Modification of the mean-square error principle to double the convergence speed of a special case of Hopfield neural network used to segment pathological liver color images

Segmentation is an important step in most applications that use medical image data. For example, segmentation is a prerequisite for quantification of morphological disease manifestations and for radiation treatment planning [1, 2], for construction of anatomical models [3], for definitions of flight paths in virtual endoscopies [4], for content-based retrieval by structure [5], and for volume visualization of individual objects [2].

A Number of algorithms based on approaches such as histogram analysis, regional growth, edge detection and pixel classification have been proposed in other articles of medical image segmentation. In recent years, Artificial Neural Networks (ANNs) have been proposed as an attractive alternative solution to a number of pattern recognition problems. In our previous works [6], we have explored the potential of a Special Case of Hopfield Neural Network (SCHNN) in segmenting cerebral images obtained using the Magnetic Resonance Imaging (MRI) technique.

Hopfield network for the optimization applications consists of many interconnected neuron elements. The network minimizes an energy function of the form:

where N is the number of neurons, V _k is the output of the k ^th neuron, I _k is the bias term, and T _kl is the interconnection weight between the k ^th and l ^th neurons. The energy function used in the segmentation problem is slightly different from the one defined by Hopfield and the arguments are given in [7].

The results that have been obtained in [6] were preferable to those obtained using Boltzmann Machine (BM) and the conventional ISODATA clustering technique. Also, in [8] we have shown that SCHNN is also able to make crisp segmentation of pathological liver colour images. However, during our study attempt to improve the segmentation process, we found that SCHNN segmentation results depend strongly on some parameters in the energy function formulating the classification problem. A summery of this study follows.

The segmentation problem of an image of N pixels is formulated in [8] as a partition of the N pixels among M classes, such that the assignment of the pixels minimizes a criterion function. The SCHNN classifier structure consists of a grid of N × M neurons with each row representing a pixel and each column representing a cluster. The network classifies the image of N pixels of P features among M classes, in a way that the assignment of the pixels minimizes the following criterion function:

where R _kl is the Mahalanobis distance measure between the k ^th pixel and the centroid of class l, R _kl is also equivalent to the error committed when a pixel k is assigned to a class l. The index n in is the power or weight of the considered error in the energy function of the segmentation problem, and V _kl is the output of the kl ^th neuron. N _kl is a N × M vector of independent high frequency white noise source used to avoid the network being trapped in early local minimums. The term c(t) is a parameter controlling the magnitude of noise which is selected in a way to provide zero as the network reaches convergence. The minimization is achieved by using SCHNN and by solving the motion equations satisfying:

where U _kl is the input of the k ^th neuron, and μ(t) is a scalar positive function of time, used as heuristically motivated stopping criterion of SCHNN, and is defined as in [6] by:

β(t) = t(T _s - t)     (4)

where t is the iteration step, and T _s is the pre-specified convergence time of the network which has been found to be 120 iterations [6]. The network classifies the feature space, without teacher, based on the compactness of each cluster calculated using Mahalanobis distance measure between the k ^th pixel and the centroid of class l given by:

where X _k is the P-dimensional feature vector of the k ^th pixel (here P = 3 with respect to the RGB color space components), is the P-dimensional centroid vector of class l, and Σ _l is the covariance matrix of class l. The segmentation algorithm is described as follows [8].

Step 1 Initialize the input of the neurons to random values.

Step 2 Apply the following input-output relation, establishing the assignment of each pixel to only and only one class.

Step 3 Compute the centroid and the covariance matrix Σ _l of each class l as follows:

where n _l is the number of pixels in class l, and the covariance matrix is then normalized by dividing each of its elements by .

Step 4 Update the inputs of each neuron by solving the set of differential equations in (2) using Eulers approximation:

Step 5 if t <T _s , repeats from Step 2, else terminated.

For this study, a total of 20 liver tissue sections were provided by the pathological division of National Cancer Center in Tokyo. These sections were taken using needle biopsy, stained with hematoxylin and then magnified with an optical microscope. Figure 1 shows a true RGB color image of liver tissue of 768 × 512 pixels. We have used the above described SCHNN classifier with the image components in the R.G.B color space. The number of classes is fixed to five based on medical information. These classes are the contour of the image, the cell's nuclei, the cytoplasm, the fibrous tissues, and the class of both blood sinus and fat cells.

Figure 2 shows the curves of SCHNN energy function during the segmentation of the sample shown in Figure 1 with T _s values between 30 and 120 iterations. Similar curves were obtained for the rest of the images of the dataset. As it is illustrated in Figure 2. The curve corresponding to T _s = 120 iterations gives the optimal solution, the same as it is with MRI data [6].

In order to study the effect of the weight of the Mahalanobis distance R _kl in the cost function (2), we have provided a simple modification to the above algorithm as follows:

Step 1 Use the same random initialization N × M matrix, as input of the neurons, when minimizing the energy function (1) with different error's weight n.

This condition is added to the algorithm in order to make sure that the random field does not have any effect on the generated results.

Step 2 trough Step 5 remain the same.

We analyzed the effect of considering the mean-square error in formulating the segmentation problem of multidimensional medical images. We have shown, empirically, that considering an integer power equal to six, of the error in the energy function of the problem, helped SCHNN to converge twice as fast as the same optimal solution obtained with the mean-square error algorithm. This result is promising to make our segmentation method useful for a Computer Aided Diagnosis (CAD) system for liver cancer and the like. In our future work, we will study deeply the effect of the random initialization and its effect on the segmentation result and on the SCHNN classifier.