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Vision Research 45 (2005) 2436–2448

Non-monotonic changes in performance with eccentricitymodeled by multiple eccentricity-dependent limitations

Frederic J.A.M. Poirier a,*, Rick Gurnsey b

a Neurodynamics and Vision Lab—Centre for Vision Research, York University, Computer Sciences and Engineering Building,

Room B0002E, 4700 Keele Street, Toronto, ON, Canada M3J 1P3b Department of Psychology, Concordia University, Montreal, Canada H4B 1R6

Received 24 July 2004; received in revised form 10 March 2005

Abstract

Eccentricity-dependent resolution losses are sometimes compensated for in psychophysical experiments by magnifying (scaling)

stimuli at each eccentricity. The use of either pre-selected scaling factors or unscaled stimuli sometimes leads to non-monotonic

changes in performance as a function of eccentricity. We argue that such non-monotonic changes arise when performance is limited

by more than one type of constraint at each eccentricity. Building on current methods developed to investigate peripheral perception

[e.g., Watson, A. B. (1987). Estimation of local spatial scale. Journal of the Optical Society of America A, 4 (8), 1579–1582; Poirier, F.

J. A. M., & Gurnsey, R. (2002). Two eccentricity dependent limitations on subjective contour discrimination. Vision Research, 42,

227–238; Strasburger, H., Rentschler, I., & Harvey Jr., L. O. (1994). Cortical magnification theory fails to predict visual recognition.

European Journal of Neuroscience, 6, 1583–1588], we show how measured scaling can deviate from a linear function of eccentricity in

a grating acuity task [Thibos, L. N., Still, D. L., & Bradley, A. (1996). Characterization of spatial aliasing and contrast sensitivity in

peripheral vision. Vision Research, 36(2), 249–258]. This framework can also explain the central performance drop [Kehrer, L.

(1989). Central performance drop on perceptual segregation tasks. Spatial Vision, 4, 45–62] and a case of ‘‘reverse scaling’’ of

the integration window in symmetry [Tyler, C. W. (1999). Human symmetry detection exhibits reverse eccentricity scaling. Visual

Neuroscience, 16, 919–922]. These cases of non-monotonic performance are shown to be consistent with multiple sources of reso-

lution loss, each of which increases linearly with eccentricity. We conclude that most eccentricity research, including ‘‘oddities’’,

can be explained by multiple-scaling theory as extended here, where the receptive field properties of all underlying mechanisms

in a task increase in size with eccentricity, but not necessarily at the same rate.

� 2005 Published by Elsevier Ltd.

Keywords: Peripheral vision; Spatial scaling; Hyper-acuity

1. Introduction

1.1. Eccentricity-dependent sensitivity losses

Retinal positions are typically described in polar coor-

dinates and the term eccentricity denotes distance from

the centre of the retina expressed in degrees of visual

0042-6989/$ - see front matter � 2005 Published by Elsevier Ltd.

doi:10.1016/j.visres.2005.03.007

* Corresponding author. Tel.: +1 416 736 2100x33325; fax: +1 416

736 5857.

E-mail address: [email protected] (F.J.A.M. Poirier).

angle. It is evident to anyone in possession of a normally

functioning visual system that our ability to resolve the

details of visual patterns decreases as the eccentricity ofstimulus presentation increases. Many studies have dem-

onstrated that performance drops when stimuli of fixed

size are presented at greater eccentricities (for examples,

see Berkley, Kitterle, & Watkins, 1975; Carrasco, Evert,

Chang, & Katz, 1995; Dakin & Hess, 1997; Herbert,

Faubert, & Humphrey, 1997; Hess & Dakin, 1997).

There are many reasons why resolution is poorer in

the retinal periphery than at fixation; viz., (i) there is

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F.J.A.M. Poirier, R. Gurnsey / Vision Research 45 (2005) 2436–2448 2437

increased achromatic aberration (Ogboso & Bedell,

1987) at greater eccentricities; (ii) both cone size

(Young, 1971) and cone spacing increase with eccen-

tricity (Hirsch & Curcio, 1989); (iii) the receptive fields

of both magnocellular and parvocellular retinal gan-

glion cells increase with eccentricity (Croner & Kaplan,1995); (iv) the number of striate cells available to pro-

cess a region of unit size on the retina decreases with

eccentricity (see Grusser, 1995); (v) peak spatial fre-

quency preference in areas V1, V2, V3/VP and V4v de-

creases with eccentricity (in Tootell, Hadjikhani,

Mendola, Marrett, & Dale, 1998). It is important to

note, however, that the gradients of these eccentricity-

dependent resolution losses may differ. For example,as one moves from fovea to periphery, there are fewer

cortical V1 cells per retinal ganglion cell (Azzopardi &

Cowey, 1993, 1996a, 1996b; Rolls & Cowey, 1970),

there is less overlap of cortical receptive fields (Dow,

Snyder, Vautin, & Bauer, 1981; Hubel & Wiesel,

1974), and there is a decrease of the ratio of parvocel-

lular to magnocellular contributions to visually evoked

potentials (Baseler & Sutter, 1997). In other words,there are many different sources of eccentricity-depen-

dent resolution loss, and each may have a unique

gradient.

