Weighted averaging of a set of noisy images for maximum signal-to-noise ratio

The problem of estimating a signal from a weighted average of N registered noisy observations is considered. A set of optimal weighting coefficients is determined by maximizing a signal-to-noise ratio criterion. This solution can be computed by first standardizing each observation with respect to its first and second moments and then evaluating the first eigenvector of the corresponding N× N inner-product matrix. The resulting average is shown to be proportional to the first basis vector of the Karhunen-Loeve transform provided that the data has been standardized in an appropriate fashion. The low sensitivity of this approach to the presence of outliers is illustrated by using real electron micrographs of ostensibly identical virus particles