Nonlinear analysis of spatial vision using first-and-second-order volterra transfer functions measurement

The harmonic input method of nonlinear system identification is modified to allow the Volterra series approach to be used for psychophysical investigation of various aspects of human pattern vision in the spatial frequency domain. While it is well known that only one modulation transfer function provides a complete characterization of a linear system, a number of multidimensional transfer functions are needed to identify a nonlinear system. We have shown, that so far as the contrast sensitivity to sine-wave gratings may be used for an empirical estimate of the first-order modulation transfer function of the human visual system, the contrast sensitivity to difference harmonics may be used as an empirical estimate of the second-order modulation transfer function.

A difference harmonic arises from a mixture of two sine-wave gratings resulting from the nonlinearity of the visual system. Difference harmonic, experienced as some periodic beatlike structure, may still be observed if frequencies of the component gratings are higher than the maximum visual acuity.

The visibility of the low-frequency beatlike pattern produced by pairs of sine-wave gratings, which themselves are of spatial frequencies too high to be resolved, could be accounted for either by a difference frequency distortion product (Burton, 1973) or by a special beat detector (Derrington & Badcock, 1985). We found that increasing the contrast of one component grating may be compensated for by reducing the contrast of the other component grating, the beatlike pattern being at threshold. This is exactly what would be expected if the beatlike pattern is detected because of the difference harmonics produced by nonlinearity of the visual system.

We have determined contrast thresholds for the difference harmonics which occur between two unresolved different spatial frequencies. The contrast sensitivity function for difference harmonics was found to have a marked similarity both in the shape and position of peak sensitivity to the contrast sensitivity function for single sine-wave gratings. Another important characteristic of the contrast sensitivity function for difference harmonics is that it depends only on the frequency difference, Δf = f₁ ? f₂, rather than on the value of either f₁ or f₂.

All this indicates that a difference harmonic arises from local nonlinearities in the visual system. More specifically, the visual system may be represented as a cascade system, composed of a linear system with transfer function O(f) followed by a nonlinear element, r(·), without spatial spread in cascade with another linear system with transfer function P(f). The nth order transfer function of this cascade system, H_n(f₁,...,f_n) can be expressed in the following way: H_n(f₁,...,f_n) = a_nO(f₁)P(f₁ + ... + f_n) where a_n is the nth coefficient in the Taylor series expansion for the nonlinear function r(·). It follows from this that the measurement of the first- and second-order transfer functions is sufficient to determine O(f) and P(f). We have derived the estimates of cO(f) and P(f) from contrast sensitivity functions for single sine-wave gratings and difference harmonics by the least squares method.