1.2. Fixed scaling

Because eccentricity-dependent resolution losses arise

from undersampling of one sort or another, perfor-mance in psychophysical tasks can be approximately

equated at all eccentric positions by increasing the size

of the stimulus linearly with eccentricity. This size

increase can compensate for peripheral undersampling

in many tasks, including grating acuity (Cowey & Rolls,

1974; Hirsch & Curcio, 1989), contrast sensitivity (Dras-

do, 1991; Hilz & Cavonius, 1974; Koenderink, Bouman,

Bueno de Mesquita, & Slappendel, 1978; Rovamo &Virsu, 1979; Rovamo, Virsu, & Nasanen, 1978), orienta-

tion discrimination (Paradiso & Carney, 1988), temporal

contrast sensitivity (Virsu, Rovamo, Laurinen, & Nasa-

nen, 1982), feature and conjunction visual search tasks

(Carrasco & Frieder, 1997), orientation discrimination

visual search (Poirier & Gurnsey, 1998), 2-dot Vernier,

grating, Snellen E, 3-dot separation, and 3-dot direction

acuities (Virsu, Nasanen, & Osmoviita, 1987). Whenstimuli have been size-scaled to overcome undersam-

pling on one dimension, performance curves along other

dimensions can be compared (e.g., orientation, contrast,

curvature, vernier offset, etc.). However, studies that use

a preset function to magnify peripheral stimuli do not

measure the gradient of resolution loss for a particular

task, rather, they embody an assumption about the gra-

dient of resolution loss. At best, such tasks can ensurethat resolution loss does not affect discrimination

thresholds along some other dimension.

1.3. Single scaling

Watson (1987) was among the first (see also John-

ston, 1987; Johnston & Wright, 1986; Wright, 1987) to

describe an assumption-free method for characterizing

the gradient of resolution loss with eccentricity. Hismethod consists of first measuring performance levels

as a function of some spatial dimension at the fovea

(e.g., the contrast sensitivity function) and then repeat-

ing those measurements at different eccentricities. Local

scale is defined as the constant divisor of the spatial

measure that shifts a peripheral performance curve onto

a foveal one. Using such a procedure it is often found

that local scale changes at each eccentricity accordingto the following linear scaling function

/ðEÞ ¼ /ð0Þ � ð1þ E=E2Þ; ð1Þin which /(E) is the scaling at a given eccentricity E,

/(0) is foveal scaling, and E2 is the eccentricity at which

the peripheral scaling must double to achieve perfor-

mance levels equivalent to foveal performance (Levi,Klein, & Aitsebaomo, 1985). In theory E2 is independent

of foveal scaling or performance levels. It has been ar-

gued that linear scaling removes most eccentricity-

dependent variability in a variety of tasks, including

the size of Panum�s area (Hampton & Kertesz, 1983),

curvature discrimination (Whitaker, Latham, Makela,

& Rovamo, 1993), spatial phase discrimination (Mor-

rone, Burr, & Spinelli, 1989), orientation discrimination(Makela, Whitaker, & Rovamo, 1993; Scobey, 1982),

Vernier acuity (Levi & Waugh, 1994; Whitaker, Rov-

amo, MacVeigh, & Makela, 1992b), position acuities

and movement acuities (Whitaker, Makela, Rovamo,

& Latham, 1992a).

1.4. Multiple scaling

A recent debate in the eccentricity literature focuses

on whether more than one scaling factor may influence

performance within a given task. For example, when

stimuli vary along two dimensions (such as spatial fre-quency and bandwidth), sensitivity to changes on the

two dimensions may not scale at the same rate with

eccentricity relative to foveal limits (see Fig. 1). Unequal

scaling of two dimensions has been found in tasks of

two- and three-dot separation (Yap, Levi, & Klein,

1989), intrinsic blur (Levi & Klein, 1990a), position acu-

ity (Levi & Klein, 1990b), centre, surround, and end-

stopped sections of end-stopped perceptive-field profile(Yu & Essock, 1996), word recognition (Melmoth &

Rovamo, 2003), face recognition (Melmoth, Kukkonen,

Makela, & Rovamo, 2000), interaction zones (Toet &

Levi, 1992), letter recognition (Strasburger, Harvey, &

Rentschler, 1991; Strasburger & Rentschler, 1996), crit-

ical flicker frequency (Raninen & Rovamo, 1986; Rov-

amo & Raninen, 1988), and subjective contours

0 1 2 3 4 50

2

4

6

8

Eccentricity

Sca

ling

Fac

tor

Fig. 1. Scaling functions with E2 values of 0.7 (solid line) and 2.5

(dotted line).

2438 F.J.A.M. Poirier, R. Gurnsey / Vision Research 45 (2005) 2436–2448

(Poirier & Gurnsey, 2002). Poirier and Gurnsey (2002)

presented a unified methodology (including testing

method, fitting technique (see also Strasburger, Rentsch-ler, & Harvey, 1994) and objective statistical procedure)

for assessing the presence of multiple-scaling functions

within a task. They include several statistical tests that

can distinguish whether a single scaling factor is suffi-

cient to account for eccentricity-dependent variability

in a data set. In what follows, we review and expand

upon the Poirier and Gurnsey methodology and use it

to explain some odd phenomena in the eccentricity liter-ature (see Section 2, and the original paper for a full

description of the method).

1.5. Non-monotonic changes in performance with

eccentricity

One would expect that if resolution loss increases

with eccentricity, then moving a stimulus of fixed sizeinto the periphery should produce a monotonic decrease

in sensitivity to the high spatial-frequencies. One would

also expect that when a stimulus is scaled according to

Eq. (1), performance should be monotonic: either flat

(if the scaling is about equal to that required to compen-

sate for resolution loss), or monotonically decreasing or

increasing with eccentricity (if the scaling either under-

or over-compensates the resolution loss). In fact, excep-tions to these expectations can be found in the literature.

In the first case, Kehrer (1987, 1989); (see also Gurnsey,

Pearson, & Day, 1996; Kehrer & Meinecke, 2003; Mei-

necke & Kehrer, 1994; Potechin & Gurnsey, 2003;

Yeshurun & Carrasco, 1998, 2000) showed that when

an unscaled texture boundary is moved into the periph-

ery, detection performance actually improves until a

performance peak is reached and then performance de-creases again; Kehrer has referred to this effect as the

‘‘central performance drop’’. In the second case, when

a fixed scaling function is applied, it is sometimes found

that performance shows a quadratic dependence on

eccentricity (for example, see Poirier & Gurnsey, 1998;

Saarinen, Rovamo, & Virsu, 1987) and peaks at a

non-foveal location. In a further example, a ‘‘reverse’’

scaling effect was reported where the scaling factors de-

crease with increases in eccentricity, contrary to whatmight be expected (Tyler, 1999).

