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Jiri Najemnik, Wilson Geisler; Optimal continuous-time control of eye movements during visual search. Journal of Vision 2008;8(6):106. doi: https://doi.org/10.1167/8.6.106.
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We derive an expression for the amount of information about target location obtained in an infinitely small time period during visual search for a target embedded in a background of dynamic white noise. We show that as the target detectabilities (d-primes) across the visual field approach zero, the information collected by the ideal searcher becomes monotonically related to the weighted sum across the current posterior probability distribution of the target's location (weights = squares of the d-primes). Hence to continuously control the flow of information optimally a searcher that can instantly move its eyes anywhere without cost should, at each instant in time, move its eyes to maximize the dot product of the current posterior distribution with the square of the d-prime map. Interestingly, with human-like d-prime maps, such a searcher sometimes makes extended fixations lasting up to several hundred ms in addition to many short fixations (dozens of ms). Human eye-movements are not cost-free, it takes time to initiate the movements and travel, and during travel visual information collection is interrupted. The human eye-movement controller is also uncertain about where the eyes will land because of random scatter around the intended landing point. When the optimal eye movement controller with human d-prime maps takes these costs of moving the eyes into account, it can be made to exhibit a distribution of fixation durations qualitatively similar to humans by adjusting a single free parameter that specifies the length of a look-ahead time over which the expected information gain (for a given considered eye movement) is computed. The look-ahead time that produces human-like fixations is consistent with a simple neural computation (one taking a few dozen ms). Apparently, large saccadic eye movements and long fixation durations may be rational even for an eye movement system that could move its eyes continuously.
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