Observations on morphological associative memories and the kernel method

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 neural network models that act as associative memories. Several researchers have had signiicant success in retrieving complete stored patterns from noisy or incomplete input pattern keys by using morphological associative memories. Thus far morphological associative memories have been employed in two diierent ways: a direct approach which is suitable for input patterns containing either dilative or erosive noise and an indirect one for arbitrarily corrupted input patterns which is based on kernel vectors. In a recent paper 22], we suggested how to select these kernel vectors and we deduced exact statements on the amount of noise which is permissible for perfect recall. In this paper, we establish the proofs for all our claims made about the choice of kernel vectors and perfect recall in kernel method applications. Moreover, we provide arguments for the success of both approaches beyond the experimental results presented up to this point. The concept of morphological neural networks grew out of the theory of image algebra developed by G.X. Ritter 20]. A sub-algebra of image algebra can be viewed as the mathematical background not only for morphological image processing but also for morphological neural networks 16, 6]. A number of researchers devised morphological neural networks for very specialized applications. J.L. Davidson employed morphological neural networks in order to solve template identiication and target classiication problems 5, 4]. Suarez-Araujo applied morphological neural networks to compute homothetic auditory and visual invari-ances 21] 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 24, 25]. 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 results. A fairly comprehensive and rigorous basis for computing with morphological neural networks appeared in 17]. …