In the present paper, we show how the central perfor-

mance drop, quadratic performance effects, and reverse

scaling effects can arise from both unscaled stimuli and

scaled stimuli; i.e., stimuli that vary in size with eccen-

tricity according to Eq. (1). We propose that these effects

may occur when there are multiple, linear eccentricity-

dependent limitations on performance that change theirrelative contributions to performance as a function of

eccentricity.

2. Analysis

2.1. Representations of response surfaces

We begin with some necessary definitions and termi-

nology concerning performance in various psychophys-

ical tasks performed at a fixed location in the visual

field (e.g., at fixation). We represent a stimulus contin-

uum with the term ki. For example, k1 could be grating

wavelength, k2 could be window width and k3 could be

orientation. Variations in performance (P) arising from

increases in stimulus parameter ki may be captured by asigmoid/logistic function as in Eq. (2):

P iðkiÞ ¼ f1þ 10½�rðlog ki�log ki;50%Þ�g�1; ð2Þ

where ki,50% corresponds to the mid-point of the func-tion and r determines the slope of the psychometric

function. This function describes the probability that a

HIT would be made to stimulus level ki where partici-

pants are asked to report if the stimulus was present in

a detection task, or different in a discrimination task

(hit rate ranges from 0% to 100%), assuming false alarm

rates are minimized experimentally (see Fig. 2(A)). Un-

der these conditions, performance in a 2AFC task wouldbe 50% + 50% * P (percent correct ranges from 50%

(chance) to 100%).

In many psychophysical tasks, performance may be

based on multiple mechanisms operating simulta-

neously. In what follows we will deal with the special

case in which two mechanisms limit performance. How-

ever, the method we present generalizes to more than

two mechanisms.In some cases stimulus discrimination requires non-

zero responses from both mechanisms. Consider an ori-

entation discrimination task in which orientation (k1)and spatial frequency (k2) are varied. Discrimination

requires grating spatial frequency to be low enough to

be visible and the orientation difference great enough

Fig. 2. (A) Probability of a correct detection for the two independent stimulus dimensions (k1 and k2) used to generate the curve in (B). (B)

Probability of a correct detection for each configuration (X) and scale (Y) derived from Eqs. (3)–(5) (white and black represent 100% and 0% hit rates

respectively). It is clear that for a particular configuration the probability of a correct detection increases monotonically with stimulus scale.

1 From this space it is possible to compute all combinations of k1 andk 2 tha t e l i c i t th r e sho ld pe r fo rmance . The func t ion

n2 = (k1 � k1,min)(k2 � k2,min) can be used to approximate the shape

of such a locus of points (Poirier & Gurnsey, 2002; Strasburger et al.,

1994).


to be detectable. Orientation discrimination is not possi-

ble if the spatial frequency is too high to be resolved or if

the orientation difference is too small to be discrimi-

nated. In a 2AFC task we can represent the probability

of hits arising from these requirements (assuming near-

zero false alarm rates) as follows.

P ðk1; k2Þ ¼ P ðk1Þ � Pðk2Þ. ð3Þ

In other words, performance will be at chance (i.e., no

hits, no false alarms; a conservative response strategy

would produce only misses and correct rejections) when

either P(k1) or P(k2) is 0 (Green & Swets, 1966). Eq. (3)

permits one to compute the probability of a correct

detection/discrimination for all combinations of k1 and

k2.Poirier and Gurnsey (2002) found it convenient for

theoretical and practical reasons to consider a transfor-

mation of the [k1,k2] space of responses, defined in Eq.

(3), into an [X,Y] space of responses, defined in Eqs.

(4) and (5)

X ¼ a logðk2=k1Þ; ð4Þ

Y ¼ a logðk2k1Þ; ð5Þ

where a = cos45�. In this representation X denotes a

configuration; i.e., all values of k1 and k2 that produce

the same ratio, and Y denotes stimulus scale. For illus-

tration, consider subjective contours that vary in carrier

grating wavelength and contour length (Poirier & Gurn-sey, 2002). The ratio of contour length to carrier grating

wavelength defines a configuration (X). In [k1,k2] spacethis defines a line emanating from the origin whose slope

is given by the ratio k2/k1. Stimuli of a particular config-

uration can vary in scale (Y), which is related to the dis-

tance of point [k1,k2] from the origin. As Eqs. (4) and (5)

show, the transformation of [k1,k2] space into [X,Y] in-

volves a simple logarithmic transformation of the units,

a 45� counter-clockwise rotation of the resulting axes,

followed by a reflection around the X-axis.

Fig. 2(B) shows the probability of a correct detection

for each configuration (X) and scale (Y) derived from

Eqs. (3)–(5). It is clear for a particular configuration,

the probability of a correct detection increases mono-

tonically with stimulus scale.1

Alternatively, there are tasks in which threshold canbe achieved if either of two mechanisms are sufficiently

activated, in which case probability summation is used:

P ðk1; k2Þ ¼ 1� ½1� Pðk1Þ� � ½1� P ðk2Þ� ð6Þ(Green & Swets, 1966). For example, a test stimulus may

differ from a standard stimulus in orientation or spatial

frequency and the subject�s task is to discriminate thetest stimulus from a standard. Fig. 3 shows the probabil-

ity of a correct discrimination for each configuration (X)

and scale (Y) derived using Eqs. (4)–(6). Again, for a

particular configuration the probability of a correct

detection increases monotonically with stimulus scale.

