Quantifying the Ebbinghaus figure effect: Target size, context size, and target-context distance determine the presence and direction of the illusion

Abstract

Over the last 20 years, visual illusions, like the Ebbinghaus figure, have become widespread to investigate functional segregation of the visual system. This segregation reveals itself, so it is claimed, in the insensitivity of movement to optical illusions. This claim, however, faces contradictory results (and interpretations) in the literature. These contradictions may be due to methodological weaknesses in, and differences across studies, some of which may hide a lack of perceptual illusion effects. Indeed, despite the long history of research with the Ebbinghaus figure, standardized configurations to predict the illusion effect are missing. Here, we present a complete geometrical description of the Ebbinghaus figure with three target sizes compatible with Fitts’ task. Each trial consisted of a stimulus and an isolated probe. The probe was controlled by the participant’s response through a staircase procedure. The participant was asked whether the probe or target appeared bigger. The factors target size, context size, target-context distance, and a control condition resulted in a 3×3×3+3 factorial design. The results indicate that the illusion magnitude, the perceptual distinctiveness, and the response time depend on the context size, distance, and especially, target size. In 33% of the factor combinations there was no illusion effect. The illusion magnitude ranged from zero to (exceptionally) ten percent of the target size. The small (or absent) illusion effects on perception and its possible influence on motor tasks might have been overlooked or misinterpreted in previous studies. Our results provide a basis for the application of the Ebbinghaus figure in psychophysical and motor control studies.

Full text

ORIGINAL RESEARCH
published: 04 November 2015
doi: 10.3389/fpsyg.2015.01679
Frontiers in Psychology | www.frontiersin.org 1November 2015 | Volume 6 | Article 1679
Edited by:
Laurence T. Maloney,
Stanford University, USA
Reviewed by:
Carl M. Gaspar,
Hangzhou Normal University, China
Matthieu M. De Wit,
The University of Hong Kong,
Hong Kong
*Correspondence:
Hester Knol
hester.knol@univ-amu.fr;
Viktor K. Jirsa
viktor.jirsa@univ-amu.fr
Specialty section:
This article was submitted to
Perception Science,
a section of the journal
Frontiers in Psychology
Received: 18 December 2014
Accepted: 19 October 2015
Published: 04 November 2015
Citation:
Knol H, Huys R, Sarrazin J-C and
Jirsa VK (2015) Quantifying the
Ebbinghaus figure effect: target size,
context size, and target-context
distance determine the presence and
direction of the illusion.
Front. Psychol. 6:1679.
doi: 10.3389/fpsyg.2015.01679
Quantifying the Ebbinghaus figure
effect: target size, context size, and
target-context distance determine
the presence and direction of the
illusion
Hester Knol 1, 2, 3*, Raoul Huys 4, 5, Jean-Christophe Sarrazin 3and Viktor K. Jirsa 1, 2, 4*
1Institut de Neurosciences des Systèmes, Aix-Marseille Université, Marseille, France, 2Institut de la Santé et de la Recherche
Médical, UMR_S 1106, Marseille, France, 3Systems Control and Flight Dynamics Department, Office National d’Etudes et de
Recherches Aérospatiales (ONERA), Salon de Provence, France, 4Centre National de la Recherche Scientifique, Paris,
France, 5Centre de Recherche Cerveau & Cognition - UMR5549, Université Toulouse III - Paul Sabatier, Toulouse, France
Over the last 20 years, visual illusions, like the Ebbinghaus figure, have become
widespread to investigate functional segregation of the visual system. This segregation
reveals itself, so it is claimed, in the insensitivity of movement to optical illusions. This
claim, however, faces contradictory results (and interpretations) in the literature. These
contradictions may be due to methodological weaknesses in, and differences across
studies, some of which may hide a lack of perceptual illusion effects. Indeed, despite
the long history of research with the Ebbinghaus figure, standardized configurations
to predict the illusion effect are missing. Here, we present a complete geometrical
description of the Ebbinghaus figure with three target sizes compatible with Fitts’ task.
