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Vision Research 40 (2000) 3121–3144
Signal detection theory in the 2AFC paradigm: attention,
channeluncertainty and probability summation
Christopher W. Tyler *, Chien-Chung ChenSmith–Kettlewell Eye
Research Institute, 2318 Fillmore St., San Francisco, CA 94115,
USA
Received 17 March 1999; received in revised form 3 December
1999
Abstract
Neural implementation of classical High-Threshold Theory reveals
fundamental flaws in its applicability to realistic neuralsystems
and to the two-alternative forced-choice (2AFC) paradigm. For 2AFC,
Signal Detection Theory provides a basis foraccurate analysis of
the observer’s attentional strategy and effective degree of
probability summation over attended neuralchannels. The resulting
theory provides substantially different predictions from those of
previous approximation analyses. Inadditive noise, attentional
probability summation depends on the attentional model assumed. (1)
For an ideal attentional strategyin additive noise, summation
proceeds at a diminishing rate from an initial level of fourth-root
summation for the first fewchannels. The maximum improvement
asymptotes to about a factor of 4 by a million channels. (2) For a
fixed attention field inadditive noise, detection is highly
inefficient at first and approximates fourth-root summation through
the summation range. (3)In physiologically plausible
root-multiplicative noise, on the other hand, attentional
probability summation mimics a linearimprovement in sensitivity up
to about ten channels, approaching a factor of 1000 by a million
channels. (4) Some noise sources,such as noise from eye movements,
are fully multiplicative and would prevent threshold determination
within their range ofeffectiveness. Such results may require
reappraisal of previous interpretations of detection behavior in
the 2AFC paradigm.© 2000 Elsevier Science Ltd. All rights
reserved.
Keywords: Psychophysics; Summation; Probability summation; 2AFC;
Attention; Uncertainty; Signal detection theory; Additive noise;
Multiplica-tive noise
www.elsevier.com/locate/visres
1. Introduction
A principal function of early human vision is toanalyze the
spatial structure of images of the visualworld. This information is
then used to develop arepresentation of the properties of the
objects before usand their layout in 3D space and in time.
Despiteprevious attempts, a valid analytic framework has yetto be
applied to the variety of spatial integration phe-nomena measured
in laboratory studies. The analysisprovided in this paper will
demonstrate the deficienciesin previous approaches and form the
basis for a com-prehensive analysis of spatial summation based on
thetenets of Signal Detection Theory, specifically in thecontext of
detection and discrimination tasks measuredby the two-alternative
forced-choice (2AFC) paradigm.
The analysis is valid for summation in any stimulusdomain, but
it will be illustrated with specific referenceto summation in one
and two-dimensional spatialvision.
Detailed analysis of summation behavior requiresaccurate models
of the kinds of summation principlesthat can operate in
psychophysics. The kind of summa-tion performed by physiological
receptive fields will betermed physiological summation (whether
linear ornonlinear), to distinguish it from probability summa-tion
performed on the outputs of a set of decisionvariables (even though
the latter operation must alsoultimately be a physiological process
in the brain). Theprimary theoretical analysis will be developed
under theassumptions of Signal Detection Theory: that the
mainsource of noise is external, Gaussian and independentof
stimulus contrast. The theory also encompasses con-ditions where
threshold is dominated by internal Gaus-sian noise and other forms
of the noise distribution.
* Corresponding author. Fax: +1-415-3458455.E-mail address:
[email protected] (C.W. Tyler).
0042-6989/00/$ - see front matter © 2000 Elsevier Science Ltd.
All rights reserved.PII: S0042-6989(00)00157-7
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C.W. Tyler, C.-C. Chen / Vision Research 40 (2000)
3121–31443122
Summation is quantified over arrays of processingmechanisms that
are equal in sensitivity, although thetheory could be extended to
arbitrary sets of processes.Extension of the analysis to cases
where the internalnoise properties are some function of internal
signalstrength reveals major departures from the behaviorwith
independent noise.
No complete account of summation behavior underthe 2AFC paradigm
has been published despite itswidespread use for several decades.
The most extensivepublished analysis of these issues is by
Pelli(1985), which provides the basis for much of thepresent
treatment, although many of our conclusionsdiffer from the
approximations derived in that paper.One of Pelli’s goals was to
show that WeibullHigh-Threshold Theory could approximate the full
pre-dictions of Signal Detection Theory for the 2AFCparadigm. The
approximations were valid over a lim-ited range under the
assumptions Pelli made, but he didnot develop the theory in more
general cases. A keyassumption was that the human observer is
alwaysoperating under conditions of high uncertainty.
Thisinterpretation seems inherently implausible inpracticed
observers and we show that there are condi-tions under which this
assumption is violated. Hencethe 2AFC predictions need to be
developed in accurateand usable form for a full treatment of
psychophysicaldata.
The paper is divided into four main sections. Thefirst section
considers the implications of previousanalyses of 2AFC probability
summation throughHigh Threshold Theory and finds these approaches
tobe fundamentally flawed in several respects. Thesecond section
develops the analysis of 2AFCsummation through Signal Detection
Theory limited byadditive noise (from either external or
internalsources). In the third section, the implications of
avariety of non-ideal attentional strategies arespelled out for
this additive noise case. The finalsection expands the analysis to
cases where theinternal noise properties are some multiplicative
func-tion of internal signal strength, revealing major depar-tures
from the behavior with signal-independentnoise.
1.1. Assumptions of the 2AFC analysis
The assumptions of the main 2AFC analysis (Sec-tions 3–5) are
generally straightforward. There are alsosubsidiary issues that
arise from considering alterna-tives to some of the assumptions.
These alternatives arenoted in brackets (A note on terminology; The
term‘distribution’ is used here to imply a probability
densityfunction, PDF, to which some noise variable conforms,as in
‘Gaussian distribution’. The cumulative integral of
such a function is termed its ‘cumulative distributionfunction’,
or CDF).
1. In the 2AFC paradigm, the observer is presentedwith two
defined stimulus events, both containingsome background condition,
while one also con-tains a test stimulus to be detected. The
observer’stask is to indicate which of the two events includedthe
test stimulus.
2. There are sources of noise present in the stimulusevents. Any
component of the noise that is corre-lated between the two events
forms part of thebackground from which the test is to be
discrimi-nated. We therefore consider ‘noise’ to include allsources
of trial-to-trial variation that are uncorre-lated between the
stimulus events.
3. The noise is assumed to be white in space and time(for a
fixed stimulus level) and Gaussian in itsprobability density
function (PDF). The Gaussianassumption is plausible because of the
CentralLimit Theorem that the PDF for combinations ofnon-Gaussian
noise is asymptotically Gaussian. Ifthere are many sources of
external and internalnoise impinging at the decision site,
therefore, theresulting noise is most likely to be Gaussian.
[Al-ternatively, the PDF is assumed to take the form ofa Poisson
noise distribution.] [The Ideal Observerformulation makes the
restrictive assumption thatthere are no noise sources except those
present inthe stimulus.]
4. The noise is assumed to be additive and indepen-dent of the
strength of the test stimulus. [Alterna-tively, the noise variance
is assumed to vary assome function of stimulus strength.]
5. Without the noise, the internal signals for eachmechanism on
which a decision is based are as-sumed to vary linearly with
stimulus strength. [Al-ternatively, the internal signal is assumed
toincrease directly with stimulus strength above somelevel but be
limited by a threshold such that theinternal signal remains at zero
below that level. Ifthe threshold occurs at or above the level of
thesystem noise in the absence of a test stimulus, it isknown as a
‘high threshold’.]
6. The visual system is assumed to consist of some(large) number
of local mechanisms that transmitindependent signals concerning the
state of theoutside world. The mechanisms are independent inthe
sense that their noise sources are statisticallyindependent.
7. Each local mechanism is assumed to summate lin-early over
space within some weighting functionknown as its summation field.
The summation maybe over signals that are preprocessed for
somestimulus attribute (such as orientation) by earlierneural
mechanisms. [The Ideal Observer formula-
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C.W. Tyler, C.-C. Chen / Vision Research 40 (2000) 3121–3144
3123
tion assumes that there is a summation field match-ing the
profile of each stimulus presented.] [Theinefficient Ideal Observer
formulation assumes thateach summation field is incompletely
sampled to asimilar extent, with the loss of a constant propor-tion
of information for all fields.]
8. The local mechanisms are assumed to draw fromthe same local
noise sources at all sizes of summa-tion fields.
9. The signals from the local mechanisms are as-sumed to be
combined by some nonlinear processknown as an ‘attention field’
that is able to surveythe local signals and isolate the largest
signal. Theimplications of several types of control over thesize of
the attention field are considered. [The IdealAttention formulation
assumes that the attentionfield matches the stimulus extent, even
when thelocal summation fields do not.]
10. The observer’s 2AFC decision is assumed to derivefrom the
larger of the signals from the attentionfield for the two stimulus
events.
2. Problems with High Threshold Theory in thepresence of
additive noise
This section considers the implications of previousanalyses of
2AFC probability summation in relation toHigh Threshold Theory and
finds inherent problemswith such approaches in several respects.
These flawsindicate that High Threshold Theory does not providea
firm basis for the analysis of attentional integration ofneural
information in the presence of additive noise. Toexplain these
problems, we first review High-ThresholdTheory, but the source
references should be consultedfor full details.