In Eqs. (3) and (6), performance is depicted as

increasing with increases in the psychometric variables

(ki). One could think of stimulus size as an example ofthis situation; performance improves as stimulus size in-

creases. There are other situations, however, in which

performance increases as ki decreases. For example,

the detection of bilateral symmetry is best when the sym-

metrical regions abut (i.e., have zero separation) and be-

comes more difficult as the separation increases (Tyler,

1999). Similarly, texture discrimination becomes more

difficult as the separation between texture elements in-creases (Nothdurft, 1985; see also Gurnsey & Laundry,

Fig. 3. There are tasks in which thresholds can be achieved if either of two mechanisms responds to the task, in which case probability summation is

used. As with Fig. 2, for a particular configuration the probability of a correct detection increases monotonically with stimulus scale.


1992). This kind of dependence can be captured by Eq.

(2) by simply changing the sign of the slope parameter

(r). If k2 has a negative slope and k1 has a positive slope

and performance depends on both variables (as in Eq.

(3)) the resulting performance surface in XY space

would have the form shown in Fig. 4. In this case, per-

formance for each configuration (X) changes non-mono-

tonically as scale (Y) increases.

2.2. Response surfaces and spatial scaling

Eq. (2) defines how the proportion of hits changes as

a function of the psychometric variable (ki). If we as-

sume that Eq. (2) defines performance at fixation, then

scaling theory (e.g. Eq. (1)) says that performance at

some eccentricity (E) should be characterized by a psy-chometric function that is identical in all respects to that

at fixation, except for the placement of the function

along the stimulus axis. In other words, at eccentricity

E, the performance curve should be a shifted version

Fig. 4. If k1 has a positive slope and k2 has a negative slope and performa

performance surface in XY space would have the form shown here. In this ca

scale increases, but instead changes monotonically as configuration increase

of the foveal curve, centred on ki,50% as specified in

Eq. (1), thus the logistic Eq. (2) becomes:

P iðE; kiÞ ¼ f1þ 10½�riðlog ki�log/iki;50%Þ�g�1; ð7Þ

where /i is the scaling function (defined in Eq. (1)) for

stimulus parameter ki. With Eq. (1), then:

ki;50%ðEÞ ¼ ki;50%ð0Þð1þ E=E2iÞ; ð8Þ

where E2i is an empirically derived constant associatedwith a particular stimulus continuum. As described in

the introduction, each stimulus variable ki that limits

performance in a given task may scale differently with

eccentricity. Therefore, E2i may be different for each va-

lue of i. Whatever the case, applying Eq. 8 to k1,50% and

k2,50% will result in shifts of the response surfaces in

Figs. 2–4. We use the shifts in these response spaces to

explain a number of apparently anomalous finding inthe eccentricity literature. We find that all of these

anomalous results can be explained as direct conse-

quences of multiple-scaling theory.

nce depends on resolving both variables (as in Eq. (3)) the resulting

se, performance for each configuration changes non-monotonically as

s.

2 The Matlab code used to fit the data is available upon request to

the first author.


3. Results

3.1. Non-linear scaling in grating detection

Thibos, Still, and Bradley (1996) studied the detection

and resolution of gratings across the visual field. Detec-tion thresholds refer to the highest spatial frequency that

participants can discriminate from an equiluminant

blank field. Resolution thresholds refer to the highest

spatial frequency at which participants can identify the

grating�s orientation (vertical vs. horizontal). The high

frequency cutoffs for detection and resolution are similar

at the fovea (63.6 cpd) but diverge as the stimuli are

moved into the periphery. At eccentricities of 10�, 20�and 30� detection cutoff frequencies are 26.4, 25.1 and

22.9 cpd respectively and resolution cutoffs are 9.9, 6.4

and 4.9 cpd respectively. Detection beyond the resolu-

tion limit is attributable to aliased frequencies; the stim-

uli are not perceived veridically.

The detection and resolution data are plotted in Fig.

5(A) as black and white dots respectively. They repre-

sent grating frequency at threshold. The resolutionthresholds (Fig. 5(A) white dots) can be fit with a linear

scaling function (using Eq. (1)) and yield an E2,2 of 2.48.

The dotted line plotted through the resolution data rep-

resents the reciprocal of the best fitting linear scaling

function. Detection thresholds (Fig. 5(A) black dots)

clearly deviate from a linear scaling function (its recipro-

cal is represented here as the dashed line).

Detection thresholds reflect two limitations (aliasedand veridical visual limitations) affecting performance

at different eccentricities. This is a case in which perfor-

mance depends on either of two mechanisms responding

appropriately, therefore, detection threshold data can be

fit by combining Eqs. (6) and (7). In the model, aliased

and veridical limits were represented with k1 and k2respectively, and the slope parameters of these functions

which are represented by r1 and r2 respectively (bothslope values were positive and were taken from Thibos

et al., 1996). The E2 parameters which influence the rate

at which aliased and veridical limits scale with eccentric-

ity are represented E2,1 and E2,2 respectively. Eqs. (6)

and (7) (parameterized in the manner just described)

specify detection performance for any given eccentricity,

stimulus configuration and stimulus scale, as a combina-

tion of the two limits.The actual fitting was done simultaneously for detec-

tion and resolution data. For each task, we recovered

threshold scaling factors over a range of eccentricities

for a fixed stimulus configuration which replicates the

sampling methods used by Thibos et al. (1996). The free

parameters (k1, k2, E2,1, and E2,2) were varied to mini-

mize the sum of squared deviations between the data

and the model simultaneously for detection and resolu-tion data (see Appendix A for recovered values). All

data were fit using the error minimization routine pro-

vided in MATLAB (Mathworks Ltd.); this routine

(fmins) used the Nelder–Mead simplex (direct search)

method.2

Fig. 5(A) shows the detection and resolution data sets

(black and white dots respectively; foveal detection and

resolution thresholds overlap), and the model fit to thedetection data (solid line). We modeled detection thresh-

olds as the optimal combination of two limits (dotted

lines): (1) the aliased limit which drops slowly with

eccentricity (e.g. from 30.7 cpd foveally to 22.3 cpd at

30�), and the veridical limit which follows the rapid res-

olution drop over the range of eccentricities tested. It is

clear that the limitations governing performance in the

detection task changes with eccentricity of presentation,even though both limits are present at all eccentricities.

