The classification image was obtained by a linear regression method. The observer’s internal response is assumed to be given by the linear relationship
where
rk,s is the internal response on trial
k of a given stimulus level,
s;
nk,s,i is the external noise amplitudes, where the subscript
i goes from 1 to 11 for the 11 spatial frequencies, the subscript
s goes from 0 to 3 (detection task) or −1 to +1 (position task), and
qk,s is the internal noise plus the truncation noise that is needed to make
rk,s an integer.
Equation 10 is based on the assumption that higher-order nonlinearities are negligible. We intend to investigate this assumption in future studies. The term f
s in
Equation 10 is a constant that depends on the stimulus level. Because it is a constant, it will cancel when the response is cross-correlated with the zero mean noise. The subscript k indicates the trial number for a given level, and goes from 1 to about 50 (200/4) for the detection task and about 67 (200/3) for the position task. As will be discussed in “
Results,” we separately analyzed each stimulus level,
s, to minimize bias. The coefficients
wi,s are the regression coefficients that correspond to the template weighting used by the observer. These coefficients are the classification image.
In order to calculate the classification image, we make the approximation that for each stimulus level,
s, the internal response,
rk,s is linearly related to the observer’s response. This assumption is equivalent to an assumption that the criteria were uniformly spaced. This assumption seems reasonable because the observers were encouraged to distribute their responses uniformly. The subscript,
s, enables the constant of proportionality to be included in the coefficient
wi,s so that
rk,s can be taken as the observer’s response. How the constant of proportionality depends on the placement of criteria is considered elsewhere (
Klein & Levi, 2002). The standard method to obtain the coefficients
wi,s is to cross-correlate the responses with the external noise
where
ntrials is the number of trials at a given stimulus level, and from
Equations 10 and
11,
Equation 11 can be solved by multiplying both sides by the inverse of the square matrix
N:
where the second term is noise that is of order
ntrials−0.5 and will be neglected in the present analysis.
N, the noise variance-covariance, is approximately a diagonal matrix with the diagonal elements being close to
n2. In that case,
Equation 14 is approximately
.
Equation 15 gives the estimate obtained by the cross-correlation method that is the most common method for estimating the classification image. All of our results will use the linear regression method of
Equation 14 that provides estimates with variance lower than the cross-correlation method. Our forthcoming plots of
wi,s will have an ordinate with units. It is useful to consider the meaning of the magnitude of
wi,s. The numerator of
Equation 15 has units of response times noise and the denominator has units of noise squared. Thus
wi,s has units of response divided by noise. Because the noise is n = 0.04, w
i,s is 25 times the response variability. Consider, for example,
w6,s in
Figure 2, whose value is
w6,s = 5. That means the 6 c/degree component of the noise contributes a variation of 5/25 = 0.2 to the response
rk,s. A larger value of
wi,s means a greater variability of responses, which would produce a lower d′. Thus we have the counterintuitive result that a larger classification image is correlated with reduced d′ (see discussion preceding
Figure 9).
Klein & Levi (2002) provide further details on the meaning of the magnitude of the classification components,
wi,s, including a redefinition of
wi,s that removes the response variance so that
wi,s becomes the correlation between the stimulus and response.
In “
Results,” we will be plotting the 11 coefficients
wi,s versus the 11 spatial frequencies. We will also plot the classification images given by
for the detection task, and
for the position task