2.1. O6er6iew of High Threshold Theory
High Threshold Theory (Quick, 1974) is an analysisof the
detection of signals that assumes that detection islimited by a
noise-free, or fixed, threshold, below whichno stimulus information
is transmitted (Fig. 1a). Thetheory gets its name because the
threshold is assumedto be high with respect to any noise in the
signalarriving at the decision site. The goal of HighThreshold
Theory is to define the properties of summa-tion over independent
channels, which has come to beknown as ‘probability summation’. In
spatial vision, theprobability summation hypothesis implies that
themechanism of attention is distributed over many spatialchannels
rather than being focal, since one cannotmonitor many channels
without attending to them. It isthen assumed that, on every trial,
the attention mecha-nism can select the maximum channel response
over themonitored range for use in the detection decision andignore
all other channels. Probability ‘summation’ isthus a max operator
rather than a summing operator inthe normal sense, and has
generally been considered asthe minimal combination rule among
independentmechanisms.
The psychometric function C is the theoretical formof the
observer’s proportion correct in a detection taskas a function of
stimulus strength. In Quick’s (1974)version of High Threshold
Theory, the psychometricfunctions Ci for each individual channel
with meanresponse Ri are given by the Weibull function:
Ci=1−e− (Ri )
b
,
where Ri= f� s
ai
�for stimulus strength s (1a,b)
with f being in general any monotonic function, aidetermining
the sensitivity of the ith local mechanismand b controlling the
steepness of the psychometric
Fig. 1. (a) High-threshold analysis, that noise distribution in
the absence of signal (left distribution) lies below some threshold
level (Ru). Thesignal distribution varies in its position as the
signal varies (arrow), and passes across the threshold as R
increases to reveal some proportion ofthe signal distribution of
correct responses (shaded area). (b) Weibull predictions for
probability summation over number of samples (in time, areaor any
other stimulus parameter) for assumed psychometric exponents of b=4
and 1.3 (d % powers of 3.2 and 1; see below).
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C.W. Tyler, C.-C. Chen / Vision Research 40 (2000)
3121–31443124
function. For the remainder of this treatment, we willassume
that f is a linear function, such that
Ri=sai
, s\0. (1c)
How the function behaves for negative s depends onthe stimulus
domain. For luminance, and for (Michel-son) contrast, there are no
negative signals, so thefunction does not exist in the negative
region. For otherstimulus domains, the negative portion will have
to beanalyzed according to its particular properties.
The Weibull function is derived from the theory ofFailure
Analysis and represents the combination ofexponentially decaying
failure functions. The effect onthe overall psychometric function
of probability sum-mation over the set of channels (assuming equal
sensi-tivities ai for the mean responses Ri of the
individualchannels to the stimulus) is based on the
standardstatistical formula that probabilities of not
detectingmultiple events should be multiplied together:
C=1−e− (R)b
=1−5n
(1−Ci)=1−5n
(1−e− (Ri )b
)
=1−e− (n1/bRi )
b
(2)
Psychophysical threshold is estimated by solving Eq. (2)for
C=0.5 (This basic version of the theory assumesthat the observer’s
guessing rate is zero). For a criterionlevel of the output function
C, the similarity in form ofthe first and last expressions makes it
clear that theeffective mean response over the set of channels
isR=n1/bRi (see Robson & Graham, 1979, for details).Thus, as
the stimulus extent is increased to sample moreof the local
mechanisms, the internal response increasesin proportion to the bth
root of the number of mecha-nisms sampled by the attention
mechanism.
Fig. 1b depicts the degree of probability summationover the
number of mechanisms sampled for the typicalcase of b=4 and for the
hypothetical case of a linearpsychometric function, when b=1.3
(Pelli, 1987). Em-pirically, the exponent b of the Weibull
approximationto psychometric data may take values from 1.3 to
6(Mayer & Tyler, 1986). Under the assumption that f isa linear
function (Eq. (1c)), the low value represents thetheoretical low
limit on the expected slope if the stimu-lus is present in all the
channels that the visual systemis monitoring (and there is no phase
uncertainty; Pelli,1985). A high value for b represents a high
degree ofchannel uncertainty. When there is minimal uncer-tainty,
probability summation effects are predicted tobe large relative to
the possible contrast measurementrange (Fig. 1). If b=1.3, for
example, sensitivity im-provement of as much as a factor of 200 is
predicted forprobability summation over n=1000 equally stimu-lated
channels. Such a result would be predicted by anincrease in
stimulus diameter by a factor of �30 onhomogeneous retina, if n
represents the number of local
retinal filters). Under such conditions, probability sum-mation
could not be dismissed as a minor, near-threshold effect. The
generation of such largesummation effects from purely attentional
processeswould cloud the issue of what physiological summationmight
be taking place because the two effects are ofcomparable
magnitude.
2.2. High threshold analysis of probability summationassumes
non-Gaussian additi6e noise
Suppose a signal with intensity s can produce aninternal
response distribution D(r ;R,s), where r repre-sents the dimension
of the random internal responsevariable, with mean R (which is
assumed to be amonotonic function of s) and standard deviation
s.Under the assumption of a high threshold, this noisedistribution
is progressively revealed as the signal inten-sity moves up beyond
the threshold level. Thus, if thenoise distribution is additively
independent of the meanresponse, the probability of detecting
signal s is theintegral of the internal signal-plus-noise
distributionfrom the threshold Ru to infinity. The Weibull
formula-tion of the psychometric function (equation 1a)
mustcorrespond to this integral for some particular
noisedistribution Db(r−R) around the mean signal R,
C=1−e−Rb=&�
Ru
Db(r−R)dr (3)
where Ru is the mean internal response level atthreshold.
For the assumption that the PDF of the signal+noise distribution
generating the Weibull function is offixed form, Db(r), it can be
solved by taking thederivative of both sides of Eq. (3) for each
integrationlimit
bRb−1 e−Rb
= limo�
Db(o−R)−Db(Ru−R) (4a)
from which,
Db(r)=brb−1 · e−rb, with r=R−Ru−r,
for rBR−Ru (4b)
On the assumption that the mean response R is linearwith the
external stimulus strength s, equation (4) defi-nes the implied PDF
that would have generated theWeibull expression for the measured
psychometricfunction. The forms of the psychometric functions
andthe implied noise distributions Db(r) for values of bfrom 1.3 to
8 (corresponding to d % exponents from 1 to6.5; see following
sections) are shown in Fig. 2 for aYes/No experiment (assuming zero
false alarm rate). Itis evident that the implied noise
distributions in thelower panel are generally far from
approximating aGaussian form except in the mid-range of
parametervalues, the special case where b:4 (the value for
which
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C.W. Tyler, C.-C. Chen / Vision Research 40 (2000) 3121–3144
3125
Fig. 2. (a) Psychometric functions with Ru=1 predicted by
HighThreshold Theory for values of 1.3, 2, 4 and 8 for the exponent
b.Functions are corrected for guessing. (b) Implied noise
distributionfunctions according to equation (4), plotted relative
to the meanresponse (i.e. as the additive noise distribution). Note
the markedchange in distribution shapes as the exponent varies.
provement in sensitivity with an increasing number ofchances to
detect the presence of a signal. This theoryassumes that
probability summation occurs because theobserver can identify the
max of the samples of sig-nal+noise distributions provided by each
of n stimu-lated channels. If we assume a physiological version
ofthe high-threshold system that has Gaussian noiseadded to the
signal, the internal response after proba-bility summation is
provided by the distribution of suchmax values over trials. Note
that, for such probabilitysummation to occur, the threshold has to
be appliedafter the max operator. For this analysis, the
maxoperator is assumed to function like an ideal
attentionmechanism, in that it samples from all of, and onlyfrom,
the relevant channels.
In general, it is a well-known statistical rule that
thecumulative distribution of maximum values for a set ofsamples
D(ri) from a parent distribution D(r) (where ris the instantaneous
internal response) is given by theintegral of the parent
distribution to the power of thenumber of values within each
sample:& r
−�
max[D(ri)]i=1:n
dr=�& r
−�
D(r %)dr %nn
(5)
Thus, the expected distribution Mn(r ;R,s) of the maxof a set of
samples is given by taking the derivative ofboth sides of Eq. (5)
with respect to their independentvariables:
Mn(r ;R,s)= maxi=1:n
[D(ri)]=ddr�& r
−�
D(r %)dr %nn
(6)
The mean R and standard deviation s parameters inthe expression
for the max distribution Mn imply thatwe are deriving the form of
the expected function of theresulting probability distribution,
which may be charac-terized by the parameters of its location and
spread. Itdoes not imply that these are the only parameters of
thedistribution (as they would be for a Gaussian distribu-tion),
merely that we restrict our consideration to thesetwo
parameters.
To obtain the new threshold signal level, the signalcan be
reduced until the max distribution Mn reachesthe original threshold
criterion again. The extent towhich the signal has to be reduced
constitutes theimprovement in sensitivity attributable to
probabilitysummation on the basis of the max rule. If the noise
isassumed to be additive, however, this process createsthe fatal
problem that, for a large enough number ofchannels, the mean signal
needs to be set to a negativevalue in order to bring the
signal+noise distributiondown to threshold. Fig. 3a depicts the
case for suchsummation over 100 channels, where the initial signal
isassumed to have a mean of two times the internalthreshold level
and a s of 0.67 (so as to provide 75%correct performance at this
signal level). The max distri-bution for 100 channels from Eq. (6)
has a mean of
Pelli, 1983, established that the Gaussian is a
goodapproximation). Thus, Weibull analysis is not an accu-rate
theory for the description of systems with a highthreshold and
Gaussian noise unless the psychometricslope happens to fall at this
mid-range value. In prac-tice, empirical slopes have been found to
approximatethis value in many situations (Robson & Graham,
1981;Williams & Wilson, 1981; Pelli, 1985), but there may
besubstantial inter-observer differences (Mayer & Tyler,1986)
and large changes in slope under certain circum-stances (Tyler,
1997). Thus, there is a need for acomprehensive and accurate theory
of probability sum-mation when the assumptions of High Threshold
The-ory are violated.
