When performance is limited by stochastically defined masks, (psychophysical) reverse correlation has proven to be an especially efficient tool for estimating the templates used by detection and discrimination mechanisms. Here I describe a maximum-likelihood approach to quantifying the significance of differences between estimates of template. Four methodologically related experiments illustrate the versatility of reverse correlation. Experiment 1 shows significant differences between the templates used by different observers when detecting a bright Gaussian blob. The results of Experiment 2 are consistent with observers not using information about the phase of a parafoveal wavelet when detecting it. Experiments 3 and 4 reveal not only the templates used by detection mechanisms but also aspects of their response functions. Both results are consistent with a sensory threshold. Experiment 3 shows that 2-alternative forced-choice detection errors are caused when the target’s effective contrast is reduced, not when the mask looks more like the expected target+mask than the actual target+mask. Experiment 4 suggests that observers use optimally tuned detection templates for orientation discrimination.

*i*of this procedure, observers must select the one of two displays of visual noise that also contains a target pattern. In Equation 1, the target

**t**and both samples of noise

**n**

_{i1}and

**n**

_{i2}are represented by vectors. For simplicity, assume that both target and noise are gray-scale (as opposed to color) images. In that case, each entry in each vector describes the intensity of a particular pixel. The template, represented by the vector

**w**, describes the detection mechanism’s relative sensitivity to each pixel. specifies the rule for a correct response: the match (i.e., the inner product) between the target-present display and the template must exceed the match between the target-absent display and the template. To account for the fact that observers do not always respond in the same way to the same stimuli, each match is thought to be perturbed by internal noise.

*η*

_{i1}and

*η*

_{i2}represent two samples of that noise.

**s**

_{i}represents the display on trial

*i*, which may or may not contain the target.

*c*is an arbitrary constant called the internal criterion. Each of these equations makes a prediction that has been confirmed by numerous conventional experiments (Pelli, 1990): threshold contrast increases linearly with the contrast of the noise mask.

**s**

_{i}and constants r

_{i},

*i*=1,…,

*N*, multiple regression is the conventional statistical technique that finds that

**w**which minimises in (Mood, Graybill, & Boes, 1974).

*σ*.

*i*:

**Φ**(x) is the standard normal CDF (cumulative density function). The likelihood of an incorrect response is 1 −

*ψ*

_{i}. Similarly, the assumption of Gaussian internal noise, when coupled with the response rule in Equation 2, implies the following formula for the likelihood of a “yes” response on any trial

*i*:

*ψ*

_{i}. For clarity, any parameters appearing in the likelihood formula will be called response parameters. Note that Equation 4 has one response parameter:

*σ*. Equation 5 has two:

*σ*and

*c*.

**w**, one needs only to assign values to the response parameters in order to calculate the joint likelihood of all responses collected. For example, if the responses to trials 1, 2, and 3 were “yes,” “yes,” and “no,” respectively, then the joint likelihood of these responses would be

*ψ*

_{1}

*ψ*

_{2}(1−

*ψ*

_{3}). The second step of the MLA is to find those values for the response parameters that maximize the joint likelihood of all responses collected by each observer, given no difference between the ideal and visual templates, (i.e., for target detection in white noise)

**w**=

**t**. I use Mathematica’s FindMinimum routine (Wolfram, 1999) to find these parameter values.

*σ*is a free parameter, template amplitude can be assigned any arbitrary value.) In step three of the MLA, the maximum joint likelihood of all responses is again determined, this time with some subset of the template parameters allowed to vary from their ideal values along with the response parameters.

^{2}distribution, with degrees-of-freedom equal to the number of freely varying template parameters in step three (Mood et al., 1974). The significance of the difference between ideal and visual templates can therefore be quantified.

*f*(

*x*) increases faster than

*x*. The simplest such function is a “hard” threshold (see “2” for details): Note that both displays in the 2AFC procedure now contain the same mask

**n**

_{i}.

**e**should therefore be positively correlated with the template (i.e.,

**w**

^{t}

**e**>0). Thus, if the template has any similarity to the target (i.e.,

**w**

^{t}

**t**>0), Then

**t**

^{t}

**e**>0.

**t**

^{t}

**e**should be less than zero. Simulations of these models are shown in conjunction with Experiment 3, which corroborates the prediction of sensory-threshold theory.

