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Davis Cope, Barbara Blakeslee, Mark McCourt; Structural theorems for simple cell receptive fields. Journal of Vision 2008;8(6):802. https://doi.org/10.1167/8.6.802.
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Theorists interested in developing mathematical descriptions and computational models of human spatial vision frequently employ Gabor filters as basis functions. Due to some undesirable features of the Gabor function, in particular the fact that it does not integrate to zero for all spatial phases of the carrier wave, a variety of competing mathematical formulations has been proposed, which include the difference-of-Gaussians, the Laplacian-of-a-Gaussian, the log-Gabor and the Cauchy function, to name a few. Here, we mathematically derive a class of functions based on three cardinal response properties of simple cells. Specifically, receptive field models must satisfy the following criteria: 1) Their response to a spatially homogeneous stimulus must be zero; 2) There must exist a direction (orientation) which produces zero response to sinusoidal grating stimuli; and 3) The response to sinusoidal grating stimuli must be a monotonic function of orientation, increasing from zero to a maximum as orientation changes from the zero response direction. The class of receptive field functions satisfying these criteria includes, as its most representative case, a novel variation of the Gabor filter - the balanced-Gabor - which integrates to zero for all spatial phases of the carrier wave. The balanced Gabor filter possesses all of the desirable features of the standard Gabor model of simple cells and, by virtue of integrating to zero, obviates its chief deficiency.
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