2.3. High-threshold probability summation fails foradditi6e
noise
High Threshold Theory has been widely used topredict the effects
of probability summation, the im-
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C.W. Tyler, C.-C. Chen / Vision Research 40 (2000)
3121–31443126
about 4.5, giving essentially 100% correct performance.Fig. 3b
shows how the signal has to be readjusted tobring the tail of the
max distribution down belowthreshold so as to reattain 75% correct
performance.On the linear assumption of Eq. (1c), the level of
themean internal response in the null interval correspondsto an
external signal of zero (The internal scale isarbitrary, so we
choose zero to represent the mean nulllevel for analytic
convenience). Thus, since the internalsignal needs to be reduced
from 4.5 back to thethreshold level of 1.0 (dashed arrow) the
external signalrepresented by the filled arrow has to go
substantiallynegati6e before 75% correct performance is
achieved.
The problem is fatal in some domains, such as theamplitude of
light, because negative signals do notexist. Other domains, such as
contrast, may be definedin such a way that there are negative
signals, but theproblem reasserts itself because the system
containsnegative-sensitive elements (e.g. off-center cells)
thatrespond positively to the negative signal. Thus, ratherthan
becoming less detectable by its max value, thesignal becomes more
detectable as the correspondingminimum of the set of samples (at
the left-hand tail ofthe distribution in Fig. 3) passes above the
correspond-ing negative threshold before the max falls below
thepositive threshold. Once again, therefore, it is impossi-ble to
return to the 75% performance level after themax operator has taken
effect.
High-threshold analysis is immune to this problemonly if the
noise on the signal is multiplicati6e withsignal strength rather
than additive, and hence can bereduced indefinitely by appropriate
signal reductionswithout the signal going negative. Thus, if the
noise ispurely multiplicative, the max level on the noise
distri-bution may be freely reduced to the threshold level to
provide a measure of the threshold sensitivity for theinput
signal. High-threshold analysis is self-consistentin that the noise
implied by the Weibull formulationhas the property of being
multiplicative. Because thisproperty is rarely made explicit, it
should be mentionedthat the property follows from Eq. (2), which
showsthat the Weibull psychometric function has a constantform when
plotted on log coordinates, i.e. is scaled inproportion to signal
amplitude. This implies that thelimiting noise is similarly scaled
through the probabilitysummation operator. To reiterate, Fig. 3
goes further inshowing that the assumption of additive noise is
incom-patible with Weibull analysis in general.
Note that, for mixed additive and multiplicative noisesources,
reducing the signal will tend to reduce themultiplicative noise to
the point where additive noisedominates. Since there are always
sources of additivenoise in any physical signal-detection system
(e.g. ther-mal noise, and quantal noise considered with respect
tomodulation variables, such as a sinusoidal grating,which keep the
mean signal constant), any noise-limitedthreshold is likely to be
limited by its additive compo-nent. The only amelioration of this
problem is if thehigh threshold is so high that it sits at or above
the levelfor the max of the additive noise from all
monitoredchannels (which one might term an ‘ultra-high’threshold).
Were it any lower, the negative signal prob-lem would be
encountered. Thus, for Weibull analysisto operate, the system must
be functioning withthresholds so high as to be quite inefficient,
especiallyconsidering that the degree of probability
summationrequired by the quantitative application of
UncertaintyTheory may be of the order of many thousands or
evenmillions of channels (Pelli, 1985). The Weibull analysisof
probability summation is thus implausible in realisticthreshold
systems.
Fig. 3. (a) Threshold signal-plus-noise distribution for 75%
correct detection (left distribution, with 75% of the area above
the threshold level of1) together with the distribution of maxes
over 100 channels (right distribution). Since the max distribution
is effectively all above the thresholdlevel the signal would be
detectable close to 100% of the time if probability summation were
in operation. (b) Thus, the signal level has to bereduced (dashed
arrow) until the max distribution sits at the 75% level above
threshold. The problem is that this reduction produces a
negativevalue for the mean signal in each channel (filled arrow),
which is likely to be unobtainable in typical vision paradigms.
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C.W. Tyler, C.-C. Chen / Vision Research 40 (2000) 3121–3144
3127
The conclusions from the analysis of High ThresholdTheory are:1.
The high-threshold analysis developed by Quick
(1974) implies a reciprocal relationship between theexponent of
the psychometric function and the logslope of its probability
summation behavior (Fig.1b). If there are conditions where
psychometricfunctions are empirically found to be shallow (andthe
noise sources locally independent), steep summa-tion slopes would
be predicted. In practice, suchconditions have been found to show
shallow sum-mation slopes, calling the theoretical frameworkinto
question. For suprathreshold masking condi-tions, a range of
studies such as Foley and Legge(1981) and Kersten (1984) report
exponents of the d %function close to 1 for the 2AFC paradigm,
evenwhen the mask is a noise background that is ran-domly
independent at all locations within the stimu-lus. Nevertheless,
Kersten (1984) showed thatsummation is negligible under
suprathreshold,noise-masked conditions. Both because such
near-unity exponents imply strong summation behavior,and because it
is hard to conceptualize a thresholdoperating under
‘suprathreshold’ conditions, HighThreshold Theory cannot be applied
to suchdata. There is thus need for a theory that can beused to
analyze suprathreshold discrimination ex-periments.
2. The form of the Weibull function implies bizarrevariations in
the noise distribution (Fig. 2b) if it isassumed that the neural
noise is additive in thethreshold range. Since noise asymptotically
Gaus-sian (such as quantal noise in the light, thermalnoise in the
photoreceptors or retinal noise in theganglion-cell outputs), High
Threshold Theory isincompatible with plausible assumptions about
thenoise distribution.
3. Quick’s High Threshold analysis through theWeibull function
assumes the performance islimited by a high threshold rather than
by noise ofany kind. However, noise is an unavoidablecomponent of
the analysis of the 2AFC paradigm.In order to adapt the
high-threshold analysisto the 2AFC paradigm, Pelli (1985) made the
as-sumption that the observer was monitoringa much larger number of
channels than were stimu-lated as a means of obtaining a steep
psychometricfunction that approximated threshold behavior.Thus,
Pelli’s approximation fails if the stimulus isstructured so as to
stimulate as many channels asthe observer is monitoring, because
thepredicted psychometric function is then shallowand violates the
high-threshold assumptions. Notheoretical analysis for these
conditions has beenpublished.
3. Signal Detection Theory for the Ideal Observer andits
Bayesian approximation
Following from the inadequacies of High ThresholdTheory, this
section develops the analysis of summationproperties in the
two-alternative forced-choice (2AFC)task. The analysis is
approached by specification of thepsychometric function through
Signal Detection The-ory as limited by additive noise. When the
only sourceof this additive noise is quantum fluctuations (in
acontrast detection task), the Signal Detection Theoryanalysis
amounts to a single-channel Ideal Observermodel. The implications
of an attentional strategy ap-proximating Ideal Observer behavior
are also spelledout for this additive noise case.
3.1. Specification of the psychometric function
The first step to understanding psychometric functionin a 2AFC
task is to specify the proportion correct ofthe observer’s
responses. The 2AFC task typically in-volves the presentation of
two stimulus intervals (orspatial stimulus regions), one of which
contains thestimulus to be detected while both contain the
back-ground condition from which the stimulus is to
bedistinguished. The observer’s task is to estimate whichinterval
contains the discriminative stimulus. Tradition-ally, the observer
is assumed to exhibit ideal behaviorin three ways:1. to have exact
knowledge of the stimulus and to view
it with a matched filter, excluding all
irrelevantinformation
2. to be noise-free; performance is limited only bynoise in the
physical stimulus
3. to respond according to the maximum output of thefilter in
the two intervals, with no confusion.
When the first assumption is violated by ignorance ofthe correct
filter, the observer may still adopt an idealattentional strategy
across a set of filters, to make thebest guess as to which is the
optimal filter to select oneach trial. The second assumption may be
violated bythe introduction either of early noise before the filter
orof late noise at the decision stage. In the case of earlynoise,
the observer’s performance will still reflect theform of the Ideal
Observer, but at reduced efficiency. Inthe case of late noise, the
threshold will become inde-pendent of stimulus extent as long as
the late noisedominates other sources of noise.
In Signal Detection Theory (SDT), the proportioncorrect is
conceptualized through an imaginary ROC(receiver operating
characteristic) curve of proportionof hits versus proportion of
false alarms (Green &Swets, 1966), treating each trial as a
separate Yes/Notask with a different criterion. Not only is this
instanta-neous criterion inaccessible, but the 2AFC
proportioncorrect is defined as the area under the ROC curve,
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Fig. 4. Derivation of the 2AFC psychometric function. (a)
Internalresponses for signal levels from zero (distribution xn)
through variousmean levels R1−Rm. (b) Difference distributions
between the distri-butions for each signal level and that for no
signal. Proportioncorrect is derived by integrating areas p1 to pm
to the right of thevertical line at zero difference indicating the
response criterion. (c)Psychometric function derived from plotting
the area of each differ-ence distribution lying above the zero
criterion from (b). (d) Log (z)transform of the cumulative function
to provide a straight line inprobit coordinates.
For signal intensity s, let r1 be the internal responseto the
first observation interval, r2 be the internalresponse to the
second interval and k be the interval theobserver chose as the
signal interval. (Note that here sserves as a scalar for signal
strength and s as an indexfor the signal interval. Similarly, n is
the index for thenull interval whereas n defines the number of
stimulatedchannels elsewhere.) We assume that there is a
fixedsignal level throughout the test interval.
