The history of dipper functions

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					Attention, Perception, & Psychophysics
2009, 71 (3), 435-443
doi:10.3758/APP.71.3.435




                                                 TuTorial review
                                  The history of dipper functions
                                                          Joshua a. solomon
                                                    City University, London, England

                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                                                             Figure 2A shows an example of their stimuli. The ob-
   Did you ever have your walls dusted? I did, and the                  servers had a two-alternative forced choice (2AFC): They
duster (who shall remain nameless) managed to knock                     were shown two textures and asked to select the one hav-
all of my pictures out of alignment. You can try, but it                ing greater variance. This variance was manipulated until
is impossible to hang a picture perfectly straight. Some-               observers were responding with an accuracy of 82%.
times the right side seems a little too high, sometimes                    The horizontal position of each point in Figure 2B
the left. I would change my mind about the tilt without                 shows the standard deviation of orientations in the less
even moving the picture. This uncertainty is manifest in                variable, or “pedestal,” texture. The vertical position of
psychometric functions for orientation discrimination.                  each point shows how much greater the standard devia-
An example is shown in Figure 1, which summarizes                       tion of the other texture needed to be for M.M. to iden-
observer M.M.’s responses to visual targets having dif-                 tify it with 82% accuracy. This is the JND, or just notice-
ferent tilts. Consider the large data point: When forced                able difference, in standard deviation. The JNDs form a
to choose, observer M.M. said “clockwise” on 14 of 58                   dipper-shaped function of pedestal magnitude; that is, as
trials in which the true stimulus orientation was 1º anti-              the pedestal increases from zero, the JNDs first decrease
clockwise of vertical.                                                  and then increase. This result may seem counterintuitive,
   Notice that the psychometric function is well fit by a               but dipper functions of this general shape, with a “dip” in
Gaussian distribution. That is,                                         the middle and a “handle” on the right, are ubiquitous in
                                                                        contemporary psychophysics. The historical review that
	                          Ψ(θ)  Φ(θ/σ),                        (1)
                                                                        follows should help to steer interested empiricists away
where Φ is the normal cumulative distribution function                  from particularly contentious issues.
(CDF) and σ 5 1.2º. This fit suggests a Gaussian source
of noise, intrinsic to the visual system, which corrupts the            Weber’s Law
ability to estimate orientation.                                           Dipper functions are a subset of those functions that de-
                                                                        scribe the relationship between I, a baseline magnitude,2
Discriminating Different Levels of                                      and DIJND , a just noticeable increment. An even simpler
Gaussian Noise                                                          relationship, attributed to Weber (by Fechner, 1860/1912),
   Although persuasive, this suggestion is at odds with the             can be written
way my pictures used to look—that is, perfectly straight.
                                                                        	                       DIJND /I 5 k,                      (2)
If the apparent orientation of each picture was corrupted
by Gaussian noise, then how could they ever have seemed                 where the “Weber fraction” k does not vary with I. So
to be aligned? This puzzle prompted Morgan, Chubb, and                  much has been written about Weber’s law (and the “near-
Solomon (2008)1 to speculate that maybe the visual sys-                 misses” thereto) that I feel unable to add anything intel-
tem squelches its own noise. To test this possibility, they             ligent on the topic. Instead, I direct the reader to chapter 1
measured how well observers could discriminate between                  of Laming (1986). His chapter 2 contains an unbeatable
two textures whose otherwise parallel elements were tilted              review of the psychophysical methods appropriate for ob-
with different amounts of Gaussian noise.                               taining Weber’s law.


				
DOCUMENT INFO
Description: 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. [PUBLICATION ABSTRACT]
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