A novel regularization technique is proposed for the restoration of signals distorted by a stochastic, shift-variant impulse response (blur) function and additive noise. The generic ill-posed inverse problem is formulated as a constrained optimization problem wherein a new cost function, termed stochastic constrained restoration error energy, is minimized. For signals that are quantized in amplitude and sampled in time or space, an artificial neural network is developed and implemented for computational efficiency. By matching the cost function with the energy function of the neural network, the interconnection strengths and bias inputs of the neural network are related to the degraded signal, blur statistics, and constraint parameters. The solution that minimizes the energy function of the neural network is then obtained by the simulated annealing algorithm. Simulation results are also provided to demonstrate the effectiveness of the proposed approach.