The observer indicates the first interval as the signalinterval
(6=1) if r1−r2\0 and indicates interval (6=2) if r2−r1\0. The
response is correct if either 6=1when the signal is the first
interval, denoted by Bsn\or 6=2 when the signal is the second
interval, denotedby Bns\ . The proportion correct in terms of
theinternal difference response d is:
pcorr(d)=p(6=1�Bsn\ )*p(Bsn\ )+p(6=2�Bns\ )*p(Bns\ )
=p(r1−r2\0�Bsn\ )*p(Bsn\ )+p(r2−r1\0�Bns\ )*p(Bns\ ) (7)
If rn is the internal response to the null interval and rsis the
internal response to the signal interval, Eq. (7)can be rewritten
in terms of the psychometric function:
C(s)=p(rs−rn\0�Bsn\ )*P(Bsn\ )+p(rs−rn\0�Bns\ )*p(Bns\ )
=p(rs−rn\0)=p(d\0)=&�
0
Zs(d ;D)dd (8)
where d=rs−rn is the difference between signal andnull interval
internal responses and Zs(d ;D) is the PDFof the difference
distribution for signal strength s (nor-malized in units of its
standard deviation), with meanD.
Eq. (8) describes the relation between the proportioncorrect and
the observer’s internal responses to signaland null intervals. The
psychometric function can beobtained by repeating the computation
of Eq. (8) for allrelevant signal intensities. Fig. 4 illustrates
the relationsbetween the internal responses and psychometric
func-tion based on Gaussian additive noise.
The probit transform (Finney, 1952) is the appropri-ate
representation of the psychometric function, on thebasis of the
additive Gaussian assumption. It normal-izes proportion correct to
its standard deviation unit(z-score) through the inverse cumulative
Gaussian func-tion F−1. That is
Zs(d ;D)=F−1(C(s)) (9a)
In the psychophysical literature, the normalized signalZs(d ;D)
represents the detectability of the signal atstimulus level s,
defined by
d %=Zs(d ;D) (9b)
which requires a further level of abstraction from
thistwo-dimensional distribution of signal strength andcriterion
level (see Macmillan & Creelman, 1993 fordetails).
The 2AFC task is amenable to a simpler form ofanalysis based on
the difference distribution of theinternal responses (MacMillan
& Creelman, 1993). Oneach trial, the observer responds by
indicatingwhichever observation interval produces a larger
inter-nal response, which amounts to taking the differencebetween
the two internal signals and picking the inter-val according to the
sign of this difference signal (seeFig. 4). The criterion is
therefore fixed in this differencespace, at a difference of zero
(whereas it can range fromtrial to trial over the whole extent of
the internalresponse distribution). Stated formally:
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3.2. Comparison of a Gaussian and a non-Gaussianexample
Suppose the system response is dominated by onechannel whose
internal response distribution DN(r) inthe null interval is added
Gaussian noise with expectedvalue 0 and standard deviation s,
denoted as DN(r):G(r ;0,s). At signal intensity s, the internal
response inthe signal interval will also have a Gaussian
distribu-tion but with mean R and standard deviation s, de-noted as
DR(r):G(r ;R,s). From the properties of theGaussian distribution,
the difference distribution ofd=rs−rn is another Gaussian
distribution with meanR and standard deviation 2s. From Eq. (8),
theproportion correct is
C(s)=1−F(d,D,sN+sR)=1−F(−r ;R,2sN)=F(r ;R,2sN) (10)
where F denotes the Gaussian cumulative distributionfunction.
Eq. (10) is commonly used in fitting thepsychometric function to
2AFC data (MacMillan &Creelman, 1993).
In general, however, it is important to avoid theimplication
that the psychometric function matches thecumulative distribution
function of its underlying prob-ability distribution. The match is
valid only if the noiseis additi6e to the mean internal signal
strength R and ifits distribution is symmetric (as revealed by Eq.
(10)).In general, different signal levels may produce differentrs
distributions if the noise is non-additive, and in turnaccess
different DR(d). Thus, the general 2AFC psycho-metric function
would not be a cumulative function ofany particular difference
distribution. Only when thenoise is additive and symmetric (e.g.
Gaussian) will the
difference distributions at different signal levels all havethe
same variance and the psychometric function isequivalent to its CDF
(e.g. the cumulative Gaussian orerf ). On the other hand, if the
noise distribution isPoisson rather than Gaussian (a common
alternativeassumption) the noise is no longer additive but
varieswith the mean level, and also is asymmetric. Thus,
thepsychometric function derived from Eq. (8) will notexactly match
the cumulative distribution function (Fig.5).
3.3. O6er6iew of Ideal Obser6er analysis
The Ideal Observer formalism assumes that the ob-server has
complete knowledge of the stimulus and usesa single matched filter
to detect its presence (Wiener,1949). The Ideal Observer therefore
is effectively aBayesian detector with a prior probability of 1.0
on thematched filter and zero elsewhere. Optimal performancewith an
ideal filter is assumed to occur with linearsummation over the
noisy filter inputs sampled by thefield. The summation properties
of the filters will varywith respect to a large number of stimulus
attributes.For simplicity, we consider the case of spatial
summa-tion over two-dimensional stimuli S(x,y) varying in
onedimension of overall size. This variable size dimensioncould be
the height, the width, the area, or any parame-ter that is linear
with the number of sources of input toeach summing field over the
domain (x,y). The inputfor the matched filter is provided by
discrete sensorswith independent noise sources drawn from the
sameunderlying distribution. When the local regions haveidentical
sources of independent Gaussian noise withstandard deviation s, the
summed output of each fieldis given by summing over the product of
the stimulusprofile and the matching ideal filter. We can show
thatthe signal-to-noise ratio in such a matched filter
isproportional to the square root of the stimulus area.
In general, the response of the matched filter can
beapproximated as the weighted sum of its responses tothe
samples
R=% S(x,y) · I(x,y) (11a)
and the signal variance as the weighted sum of the
localvariances
sR2 =% S(x,y) · I(x,y) · s2, (11b)
Hence
sR=s�% S(x,y) · I(x,y)�1/2 (11c)
The discriminability of the ith stimulus, d %i, therefore,can be
approximated as the reciprocal of the standarddeviation times the
sampling interval. Appendix A
Fig. 5. Theoretical psychometric function for Poisson noise
(fullcurve), showing failure to match the cumulative distribution
in thiscase of non-Gaussian noise (dashed curve).
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3121–31443130
Fig. 6. Physiological implementation of ideal observer behavior.
Thincurves: summation behavior for five individual Gaussian filters
as theextent of a Gaussian test stimulus is varied (arrowheads
indicate filterextents or number of input samples in each filter at
half-height).Thick curve: fourth power attentional summation over
the individualchannels approximates ideal observer summation
behavior (a logslope of −0.5, dashed line) in the range where
physiological filtersare available, with departures below and
above. The ideal strategy isto read out from that filter only when
it matches the stimulus extent.At the two ends of the range, only
one filter dominates detectionbehavior and hence system performance
(thick line) departs from theideal slope of −0.5 to follow the
function for the most sensitive filterin that region.
If the task is summation over a range of stimulussizes, the
Ideal Observer model requires a summingreceptive field matching
every size of stimulus for whichthe summation behavior is
exhibited. A physiologicalimplementation of such behavior is
depicted in Fig. 6,where the attention mechanism is assumed to
switch tothe receptive field size matching the stimulus presentedin
each condition. This behavior is possible only if thestimuli are
presented in blocks of trials, so that theform of the next stimulus
on each trial is known. Thus,if human observers exhibit a log-log
summation slopeof −1/2 (dashed curve in Fig. 6) they may be said
tomanifest Ideal Observer behavior, in the sense of usingideal
matched filters to improve in the way an IdealObserver would, even
if the absolute sensitivity is lessthan predicted for an Ideal
Observer (i.e. lower thanideal efficiency). Such (inefficient)
Ideal Observer be-havior may be taken as evidence that the brain
hasaccess to summing fields matching the sizes of all thetested
stimuli, either present and selectable by attentionas in the
central region of Fig. 6, or alternatively as anadaptive mechanism
re-forming itself for each newstimulus condition.
If the system has access to only a limited range ofsumming field
sizes, the summation slope shouldasymptote to −1 for stimulus sizes
below that of thesmallest summing field size and should asymptote
to 0for sizes above that of the largest summing field, asdepicted
by the bold curve in Fig. 6. Thus, the form ofthe summation
function in any stimulus domain carriesimportant information about
the range of summingfield sizes operating in that domain (see Gorea
& Tyler,1997, for an example in the temporal domain andKersten
(1984), for an example in the spatial domain).The model that the
brain contains an adaptive filterre-forming itself for each new
stimulus condition seemsto be incompatible with the occurrence of a
limitedsummation range, for why would such adaptive capa-bility
fail at a particular point?
3.4. The concept of probability summation
Probability summation is an option available to adecision
mechanism with access to a number of inde-pendent signals
reflecting the occurrence of a stimulus.The analogy is with a group
of human monitors look-ing out for an approaching plane, for
example. Theprobability of detecting the plane is higher if
detectionis considered to have occurred when any one of themonitors
spots the plane than by relying on a loneobserver. In other words,
probability summation corre-sponds to a decision rule in which the
group decision isdefined by a response from any single member of
thegroup. This decision rule corresponds to defining adetection
event when the signal in any one of m moni-tored channels reaches a
criterion level. This decision
shows how characterization of the stimulus size interms of the
sampling density within the stimulus envel-ope allows the
discriminability to be expressed in termsof the effective area Ai
of the stimulus
d %i=RisRi�8
A i1/2
s, (12)
showing that ideal discriminability is proportional tothe square
root of stimulus area.
Note, however, that there is a problem with applyingthis model
in practice, since the psychometric functionin this model is based
on a linear relation between d %and signal strength. This linear
relation is violated bymost d % measurements, which typically show
an expo-nent of about 2 (e.g. Stromeyer and Klein, 1974).Similarly,
translation of this prediction into the Weibullformat yields a
predicted Weibull exponent of 1.3 inEq. (2) (whereas most
measurements show exponents of3–4). Extension of the theory to
non-ideal attentionbehavior, which encompasses steeper exponents,
is leftto the next section. First we consider an approximationto
ideal behavior that can be used if the observer knowsthe set of
stimulus types that may be presented in ablock of trials, even if
the particular stimulus is notknown in advance on each trial.