There is also a transition area near 2.5� of eccentricity,where both mechanisms contribute equally. Thus, multi-

ple-scaling theory can be used successfully to fit the data

of Thibos et al. (1996) in agreement with their conclu-

sions. Multiple-scaling theory could also be used to

model any situation in which multiple mechanisms pro-

vide independent cues for task performance (e.g. Levi &Klein, 1990a, 1990b).

3.2. Central performance drop in texture segregation

In contrast to the grating detection and resolution

example above, stimulus discrimination may require

that both independent mechanisms are properly stimu-

lated. The texture segregation task seems to provide anexample of this condition. Many models of texture seg-

regation involve two stages of spatial filtering. In the

first stage the input image is convolved with a set of

band-pass filters, creating a set of ‘‘neural images’’

(Robson, 1980). The responses within each neural image

are rectified and put through a second filter designed to

detect local differences in first-layer responses. The sal-

ience of a texture boundary depends, therefore, on twolevels of spatial analysis. Texture segregation will be dif-

ficult if the first-layer filters are not sufficiently stimu-

lated; this might happen if the spatial frequency

content of the textures is too high to be resolved. Tex-

ture segregation will also be difficult if there is a poor

match between the content of a rectified neural image

and the properties of the second-layer filter. For exam-

ple increasing the distance between texture elements (tex-els) may reduce the salience of a texture edge because the

texels of adjacent textures are not sufficiently close to

engage the second-layer filter. So, texture segregation re-

quires appropriate activation of both first- and second-

layer mechanisms (Eq. (3)).

0 10 20 300

10

20

30

40

50

60

Gra

ting

freq

uenc

y (c

pd)

Eccentricity(A) -0.5 0 0.5 1

-0.5

0.0

0.5

1.0

1.5

0˚

2˚

10˚

20˚30˚

Configuration (X)

Sca

ling

(Y)

(B)

Fig. 5. (A) Threshold grating spatial frequency for detection (black dots) and resolution tasks (white dots) as a function of eccentricity (from Thibos

et al., 1996) and model fit (solid line for detection thresholds, steeper dotted line for resolution thresholds; dotted lines represent the two limits used to

generate the solid line; see text). Also shown is the best linear scaling function fit to the detection data (its inverse is represented as a dashed line). (B)

Isoperformance curves associated with the fit, for different eccentricities, where exceeding either limit leads to correct performance in the detection

task (solid line), but only the veridical perception limit leads to correct performance in the resolution task (dotted line).


If a texture display of fixed configuration is moved

from fovea to periphery it will engage segmentation

mechanisms of different structures and, according to

the above, this may lead to non-monotonic changes in

performance. Exactly this kind of result has beenreported by Kehrer (1987, 1989, 1997), Scialfa and Joffe

(1995), Gurnsey et al. (1996), Yeshurun and Carrasco

(1998, 2000), Morikawa (2000), Potechin and Gurnsey

(2003), Gurnsey, Di Lenardo, and Potechin (2004).

These studies show that segregation performance peaks

at some point in the periphery and declines as the dispa-

rate texture is moved further into the periphery or closer

to fixation. We modeled the central performance dropby assuming two underlying, eccentricity-dependent lim-

itations characterized by opposite signs on the slopes (r)in the psychometric functions defined in Eq. (7); viz, tex-

ture discrimination improves with increases in texel size

(i.e. r1 > 0) and decreases in texel separation (i.e. r2 < 0).

Eqs. (3) and (7) specify performance for any given eccen-

tricity, stimulus configuration and stimulus scale, which

could be compared directly to data. For each limitationwe solve for ki, E2i and ri to minimize the sum of

squared deviations between the data and the model.

We fit data from Kehrer (1989) and Gurnsey et al.

(1996).

Fig. 6(A) shows the results reported by Kehrer

(1989). Subjects were to discriminate a small region of

left-oblique lines (texels) embedded in a larger surround

of right-oblique lines (and vice versa) at a range ofeccentricities. The texel size was kept approximately

constant (9–10 pixels), but the inter-element spacing

could be narrow, medium, or wide (11, 14, or 18 pixels,

respectively).

Fig. 6(B) shows the fit of the model to these data. The

conventions for labelling the different conditions are the

same as in Fig. 6(A) and (C). The fits capture the main

features of the data but are compromised, to some ex-

tent, by the fact that the original experiments included

variations in presentation duration. The effect of this

variable on performance cannot easily be equated using

scaling or percent correct adjustments, so we were un-able to include it in the model.

According to our theory, decreases in performance in

the far periphery is explained by reduced resolvability of

the texels, whereas reduced performance near the fovea

is explained by reduced ability to compare texture

information at the texture edge because texels are too

distant to engage the available texture edge mechanisms

(Gurnsey et al., 1996). The stimulus that elicits the bestresponse to the texture boundary is defined by a specific

ratio of texel separation to texel size. According to this

framework, the location of peak performance depends

on the relative activation in the two levels of processing.

Performance will be monotonic with eccentricity if the

particular configuration used consistently challenges

one mechanism yet is easily resolved in the other at all

eccentricities. Depending on which mechanism ischallenged, performance will either increase or decrease

with eccentricity: if texture resolvability controls perfor-

mance, then performance will drop with eccentricity,

whereas if texels are too distant to engage the texture

edge mechanism, then performance will increase with

eccentricity.