Each trial consisted of a stimulus and an isolated probe. The probe was controlled by the
participant’s response through a staircase procedure. The participant was asked whether
the probe or target appeared bigger. The factors target size, context size, target-context
distance, and a control condition resulted in a 3 ×3×3+3 factorial design. The results
indicate that the illusion magnitude, the perceptual distinctiveness, and the response time
depend on the context size, distance, and especially, target size. In 33% of the factor
combinations there was no illusion effect. The illusion magnitude ranged from zero to
(exceptionally) 10% of the target size. The small (or absent) illusion effects on perception
and its possible influence on motor tasks might have been overlooked or misinterpreted
in previous studies. Our results provide a basis for the application of the Ebbinghaus
figure in psychophysical and motor control studies.
Keywords: perception, visual illusions, ventral stream, dorsal stream, Ebbinghaus illusion
INTRODUCTION
Optical illusions evoke a perceived image, color, contrast, lightness, brightness, or size that differs
from the physical “reality” of the figure. These illusions have mainly been used to test theories
predicting the successes and failures of the perceptual system, particularly by the Gestalt school
(Robinson, 1998). Optical illusions have been classified based on the behavioral manifestation of

Knol et al. Quantifying the Ebbinghaus figure effect
FIGURE 1 | (A) The parameters of the Ebbinghaus figure with the radius of the target (a) and the context (c), and the distance from the target center to the context
center (b). (B) Example of the Ebbinghaus stimulus with the scaling probe (not scaled to real size). The distance between the center of the probe and the center of the
target was 16 cm. The context circles covered approximately 75% of the circumference.
45 illusions (e.g., Coren et al., 1976). One commonly mentioned
class is the one of size-contrast illusions, in which the size of
an element is affected by its surrounding elements. A famous
size-contrast illusion is the Ebbinghaus figure (see Figure 1), also
called Titchener circles.
For over a century the Ebbinghaus figure has been used
in experimental psychology to evoke an optical illusion of the
perceived circle size. The Ebbinghaus figure consists of a target
circle (ain Figure 1A) that is surrounded by multiple context
circles (bin Figure 1A). It is thought that by surrounding the
target with small or big circles, the target will appear bigger
or smaller, respectively (Obonai, 1954; Massaro and Anderson,
1971). More than 10 theories have been trying to explain the
physiological mechanism(s) responsible for the over- and under-
estimation of the target (for a review see Robinson, 1998).
However, attempts to quantify the illusion magnitude of this
widely used geometrical visual illusion have not resulted in a
(complete set of) geometrical rule(s), which is in all likelihood
at least partly due to the broad spectrum of parameters involved.
Several rules have been developed to identify the principal factors
influencing the perceptual judgment evoked by the Ebbinghaus
figure (e.g., Massaro and Anderson, 1971; Roberts et al., 2005;
Nemati, 2009). Principle factors that have been identified are
the size of the target (ain Figure 1A), the context circle
size (cin Figure 1A), the number of context circles (Massaro
and Anderson, 1971; Roberts et al., 2005), the target-context
distance (bin Figure 1A;Roberts et al., 2005; Im and Chong,
2009) and the size of the area of empty space between the
context circles (Nemati, 2009). However, these proposed rules
do not specify the exact interplay between the three parameters
specified in Figure 1A, which makes utilization of these rules
for parameter selection and the prediction of the corresponding
illusion effect tricky if not impossible. Furthermore, these rules
have barely been validated. Indeed, Franz and Gegenfurtner
(2008) concluded their review stating that: “. . . currently not
much is known on the exact sources of the Ebbinghaus
illusion.”
This lacuna did not withhold experimentalists to employ this
figure to shed light on the so-claimed distinction between the
ventral and dorsal visual pathway (see the review of Franz and
Gegenfurtner, 2008). Accordingly, the visual system contains two
distinct streams: the ventral pathway is specialized in processing
information leading to conscious perception whereas the dorsal
pathway is specialized in processing information for sensory-
motor action (Goodale and Milner, 1992; Milner and Goodale,
1995). The dorsal stream encodes visual information into the
required coordinates for skilled motor behavior, and does this in
absolute metrics determined relative to the observer (egocentric
frame of reference), whereas the ventral stream encodes the
information into object properties relative to the properties of
other objects (scene based frame of reference), and therefore
provides a rich and detailed representation (Goodale, 2014).
