In this chapter, we introduce our chromatic symmetry detection model and describe how we apply it to the 2AFC noise masking task used in the experiments.
Note that in each trial of the task, the stimuli consisted of either a symmetric target or a non-symmetric random-dot control superimposed on a random-dot mask. All stimuli in a trial contained the same number of the colors with equal probability, in which the number of the color (n) was from 1 to 4.
As Figure 1.2 shown, the chromatic symmetry detection model contains two stages: a perception stage and a decision stage. The perception stage concerns the noise-limited sensitivity of a visual mechanism to the stimuli limited by both internal and external noise, while the decision stage concerns the effect of uncertainty on the decision criterion.
The first step of the perception stage is a band of color-orientation selective symmetry encoders that are sensitive to symmetry in an image. Each encoder is sensitive to the mirror symmetry about one axis with a certain color. As mentioned in Chapter 1, each encoder contains two steps, matching and pooling. The matching stage extracts the corresponding color features in an image while the pooling stage analyzes those color pairs to determine whether their equal-distance points form a symmetry axis. In other words, these symmetry encoders are long-range pairs of local multiplicative color detectors that register a signal whenever there is their target color at two locations in the field equidistant from a symmetry axis. The outputs of all such pairs of detectors relative to a given symmetry axis are linearly summed to form the symmetry signal relative to that location. Only when a number of them line up with
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respect to a particular symmetry axis, the symmetry encoders regard the chromatic pattern as symmetry.
In the 2AFC noise masking task, the image in the interval that contains target plus mask can be considered to consist of two components: the symmetric target and the noise mask, while the image in the interval that contains noise control plus mask can be considered to consist of just one component with a density that is the sum of the control and the mask.
For a sparse n-color random-dot pattern, the excitation of the j-th color-selective symmetry encoder to the i-th image component, Eji, is
i i j i
j
D
Se n
E
1,
, = ⋅ (1)
where Sej,i is the sensitivity of the j-th symmetry encoder to i-th image component, while 1/n*Di, is the dot density of the pattern of the j-th encoder’ target color in i-th image component. The total excitation of j-th encoder, Ej, is the sum of excitations produced by all image components,
∑
= i ji
j
E
E
, . (2)The response of the perception stage is the excitation of the j-th symmetry encoder, Ej, raised by a power p, and then divided by a divisive inhibition term Ij plus an additive constant z,
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where Ij is the summation of a non-linear combination of the inhibition from all image components to mechanism j. This divisive inhibition term Ij can be represented as
∑
image components consisting of the target color and of the non-target color respectively.The contribution of each channel to the visual performance is limited by both internal noise of that channel and the external noise provided by the noise patterns.
The variability of the internal noise, σa2
, is a constant for all symmetry channels. The variability of external noise, σe2
, is proportional to the square of the density of random noise mask, that is, σe2
= v * Db2
in which v is a scalar constant and the index b denotes the noise mask. Pooled together, in each channel the standard deviation of the response distribution is
(
b2 a2)
1/2r v D σ
σ = ⋅ + . (5)
The output of the perception stage is then sent to the decision stage. The decision stage monitors more channels than those that are relevant to the visual tasks (Pelli, 1985). The performance of the system is limited not only by the noise in the relevant
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channels but also by that in the irrelevant channels. In our experiment, the task of the observer was to detect the symmetry component in an image. Hence, a relevant channel is the one whose color-orientation selectivity matches that of the image. The observer detects a symmetric pattern if the maximum response of all monitored channels to an image is greater than the response of a random-dot pattern by an amount that exceeds the level of noise in the system (Green & Swets, 1966).
When there are m channels, in which n channels are relevant while m-n channels are irrelevant, to be monitored, the maximum response of these channels can be described by a distribution whose mean approximates a fourth-power summation over these m channels (Graham, Robson, & Nachmias, 1978; Quick, 1974; Pelli, 1985), though the Gaussian distribution theory of Tyler and Chen (2000) shows that the fourth power exponent is valid only for the restricted conditions of a particular attention model and a linear signal transducer. Hence, for the target plus mask images where there are n channels responding the symmetry image component, the mean of the response R’ can be expressed as
(
1 1 4,( ))
1/4where the subscript b and t denote the noise control pattern and the symmetry component in the images respectively. Instead, the mean of the response R’ for the noise control plus noise mask images is
(
1)
1/421
in which the subscript b+c indicates that the image contains both the noise mask and a control pattern with the same number of dots as the corresponding symmetry target.
The decision variable, d’, is the difference of the response to the image with the symmetry component and the response to the random-dot control image divided by the standard deviation of the max distribution, σp. That is,
( R
b tR
b c)
pd
'= ' + − ' +σ
(8)The threshold is defined when d’ reaches unity. Note that the standard deviation of the max distribution of multiple independently and identically distributed samples is k times the standard deviation of the original distribution, in which the variable k can be estimated by the method Chen and Tyler (1999) proposed. Thus, σp = σr for 1-color condition while σp = k*σr for n-color conditions.
The above is the description of our chromatic symmetry detection model that applies to a 2AFC noise masking task. In Chapter 5 and 6, we examined the color-selective property of the symmetry encoders in the model. In Chapter 7 to 9, we manipulated the number of the colors in the images and measured the symmetry detection threshold to get the TvD functions, to investigate the integration of these symmetry channels in different conditions. The details of the model implementation are described in each chapter.
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