Fig. 6(C) shows the configuration/scale interpretation

of this result. Each line in Fig. 6(C) represents an iso-re-sponse curve corresponding to 50% hit rate on a surface

such as shown in Fig. 4; performance increases from

right to left. Each curve is associated with a performance

surface at selected eccentricities—the lowest curve repre-

sents foveal response and higher curves represent

increasing eccentricities. That is, all combinations of

configuration and scale falling on these contours elicit

0 2 4 6 8 100

50

100

Per

cent

Hits

(A) 0 2 4 6 8 100

50

100

0 2 4 6 8 100

50

100

Eccentricity

Per

cent

Hits

(B)0 2 4 6 8 10

0

50

100

Eccentricity0.0 0.5 1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0˚

1˚

2.5˚

5˚ 10˚

Configuration (X)

Sca

ling

(Y)

(C)

NarrowMediumWide

Fig. 6. (A,B) Percent correct as a function of eccentricity for different stimulus configurations (e.g. foreground texel spacing could be narrow,

medium, or wide; see legend in panel C; note that all filled symbols represent medium-spaced foregrounds, thus the filled circle would overlap the

filled square in panel C) in a texture discrimination task by Kehrer (1989) for actual data (A) and model fits (B). (C) Isoperformance curves at

eccentricities of 0�–10� associated with the fit, where correct performance only occurs if both task limitations are exceeded.


the same level of performance. These contours each

specify the location of a psychometric surface from

which it is possible to calculate the probability of a hit

for stimuli of specific configurations and scales at each

eccentricity. In essence we have solved for E2k1, E2k2

,

k1, k2, r1, and r2 that specify the probability of a correct

response at all eccentricities. All data in Fig. 6(B) weredetermined by the surfaces characterized by the iso-

response lines in Fig. 6(C) (samples for the different con-

figurations are indicated by the same symbols as used in

Fig. 6(A) and (B)).

Fig. 6(C) is conceptually similar to Fig. 4(B), where

performance increases from right to left, and where

curves are iso-response curves corresponding to 50%

hit rate. If we consider the medium condition (see filledsquare and circle in Fig. 6) used by Kehrer (1989), then

it maps to a position on the foveal surface (0�) that cor-responds to a hit rate just below 50%. At the same time,

the medium condition falls near the peak of the 5� sur-face (since it is at the same Y position where the bend

in the 5� eccentricity curve is located) and hence repre-

sents a much higher hit rate (perhaps close to 100%). Fi-

nally, the medium condition is close to the iso-contourline of the 10� curve and would elicit a hit rate close

to 50%. In other words, a stimulus of fixed configuration

presented at different eccentricities would elicit greater

accuracy at 5� than at 0� or 10�.A corresponding analysis was performed on data re-

ported by Gurnsey et al. (1996) who used a texture dis-

crimination task similar to Kehrer�s (1989), except

stimulus parameters were kept physically constant whileviewing distance was varied. To fit these data it was nec-

essary to include a free parameter representing the dis-

tance between the centre of the texture patch and the

texture edge nearest to the fovea; i.e., Gurnsey et al.

(1996) specified eccentricity in terms of distance to the

centre of the disparate region whereas we specify eccen-

tricity in terms of distance to the nearest edge. The stim-

uli were similar to those used by Kehrer (1987) except

that only one configuration was used and stimuli were

viewed from three distances. In our terminology, the

stimuli differed in scale (and different eccentricities weretested at each scale). The results are shown in Fig. 7(A)

(filled symbols) along with the best fits of the model

(empty symbols). The symbols in Fig. 7(B) indicate that

all stimuli had the same configuration (i.e., project to the

same value on the X-axis) but had different scales

(i.e., project to different values on the Y-axis). The black

triangle in Fig. 7(B) corresponds to a stimulus viewed

from 114 cm. It falls near the iso-response line for the0� and 10� curves and near the peak of the 5� curve,

yielding the classic central performance drop with a

peak at approximately 5� + stimulus width.

According to the above analyses, the central perfor-

mance drop effect does not represent a failure of the

scaling theory as extended here. The model we fit has

two sources of eccentricity-dependent limitation; the

two sources could be first- and second-layer filters thatrespond to individual texels and texel differences, respec-

tively. Even though both mechanisms scale linearly with

eccentricity, it is possible for stimuli of fixed configura-

tion and fixed scale to elicit non-monotonic changes in

performance as it is moved from fixation to more

peripheral locations.

3.3. Reverse scaling in symmetry

A third example of an apparent anomaly in the size

scaling literature is found in the results of Tyler

(1999). Tyler had subjects discriminate symmetrical

0 10 20 300

10

20

30

40

50

60

70

80

Eccentricity

Per

cent

Hits

(A) -0.5 0 0.5

-0.5

0.0

0.5

1.0

1.5

0˚2.5˚

5˚ 10˚

30˚

Configuration (X)

Sca

ling

(Y)

(B)

LargeMediumSmall

Fig. 7. (A) Percent correct as a function of eccentricity for different stimulus scales in a texture discrimination task by Gurnsey et al. (1996, solid

shapes) and model fits (empty shapes). (B) Isoperformance curves associated with the fit, and the three stimuli used (configuration = 0; scale varied).


from non-symmetrical stimuli composed of dense black

and white checks. For symmetrical displays the left and

right halves of the display were mirror reflections ofeach other. Symmetry was degraded by replacing a re-

gion of checks spanning the axis of symmetry with ran-

dom black and white checks and the width of this

occluder was varied.3 Stimuli were presented for vari-

able durations and sensitivity was defined in terms of

the presentation duration required to elicit a d 0 of 0.5.

The symmetry integration region was defined as the oc-

cluder width that lead to a fixed reduction in sensitivityrelative to peak sensitivity. The procedure was carried

out with the axis of symmetry placed at eccentricities

of 0�, 2.5�, 5� and 10� from fixation. For all three sub-

jects tested, the size of the symmetry integration region

was found to decrease with eccentricity. This ‘‘reverse

eccentricity scaling’’ is prima facie inconsistent with

standard scaling theory.