Based on this hypothesis, online control, and the programming
of movements would recruit the dorsal stream and, since
absolute metrics are determined relative to the observer and
not relative to the context of the object, would therefore
be insensitive to visual illusions (Milner and Goodale, 2008;
Goodale, 2014).
Several studies have reported evidence for the illusion
insensitivity during grasping movements (Aglioti et al., 1995;
Haffenden et al., 2001; Milner and Goodale, 2008; Stöttinger
et al., 2010, 2012). However, these findings seem to mismatch
with studies that show a clear effects of visual illusions on
grasping (Pavani et al., 1999; Franz et al., 2000) and pointing
(Gentilucci et al., 1996; van Donkelaar, 1999). These seemingly
contradicting results led to the hypotheses (for a review see Franz
and Gegenfurtner, 2008) that a clear functional dissociation
between perception and action cannot be made (Gentilucci
et al., 1996; Franz et al., 2000), that the ventral pathway
would have to be partially involved (Aglioti et al., 1995;
Carey, 2001), and that two dorsal pathways (e.g., the use and
grasp system) exist in stead of one (Binkofski and Buxbaum,
2013).
Seemingly conflicting results of studies that quantified
the illusion effect in perception and movements tasks may
be explained in various methodological ways (Bruno et al.,
2008; Bruno and Franz, 2009). Franz (2001) classified two
measurement types, to which he referred as the standard
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Knol et al. Quantifying the Ebbinghaus figure effect
and the non-standard perceptual measures. In the standard
method, participants either compare the size of two illusion
stimuli or of one probe and one illusion stimulus. In the
non-standard method, participants scale the aperture (with or
without vision of the hand) to indicate the perceived size.
Potential problems arising in the standard method are: First,
by changing the size of the inner circle of an Ebbinghaus
figure, as in Aglioti et al. (1995), it is not just the target
size that is changed but also the distance from the target to
the context circle, and therefore also the illusion magnitude
(Roberts et al., 2005). Second, sometimes a stimulus-stimulus
configuration is used in the perceptual task whereas a stimulus-
probe configuration is used in the motor task (as in Aglioti
et al., 1995). Third, if a task consists of comparing stimulus
A with stimulus B, the question comes up which stimulus
evokes an illusion effect (if any). For the non-standard method,
a potential problem is that it is questionable that studying
the perceptual illusion effect by asking participants to scale
their aperture indeed provides a “pure” perceptual measure.
Note that this method has generated conflicting results (Daprati
and Gentilucci, 1997; Haffenden and Goodale, 1998). Across
methods, if graspable targets are used (in the perceptual task),
the minimum stepsize of the target or probe might be relatively
big compared to the illusion magnitude. Furthermore, Franz
and Gegenfurtner (2008) identified methodological biases and
statistical corrections in the comparison of perception and
movement task data. There are, however, also studies that have
not quantified or reported the illusion effect on perception (e.g.,
van Donkelaar, 1999; Jackson and Shaw, 2000; Westwood et al.,
2000; Ellenbürger et al., 2012), or have not used a control
condition (Ellenbürger et al., 2012). To recapitulate, the conflicts
in the reported results may well be due to the various methods
used, and potential weakness therein as discussed here above.
Consequently, it is hard, if possible at all, to draw strong
conclusions about the proposed dissociation of the ventral and
dorsal stream in perceptuomotor tasks based on research using
optical illusions.
With the aim to (partly) fill this gap, we here provide a fully
parameterized Ebbinghaus figure, and systematically quantified
the illusion effect for parameter ranges that are relevant for
behavioral experiments. Thereto, we used a methodology that
is well-established in the psychophysics literature, namely, the
staircase procedure. We predicted that target size, context
size, and target-context distance would affect the perceived
target size of the Ebbinghaus figure, but that some parameter
combinations, in particular those involving small target sizes
(Massaro and Anderson, 1971), would fail to elicit a significant
illusion effect. Intuitively, we further expected that some stimulus
configurations, in particular those evoking a strong illusion
effect, would be perceptually more distinct than others, and
that this would affect the decision making as expressed in the
response times. That is, we expected response time to scale
inversely with perceptual distinctiveness. Our results will be able
to guide future experimentalists, which, we hope, will contribute
in clarifying the role of the ventral stream in the guidance of
motor behavior.
