Seeing Beyond the Nyquist Limit

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Abstract
In many biological systems the primary transduction of sensory stimuli occurs in a regular array of receptors. Because of this discrete sampling it is usually assumed that the organism has no knowledge of signals beyond the Nyquist frequency. In fact, higher frequency signals are expected to mask the available lower frequency information as a result of aliasing. It has been suggested that these considerations are important in understanding, for example, the design of the receptor lattice in the mammalian fovea. We show that if the organism has knowledge of the probability distribution from which the signals are drawn, outputs from a discrete receptor array can be used to estimate signals beyond the Nyquist limit. In effect, a priori knowledge can be used to de-alias the image, and the estimated signal above the Nyquist cutoff is in fact coherent with the real signal at these high frequencies. We address initially the problem of stimulus reconstruction from a noisy receptor array responding to a Gaussian stimulus ensemble. In this case, the best reconstruction strategy is a simple linear transformation. In the more interesting (and natural) case of nongaussian stimuli, optimal reconstruction requires nonlinear operations, but the higher order correlations in the stimulus ensemble can be used to improve the estimate of super-Nyquist signals.