The same general model used to fit the central perfor-mance drop data can be used to fit Tyler�s (1999)

reverse-scaling data. We assumed that correct detections

depend on both the resolvability of the elements com-

prising the symmetrical patterns and the proximity of

the two halves of the symmetrical display. In other

words, this is another example of a discrimination that

requires the joint activation of two mechanisms, as

embodied in Eq. (3). We used a negative slope in Eq.(7) to capture the fact that performance drops as occlu-

der size increases (r2 < 0) and a positive slope in the

other function to represent that performance increases

as the second psychometric variable increases (r1 > 0).

We acknowledge that Tyler�s (1999) stimuli and data

pose challenges to our modeling efforts because it is dif-

3 We note that the central occluder not only increases the distance

between the nearest symmetrical checks but also reduces the number of

checks that are symmetrically paired.

ficult to specify precisely the nature of this second psy-

chometric variable.

The idea that symmetry is integrated within a re-stricted region is clear (Dakin & Herbert, 1998; Rainville

& Kingdom, 2002) and hence the role of the occluder is

well defined. However, the second source of limitation is

complicated by the fact that the stimuli are broadband

and so symmetry information is available through many

scales. Furthermore, sensitivity is defined in terms of

stimulus presentation duration, which leads to the possi-

bility that different mechanisms underlie performance atdifferent times. In view of these complications we deal

with a level of abstraction beyond that described for

the central performance drop and texture discri-

mination.

To derive thresholds, we varied configuration and

kept scale constant. We solved for the configuration that

elicited constant threshold performance, thus approxi-

mating Tyler�s measured occluder width. We then variedthe free parameters to find the best fit to Tyler�s occluderwidth data.

Fig. 8(A) shows the fit to the data of Tyler (1999) and

Fig. 8(B) plots our interpretation of the data in a fashion

similar to Figs. 6(C) and 7(B). Changes in occluder

width correspond to changes in configuration; i.e., when

occluder width increased the value of the second vari-

able decreased proportionally. According to our inter-pretation the second psychometric variable imposes

little or no limit on performance at fixation and hence

the �thresholds� are only determined by occluder width.

However, when stimuli are moved into the periphery,

the second variable starts to limit performance (e.g.,

check size gets too small, or, the total Fourier energy

passed through the system is decreased because of

increasing low-pass filtering) and hence a combinationof the two variables limits performance. By our interpre-

tation, it is not that there is ‘‘an increase of long-range

connectivity in the fovea, resulting in a larger spatial

0 5 100

1

2

3

4

5

Eccentricity

Ape

rtur

e W

idth

at T

hres

hold

(A)

ModelData

0.0 0.5 1.0 1.5 -1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0˚ 1˚2.5˚ 5˚ 10˚

Configuration (X)

Sca

ling

(Y)

(B)

0 5 100

20

40

60

80

100

Aperture Width

Per

cent

cor

rect

(C)

0˚2˚5˚10˚

Fig. 8. (A) Occluder width as a function of eccentricity in a symmetry detection task (circles; Tyler, 1999) and model fit (line). (B) Isoperformance

curves associated with the fit. (C) Percent correct as a function of occluder size, for different eccentricities. These functions replicate the main

observations of Tyler (1999, see text).


integration area’’ (Tyler, 1999, p. 922), but rather thatthe experimental conditions are more demanding

peripherally, forcing the symmetry mechanisms to make

better use of the remaining stimulus information.

Fig. 8(C) shows percent correct responses (Y-axis) for

different occluder sizes (X-axis) and eccentricities.

Tyler�s main observations are replicated here: (1) peak

performance is roughly independent of eccentricity,

and (2) occluder size at threshold gets narrower witheccentricity. Translating our percent correct measures

to exposure duration will change the shape of the curve,

but we expect that those two main findings will remain

unchanged.

Our interpretation of the reverse scaling differs from

Tyler�s (1999). At this point, the available data do not

allow for an empirical test of which of the two alterna-

tives is correct. An experiment designed to address thisissue would have to sample independently a range of

texture scalings (either covarying texel size and patch

size, or measuring over a range of texel sizes and patch

sizes) and a range of occluder widths, such that the

whole performance curve is recovered foveally and at

peripheral locations. Then multiple-scaling could be ap-

plied to recover independently the scaling functions for

the texture properties, and the scaling function for thesymmetry occluder width. Our prediction is that occlu-

der width, measured under these conditions, would scale

linearly upwards with eccentricity, in contrast to Tyler�sconclusion.

4. General discussion

Most physiological scaling functions are nearly linear

with eccentricity (for examples: cone spacing, receptive

field size in V1, preferred grating wavelength). The onlyclear deviation from this rule concerns the distribution

of rods in the retina, i.e., there are no rods foveally be-

cause of a high concentration of cones there, with the

rod distribution showing a peak at an eccentricity of

about 15� from the fovea. Non-linear eccentricity effects

could be natural consequences of rod distribution when

testing is conducted under scotopic conditions. However,

no such simple account could be given for non-linear ef-fects that are obtained under photopic testing conditions.

We have extended multiple-scaling theory to include

probability summation (e.g. Eq. (6)) and reverse scaling

(e.g. negative slope (r) in Eq. (2)). Using the extended

multiple-scaling theory, we provided accounts of the cen-

tral performance drop and reverse scaling using multiple

mechanisms that scale linearly with eccentricity. We have

shown that the pattern of non-linearities is different forcases where performance depends on the simultaneous

satisfaction of two limitations, compared to cases where


two independent mechanisms can each lead to correct

performance, and for cases where performance increases

or decreases with increases in scaling. We have also ar-

gued that proper multiple-scaling methods can differenti-

ate between these cases, and can provide even more

useful information about the underlying mechanisms.Because this theory is not specific to any given mecha-

nism, it may be generalized to other cases where multiple

mechanisms are used in a task.

