The classical (Fechnerian) threshold

Fechner is said to have borrowed a concept from Hebart, that of a limen or limit below which intensity is too small to be seen (Boring, 1942). Thus energy at or below the limen is unseen, while that above the limen is seen. This concept is shown schematically in Figure 51.1A. Measurement, by techniques that Fechner himself introduced, disproved the elegance of this concept immediately. No such unique limit can be found experimentally. Instead, the limit gives the appearance of whimsy: it is sometimes at one value, sometimes at another, and always both changing and unpredictable. Figure 51.1B illustrates this modified classical threshold for three different instances: low, medium, and high. The evidence for this modified theory is that some intensity that is not (reported as) “seen” on one occasion may be seen on the next, and one that was seen might not be seen later.

Figure 51.1..  

A, Schema for Fechner's idea of (classical) threshold. The abscissa is contrast, and the ordinate is a dichotomous variable indicating “seen” or “not seen.” A contrast above the threshold level is seen; otherwise it is not seen. B, Modified threshold schema for the case where the threshold can vary from time to time. Coordinates are as in A. Three arbitrary instances are illustrated. The (traditional) threshold theory holds that the level of contrast above which a target is seen varies at random.

One can imagine that for repeated examples, the measured frequency seen will equal the fraction of occasions on which threshold, defined in this way, is exceeded. Thus, the measured frequency seen averages over a multiplicity of instances, and the result is a curve that rises from zero or a few instances seen and asymptotes at 100% seen. Instances of reports of seen intensity when no stimulus has been delivered have been known since the time of Fechner. Thus the frequency of reports seen may rise from a nonzero value at zero contrast. As will be seen below, there are two different ways of dealing with such events. One assumes that they are guesses and then modifies all seen events by a putative guess rate. The other approach treats these as noise-caused events and uses them to gain a greater understanding of the relevant noise.

Figure 51.2 shows an example of such a frequency seen curve (also termed a psychometric function—see the later section “The Psychometric Function”) and, on the same abscissa, the putative probability density of the threshold value. In a sense, then, this traditional concept of the threshold incorporates a sort of noise, a random variability of the value of the threshold. The conclusion that Fechner and the entire vision community reached, and that held for 100 years, was that of a randomly varying limen or threshold. The inclusion of randomness in this model may have been the first acknowledgment that noise had to play a role in threshold, though this was not explicit. How then to quantify or estimate the threshold?

Figure 51.2..  

Frequency of the seen curve (solid) for a traditional threshold observer. Another name for this function is psychometric function. The abscissa is target contrast. The ordinate is frequency seen, the fraction of trials on which the observer reports that the target is seen. The dashed curve shows the presumed underlying probability distribution of the threshold level that would have led to the solid frequency seen curve.