**w**) on every trial of a particular experiment. For many experiments, this assumption may not be correct. Indeed, one popular model of detection posits that some responses are based on the outputs of mechanisms that are completely insensitive to the target (Pelli, 1985). Support for this theory can be demonstrated (see Ahumada & Beard, 1999 and below) using a Y/N procedure. Hits (i.e., “yes” responses to targets of non-zero contrast) are likely to be caused by stimulation of mechanisms that actually are sensitive to the target, whereas false alarms (i.e., “yes” responses to targets of zero contrast) are just as likely to be caused by stimulation of mechanisms that are not. Both the total number and the distribution1 of responses can affect the clarity of an estimated template, but any difference between the shapes of templates estimated from trials with zero-contrast targets and trials with nonzero-contrast targets may be due to uncertainty regarding the mechanisms most sensitive to the target and/or an inability to ignore mechanisms insensitive to the target.

*l*

_{max}, decreased from 40 to 26 cd m

^{−2}. Every few months the monitor was re-calibrated and the background luminance was set to one half of the maximum luminance. Minimum luminance

*l*

_{min}, was always < 0.1 cd m

^{−2}.

*m*is the maximum available contrast, then an

*x*dB stimulus is one that has a contrast of

*m*10

^{x/20}. I also frequently cite the correlation

*r*

_{x,y}between two vectors

**x**and

**y**(usually representing a target and an estimated template): A correlation of 1 indicates a perfect match; −1 indicates that the two vectors represent photographic negatives of each other.

*σ*= 0.15 degrees). This target was added, pixel-by-pixel, to one of the displays in each trial. An adaptive staircase (Watson & Pelli, 1983) determined the target contrast required for observers to identify the target display with 75% accuracy. The accuracy of each response was indicated with a tone. Viewing was binocular and there were four observers: J.A.S. (the author), A.J. and S.Y.A. (experienced psychophysical observers, naïve to the purpose of this experiment) and N.E., who had no previous experience with psychophysical experiments.

*p*< 3 × 10

^{−8}).

*p*< 0.03). (The other observers’ responses would have been better fit by the general model had it allowed bright backgrounds.) Templates satisfying the constraints of the more general model, which maximize the joint likelihood of each observer’s responses, are shown Figure 1 (blue curves).

*p*values for each of these 12 tests are given in Table 1. As a guide to reading Table 1, consider the entry in the second column of the first row. This entry denotes that the parameter values describing S.Y.A.’s template are significantly different from those derived from A.J.’s responses (

*p*< 10

^{−16}). Because the parameter values describing A.J.’s template are significantly different from those derived from the other three observers’ responses and the parameter values describing each of their templates are significantly different from those derived from A.J.’s responses, it is reasonable to conclude that A.J.’s template is significantly different from those used by the other three observers. On the other hand, because the parameter values describing S.Y.A.’s template are not significantly different from those derived from J.A.S.’s responses (

*p*≈ 0.12) nor are the parameter values describing J.A.S.’s template significantly different from those derived from S.Y.A.’s template (

*p*≈ 0.23), it is reasonable to conclude that S.Y.A. and J.A.S. use templates that are not significantly different.

Responses | ||||
---|---|---|---|---|

Templates | S.Y.A. | A.J. | J.A.S. | N.E. |

S.Y.A. | <10^{−16} | 0.12 | 3×10^{−4} | |

A.J. | 3 × 10^{−13} | 10^{−13} | 10^{−10} | |

J.A.S. | 0.23 | <10^{−16} | 0.09 | |

N.E. | 0.06 | 3 × 10^{−9} | 0.4 |

*r*≈ 0). In Experiment 2, I replicate this result using a 3 cycle/degree Gabor presented at an eccentricity of 3 degrees. Ahumada and Beard concluded that observers must harbor some uncertainty as to which detection mechanism is most sensitive to high-frequency targets. My results demonstrate that observers are unable to ignore the activity of mechanisms having templates mismatched to the phase of peripheral detection targets.

*σ*= 0.26 degrees) centered on a bright stripe. Its contrast was 0.63 times that required for a hit rate of 82%, as determined by the adaptive staircase. All other methods were identical to those of Experiment 1.

*p*> 0.16, J.A.S.;

*p*> 0.74, M.J.M.). Thus, although standard analyses of target-present trials produced templates that were similar to the target, no pattern emerged from standard analyses of target-absent trials.

*σ*= 0.14 degrees) centered on a bright stripe. This target was added, pixel-by-pixel, to one of the displays in each trial. The adaptive staircase determined the target contrast required for observers to identify the target display with 75% accuracy. All other methods were identical to those of Experiment 2.