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structure is implemented by applying a max rule to allthe
channel outputs and defining the detection eventwhen the max
reaches some preset criterion level (Pelli,1985; Kontsevich &
Tyler, 1999a). The neural imple-mentation of such a decision rule
may be designated as‘attentional summation’.
Although probability summation is often consideredas a purely
mathematical operation, it is meaningless inthe context of the
human vision (in a single observer)unless it is mediated by some
neural hardware. Thisraises the issue of whether there are
independent neuralchannels and what is meant by a max operator in
aneural system. In terms of detection theory, two chan-nels are
considered independent when they are gov-erned by sources of noise
that are statisticallyindependent. There is plenty of evidence for
a highdegree of statistical independence among even neigh-boring
cortical neurons (e.g. Freeman, 1994, 1996;Shadlen & Newsome,
1998), so cortical neurons can beconsidered to be separate channels
for this purpose.What would constitute the probability summation
ormax operator? It needs to be a neural system receivingsignals
from an array of channels (or axons) that havestatistically
independent noise up to that point. It thenneeds to respond when
any of these inputs manifests asignal but not otherwise. Such a
neural system wouldhave this property if it would transmit a spike
thatinitiated a detection response on receiving a spike fromany one
of its inputs. The threshold characteristic ofcortical neurons with
wide-field input sampling thusprovides the requisite hardware for a
max operator.
In terms of the detection of signals in additive noise,the
optimal strategy is to use a matched filter, toconvolve the
stimulus input with a linear filter exactlymatching the stimulus
profile. It is possible to approxi-mate ideal observer strategy by
performing probabilitysummation over the full set of filters in the
form of themax of the signal-to-noise ratios (Pelli, 1985).
Thisapproach may be considered an ideal (or Bayesian)attentional
strategy in that the observer knows the setof likely filters to
survey on each trial. This strategy willhave the effect of
isolating the most efficient filter underany condition, and hence
mimic ideal observer behaviorwithout requiring prior knowledge of
the stimulus.However, implementation of this strategy does
requirethe neural system to have an accurate representation ofthe
noise level, in order to compute the signal-to-noiseratios. Simply
taking the max over raw signals will tendto emphasize the noisiest
fields. But if it is plausible thatthe neural system normalizes to
the prevailing (long-term) noise level, then a max operator would
provide amechanism for implementing Ideal Observer behavior.
It is common practice to combine the response out-puts in neural
network models by a Minkowski summa-tion rule:
R=�%
n
(Rip)n1/p
(13)
where the summing exponent is often set at p=4. Notethat such
fourth-power summation (thick curve in Fig.6) produces a completely
smooth curve in the rangewhere the filters are present even though
in this exam-ple the assumed filters are separated by factors of
twoin size. It is thus possible to approximate Ideal Ob-server
behavior with relatively coarse physiologicalsampling in a
particular domain if there is some way toimplement in the cortex
the Minkowski summation ofEq. (13) with a high summation
exponent.
3.5. Attentional summation in 2AFC experiments doesnot conform
to high threshold analysis, but deri6esfrom the s of the difference
distribution
For 2AFC detection using more than one channel,attentional (or
‘probability’) summation effects shouldbe analyzed through Signal
Detection Theory. For atractable analysis, we assume n stimulated
channels ofequal sensitivity with additive Gaussian noise. For
thefull analysis, we will consider the situation where theobserving
system is monitoring more channels (m) thanare being stimulated.
The statistical combination rulefor attentional summation of the
responses over chan-nels is derived again from the maximum value of
the setof m monitored channel responses in each stimulusinterval
(Pelli, 1985; Palmer, Ames, & Lindsey, 1993).For the null
stimulus of the pair, which by definitioncontains only noise, the
combined response distributionMm(R,sR) is based on the noise-alone
distributions inthe responses of all m channels. Mathematically,
thiscombined distribution is given in terms of the expectedvalues
of the distributions by the derivatives in a similarfashion to Eq.
(6), omitting the distribution variablesfor clarity:
Mm(R,sR)= maxi=1:m
[DN ]=ddr�& r
−�
DN dr %nm
=mDN�& r
−�
DN dr %n(m−1)
(14)
The two parameters in the expression Mm(R,sR) for themax
distribution imply that we are deriving the form ofthe expected
function of the resulting probability distri-bution, which may be
characterized by the parametersof its location and spread (as for
the High ThresholdTheory of Eqs. (5) and (6)).
With the inclusion of n signal channels for the signalinterval
of the stimulus pair, the max must be takenover the maxes of the
separate n signal+noise andm−n noise-alone distributions:
Mn,m(R,sR)=max�
maxi=1:n
[DR(ri)] maxi=n+1:m
[DN(ri)]n
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3121–31443132
Fig. 7. Max distributions for a Gaussian probability density
functionfor numbers of samples increasing in factors of 10 from n=1
to 1million. Note the decreasing standard deviation and small
asymmetryof these max distributions.
suming that the observer employs an ideal attentionwindow that
always matches the stimulus extent, so thatno unstimulated channels
are monitored. Nevertheless,it is assumed that the observer cannot
perform idealsummation over the stimulus area, but is forced
tomonitor a set of n local channels to find which gives themax
response in any test interval (Pelli, 1985).
Fig. 7 shows the numerical distributions for samplesof maxes
computed according to the derivation of Eq.(14) for noise alone (or
Eq. (15) for signal+noise withm=n) in factors of ten from n=1 to 1
million chan-nels of equal sensitivity. The s of these max
distribu-tions decreases by a factor of about four (in contrast
tothe factor of 200 decrease predicted for only 1000channels under
High-Threshold Theory with no uncer-tainty). In each case, the
observer’s task is to distin-guish between sample stimuli drawn
from the maxdistributions of noise-alone and signal+noise for
sum-mation over a given number of channels. Discriminabil-ity
therefore improves with the reciprocal of thereduction in s in
these max distributions (Fig. 7), asshown in the leftward shift of
the d % functions of Fig.8a. The consequent improvement in
sensitivity at thelevel of d %=1 is depicted in Fig. 8b. Because
thefunction in Fig. 8b defines ‘ideal’ probability summa-tion for
the 2AFC paradigm, we provide the values intabular form in the
Appendix for ready reference. Notethat the signal+noise max
distributions have to becomputed by time-intensive numerical
integration. Wehave therefore developed an approximation
method(Chen & Tyler, 1999) that captures this function within1%
accuracy. (Pelli, 1985, had also considered this
=ddr��& r
−�
DR dr %nn
·�& r
−�
DN dr %nm−n�
(15)
In the general case, Eq. (15) does not simplify in themanner of
Eq. (14).
The simplest case of 2AFC attentional summation isthe case where
m=n, so there is no uncertainty as towhich of the monitored
channels contain the stimulus,and the two distributions differ only
in their mean levelof internal response. This situation corresponds
to as-
Fig. 8. (a) Theoretical d % functions under 2AFC probability
summation assumptions. Note that the exponent (or steepness) is
almost invariant withnumber of equally-sensitive channels monitored
from n=1 to 1 million (assuming no uncertainty). 2AFC summation
behavior is thereforeessentially invariant with the d % criterion
selected. (b) 2AFC probability summation over six decades on
(unequal) double-log coordinates,compared with summation slopes for
full summation (−1), for ideal observer summation (−0.5) and for
Weibull summation assuming b=4(−0.25). Note that the 2AFC summation
function is never steeper than a slope of −0.25, and becomes
extremely shallow for more than aboutten samples.
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C.W. Tyler, C.-C. Chen / Vision Research 40 (2000) 3121–3144
3133
function and provided an approximation formula thatis accurate
to within 20%.)
Thus, the complete analysis of 2AFC attentionalsummation over
channels of equal sensitivity showsthat the Ideal Attention
operator provides dramaticallydifferent ‘probability summation’
behavior than thatimplied by Pelli’s (1985) high-uncertainty
approxima-tion to High-Threshold Theory. At its steepest, this2AFC
function exhibits a slope of only about −0.25(from one to four
samples) and soon produces negligi-ble summation for larger numbers
of samples. The keyreason for the difference between this
prediction andthat for the Weibull approximation is that the tails
inthe Gaussian distribution fall much more rapidly thanexponential
tail of the Weibull distribution. A summa-tion mechanism that
focuses on the information in thistail region will necessarily give
different results for thetwo distributions. Justification for the
ubiquity of theGaussian distribution is discussed in the
assumptionssection of Section 1.
Consider the practical implication of the summationfunction of
Fig. 8b. For most reported psychophysicaltasks, the smallest
stimulus might plausibly stimulatemany local mechanisms. The
expected starting point fora probability summation prediction would
then besome way down this curve, say at the 102 level, beyondwhich
little improvement is evident. Under the idealattention assumption,
the only way to achieve summa-tion exponents even close to the
reported values ofaround −0.25 (Watson, 1979; Robson &
Graham,1981; Williams & Wilson, 1981; Pelli, 1985) would be
toassume that attention can be focused onto a singleneural channel
for the smallest stimulus in the series.
A major prediction of the High Threshold theory ofprobability
summation is that the summation exponentcan be predicted from the
empirical exponent of thepsychometric function measured during the
summationexperiment (Quick, 1974). This prediction has beenborne
out in several studies (Watson, 1979; Robson &Graham, 1981;
Williams & Wilson, 1981; Pelli, 1985),but the result may be
coincidental because none have6aried the psychometric exponent to
determine whetherthe summation exponent varies as predicted.
