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We develop a provably efficient importance sampling scheme that estimates exit probabilities of solutions to small-noise stochastic reaction-diffusion equations from scaled neighborhoods of a stable equilibrium. The moderate deviation scaling allows for a local approximation of the nonlinear dynamics by their linearized version. In addition, we identify a finite-dimensional subspace where exits take place with high probability. Using stochastic control and variational methods we show that our scheme performs well both in the zero noise limit and pre-asymptotically. Simulation studies for stochastically perturbed bistable dynamics illustrate the theoretical results.

The goal of this paper is to study the Moderate Deviation Principle (MDP) for a system of stochastic reaction-diffusion equations with a time-scale separation in slow and fast components and small noise in the slow component. Based on weak convergence methods in infinite dimensions and related stochastic control arguments, we obtain an exact form for the moderate deviations rate function in different regimes as the small noise and time-scale separation parameters vanish. Many issues that come up due to the infinite dimensionality of the problem are completely absent in their finite-dimensional counterpart. In comparison to corresponding Large Deviation Principles, the moderate deviation scaling necessitates a more delicate approach to establishing tightness and properly identifying the limiting behavior of the underlying controlled problem. The latter involves regularity properties of a solution of an associated elliptic Kolmogorov equation on Hilbert space along with a finite-dimensional approximation argument.

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Noise due to the sensor and the electronics of a camera is an undesirable issue in any machine vision application. Such noise tends to corrupt images and to obstruct any further analysis. An algorithm to detect and cancel such noise, using statistical methods, is presented in this paper. The proposed algorithm is an adaptive mean filter, which filters out image regions that are found to be noise corrupted. The efficiency of the proposed filter was examined both qualitatively and quantitatively, by software simulation in several noisy conditions. The main advantage of the filter in hand is that it is appropriate for hardware implementation and can be easily incorporated to smart cameras. The hardware implementation of the filter is also presented in this paper. This implementation aims at time critical applications such as machine vision, inspection and visual surveillance.