Image restoration using a modified Hopfield network

In this paper a modified Hopfield neural network model for regularized image restoration is presented. The proposed network allows negative autoconnections for each neuron. A set of algorithms using the proposed neural network model is presented, with various updating modes: i) sequential updates, ii) n-simultaneous updates, and iii) partially asynchronous updates. The sequential algorithm is shown to converge to a local minimum of the energy function after a finite number of iterations. This local minimum is defined in terms of a unit distance neighborhood in the discrete intensity value space. Since an algorithm which updates all n neurons simultaneously is not guaranteed to converge, a modified algorithm is presented, which is called a greedy algorithm. Although the greedy algorithm is not guaranteed to converge to a local-minimum, the 11 norm of the residual at a fixed point is bounded. It is also shown that the upper bound on the 11 norm of the residual can be made arbitrarily small by using an appropriate step size. Finally, a partially asynchronous algorithm is presented, which allows a neuron to have a bounded time delay to communicate with other neurons. Such an algorithm can eliminate the synchronization overhead of synchronous algorithms. It is shown that the 11 norm of the residual at the fixed point of this algorithm increases as the upper bound on the delay increases. Experimental results are shown testing and comparing the proposed algorithms.