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One of the basic problems in pattern recognition is the detection of a pattern in noise. This problem becomes particularly difficult if the pattern varies in position and size. A system necessary to achieve this result can be modeled in a number of different ways. One currently popular approach is to use a neural network.(1,2) The advantage to using a neural network is that once the basic structure is assumed the characteristics of the network, described by it's weights, can then be learned. The learning or training process involves developing a training set of known inputs and outputs for the system and adapting the internal weights of the network so that the inputs will yield the desired outputs. The weights are adjusted to minimize the error, according to some criteria, between the actual outputs and the desired outputs. Most neural networks are composed of first order terms, that is, z = f{ w0 + wij xj } where xj are the inputs, z are the first level (or hidden) outputs, w are weight terms and the functional relationship is normally a sigmoid function for inputs between zero and one. Usually, there are at least two levels of this type. In other words, the output, yi, would be given by yk = f{ u0 + uik Zj } where y are the final outputs, u are the weights and the other terms are as before. This type of network is trained using a back-propagation technique. Neural networks offer hope in the possible solution of detecting an object in noise by proper training of the network to recognize the characteristics of the object and ignoring the noise. Unfortunately, most neural networks cannot be trained to detect an object that appears in different positions. In other words, most neural networks are not translationally invariant. However, some special higher order neural networks have been shown to posses translational invariance
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