns. So, less frequent patterns are given a larger learning rate than more frequent ones do. In practice, a suitable function must be chosen in order to improve learning. Neural networks are able to aproximate Bayesian posterior class probabilities provided there is sufficient training data and the network has one output for each class [²⁰-²¹]. This implies that the posterior probabilities obtained by the neural network can be used. However, suitable posterior probabilities are only available after proper training of the neural network. In this paper, the following function is used:

where N is the dimension of the input vector X (for the second hidden layer, X is substituted by the output vector of the first hidden layer, and so on) and A,B∈ℝ+ are empirically determined parameters. This function corresponds to the assumption that the input patterns are normally distributed.