MATERIALS AND METHODS
Participants
Twelve participants (6 females and 6 males, age mean ±SD
=28.9 ±3.5) with normal or corrected to normal vision
volunteered in the experiment. The experiment was performed
in accordance with the Helsinki Declaration and all participants
gave a written informed consent prior to their participation.
Apparatus
The visual stimuli were drawn and generated using the
Psychophysics Toolbox in Matlab R2009b (The MathWorks
Inc., Natick, MA) (Brainard, 1997; Kleiner et al., 2007). Black
stimuli were presented against a white background (see Figure 1)
and multisampled to control for aliasing effects. To prevent
interference from previous trials and to control hemispace bias
the stimuli were randomly presented on the left or the right
side of the screen while an isolated probe (i.e., target without
context circles) was presented simultaneously on the opposite
side of the screen at a distance of 16 cm from the stimulus
(and at the same height). The stimuli were displayed with a
Dell Precision T3610 and Nvidia Quadro K2000 video card on
a Dell P2714H monitor with a resolution of 1920 ×1080 pixels
(597.9 ×336.3 mm, 52.96 ×29.27◦) and a frame rate of 60 Hz.
The participants sat at a 60 cm distance from the monitor and
their heads were supported with a chin-rest so as to ensure
that the distance between the head and the monitor remained
fixed.
Procedure
Based on a fully geometrical description (Figure 1A), three target
sizes (2 ×ain Figure 1A), three target—context distances (b
in Figure 1A), three context sizes (bigger, equal, and smaller
than the target; cin Figure 1A), and three control conditions
(isolated targets) were selected, resulting in a 3 ×3×3+3
factorial design. The equidistantly spaced context circles covered
approximately 75% of the circumference in all conditions to
control for the completeness of the surround (Roberts et al.,
2005). Consequently, the number of context circles varied as a
function of context size and target-context distance. The stimuli
diameters were 0.5, 1.0, and 2.0 cm [These sizes were chosen
with an eye on planned future studies involving Fitts’ task; the
corresponding indices of difficulty (i.e., ID =log2(2D/W), where
Dand Wrepresent the distance between the targets and the target
width, respectively; Fitts, 1954 were 6, 5, and 4, respectively.)
Context sizes were 20, 100, and 180% of the target size; i.e.,
0.1, 0.5, and 0.9 cm for the small target, 0.2, 1.0, and 1.8 cm for
the medium target, and 0.4, 2.0, and 3.6 cm for the big target.
Three distances from the center of the target to the center of
the context circles (i.e., bin Figure 1A) were calculated based
on the smallest distance being 10% bigger than the radius of the
target plus the radius of the biggest context; i.e., 0.8 cm for the
small target, 1.6 cm for the medium target, and 3.0 cm for the big
target. The other two distances were incremented with 0.6 cm for
each distance. All dimensions were corrected for pixel size and
rounded to the nearest integer.
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Knol et al. Quantifying the Ebbinghaus figure effect
A two-down, one-up staircase procedure was used to find
the perceptual threshold between the probe and the target in
which the probe size was adjusted (García-Pérez, 1998). Two
staircases per condition were used, one in which the initial
condition of the probe was 0.4 cm bigger than the target size,
and one in which it was 0.4 cm smaller. Each staircase started
with a probe diameter step size of four pixels (i.e., 0.12 cm).
The participants were tasked with pressing a key (Aor L)
for the bigger appearing target or probe on the left (A) or
right side (L) of the keyboard corresponding to the target and
probe location on the screen. Depending on the response of the
participant, the probe size was adjusted according to the two-
down, one-up staircase procedure. In a sequence of responses,
a reversal is the event where the response to probe n deviates
from that at n-1. After each reversal the step size was halved,
until the minimum of one pixel (i.e., 0.03 cm) was reached, which
was then retained. The participants were instructed to respond
as soon as they had decided which key to press, but it was
made clear that it was not a reaction time task. After each key
press, the stimulus disappeared and a random noise window
was displayed for 1 s followed by a fixation cross (duration:
0.5 s). Then the next stimulus with the adjusted probe appeared.
