It is important that the motion model developed accounts for the known biology and physiology of the cortical motion pathway. Our model is based on a framework of spatial and temporal differential operators, is there evidence that the visual system has access to such operations?

*Spatial Derivatives of Gaussians
(Young, 1985)*

In the spatial domain the form of the cell receptive fields in early visual cortex shown in the figure above may be accurately modeled with blurred derivative of Gaussian functions (Young, 1985). As the order of differentiation increases these Gaussian functions become tuned to higher spatial frequencies giving a range of independent spatial channels as has been verified experimentally. Young also presents evidence that the distribution of zero-crossings of spatial receptive field shapes in monkey and cat are described better by Gaussian derivatives than by Gabor functions (Young, 1985). [Further details]

It is also interesting to note that all orders of the derivative receptive fields can be achieved by the hierarchical superposition of spatially offset circularity symmetric Gaussian receptive fields, similar to those recorded in the LGN (Young, 1985)

*Temporal Derivatives of causal Log
Gaussian (Johnston & Clifford,1995 )*

The psychophysical data points on the graph are from the work of Hess and Snowden. Hess and Snowden (1992) investigated the visual systems temporal processing using a masking paradigm. A probe (a grating reversing contrast at a particular temporal frequency) was set to be just above detection threshold. They then measured the contrast of a mask (narrow band filtered spatial noise reversing in contrast at one of a range of temporal frequencies) at which the probe could just be detected. The rational of the experiment is that if the probe and mask are detected by separate temporal channels the mask will not interfere with the detection of the probe. They found evidence for three temporal channels - a low pass channel, a band-pass channel peaking at around 10Hz and another peaking at around 18 Hz. It can be seen from the above graph that these three channels can be modeled by a process of temporal differentiation of a causal log Gaussian function (Johnston & Clifford,1995 )

This physiological and psychophysical data would seem to provide evidence that the visual system implements a filtering process using fuzzy derivatives, evidence which supports our model framework. In the spatial domain the fuzzy derivative model has only one parameter to fit, the variance of the underlying Gaussian. Gabor functions, often used in Fourier energy methods, require two fitting parameters, the spread of the Gaussian and the spatial frequency of the underlying sinusoid and therefore may have a greater opportunity to fit the experimental data. The form of the log Gaussian temporal filters is fixed with only two parameters, the individual curves being derived by simple differentiation of the zero order log Gaussian function.

**References and other papers discussing fuzzy derivatives in visual cortex
**

- Hess R. F. & Snowden R. J., (1992) Temporal properties of human visual filters: Number, shapes and spatial covariance, Vision Research. 32, 47-60
- Johnston, A., McOwan, P.W. & Buxton, H. (1992a). A computational model of the analysis of some first-order and second-order motion patterns by simple and complex cells. Proc. R. Soc. Lond. B. 250, 297-306.
- Johnston, A., P. W. McOwan & H. Buxton. (1992b) A biologically plausible scheme for measuring image velocity. J. Physiol. 452, 288.
- Johnston A. & Clifford C. W. G. (1995) A unified account of three apparent motion illusions Vision Research, 35, 1109-1123.
- Johnston A., McOwan P. W. & Benton C. P. (1995),Nonlinear interactions are in direction selective complex cells are predicted by a gradient motion model Investigative Ophthalmology and Visual Science (Supplement) 36 277
- Koenderink, J. J. & Van Doorn A. J. (1987) Representation of local geometry in the visual system. Biological Cybernetics 55 367-375
- Young R. (1985). The Gaussian derivative theory of spatial vision: analysis of cortical cell receptive field weighting profiles. General Motors Research Report GMR 4920.
- Young, R.A. & Lesperance, R.M. (1993) A physiological model of motion analysis for machine vision Technical Report General Motors Research Laboratories. GMR-7878. 1-76.