Multiple-scaling theory provides a convenient way to

fit data recovered from eccentricity experiments. Unlike

computational models, multiple-scaling theory makes

few assumptions about the mechanisms underlying

performance, and provides a way to recover scalingfunctions for multiple stimulus dimensions within mini-

mum computational time.

In texture segregation tasks, some stimulus configura-

tions lead to the central performance drop phenomenon

because different limitations come into play as a func-

tion of eccentricity (Gurnsey et al., 2004; Gurnsey et

al., 1996; Kehrer & Meinecke, 2003; Meinecke & Keh-

rer, 1994). At fixation performance may be limited bythe ability of second-layer mechanisms to cover texels

across a texel boundary (texture edge mechanisms) and

in the far periphery performance may be limited by

the ability of first order mechanisms to resolve the tex-

ture elements (Gurnsey et al., 2004; Gurnsey et al.,

1996; Kehrer & Meinecke, 2003; Meinecke & Kehrer,

1994). Cast in terms of Fig. 6(C), the central perfor-

mance drop occurs when a stimulus of fixed scale andconfiguration samples one side of the psychometric sur-

face near fixation and the other side of the surface in the

periphery. Other configurations not leading to the cen-

tral performance drop sample from either side only over

the range of eccentricities tested. In the symmetry exam-

ple, all samples were constrained to one side of the

curve, but foveal samples were closer to the transitional

part of the curve.To recover unbiased scaling functions, all of the scal-

ing functions need to be properly constrained, which

means that ki,50% has to be measured at foveal and

peripheral eccentricities for each of the performance lim-

itations influencing task performance. We therefore cau-

tion against the use of a single stimulus configuration or

scale, and instead recommend sampling a range of stim-

ulus configurations and scales to recover data from thecomplete performance curve to properly constrain data

fitting. Failing that, a qualitative description of the

mechanisms used in the task is still possible, but the

quantitative data recovered can be severely biased.

Acknowledgement

This research was supported by NSERC and FCAR

Research Grants to Rick Gurnsey. Portions of this

paper were presented at VSS, 2003, Sarasota, Florida,

and CVR, 2003, Toronto, Ontario, Canada. We would

like to thank two anonymous reviewers for helpful com-

ments on the manuscript.

Appendix A

Recall that performance arising from a mechanism

responsive to a single stimulus continuum at eccentricity

E can be described using a combination of a logistic

function (e.g. Eq. (2)) and a linear scaling function

(e.g. Eq. (1)). Thus, performance (hit rate) for mecha-

nism i is fully described by Eq. (7):

P iðki;EÞ ¼ f1þ 10½�riðlog ki�logUiki;50%Þ�g�1; ð7Þ

where ki,50% corresponds to threshold, ri determines the

slope of the psychometric function, and E2 is the eccen-tricity at which the peripheral scaling must double to

achieve performance levels equivalent to foveal

performance.

Thibos et al. (1996): Resolution performance was

defined using Limit 2 alone, whereas detection perfor-

mance was defined using both limits combined using

Eq. (6):

P ðk1; k2Þ ¼ 1� ½1� P ðk1Þ� � ½1� P ðk2Þ�. ð6ÞFor a specific stimulus configuration (X = 0), we can re-

cover the scale (Y = variable) to find the threshold scale

(e.g. giving 18% hit rate, which is equivalent to Thibos

et al.�s 68% correct assuming 0% false alarms). The

model�s threshold grating spatial frequency (1/kfit) is givenby (10[cos 45� * (Y � X)]). Slope values were taken fromThibos et al. (1996): r1 = 5.6, r2 = 14. Using Matlab�sfmin function, the parameters:

Limit 1: 1=k1 ¼ 32.6726; E2;1 ¼ 78.6551

Limit 2: 1=k2 ¼ 62.1762; E2;2 ¼ 2.4772

were found to minimize error as defined as:

err ¼X

E

½kfitðEÞ � kdataðEÞ�2. ð9Þ

Kehrer (1997): Performance at [k1,k2] was defined

using both limits combined using Eq. (3):

P ðk1; k2Þ ¼ P ðk1Þ � P ðk2Þ. ð3ÞKehrer sampled (line length = k1, inter-line distance = k2)at three stimulus conditions: narrow (9,11), medium(10,14) and wide (10,18). The parameters:

Limit 1: k1 ¼ 0.1153; E2;1 ¼ 0.1880; r1 ¼ 1.4143

Limit 2: k2 ¼ 19.9286; E2;2 ¼ 2.3071; r2 ¼ �0.8054

were found to minimize error as defined as:

err ¼X

E

½P fitðEÞ � P dataðEÞ�2. ð10Þ


Gurnsey et al. (1996): The same general fitting meth-

od was used as with Kehrer (1997). Samples were taken

at k1 = k2 = U, where U = 1,2,4 for the far, medium and

near viewing distances respectively. It was also necessary

to sample model performance closer to the nearest tex-

ture edge (e.g. at eccentricity = E � U * 1.0925�). Theparameters:

Limit 1: k1 ¼ 0.5037; E2;1 ¼ 5.3954; r1 ¼ 3.3032

Limit 2: k2 ¼ 10.0096; E2;2 ¼ 21.0177; r2 ¼ �0.3487

were found to minimize error as defined in Eq. (10).

Tyler (1999): The same general fitting method was

used as with Thibos et al. (1996), except that perfor-

mance for the two limits was combined using Eq. (3),sampling followed a fixed scale (Y = 0) and variable

stimulus configuration (X = variable), and threshold

was set to 50% hit rate (equivalent to 75% correct when

assuming 0% false alarms). The model�s threshold occlu-

der size (kfit) is given by (10[cos 45� * (Y � X)]). The

parameters:

Limit 1: k1 ¼ 0.1603; E2;1 ¼ 4.8935; r1 ¼ 2.2084

Limit 2: k2 ¼ 9.1049; E2;2 ¼ 2.7900; r2 ¼ �1.8245

were found to minimize error as defined in Eq. (9).

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