A staircase was terminated and removed from the cue after
a participant had reversed the direction of the staircase 11
times. After five conditions, the participants could take a small
pause. Upon completing the first half of the experiment, the
participants took a 10–15 min break. The entire experiment
lasted for about 2 h.
Data Analysis
From the last 10 reversals, the perceptual threshold (PT) was
calculated according to Equation (1),
PT =1
mXm
j=11
nXn
i=1SCupi+1
nXn
i=1SClowi(1)
in which mcorresponds to the number of staircases (here
Equation 2), nrepresents the number of reversals taken into
account (here 10), Explicitly, the mean of SCup and SClow are
calculated based on the last 10 reversals and are referred to as
the upper and lower staircase threshold, respectively. The range
between the mean SCup and mean SClow reflected the area of
uncertainty (AU) (Equation 2).
AU =1
nXn
i=1SCupi−1
nXn
i=1SClowi(2)
To control for inter-individual differences in the judgment of
the target sizes in the control condition, and to allow for
inter-individual and inter-trial comparisons, the judgments were
corrected by subtracting the perceptual threshold of the control
trial (PTcontrol) from the corresponding perceptual threshold
of each trial (PTtrial), i.e., IM =PTcontrol –PTtrial , where IM
stands for illusion magnitude. For the statistical analyses and the
visualizations, the illusion magnitude was used.
Response time was defined as the time between stimulus
presentation onset and the participant’s response. We next
computed the average response time before a participant crossed
one of the staircase thresholds for the first time (referred to
as RTbase). For this procedure, the first response was omitted.
The average response time following this threshold crossing was
referred to as RTAU.
Three-Way repeated measures ANOVAs with target size (a
in Figure 1A), distance (bin Figure 1A), and context size (c
in Figure 1A) as within participants factors were performed to
investigate the effects on the illusion magnitude and the area
of uncertainty. If significance levels were met (α=0.05), the
tests were followed up by Bonferroni post-hoc tests (α=0.05).
A Four-Way repeated measures ANOVA with target size (a),
distance (b), context size (c), and response moment (RTbase,
RTAU) as within participants factors was used to investigate
significant effects on response time. The degrees of freedom
were corrected according to the Greenhouse-Geisser method to
control for non-sphericity of the data if necessary. If this was
the case, the adjusted degrees of freedom were reported below.
In order to examine if the perceived size of the targets of the
illusion trials were significantly different from those of the control
trials, a paired samples t-test was performed for each condition.
Pearson correlation coefficients were calculated to investigate
potential (linear) correlations between response time, area of
uncertainty and illusion magnitude, and between the response
time before the area of uncertainty for the upper and lower
staircase.
RESULTS
Illusion Magnitude
Recall, for the statistical analysis the control perceptual threshold
(PTcontrol) per target size was subtracted from the PTtrial to
control for the participants’ ability to judge targets of different
sizes. Figure 2A displays the results of the paired samples t-
tests to investigate if the illusion magnitudes were significantly
different from the control trials. There, it can be seen that a target
appeared only bigger than it was when the context and distance
were small (i.e., 20% of the target size and 110% of target plus
biggest context size, respectively) and the target size small or
medium (i.e., 05 or 1.0 cm). In 33% of the cases, there was no
significant illusion effect. For all other conditions the target was
perceived as smaller than it actually was.
Significant main effects for illusion magnitude were found
for context size [F(2,22) =40.698, p=0.000, η2
p=0.787],
distance [F(2,22) =24.181, p=0.000, η2
p=0.687] and target
size [F(1.244,13.686) =28.973, p=0.000, η2
p=0.725]. The
illusion magnitudes of all target sizes were significantly different
[all p<0.005; mean ±SD for the small (−0.01 ±0.01), medium
(−0.04 ±0.01), and big (−0.11 ±0.02) target size], as well as
for all context sizes [all p<0.001; mean ±SD for the small
(−0.01 ±0.01), medium (−0.05 ±0.01), and big (−0.10 ±0.02)
context]. For target—context distance, small distances differed
significantly from the medium (p<0.000) and big distances
(p<0.005), however, medium and big distances did not differ
significantly from each other [p>0.05; mean ±SD for distance
small (−0.03 ±0.01), medium (−0.07 ±0.01), and big (−0.06 ±
0.01)].