Neverthe-less, this analysis shows that the extent of 2AFC
atten-tional summation varies even where the exponent of
thepsychometric function is invariant at a value close toone (Fig.
8a) and provides a much smaller improve-ment in sensitivity than is
predicted by High-Thresholdanalysis for conditions yielding shallow
exponents (Fig.8b). Even the early part of the 2AFC attentional
sum-mation slope is never steeper than −0.25 (although itmust be
said that this corresponds to a value commonlyassumed for the
Weibull exponent, b). Studies thathave assumed such a slope,
therefore, would seem tohave a valid estimate of the probability
summationeffects as long as the number of elements of equal
sensitivity that they are summing remains less thanabout four.
The analysis of Fig. 8 could therefore beregarded as validating the
use of Minkowski summa-tion with an exponent of 4 as long as the
number ofchannels remains small and the other assumptions ofthe
analysis are met.
Conversely, there is a major situation in which thesummation
slope remains unaffected while the psycho-metric steepness varies.
This behavior can occur whenthe observing system monitors more
channels than arebeing stimulated. This situation is conventionally
de-scribed as the system having uncertainty as to whichchannels are
being stimulated and is the topic of thenext section.
4. Signal Detection Theory with channel uncertainty(and additive
noise)
This section develops the implications of a variety ofnon-ideal
attentional strategies for 2AFC in the addi-tive noise case.
4.1. Channel uncertainty effects and their eliminationby
rescaling
Channel Uncertainty Theory is an elaboration ofSignal Detection
Theory in which the number of neuralchannels m monitored in the
brain is greater than thenumber of channels n stimulated (by ratio
M=m/n)(derived formally in Pelli, 1985). The level of uncer-tainty
would then be defined as log10 M (assumed to be0 up to this point
in the treatment). (An equivalenttheory of attentional distraction
among the m channels,even where the observer is certain which
channel isbeing attended, has been developed by Kontsevich
&Tyler, 1999a.) For the present analysis, we assume thatonly
one channel is being stimulated and that thedecision is mediated by
attention to successively largernumbers of channels in a non-ideal
attentional strategy.Such behavior has been offered as an
explanation forthe relatively steep psychometric functions that
areoften measured in practice (Pelli, 1985; Kontsevich &Tyler,
1999a). The full lines in Fig. 9a show the d %functions obtained
through the 2AFC derivation ofEqs. (13) and (14) for the certain
condition (monitoringonly one stimulated channel) and uncertain
conditions(in which from 10 up to one million channels
aremonitored, with only one stimulated). The d % functionsget
progressively steeper in this operating range aschannel uncertainty
increases. The dashed lines in Fig.9a show an analytic
approximation to these d % func-tions that was fitted over the full
set within the rangefrom d %=0.5 to 2 (i.e. within the practical
measurementrange). The approximation is a power function whoselog
slope U (straight dashed lines) is related simply touncertainty
(log10 M) by the expression:
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C.W. Tyler, C.-C. Chen / Vision Research 40 (2000)
3121–31443134
Fig. 9. (a) Log–log d % functions under various degrees of
channel uncertainty (M=1–1 000 000 in factors of ten running left
to right) when onechannel is stimulated. Dotted lines represent the
least squares fit of equation (16) within the readily measurable
range of −0.5B log d %B0.5(horizontal dashed lines). (b). Summation
behavior with an attention window of increasing extent, due to
increase of channel uncertainty asstimulus size increases over the
array of local filter channels, at log d % levels of −0.5, 0 and
0.5 (three curves corresponding to the horizontalcriterion lines in
a). Threshold rises gradually at first as number of monitored
samples is increased, then shows little further effect.
d %=A(s/s0)U, with U=C+B log10M (16a,b)
where M is the ratio of monitored to stimulated chan-nels and A,
B, C and s0 take values of 7.9862, 0.4468,1.0779 and 9.5414.
The point of presenting this analytic approximationis that it
allows reverse inference of the level of uncer-tainty from the log
slope of the psychometric function,fitted to the data as a straight
lines on double-logarith-mic d % coordinates. Pelli (1985) had
provided a similarapproximation to a Monte Carlo simulation of
thetheoretical curves that we derive analytically, but
hisapproximation was formulated in terms of a Weibullanalysis and
consequently appeared to emphasize thelower range of d % values,
which are unmeasurable inpractice. Our reanalysis focuses on the
most accessiblerange of the psychometric function, that between log
d %values of −0.5 and 0.5 (or percent correct valuesbetween about
60 and 90%). Fitting in this rangegenerates fits at high levels of
uncertainty that aresubstantially shallower than Pelli’s. One can
use ourfitted function to derive the inferred uncertainty
directlyfrom equation (16), within the accuracy of the
slopedetermination (Empirically, slopes may be determinedwith an
accuracy of about 0.1 log units in 300 trialsusing the efficient
Bayesian maximum likelihood al-gorithm proposed by Kontsevich &
Tyler, 1999b; cf.Cobo-Lewis, 1997. This accuracy would imply a
practi-cal resolution of about 6 discriminable slopes in theslope
range from 1 to 4).
The inverse equations for sensitivity at the criterionlevel of d
%=1 are straightforward:
M=10(U−C)/B and s=s0A−1/U (17a,b)
Equivalently, channel uncertainty effects may be re-moved by
extrapolating the measured slope of the logd % function up to d
%=8, then extrapolating back downa slope of U=C to provide an
estimate of the sensitiv-ity that would have been obtained with no
channeluncertainty. The extrapolation back to the level ofd %=1 may
be approximated by dividing the measuredthreshold by a value of 8.
This simplified procedureallows compensation of channel uncertainty
effects withminimal computation, merely from knowledge of thelog d
% slope. For complete accuracy, the computed d %functions as
depicted in Fig. 9 may be used to model ofthe psychometric function
with no approximation. If athreshold estimate is required to be
more accurate thanthe proposed approximation formula, the data for
thepsychometric function may be fitted over the family ofcomputed d
% functions to refine the compensation forchannel uncertainty.
Of course, removing channel uncertainty does notimply
eliminating measurement error in the estimates,only eliminating the
bias in the threshold estimateintroduced by channel uncertainty.
The adjustedthreshold estimates are no less variable, but
thresholdchanges due to varying uncertainty levels are elimi-nated.
In situations where the channel uncertainty re-mains constant
across conditions, such bias reduction isnot needed. But in cases
where it may vary, such assummation functions over any stimulus
domain, it iscritical to partition the threshold variations between
theunderlying sensitivity variations and the effects of prob-
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3135
ability summation, as described in the followingsections.
4.2. 2AFC attentional summation with a fixed attentionwindow
The previous section considered the general case ofestimating
the degree of uncertainty from the psycho-metric function. With
this analysis in hand, we mayevaluate the particular case of the
effect on threshold ofvarying stimulus extent with a fixed
attention window.For studies that do not expend the effort required
tomeasure the psychometric slope, it is important to havea model of
the effects of uncertainty under plausibleassumptions. Clearly, if
the attention window can bematched to the stimulus extent, the
uncertainty (oropportunities for distraction, see Kontsevich &
Tyler,1999a) will remain constant at zero and have no effecton the
measured summation function. However, in thiscase the slope of the
psychometric function should below (assuming a linear transducer),
which is known tobe invalid in many situations.
Contrary to Robson and Graham’s (1981) claim forthis situation,
2AFC spatial probability summation ef-fects with a fixed attention
window are not propor-tional to 1/b (b being the exponent of the
Weibullapproximation, equation (1)). Such summation effectsare
controlled by the change in the exponent as uncer-tainty is reduced
by increasing stimulus area (Fig. 10a);as seen Fig. 10b, the
summation effects at d %=1 ap-
proximate a log slope of −1/4 over most of the rangeof ratios of
stimulated to monitored samples. Thisresult may be considered a
justification for the wide-spread use of 4th power Minkowski rule
to approxi-mate probability summation. It is a quite
differentanalysis from that developed by Williams and Wilson(1983),
Robson and Graham (1981) and even Pelli(1985), since those analyses
all assumed a fixed form ofthe log psychometric function. In
contrast, the shapevaries substantially in the
fixed-attention-windowmodel of Fig. 10a. Nevertheless, it may
correspond to aplausible set of assumptions, so tabular values for
theexample depicted in Fig. 10 are provided in Table 1.The
fixed-attention-window model is the main theoreti-cal alternative
to the probability summation effects ofthe ideal attention window
of Fig. 8.
Thus, the curve of 2AFC attentional (or ‘probabil-ity’)
summation in double-log coordinates may haveeither a concave or an
approximately linear form ac-cording to whether the attention
window is assumed tomatch the stimulus extent (Fig. 8) or to remain
fixed(Fig. 10). The two forms are empirically distinguishablefrom
threshold measurements alone. Note that, toprovide the fourth-root
approximation, the fixed atten-tion window must be at least as
large as the largeststimulus, and detection efficiency will
necessarily beextremely low for the smallest stimuli. Because
summa-tion is only probabilistic within this large attentionfield,
efficiency will still be low for stimuli filling theattention
field. Thus, the assumption of the fourth-
Fig. 10. Probability summation for varying numbers of samples
within a fixed attention window (assumed here to allow a maximum of
1000samples). (a) Psychometric functions in log d % versus log
stimulus strength. Note similarity in shape to those in Fig. 9 but
with extra shifts at highuncertainties. (b) Summation as a function
of ratio of number of samples to total number monitored, at the
three d % criteria indicated by thehorizontal lines in (a). Thick
dashed line in (b) depicts a slope of −1/4, which provides a good
approximation to fixed-window probabilitysummation over most of the
computed range.
