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ter Haar Romeny, FEV Scale-time

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ter Haar Romeny, FEV Time measurements can essentially be processed in two ways: as pre-recorded frames or instances, or real-time. Humans are real-time, they continuously perform a temporal analysis with their senses. The scale-space treatment of these two categories will turn out to be essentially different. The prerecorded sequences are treated as regularly sampled points, leading to a regular causal time-scale-space, with temporal apertures and Gaussian temporal differential operators.

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ter Haar Romeny, FEV We can never reach the present moment: A measurement aperture (operator) with infinitesimally short duration will be needed. In the real-time measurement and analysis of temporal data we have a serious problem: the time axis is only a half axis: the past. There is a sharp and unavoidable boundary on the time axis: the present moment.

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ter Haar Romeny, FEV Koenderink proposed to reparametrize the time axis in a logarithmic Fashion, thus maintaining causality. The log time axis is now a full axis, allowing diffusion, i.e. causal operators. present

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ter Haar Romeny, FEV Dimensionless elapsed time present

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ter Haar Romeny, FEV The receptive fields become skewed: time

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ter Haar Romeny, FEV It is interesting to study if the temporal receptive field (RF) sensitivity profiles match the predicted skewness for the different differential order. Many neurophysiology labs have recorded time sequences of neural RF’s, e.g. by the reverse correlation technique.

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ter Haar Romeny, FEV DeAngelis, Ohzawa, Freeman, 1991 Cat V1 simple cell, 5 ms interframe interval

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ter Haar Romeny, FEV De Valois, Cottaris, Mahon, Elfar and Wilson, 2000 space time A clear skewness is observed in the time direction.

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ter Haar Romeny, FEV A time-scale model of skewed receptive field profiles on a logarithmic axis.

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ter Haar Romeny, FEV From group theory: Any half-axis should be parameterized logarithmically to enable linear addition of a multiplication property for the measurement convolution. Examples: The scale axis (Powers of Ten: ‘orders of magnitude’); The intensity axis (retina log mapping, gamma-corrections); The time axis; The sound pressure axis (decibels); Etc.

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