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A. J. Ahumada, Jr.; Motion Compensation in Models of Cortical Cell Processing. Invest. Ophthalmol. Vis. Sci. 2007;48(13):5890.
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The Freeman and Ohzawa (1990) stereo disparity model begins with pairs of linear Gabor receptive fields in quadrature phase. They feed complex cells that sum the squared responses. Qian and Anderson (1997) use this model to compute stereo disparity and motion energy. Here it is used to reconstruct signals that would have been smeared by small unpredictable motions such as micro-saccades. For image reconstruction, we need to compute an amplitude and a phase for each quadrature pair.
As in those models, we compute the phase from the linear outputs before they are squared. However, this calculation is not done for the entire signal period, just the most recent time period, so that the linear signal is not blurred away. The amplitude is computed from the squared and summed outputs. They can be integrated over a longer period of time because the motion does not smear them away. The first question investigated was how much motion spread can be tolerated by the linear phase calculation. The calculations were done in one dimension. The input signal was an impulse in space and constant in time. Motion was modeled as a Gaussian blurring function applied to the linear receptive fields.
For linear summation, the signal energy falls 3 dB when the blur standard deviation is one eighth of the receptive field spatial period. When the blur standard deviation increases to one fourth the spatial period, the energy falls 10 dB. When the blur standard deviation is half the spatial period, the blurred signal energy is down 35 dB. Thus channels for which the blur standard deviation is more than a fourth of the channel spatial period can benefit from estimating the phase using a shorter time constant.
Many detection models arbitrarily introduce a nonlinear amplitude response to fit the results even though a linear detector should perform better. Perhaps this nonlinearity is introduced to reduce motion blur. The fast response of cortical neurons despite the slow nature of perception may be needed to reduce the linear blurring. Recent time phase estimation should result in metacontrast-like illusions.
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