In this Appendix, we examine the effect of an increase in element separation on the signal-to-noise ratio in Yen and Finkel's (
1996,
1997,
1998) association field model. The association strength falls as a Gaussian function of distance between the elements. To avoid unnecessary clutter in the equations, we assume that the distance across the visual field is measured in spatial units whose size causes the standard deviation of this Gaussian function to have a value of 1/
. Then, the association strength between two elements will be proportional to
where
x is the distance between them. Suppose, for a particular contour element, there is another contour element (the “signal”) at a distance
s from the element, and a distractor element (the “noise”) at a distance
n. Then the signal-to-noise ratio, SNR
1, for that pair of elements (assuming equal association strength in other respects due to equal levels of co-circularity, etc.) will be given by
Now, if we increase the separation between all elements by a factor
m > 1, then the new signal-to-noise ratio, SNR
2, is given by
The ratio of these SNRs is given by
Since
m > 1, SNR
2/SNR
1 > 1 if
n >
s, and SNR
2/SNR
1 < 1 if
n <
s. Thus, increasing the element separation causes the signal-to-noise ratio to increase for distractor elements further away than the neighboring contour element and causes it to decrease for distractor elements closer than the neighboring contour element. The overall effect of increasing the separation on the signal-to-noise ratio will depend on the distribution of distractor positions, and the relative strengths of the inputs from the different elements due to other factors, such as co-circularity.