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Knol et al. Quantifying the Ebbinghaus figure effect
FIGURE 2 | Illusion effects as a function of target size, context size, and target-context distance.(A) Significance levels resulting from paired samples t-tests
and the direction of the illusion magnitude (IM) for each target size as a function of the context-target distance and context size. The black and white squares indicate
a significant effect for bigger perceived targets and smaller perceived targets, respectively. The gray squares show conditions that were not significantly different from
the control trials (α= 0.05). (B) Mean IM (and standard deviation) as a function of target size and distance. (C) Mean IM (and standard deviation) as a function of target
size and context size.
The analysis further revealed a significant interaction between
target size and target-context distance [F(4,44) =3.933, p=
0.008, η2
p=0.263; see Figure 2B], as well as between target size
and context size [F(2.244,24.687) =12.822, p=0.000, ηp2=
0.538; see Figure 2C], indicating that context size and distance
influenced the illusion effect differently for the different target
sizes. When significantly different from the baseline, the big
and medium target-context distances always had a diminutive
effect on the perceived target size (Figure 2A). The illusion
magnitude under the small distance was always smaller than that
of the medium and big distance, except when the illusion had a
magnifying effect on the perceived target size. Except for the small
distance, the big context size always had a stronger diminutive
effect on the perceived target size than the medium context size,
and the medium context size always had stronger diminutive
effect than the small context size (see Figure 2C). The interaction
of the three factors distance, context and target size approached
significance [F(3.857,42.431) =2.427, p=0.065, η2
p=0.181].
The target-context distance by context size interaction was not
significant.
Area of Uncertainty
The area of uncertainty was only significantly influenced by target
size [F(1.260,13.855) =22.731, p=0.000, ηp2=0.674]. Post-hoc
tests indicated that it increased in the control conditions as well
as in the illusion trials as target size increased [for illusion trials:
big vs. medium or small target size (p<0.005), medium vs. small
target size (p<0.05)].
Response Time
The response times for the three target sizes for the baseline
(RTbase)and area of uncertainty (RTAU )control conditions
were not significantly different [p>0.05; mean ±SD RTAU
for target small (0.89 ±0.38), medium (0.89 ±0.47), and
big (0.96 ±0.57)]. Presentation of the Ebbinghaus figures,
however, provoked longer response times compared to the
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Knol et al. Quantifying the Ebbinghaus figure effect
control conditions [F(1,11) =35.795, p=0.000, ηp2=
0.765]. In addition, for the illusion trials, RTAU was significantly
higher than RTbase [F(1,11) =7.8, p=0.017, ηp2=0.415;
Figure 3A]. Further, a significant main effect of target-context
distance [F(2,22) =6.1, p=0.008, ηp2=0.356; Figure 3B] and
of target size [F(2,22) =4.9, p=0.17, ηp2=0.310; Figure 3C] on
the response time was found. Post-hoc tests revealed that response
times were significantly longer at small distances compared to
big distances (p<0.01) and in the big target size conditions
than in the small target size conditions (p<0.05). Furthermore,
an interaction effect between target size and distance was found
[F(4,44) =2.9, p=0.034, ηp2=0.207] which was mainly
caused by the medium distance. For the small and big distance,
FIGURE 3 | Response times. (A) Average response time (and standard
deviation) as a function of time; base refers to the baseline responses and AU
refers to responses in the area of uncertainty. (B, C) represent the average
response time (and standard deviation) over a small (s), medium (m), and big (b)
distance (B) and target size (C). Asterisks indicate significant effects (α= 0.05).
the response times increased with increasing target size, whereas
for the medium distance the response time was shortest at the
medium target size.
Correlations between Illusion Magnitude,
Area of Uncertainty, and Response Time
A significant but weak correlation was found between the
absolute illusion magnitude and the area of uncertainty [r(322) =
0.12, p<0.05]. Further, as the absolute illusion magnitude
increased, the response time (moderately) increased [r(322) =
0.25, p<0.001]. In contrast, if the area of uncertainty increased,
the response time decreased [r(322) = −0.41, p<0.001]. Further
examination of the relation between the area of uncertainty and
response time across participants revealed that it was exponential,
and that the exponent decreased with target size (Figure 4).
