Academic journal article Perception and Psychophysics

# The History of Dipper Functions

Academic journal article Perception and Psychophysics

# The History of Dipper Functions

## Article excerpt

Dipper-shaped curves often accurately depict the relationship between a baseline, or "pedestal," magnitude and a just noticeable difference in it. This tutorial traces the 45-year history of the dipper function in auditory and visual psychophysics, focusing on when they happen and why. Popular theories of both positive and negative masking (i.e., the "handle" and "dip," respectively) are described. Sometimes, but not always, negative masking disappears with an appropriate redescription of stimulus magnitude.

Gaussian Noise

Did you ever have your walls dusted? I did, and the duster (who shall remain nameless) managed to knock all of my pictures out of alignment. You can try, but it is impossible to hang a picture perfectly straight. Sometimes the right side seems a little too high, sometimes the left. I would change my mind about the tilt without even moving the picture. This uncertainty is manifest in psychometric functions for orientation discrimination. An example is shown in Figure 1, which summarizes observer M.M.'s responses to visual targets having different tilts. Consider the large data point: When forced to choose, observer M.M. said "clockwise" on 14 of 58 trials in which the true stimulus orientation was 1° anticlockwise of vertical.

Notice that the psychometric function is well fit by a Gaussian distribution. That is,

Ψ(θ) [asymptotically =] Φ(θ/σ), (1)

where Φ is the normal cumulative distribution function (CDF) and σ = 1.2°. This fit suggests a Gaussian source of noise, intrinsic to the visual system, which corrupts the ability to estimate orientation.

Discriminating Different Levels of Gaussian Noise

Although persuasive, this suggestion is at odds with the way my pictures used to look-that is, perfectly straight. If the apparent orientation of each picture was corrupted by Gaussian noise, then how could they ever have seemed to be aligned? This puzzle prompted Morgan, Chubb, and Solomon (2008)1 to speculate that maybe the visual system squelches its own noise. To test this possibility, they measured how well observers could discriminate between two textures whose otherwise parallel elements were tilted with different amounts of Gaussian noise.

Figure 2A shows an example of their stimuli. The observers had a two-alternative forced choice (2AFC): They were shown two textures and asked to select the one having greater variance. This variance was manipulated until observers were responding with an accuracy of 82%.

The horizontal position of each point in Figure 2B shows the standard deviation of orientations in the less variable, or "pedestal," texture. The vertical position of each point shows how much greater the standard deviation of the other texture needed to be for M.M. to identify it with 82% accuracy. This is the JND, or just noticeable difference, in standard deviation. The JNDs form a dipper-shaped function of pedestal magnitude; that is, as the pedestal increases from zero, the JNDs first decrease and then increase. This result may seem counterintuitive, but dipper functions of this general shape, with a "dip" in the middle and a "handle" on the right, are ubiquitous in contemporary psychophysics. The historical review that follows should help to steer interested empiricists away from particularly contentious issues.

Weber's Law

Dipper functions are a subset of those functions that describe the relationship between I, a baseline magnitude,2 and ΔI^sub JND^, a just noticeable increment. An even simpler relationship, attributed to Weber (by Fechner, 1860/1912), can be written

ΔI^sub JND^/I = k, (2)

where the "Weber fraction" k does not vary with I. So much has been written about Weber's law (and the "nearmisses" thereto) that I feel unable to add anything intelligent on the topic. Instead, I direct the reader to chapter 1 of Laming (1986). His chapter 2 contains an unbeatable review of the psychophysical methods appropriate for obtaining Weber's law. …

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