Since the external noise mask had little effect on the mean response of local filters, with the constraints discussed already and combining
Equations 5 to
8 within the matched filter, the decision variable
\begin{equation}\tag{9}{{d}}\kern1.5pt^{\prime}\sim = {{\left( {{{S}} * {{R}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{S}} * {{R}}} \right)} {{{\left( {{{S}} * \left( {{{{\sigma }}_{{a}}}^{2} + {{{C}}_{{m}}}^{2}} \right)} \right)}^{{{1/2}}}}}}} \right. \kern-1.2pt} {{{\left( {{{S}} * \left( {{{{\sigma }}_{{a}}}^{2} + {{{C}}_{{m}}}^{2}} \right)} \right)}^{{{1/2}}}}}} = {{\left( {{{{S}}^{{{1/2}}}} * {{R}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{{S}}^{{{1/2}}}} * {{R}}} \right)} {{{\left( {{{{\sigma }}_{{a}}}^{2} + {{{C}}_{{m}}}^{2}} \right)}^{{{1/2}}}}}}} \right. \kern-1.2pt} {{{\left( {{{{\sigma }}_{{a}}}^{2} + {{{C}}_{{m}}}^{2}} \right)}^{{{1/2}}}}}}\end{equation}
If there is no external noise (
Cm = 0), the model can be simplified as
\begin{equation}\tag{10}{{d}}\kern1.5pt^{\prime}\sim = {{\left( {{{{S}}^{{{1/2}}}} * {{R}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{{S}}^{{{1/2}}}} * {{R}}} \right)} {{{{\sigma }}_{{a}}}}}} \right. \kern-1.2pt} {{{{\sigma }}_{{a}}}}}\end{equation}
and thus becomes a typical ideal summation model that accounts for the approximately −1/2 slope of the spatial summation curve. Adding external noise does not change this slope for the size effect: as shown in the denominator of
Equation 9, its effect does not depend on noise level. However, it is noteworthy that the spatial summation curve for our gain-control model deviates from the −1/2 slope for small target sizes. This deviation is due to fact that, when the target contrast is high, the divisive inhibition in
Equation 3 outweighs the additive constant, reducing
d′ further than that imposed by the noise. Thus, as illustrated in
Figure 7A for averaged data, our multiple stage model (red solid curve), incorporating the contrast gain control that is ubiquitous in models of contrast masking, captures not just the ideal summation tendency (green dashed line) but also the deviation at the extremes. Thus, the prediction of our model is almost indistinguishable from that of the three-component model (blue curve) that incorporates mechanisms for complete, ideal and probability summation. Indeed, our model, with only five free parameters for each observer, performs much better than the conventional ideal summation model with six free parameters for each observer. Quantitatively, the differential Bayesian information criterion (ΔBIC, Wagenmaker,
2007) of the ideal summation model, relative to our model, was −14.32 (probability that this model being more likely than ours,
p = 0.0008) for YYH and −11.82 (
p = 0.0027) for DTJ; and the three component model, with 18 free parameters for each observer, had ΔBIC −22.5 (
p < 0.0001) for YYH and −86.3 (
p < 0.0001) for TJ. Thus, taking into account the number of free parameters, the two generic models performed much worse than our model.