DISCUSSION
Summarize Findings
We investigated the role of context size, target-context distance,
and (actual) target size on perceived target size using a staircase
procedure. In accordance with our hypotheses, we found no
significant illusion effect in 33% of the 27 applied parameter
combinations. Whenever there was an illusion effect, all three
factors affected the PT. A target circle appeared bigger in only
two out of 27 conditions (i.e., 7%), namely, when presenting
a small or medium target with small context circles at a small
distance. In all other cases (i.e., 60%) the target appeared smaller.
The area of uncertainty grew with a growing target size and
with a decreasing target-context distance. Furthermore, the
response time increased whenever context circles surrounded
the target, and with increasing target size. The response time
correlated positively with the illusion magnitude, but opposing
our prediction, correlated negatively (but weakly) with the area
of uncertainty.
FIGURE 4 | Response time in the area of uncertainty as a function of the area of uncertainty. The response time decays exponentially as a function of the
area of uncertainty, and the decay increases with the increasing target size (the small (A), medium (B), and big target (C) are represented in the left, middle, and right
panel, respectively).
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Knol et al. Quantifying the Ebbinghaus figure effect
Illusion Magnitude
Massaro and Anderson (1971) formulated an equation according
to which the illusion effect scales positively with target size. In
accordance therewith, the authors reported two experiments that
both showed increased illusion effects as a function of increasing
target size (more specifically, 1.3, 1.5, and 1.7 cm). Our findings
are in agreement with theirs, and we showed that this effect holds
for a wider range of target sizes (namely, 0.5, 1.0, and 2.0 cm).
Nemati (2009) argued that illusionary effects of the
Ebbinghaus figure are the result of a combination of a size
contrast effect and the area of empty space (i.e., the area of the
stimulus that is not filled by the context). The size contrast effect
holds that smaller or bigger context circles, relative to the target,
cause an over- or under-estimation, respectively, of the perceived
target size due to contrast mechanisms (Massaro and Anderson,
1971). If so, our findings should reflect only size contrast effects
since we controlled for the empty space area by covering 75% of
the circumference in all stimulus configurations. In accordance
with Roberts et al. (2005), we reported, however, that small
context circles did not always make the target appear bigger
(i.e., only in 22% of the cases a target with a small context was
perceived as being bigger). That is, the Ebbinghaus figure cannot
be reduced to “just” a size-contrast effect in which a target is
always perceived as being bigger when the context is smaller
than the target size. In other words, we oppose earlier work
describing magnifying and reducing effects of the smaller and
bigger surround on a target, respectively (Obonai, 1954; Massaro
and Anderson, 1971).
As compared to Roberts et al. (2005), fewer parameter
combinations resulted in positive illusion magnitudes (i.e.,
over-estimation of target size) and, furthermore, the absolute
maximum illusion magnitude was bigger. Differences in the
direction and size of the illusion effect could possibly be explained
by the different target sizes (Roberts et al., 2005 employed
target sizes of 1.05 and 1.4 cm whereas we used 0.5, 1.0, and
2.0 cm), since target size played a big role in the size of the
illusion magnitude, and interacted with target-context distance
and context size.
Target-context distance has been suggested to be more
important than the size-contrast effect for the illusion magnitude
(Im and Chong, 2009). This suggestion, however, is not
supported by our results: although a significant effect of
target-context distance on illusion magnitude was found, this
effect was weaker than the effect of context size and target size.