**t**was negatively correlated with sum of the samples present on trials that produced incorrect responses

**e**. Figure 5b shows one example. For J.A.S.,

*r*

_{t,e}= −0.16 (205 incorrect responses with a −20 dB target); for S.C.D.,

*r*

_{t,e}= −0.23 (276 incorrect responses with a −24 dB target); and for A.C.M.,

*r*

_{t,e}= −0.22 (308 incorrect responses with a −20 dB target). I.M.E. was the inexperienced observer, and 3,000 responses were collected from her. The first 1,000 yielded

*r*

_{t,e}= −0.11 (286 incorrect responses with a −20 dB target); the second 1,000 yielded

*r*

_{t,e}= −0.17 (217 incorrect responses with a −20 dB target); and the final 1,000 yielded

*r*

_{t,e}= −0.21 (290 incorrect responses with a −22 dB target). For J.A.S., with the high-contrast noise,

*r*

_{t,e}= −0.14 (240 incorrect responses with a −10 dB target).

*t*was set to 0, and the variance of the observer’s internal noise was assumed to be sufficiently low for a correct response from any trial

*i*in which

**t**

^{t}(

**t**+

**n**

_{i}) > 0. Therefore, on trials when

**t**

^{t}(

**t**+

**n**

_{i}) < 0, each response was effectively selected from a Bernoulli process with a 50% success rate.

**w**

_{i}and another Gabor template

**w**

_{h}, with freely varying vertical and horizontal spreads

*σ*

_{y}and

*σ*

_{x}. Although a good case could be made for a horizontally elongated

**w**

_{h}from either the results of observer J.A.S.2, neither A.C.M.’s nor I.M.E.’s responses were significantly more likely with

**w**

_{h}≠

**w**

_{i}. S.C.D.’s responses were significantly more likely with

**w**

_{h}≠

**w**

_{i}, however, the aspect ratio (

*σ*/

_{x}*σ*) of his maximally likely

_{y}**w**

_{h}was a mere 1.3; not exactly overwhelming evidence for a horizontally elongated template. (Note that the statistics described above are not inconsistent with the hypothesis that all four observers used the same template. An analysis of Experiment 3 analogous to that shown in Table 1 has not yet been performed.)

*σ*= 0.26 degrees) centered on a bright stripe. One Gabor pattern, the distracter, was horizontal. The other, the target, was tilted 11 degrees clockwise or counterclockwise from horizontal. The observer had to decide whether the target’s tilt was clockwise or counterclockwise. Figures 2 and 6 show the cued condition in which a unidirectional arrow indicated which of the two Gabor patterns was the target. In the uncued condition, this unidirectional arrow was replaced by an uninformative bidirectional arrow (see Figure 7a). Adaptive staircases converged on the contrast (applied to both Gabor patterns) required for 75% accuracy in each condition. J.A.S. and A.C.M. (a naïve but experienced psychophysical observer) performed between 1,000 and 1,100 trials in each condition.

^{p},

*p*> 1), then the only noise samples that should have affected performance are those that changed the output of the most strongly stimulated mechanism: the one stimulated by the true target. Because samples that changed the output of the mechanism preferring the false target (in the distracter location) also affected performance, there must have been a range of near-threshold inputs with which output increased linearly. (This argument is formalized in “3.”)

*t*(see Equation 8) produced panels g and h in Figures 6 and 7. These images are qualitatively similar to those produced by the human observers.

^{2}With low-contrast noise, his responses were significantly (p < 0.009) more likely with

**w**

_{h}≠

**w**

_{i}and the aspect ratio of the maximally likely

**w**

_{h}was 2.2; with high-contrast noise, his responses were significantly (p < 0.0008) more likely with

**w**

_{h}≠

**w**

_{i}and the aspect ratio of the maximally likely

**w**

_{h}was 2.3.

**n**

_{i}represent the mask present on trial

*i*.

**n**

_{i}is added to the template estimate if

*i*elicits a “yes” response; otherwise,

**n**

_{i}is subtracted from the template estimate. The expected contribution of trial

*i*toward a standard analysis is therefore <

**n**

_{i}sgn[

**w**

^{t}

**s**

_{i}+

*η*−

*c*]>

_{η}, where

**w**represents the template,

*c*represents the internal criterion,

*η*represents a sample of the internal noise and

**s**

_{i}represents the stimulus. Because for target-absent trials,

**s**

_{i}=

**n**

_{i}, and for target-present trials,

**s**

_{i}=

**n**

_{i}+

**t**, the expected contribution of any trial toward a standard analysis can be rewritten as <

**n**sgn[

**w**

^{t}

**n**+

*η*−

*ĉ*]>

_{n,η}, where

*ĉ*is a constant that is different for target-present and target absent trials. (Note that the target must have the same contrast for all trials used in the standard analysis). Thus, it remains for me to prove Where

*k*is a constant.