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Table 12AFC max summation effects in relative threshold
valuesa
Number of samples Multiplicative noiseAdditive noise
Fixed attention windowIdeal attention window Increasing
attention window Ideal attention window
3.9311 1.0000 1.00001 1.00003.2978 1.22350.8235 0.45222
0.74423 2.9836 1.3524 0.28962.7823 1.44214 0.21430.69632.6354
1.51040.6632 0.17135
0.63856 2.5202 1.5653 0.14372.4278 1.6107 0.12447 0.61912.3503
1.65000.6032 0.11038
0.59009 2.2823 1.6845 0.09952.2245 1.714210 0.09100.57871.8695
1.90460.5149 0.053020
0.484730 1.6827 2.0110 0.04021.5566 2.082640 0.03360.46581.4626
2.13770.4523 0.029450
0.442160 1.3889 2.1826 0.02651.3278 2.218370 0.02440.43391.2763
2.24920.4271 0.022780
0.421390 1.2313 2.2762 0.0213100 0.4163 1.1924 2.3005 0.0203
0.9475 2.45380.3871 0.01472000.3725300 0.8158 2.5398 0.0124
0.7275 2.6003400 0.01110.36290.6620 2.64410.3560 0.0102500
0.33671000 0.4762 2.7793 0.00800.32012000 2.9078 0.0064
3.07200.3015 0.004950000.289310 000 3.1868 0.00410.278520 000
3.3005 0.0035
3.44370.2658 0.002950 0003.5475 0.0025100 000 0.25733.65240.2495
0.0022200 000
0.2403500 000 3.7793 0.00193.8743 0.00171 000 000 0.2339
a Results for fixed attention window assume a window size of
1000 samples.
power approximation to attentional summation carriesthe
implication that the neural system is operating atlow efficiency,
and is not applicable to situations whereefficient detection
performance is demonstrable.
4.3. Two-component summation and channel analysis
A classic case in both spatial and color vision is thesummation
for the detection of two stimulus compo-nents as their intensities
are varied relative to eachother. The results of this paradigm are
plotted on adual axis plot of the contrast threshold for the pair
ofcomponents when combined in a variety of ratios(Guth, 1967;
Graham & Nachmias, 1971; Stromeyer &Klein, 1974; Yager,
Kramer, Shaw, & Graham, 1984).These references should be
consulted for the theoreticaldevelopment, but various outcomes are
summarized inFig. 11a. If the two components are detected
entirelyindependently (by a noiseless max rule), the
detectioncontour forms a square corner (independent channels);
if they are added linearly in a single mechanism limitedby late
noise (linear summation), the detection contouris a negative
oblique line; if they are combined linearlybut detection is limited
by independent sources ofGaussian noise in the two channels, the
detection con-tour is a circular arc (squaring).
Two-component summation is an important case forchannel analysis
in general, because it represents thecombination rule between
adjacent channels for detec-tion by sets of channels in any domain.
Channel sum-mation is often modeled as a fourth-power Minkowskirule
(or pth norm) for combination over channels (Gra-ham &
Nachmias, 1971; Stromeyer & Klein, 1974;Williams & Wilson,
1981; Wilson, McFarlane, &Phillips, 1983; Yager et al., 1984).
The justification forthis rule is usually expressed in terms of
Weibull analy-sis, which we show to be on shaky grounds, but
thesituation may be reanalyzed for the 2AFC paradigmwith Gaussian
noise.
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The strict 2AFC probability summation prediction isbased on the
case where the system takes the max onevery trial, for two channels
with additive Gaussiannoise, as the relative signal strength is
varied betweenthe channels (Fig. 12). The analysis of this
situation isessentially an uncertainty analysis because the
observeralways monitors both channels as the stimulation
pro-gresses from one channel alone through both togetherto the
other alone. Even at the extremes, where onlyone channel is
stimulated, the joint signal always has toexceed the max of the
noises in both channels. Theapplicable distributions are plotted in
Fig. 12 in termsof both the max response over the two channels in
eachtest interval and the difference response d between thesignals
in each pair of 2AFC intervals. The responsedistributions for the
two intervals are set so that thedetectability for an individual
channel falls at the 75%correct position. The 2AFC uncertainty
prediction maybe developed for a full range of component ratios, as
isshown by the full curve in Fig. 11a. This prediction ispart-way
between the linear and the square-law summa-tion rule of linear
summation over sources with inde-pendent additive noise. In fact,
it is well-described by apth norm (Minkowski summation rule) with a
power of1.5 in the case of linear d % functions (i.e. involving
noadditional uncertainty about the stimulus properties).
Although most reported cases of two-componentsummation under the
2AFC paradigm show less com-
plete summation than this probability summation pre-diction,
their analysis requires consideration of the freeparameter of the
slope of the psychometric function,which is rarely specified in
published studies of two-component summation. When the slope is
steep, oneinterpretation is that there is much additional
uncer-tainty, i.e. the observer is monitoring many more chan-nels
of whatever kind than are being stimulated (Fig.9a). Quantitatively
speaking, most 2AFC studies reportthe exponent of the d % function
to be close to 2 in thefovea (e.g. Stromeyer & Klein, 1974),
which implies aratio of monitored to stimulated channels of
M=116(with n=1). As can be seen in Fig. 11b, this power of2
assumption produces a curve matching a Minkowskiexponent between 3
and 4.
Another commonly reported value of the d % exponentis 3
(approximating reports of the Weibull b from 3.5to 4). This slope
requires a channel monitoring ratio ofM=20 000, but this large
increase generates a curvethat is sharper than that for the
exponent of 4. Beyondthis range, double-precision computation was
no longercapable of computing the required max distributionsfor the
Gaussian function, but we could use the analyticapproximation in
the form of the Poisson distributionthat we developed for this
purpose (Chen & Tyler,1999). The resulting curves for d %
exponents of 4 and 6show that, again, there is very little change
in the shapeof the curve. Thus, one can conclude that there is
no
Fig. 11. (a) Theoretical functions for two-component summation
under various combination rules, assuming linearity of the
psychometricfunctions. Thin curves: Minkowski summation with
exponents of one, two and four and independent channels. Thick
curve: max rule. (b) Thincurves show theoretical functions for
two-component summation assuming accelerating psychometric
functions with the levels of uncertaintyrequired to approximate
psychometric slopes of 2, 3, 4 and 6. Dashed curves show Minkowski
summation with exponents of 1 (linear), 2(squaring) and 4 for
comparison. Note that even a steep slope of six departs
substantially from the corner prediction for independent
channels,so that one can expect to determine accurately whether two
channels are fully independent or subject to some kind of
combination rule.
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3121–31443138
Fig. 12. Derivation of the 2AFC probability summation
predictionfor two-component summation. Dash-dotted curves: max
distributionfor the two channels for the noise-alone interval; full
curve: maxdistribution for the two channels for the signal+noise
interval; heavydashed curve: distribution of the differences
between the intervals. (a)Both channels equally stimulated. (b)
Only one channel stimulated.
5. Effects of multiplicative noise
5.1. Multiplicati6e noise makes the psychometricfunction
shallower
Instead of the classical assumption of additive noise,analysis
of the noise in cortical neurons suggests that itmay have a
multiplicative component, with sR in Eq.(10) increasing according
to some function of thestrength of the mean signal R (Tolhurst et
al., 1981). Ina general expression, the total noise in the signal
maybe expressed by a power relation:
sR8kRq+sN (18)
The additive constant sN represents some irreduciblelevel of
noise that is present even when there is nosignal, when the
multiplicative component kRq will fallto zero. Such additive noise
is a physiological requisitebecause no real system is
noise-free.
Other than scaling the noise according to Eq. (18),the
multiplicative analysis employs the identical ana-lytic structure
of Signal Detection Theory developed inSection 3, but the results
are very different. The pres-ence of multiplicative noise radically
alters the expectedshape of the psychometric function derived by
insertingEq. (18) into Eq. (15), in both the absence and
thepresence of channel uncertainty. Even in the absence
ofuncertainty, the log-log steepness of the psychometricfunction
changes according to the rate of increase ofnoise with stimulus
intensity. If the exponent q of thisrate of increase is 0.5, as in
Poisson noise (whichgoverns the quantal fluctuations of light, for
example),the psychometric slope for a single channel goes toabout
0.5 (Fig. 13a, bold curve), a striking deviationfrom what is seen
empirically in psychophysical mea-surements. (Eq. (18) assumes that
the noise distributionis Gaussian rather than strictly Poisson, a
good approx-imation for high mean levels of quantal events.)
Notethat a slope of q=0.5 represents a tremendously shal-low
increase of d % with stimulus strength, implying thatthe measurable
range of the psychometric functionextends over as much as two log
units, the entire visiblecontrast range.
5.2. Dramatic probability summation with multiplicati6enoise
If we evaluate the effects on sensitivity of taking themax of n
equally-sensitive channels in the presence ofroot-multiplicative
Gaussian noise, the results are alsoprofound (Fig. 13b). Summation
over the first tenchannels actually exceeds the amount expected for
lin-ear summation in additive noise (see tabular valuesspecified
for this condition in Table 1). This resultseems counterintuitive,
but it arises because any de-crease in the signal provides a
concomitant decrease in
plausible degree of steepness of the psychometric func-tion that
will push the curve to the corner of the box ifprobability
summation is operating. Finally, it shouldbe noted that the effect
of uncertainty in increasingthe exponent is essentially equivalent
to the samechange in exponent from an accelerating threshold
non-linearity.
In conclusion, the analysis of the two-componentsummation
paradigm in terms of the Minkowski sum-mation rule (Eq. (13))
provides an adequate approxima-tion to the 2AFC behavior in
additive Gaussian noise,as long as the Minkowski exponent is not
misinter-preted according to High-Threshold Theory.
