A variable criterion for seeing

The assumption from the high threshold theory that false positive responses below the xed internal threshold were simply guesses led to the practice of adjusting the psychometric functions to account for this. After all, the theory supposed, how could the visual system respond to neural activity below the xed threshold?

Therefore, a measured psychometric function, as shown by the dotted line in Figure 1.1, would be corrected to the function shown by the solid line in Figure 1.1 using the formula shown at equation 1.1. pcorr= pact− Qf p 1 − Qf p (1.1) where:

0.0

0.2

0.4

0.6

0.8

1.0

Quanta absorbed

p(’y

es’)

0

Figure 1.1: Example psychometric function for the probability of saying yes signal present against the number of quanta absorbed at the retina. The as measured function is shown by the dotted line and the corrected function is shown by the solid line.

pcorr=corrected probability of saying yes

pact =measured probability of saying yes

Qf p=false positive response at quanta=0

The concept of the high threshold theory and the notion that responses below a xed, neu- rophysiologically determined threshold were the result of guesswork, and thus requiring the psy- chometric function to be corrected, was challenged by Tanner & Swets (1954). Tanner & Swets (1954) contended that, rather than the result of guesswork, these responses were the result of the visual system responding to neural activity, which must be generated within the visual system itself rather than resulting from the stimulus. Tanner & Swets (1954) proposed that, rather than a xed, neurophysiologically determined threshold, the observer had a variable threshold, known as a criterion. A criterion refers to the observer's implicit rule for converting the internal response elicited by the stimulus into an external response or decision; for example, internal responses above the criterion will elicit a yes response and internal responses below the criterion will elicit a no response. This study was largely responsible for the introduction of the theory of signal detectability, now more commonly referred to as signal detection theory (SDT), into the domain of psychophysics (Cohn, 1993, p. 4), though it largely remained in the audition sphere until the 1970s.

0

1.0

Stimulus intensity

p(’y

es’)

Figure 1.2: The eect of shifting the observers criterion on the position of the psychometric function.

SDT will be treated in greater detail in section 1.2, however, at this juncture, the experiment of Tanner & Swets (1954) will be introduced because of its relevance in countering the concept of high threshold theory, as suggested by Hecht et al. (1942). Tanner & Swets (1954) argued that, rather than observer responses to stimuli below the high threshold limit being the result of guesswork, the response was the result of both the small number of quanta received at the retina from the presented stimulus and a noise generated within the observer that is independent of the signal. Tanner & Swets (1954) presented ashed signals on a blank background to participants in both forced choice and yes/no experiments with a range of signal intensity levels. The participants had to identify the interval in which the signal occurred and their criterion was shifted by informing them of the prior probability of signal presentation as well as with varying nancial inducements for each possible decision outcome (correct detection, correct rejection, false alarm or miss).

The results produced similar shape functions to those produced by high threshold theory but, as illustrated in Figure 1.2, by showing how changing the observer's willingness to say signal present, or not, shifted the threshold level for the same signal, they were able to refute the notion of a xed threshold as suggested by the high threshold theory (Green & Swets, 1966, pp. 127-136). Integral with the concept of SDT was the notion of an internal noise source, independent of the signal, that drove the observer's responses in the absence of a signal, rather than their responses being

guesswork, but the origin of this noise was not elucidated by Tanner & Swets (1954) and, whilst Hecht (1945) recognised the existence of internal noise, it would fall to Barlow (1957) to provide a more detailed investigation into its eects.

The results from Tanner & Swets (1954) suggest that, given an appropriate criterion (and a well trained observer) it should be possible to demonstrate the minimum light intensity that the human visual system is able to detect and, many years later, Sakitt (1972) used SDT to do just this, arguing that only a single photon (or quantum) was necessary for the experience of seeing. Sakitt (1972) used an experimental set up similar to Hecht et al. (1942), with dark adapted participants and disc signals on blank backgrounds ashed to the temporal retina, but instead of a yes/no protocol, he used a rating scale ranging from 0 (we did not see anything) to 6 (we saw a very bright light) and three signal strengths, blank (no signal), weak (average 55 photons at the cornea) and strong (average 66 photons at the cornea). Plotting the average rating score for each signal strength against the number of quanta incident on the cornea, a linear relationship was observed and for subject BS (the study author) the following linear model was derived:

¯i = 0.0274Qc + 0.36 (1.2)

where

¯i =the average rating score

Qc =average number of quanta at cornea per ash

The form of the linear model for the average rating score closely matched the linear model for the average number of rod signals:

a = f (Qc + Xc) (1.3)

where

a =the average number of rod signals

f =the fraction of incident quanta that produce rod signals Qc =average number of quanta at cornea per ash

Xc =dark light (now more commonly referred to as internal noise)

From the close similarity of equations 1.2 and 1.3, Sakitt (1972) drew the conclusion that the average rating score was equal to the number of rod signals. As shown in equation 1.3, the number of rod signal results from the average number of eective quantum absorptions plus noise events and using the Poisson distribution of the average rating scores and the known values for f and Qc, Sakitt (1972) was able to calculate the number of quanta incident on the retina as shown in

equations 1.4 to 1.6.

Assuming the average rating score is equal to the number of rod signals then:

a = 0.0274  Qc + 0.36 0.0274  (1.4) a = 0.0274 (Qc + 13.1) (1.5)

From the cumulative probability distribution of rating scores, Sakitt (1972) estimated the average number of rod signals for a criterion of 1 as a = 0.7. Thus:

Qc =

0.7

0.0274− 13.1 = 12.4 (1.6)

Sakitt (1972) actually calculated Qc = 12.6, which was consistent with the estimated display

luminance, and he therefore argued that a trained observer (such as himself), could actually count every quantum absorption such that the absolute threshold for seeing would be a single photon.

The results from Sakitt (1972) show that there is no absolute physiological threshold for seeing, rather the threshold for seeing is variable, as dictated by the observer's criterion, and, with the right conditions, training and a low enough criterion, can be as low as a single quantum incident on the retina.