Model fitting

Two of the models tested in Chapter 6 are fitted to the data here. These are the noisy energy model (NE) and the noisy energy model with the flat template (FTNE). The difference between the two is that in the NE model the observers use a template matched to the expected signal exactly (including the attenuation with eccentricity), whereas in the FTNE model the template gives an equal weighting to the entire stimulus (a flat template). These models were chosen because the versions of these models with the template based on the stimulus extent (ENE and FENE) were the best fitting models tested in the previous chapter (see Section 6.5), and in this study the predictions from the NE and FTNE models are identical to those that would be made by the ENE and FENE models (the templates in each model are the same because unlike the “Swiss cheese” stimuli tested in the previous chapter, the tiger-tail stimuli do not have “holes” in them).

The NE and FTNE models are compared here in order to demonstrate the effect of having a matched template versus a flat template, and to evaluate the prediction made by each model against the data from this study. Although the distinction between the two models is subtle, they carry quite different theoretical implications. The matched template (NE) model involves an “ideal” combination of signal and noise over area by an observer that is aware both of the expected stimulus and of how the visual field inhomogeneity in contrast sensitivity will affect that stimulus. The flat template (FTNE) model, on the other hand, can represent an observer who simply ignores input from regions of the display that will not contain signal. Formally, both models implemented here assume the observer uses a template that follows the sinusoidal luminance modulation of the stimulus (rather than simply matching a template to the stimulus envelope), however this feature makes very little difference to the summation prediction as it is the effect of adding cycles to the stimulus that is under investigation here rather than the detail of what happens within a single cycle (Meese & Summers, 2012).

In the models tested here, the stimulus image is first multiplied by the witch hat attenua- tion surface as measured for each observer (see Section 5.2.1), and filtered by sine-phase log- Gabor filter elements (see Section 5.2.2). Each pixel of the output of this filtering is taken to represent the activity of a detector at that location. The behaviour of both models is then de- scribed by summing over the detectors

ct= v u u u u u u t s n P i=1 (σ2_{· t}2 i) n P i=1 (s2 i · ti) , (7.1)

where ctis the contrast detection threshold, siis the pixel value at each location in the attenu-

ated and filtered image, σiis the standard deviation of the noise, and tiis the template at that

location. This is derived in Section 5.3.5. In the application of the model here it is assumed that the noise is constant (σi = 1) and Gaussian.

For a template which is matched to the transduced stimulus exactly (ti = s2i), Equation 7.1

is the noisy energy model (NE). The flat template noisy energy model (FTNE) has templates (ti) unaffected by the visual field inhomogeneity or witch hat compensation (the template al-

ways has a flat contrast profile). Summation slope predictions from the Minkowski summa- tion approximation to the signal detection theory (SDT) probability summation model would be identical to those from the noisy energy model in this case (see Section 5.5.1). Predictions from the high threshold theory (HTT) probability summation model based on the psychomet- ric slope would be similar, as the Weibull β for each observer was approximately 4. Regardless, as stated previously the HTT probability summation model has been thoroughly discredited (see Section 2.4.2; Swets et al., 1961; Green & Swets, 1966e; Nachmias, 1981; Laming, 2013). The two models were fitted to the thresholds from the three observers, using the downhill simplex method (see Section 3.7) to minimise the RMS error between the data and the model

Fovea & parafovea Periphery Model Observer Offset (dB) RMSe (dB) Offset (dB) RMSe (dB) Noisy energy (NE) ASBDHB SAW 13.53 14.85 11.47 0.79 0.95 1.51 1.39 -4.89 2.371.21 Flat template noisy energy (FTNE) ASB DHB SAW 13.48 14.76 11.38 0.78 0.91 1.46 1.44 -4.81 2.381.21

Table 7.2: Vertical offset parameters and RMS errors for the noisy energy (NE) model fit- ted to the data in Figures 7.6 and 7.7, and the flat template noisy energy (FTNE) model fitted to the data in Figures 7.8 and 7.9. The fitted offset parameter in the periphery is the additional offset needed on top of that derived from the fits to the foveal and parafoveal data.

predictions. The only free parameter was the vertical offset of the model prediction curves (equivalent to varying the global sensitivity). In the first instance only the data for the foveal and parafoveal conditions were fitted. A prediction was then made for the peripheral con- dition based on the offset parameter from this fit. As the peripheral stimuli were presented beyond the region where the witch’s hat was measured however, this prediction is based on extrapolating the attenuation surface beyond the region where it is empirically supported. To correct for this, an additional fit was performed solely to the peripheral data. The variation in contrast sensitivity over the stimulus display region at that eccentricity would be very minor (it was chosen for this reason by Robson & Graham, 1981), so an inaccuracy in the extrapolated witch’s hat at this eccentricity would only cause a global over- or underestimation in sensitiv- ity. This is entirely compensated for by refitting the offset parameter for this condition. The RMS errors and fitted vertical offset parameters for all of these fits are shown in Table 7.2.