**n**=

**W**

^{t}

**^n**, where

**Ww**= ‖

**w**‖

**e**

_{1}, with

**e**

_{1}defined as the vector that is 1 in its first element and 0 in all others. Note that if all the elements of

**n**have zero mean, then so will all the elements of

**^n**. Using

**^n**, 2 can be re-written as where

*^n*

_{1}is the first element of

**^n**. 3 can be expanded if we define

**e**

_{j}as the vector that is 1 in its

*j*th element and 0 in all others:

**^n**have zero mean. If

*η*were uni-variate normal and

**n**were multi-variate normal, then a precise scalar value for the expectation in 4 could be determined. However, it is not necessary to determine the precise value. As long as all the elements of

**n**have zero mean and are independent of

*η*, the expectation will be some constant

*^k*. Thus Where

*k*≠

*^k*. Q.E.D.

**a**and

**c**represent the counterclockwise target and the clockwise targets in Experiment 4, respectively. Assume that in the cued condition the observer responds “counterclockwise” when the output of a detection mechanism sensitive to

**a**exceeds the output of another sensitive to

**c**. That is, the observer responds “counterclockwise” when where

**s**represents the stimulus in the cued location,

*f*is some non-decreasing function, and

*η*represents a sample of the internal noise. The templates

**w**

_{a}And

**w**

_{c}, are constructed such that

**c**+

**n**. If

*f*were linear, then incorrect responses would occur whenever

**w**

_{a}

^{t}

**n**−

**w**

_{c}

^{t}

**n**>

*η*′, for some random

*η*′. (NB: the PDFs of the random variables giving rise to

*η*′ and

*η*′ would have the same shape.) However, the results of Experiment 4 indicate that incorrect responses occur when

**c**

^{t}

**n**(and thus

**w**

_{c}

^{t}

**n**) is negative;

**w**

_{a}

^{t}

**n**does not seem to matter. Thus

*f*cannot be linear.

*p*be some small positive number, 0<

*p*<<1. The results of Experiment 4 indicate that

**s**=

**c**−

*p*

**w**

_{c}would cause more errors than

**s**=

**c**+

*p*

**w**

_{a}. Thus, Using the identities in A7, 8 can be rewritten as and rearranged as But, , so 10 implies where

*a*<

*b*<

*c*. That is,

*f*(

*x*) increases faster than

*x*.

**c**represent that clockwise target and let

**d**represent the distracter. Thus the stimulus in the target position can be described

**c**+

**n**

_{1}and the stimulus in the distracter position can be described

**d**+

**n**

_{2}. Finally, assume that the observer responds incorrectly when where

*η*represents a sample of the internal noise and the templates

**w**

_{a}and

**w**

_{c}, conform to 7. To determine whether

*f*(

*x*) = Max{

*x,c*} or

*f*(

*x*) = Sgn(

*x*)|

*x*|

^{p}is a better representation, assume for the time being that Given the difference between

**c**and

**d**,

**w**

_{c}

^{t}(

**c**+

**n**

_{1}) >

**w**

_{c}

^{t}(

**d**+

**n**

_{2}) and

**w**

_{a}

^{t}(

**d**+

**n**

_{1}) >

**w**

_{a}

^{t}(

**c**+

**n**

_{2}) for virtually any noise samples

**n**

_{1}and

**n**

_{2}. Because

*f*is non-decreasing, we can simplify 12 to say that the observer will respond incorrectly on clockwise trials when These last two equations can be combined to yield If

*p*were much greater than 1, we would not expect to find any effect of

**n**

_{2}on response accuracy. However, the results of Experiment 4 indicate that incorrect responses occur not only when

**c**

^{t}

**n**

_{1}(and thus

**w**

_{c}

^{t}

**n**

_{1}) is negative, but also when

**a**

^{t}

**n**

_{2}(and thus

**w**

_{a}

^{t}

**n**

_{2}) is positive; thus

*p*must not be much greater than 1. In fact, when

*p*is exactly 1, and

**w**

_{a}and

**w**

_{c}are constrained to be rotated versions of

**d**, simulation indicates that

**w**

_{c}

^{t}

**c**−

**w**

_{a}

^{t}

**c**is maximized when

**w**

_{a}and

**w**

_{c}are rotated ±28 degrees, a value that conforms to the result of Experiment 4 (see main text).