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C.W. Tyler, C.-C. Chen / Vision Research 40 (2000) 3121–3144
3139
the accompanying noise level. Since the effect of proba-bility
summation is to allow a small decrease in thesignal to begin with,
the multiplicative reduction in thenoise provides a further
enhancement of the signal/noise ratio, resulting in the net
improvement depicted inFig. 13b.
This dramatic degree of probability summation un-der
root-multiplicative noise conditions has powerfulimplications for
neural processing, which seems to begoverned by this type of
principle throughout the cor-tex (Tolhurst, Movshon, &
Thompson, 1981; Tolhurst,Movshon, & Dean, 1983; Vogels,
Spileers, & Orban,1989). Shadlen and Newsome (1998) point out
that themultiplicative behavior makes the signals at
individualneurons so noisy that they cannot account for
thediscriminative behavior of the animal as a whole, evenif the
neuron’s response is optimal for the local stimula-tion employed.
They estimate that the activity must beintegrated over 50–100
neurons to account for theobserved behavior, implying that the
signal/noise ratioof the optimal neuron is about a log unit below
therequired level.
However, the plot in Fig. 13b implies that a differentstrategy
is available under root-multiplicative noiseconditions. Instead of
integrating the activity of 100neurons, and losing the potential
specificity availablefrom the elements of that assemblage, the
cortex couldmonitor the activity of just ten rele6ant neurons.
Takingthe max of the ten responses gives the required boost ofa
factor of 10 in signal/noise ratio, equivalent to sum-ming over 100
neurons. Thus, a much smaller pool isrequired for the same gain in
detectability, if the brainis capable of implementing a max rule.
Such implemen-
tation seems plausible because it is the core operationof an
attentional process, for which there is muchbehavioral and
increasing neurophysiological evidence.In fact, a simple neural
threshold has the effect ofimplementing a max rule in a
psychophysical taskwhere the stimulus is reduced until the last
response ofthe most sensitive neuron carries it. (The detailed
effectsof a hard threshold on 2AFC performance, which arebeyond the
scope of the present treatment, are dis-cussed in Kontsevich &
Tyler, 1999c.)
5.3. Disambiguating multiplicati6e noise and uncertainty
Inclusion of channel uncertainty in the case wherethe noise is
somewhat multiplicative has similar effectto the case of additive
noise (see Fig. 9a), except thatthe entire fan of uncertainty
functions is rotated tobecome shallower. Fig. 14 plots sample
psychometricfunctions for square-root-multiplicative noise (the
caseof p=0.5 in Eq. (18)). At first sight, it might seem thatthis
result implies that the empirical effects of the noisemultiplier
and uncertainty would be hard to disentan-gle. However, notice that
the steepening effects of un-certainty in Fig. 14 are much reduced
at high levels ofd %. Thus, the steepness at high d % (say, above
the level ofd %=2), are highly diagnostic of the degree to which
thenoise is multiplicative. If the fitted slope in this regionof
the unmasked psychometric function is 1 or above,as in Fig. 9a, the
implication is that the major compo-nent of the noise is additive.
A high d % slope signifi-cantly less than 1 (Fig. 14), on the other
hand, is strongevidence for multiplicative noise operating in the
near-threshold region.
Fig. 13. Effects on probability summation of assuming
square-root multiplicative noise according to Eq. (18) with p=0.5.
(a) Shallower d %functions. (b) Dramatically enhanced summation
behavior for threshold stimulation at the criterion of d %=1 that
is even supralinear for smallnumbers (Upper and lower criterion
levels are omitted for clarity).
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C.W. Tyler, C.-C. Chen / Vision Research 40 (2000)
3121–31443140
Fig. 14. Effects of uncertainty on the steepness of the d %
function withthe root-multiplicative noise assumption, with various
values of M.Unlabeled curves have monitoring ratios in factors of
ten from ten to100 000. Note that a high level of uncertainty is
required before thefitted slope approaches 1.
image being viewed. Whatever the distribution of eyemovements,
they will introduce some level of fluctua-tion in the response of a
local filter viewing any kind ofcontrast stimulus. The resulting
fluctuation is a form ofnoise that is necessarily in direct
proportion to thestimulus contrast (assuming that the eye movements
areindependent of contrast). This property of direct
pro-portionality may be shown analytically in terms of thetemporal
waveform of the signal fluctuation of theoutput of each linear
filter Iki(x,y) responding somestimulus S(x,y), such as a
sinusoidal grating, projectedon to the moving retina.
ri(t)=Iki(x,y)s · S(x−Dx(t),y− (Dy(t)))
=s · [Iki(x,y)S(x−Dx(t),y− (Dy(t)))] (19)
where is the convolution operator, Dx(t), Dy(t) isthe retinal
shift over time and s is the scaling constantof stimulus
strength.
Thus, for a given filter and eye-movement sequence,the filter
output ri(t) is directly proportional to thecontrast of the
stimulus, because convolution is a linearoperation. We may treat
the response of the filterderived from such eye movements as a
noise source bydetermining its standard deviation sE computed
oversome temporal window t1:t2 according to
sE=�& t2
t 1
�ri(t)−
& t2t 1
ri(t)dt�2
dtn0.58ri(t)8s (20)
The standard deviation of this source of noise is thusdirectly
proportional to the contrast of the backgrounddisplay. Such
proportional noise will tend to overtakeother sources of noise that
do not increase so rapidlywith contrast, and will therefore tend to
dominate athigh contrast. We are not aware of any previous
con-sideration of such noise.
The effect of proportional multiplicative noise (q=1in Eq. (18))
on the form of the psychometric functionsis shown in Fig. 15. The
fitted slopes become evenshallower than in the case of square-root
noise (Fig. 14)when the signal rises out of the additive noise
regime,where the slopes approximate unity. As stimulusstrength
increases, the effect is to make the functionsasymptote to a
constant d % level, with no further im-provement in sensitivity at
high stimulus strengths. Thishorizontal asymptote thus becomes a
conspicuous sig-nature of the presence of full multiplicative
noise. Suchbehavior has rarely been seen in psychometric
functionsfor contrast detection (e.g. it is not evident in
thehigh-contrast study of Foley & Legge, 1981), suggestingthat
this type of multiplicative noise is not a usualfeature of contrast
detection tasks. However, it is notclear that previous workers have
designed their studiesfor careful evaluation of this high d %
region of thepsychometric function, so there is room for
furtherevaluation of particular situations of interest before
the
There is a curious crossover in the functions in Fig.14 at low d
%, where the curve for no uncertainty actu-ally shows a slightly
higher threshold than the curvesfor low uncertainty. This result
may seem counterintu-itive, but it arises from the necessary
assumption thatthere is an additive component to the noise (Eq.
(18)),which tends to reduce sensitivity as the
multiplicativecomponent approaches the level of the additive
compo-nent at low contrasts. As uncertainty increases, it
de-emphasizes the role of the additive noise, effectivelyincreasing
the sensitivity. Without this additive compo-nent, the log slope
fitted to the psychometric functionwould be at 0.5 even with no
uncertainty. However,there must always be a noise component that is
additivewith respect to contrast due to the existence of
quantalnoise in the stimulus and thermal noise in the
receptors.Because these curves are governed by two free
parame-ters, the particular summation function at some level onthe
curves is not of canonical interest, and summationfunctions are
therefore not plotted for this case. If suchmultiplicative noise is
implicated in detection behavior,the role of additive and
multiplicative noise compo-nents must be estimated by measurement
of full psy-chometric functions.
5.4. Fully multiplicati6e noise introduces
psychometricsaturation
A more extreme form of multiplicative noise is thecase where the
noise sR is directly proportional to thestimulus strength (q=1 in
Eq. (18)). Direct proportion-ality is not implausible, as such a
form occurs in thecase of noise due to eye movement fluctuations
over the
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C.W. Tyler, C.-C. Chen / Vision Research 40 (2000) 3121–3144
3141
case may be considered to be settled. For example,although such
noise might not be expected in a simpledetection task, it might
plausibly be found in a difficultdiscrimination task where the
contrast threshold is be-ing measured as a function of a slight
spatial differencebetween two stimuli, with long-duration
presentationsallowing eye-movement-generated noise in the
pedestalstimulus to become a significant factor limiting
discrim-ination performance.
For summation over increasing numbers of mecha-nisms, the curves
of Fig. 15 show that the additive noiseregime (approximating a
slope of 1) tends to dominatethe domain of measurable (and
computable) range of d %functions. As a result, derivation of a
summation curvefor this case is relatively meaningless because its
formwould depend on the exact ratio of additive to multi-plicative
noise assumed. When in the domain domi-nated by the fully
multiplicative noise (horizontal leg ofcurves), summation is
indeterminate because reductionof the signal would be accompanied
by a proportionatereduction of the noise, and signal/noise ratio
(discrim-inability) would be maintained at a constant level. Inthe
full-multiplicative noise regime, therefore, discrim-inability is
insensitive to the signal level, and thresholdcannot be determined.
Only when the signal level isfinally reduced into the domain
dominated by additivenoise (the left-hand region of Fig. 15) would
summa-tion revert toward the form depicted in Fig. 8b.
6. Conclusion
Psychophysical measures of summation are widelyused as indexes
of underlying integrative mechanisms in
visual processing. The preceding analysis provides arigorous
approach to the universe of such mechanisms,detailing the
properties of physiological summation,ideal observer summation and
attentional summation inthe 2AFC detection paradigm, for situations
of bothadditive and multiplicative noise limiting the
detectiontask. The key difference between these three types
ofsummation is the type of attention process accessing anarray of
filters. If attention accesses a single filter, thephysiological
summation within that filter predomi-nates; if attention can switch
among filters matchingeach stimulus, ideal observer behavior
occurs; if atten-tion can access the max response of an array of
filters,the result has been described as probability summation,but
we fav