Kernels for Morphological Associative Memories

1 Abstract The ability of human beings to retrieve information on the basis of associated cues continues to elicit great interest among researchers. Investigations of how the brain is capable to make such associations from partial information have led to a variety of theoretical neu-ral network models that act as associative memories. Recently, several researchers have had signiicant success in retrieving complete stored patterns from noisy or incomplete input pattern keys by using morphological associative memories. For certain types of noise in the input patterns, this new model of artiicial asso-ciative memories can be successfully applied following a direct approach. If the input patterns contain both dilative and erosive noise, an indirect approach using kernel vectors is recommended, however the problem of how to select these kernel vectors has not yet been solved. In this paper, we provide suucient conditions for kernel vectors which connrm the intuitive notion of kernel vectors as sparse representations of the input vectors. In addition, we deduce exact statements on the amount of noise which is permissible for perfect recall. 2 Introduction The concept of morphological neural networks grew out of the theory of image algebra. A subalgebra of image algebra includes the mathematical formulations of currently popular neural network models 11, 9]. G.X. Rit-ter and J.L. Davidson were the rst to formulate useful morphological neural networks 10, 4]. Since then, only a few papers involving morphological neural networks have appeared. Davidson employed morphological neu-ral networks in order to solve template identiication and target classiication problems 3, 2]. Suarez-Araujo applied morphological neural networks to compute ho-mothetic auditory and visual invariances 15] Another interesting network consisting of a morphological net and a classical feedforward network used for feature extraction and classiication was designed by Won, Gader, and Cooeld 16, 17]. All of these researchers devised multilayer morphological neural networks for very specialized applications. A more comprehensive and rigorous basis for computing with morphological neural networks appeared in 12]. The properties of morphological neural networks diier drastically from those of traditional neural network models. These diierences are due to the fact that traditional neural network operations consist of linear operations followed by an application of nonlinear activation functions whereas in morphological neural computing the next state of a neuron or in performing the next layer neural network computation involves the nonlinear operation of adding neural values and their synaptic strengths followed by forming the maximum of the …