Whereas a target-context distance larger than 1.9 cm (3.5◦)
was found to decrease the perceived target size (Roberts et al.,
2005), a small target-context distance (0.3–1.2 cm in Girgus
et al., 1972) has been shown to increase the perceived target size
(Oyama, 1960; Girgus et al., 1972). That is, perceived target size
seems to reveal an inverted u-shape as a function of context
distance. In line therewith we found increased perceived target
sizes for small distances (0.8 and 1.6 cm for the small and medium
sized target, respectively) when combined with a small context
whereas a distance of 1.4 cm in combination with a small context
size did not result in an increased perceived target size. For
all other target-context distances (i.e., 2.0–4.2 cm), if there was
an effect, the perceived target size was smaller than the actual
target size. However, this was also the case for the smallest
target-context distance for the biggest target (2.4 cm). In fact,
the largest target was never perceived as being bigger, which
could be due to the target-context distances that for this target
size always exceeded the 0.3–1.6 cm range, or other protocol
variations (a, b in Figure 1). An increase in distance up to
3.6 cm (all conditions except the large distance—large target
condition) resulted in a larger illusion magnitude. A distance
greater than 3.6 cm (i.e., 4.2 cm; large distance—large target)
reduced the illusion magnitude (see Figure 2B), which could
explain the interaction effect between target size and target-
context distance. That is, these findings agree with an inverted
u-shape of the illusion magnitude over target-context distance.
This might stroke with what Sarris (2010) called Ebbinghaus’
law of relative size-contrast, in which he describes a general
inverted u-shape trend for size-contrast effects (Note, though,
that Ebbinghaus pointed at the relative size comparison of
dwarfs, men, and dolls; Ebbinghaus and Dürr, 1902; Sarris,
2010). However, to confirm this hypothesis, a broader range
with smaller and larger target-context distances should be
tested.
We did not find a significant interaction effect between target-
context distance and context size. In Roberts et al. (2005), this
interaction was tested for in two experiments. In their experiment
3, which was similar to our experiment—they reported an illusion
magnitude from 0.084 to 0.12 cm for a target size of 1.4 cm, a
context size of 0.35 and 1.4 cm, and a target-context distance
of 1.05–4.67 cm—the interaction turned out to be significant
whereas in the other (experiment 1) it did not. Our results
indicate that the illusion magnitude was affected by target-
context distance and context size in a similar fashion (illusion
magnitude of 0.1–0.13 cm for target size 1 and 2 cm, context size
0.2–3.6 cm, and a target-context distance of 1.6–4.2 cm).
Area of Uncertainty
We quantified the distance between the points as asymptotically
reached by the upward and downward staircases, and refer to
it as the area of uncertainty. The area of uncertainty represents
a measure of the perceptual distinctiveness of the illusion. We
showed that it increased as the target size increased. This might
be a simple demonstration of Weber’s law (or the Weber-
Fechner law) according to which sensitivity to changes in
perception decreases when stimulus intensity increases (i.e.,
the ratio between the “just-noticeable difference” in a physical
property and its magnitude is invariant). Schmidt et al. (1979)
proposed that variability (in force production) would increase
proportionally with the absolute magnitude (of the forces)
(Schmidt et al., 1979). Along the same line, it might be that
the variability represented by the area of uncertainty scales
linearly with target size. Interestingly, it has been shown that
internal noise increases with letter size (Pelli and Farell, 1999).
It may well be that our scaling of target size similarly increased
internal noise. In that regard, investigating variations in internal
noise as a function of the various Ebbinghaus figure parameters
(Figure 1A), as well as relative to control conditions, may well
shed novel insights into the (strength of the) illusion effect and
its perceptual distinctiveness.
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Knol et al. Quantifying the Ebbinghaus figure effect
The increase of the area of uncertainty confirms the use of a
minimum of two staircases and shows the directionality imposed
by the procedure. By taking the mean of the two staircases,
information about the distance between these two staircases is
lost, even though this contains valuable information about the
perceptual and decision-making processes, and thus the illusion
effect.
Response Time
Our response time data showed a complex effect of the illusion.
First of all, the response time in the control conditions was
unaffected by target size, which stands in contrast to reports of
an inverse relation between target size and reaction time (Payne,
1967; Osaka, 1976; Marzi et al., 2006). The illusion conditions,
however, showed two contrasting effects: the influence of target
size on the response time (response time increased with target
size; Figure 3C), and the influence of target-context distance
on the response time (response time decreased with increasing
target-context distance; Figure 3B). Furthermore, the response
time correlated positively (but weakly) with the absolute illusion
magnitude and negatively with the area of uncertainty. Since, to
our knowledge, most of the Ebbinghaus studies neglected the
response time, we can only refer to a study with schizotypal
traits in which the authors measured the illusion magnitude
and the response time of two Ebbinghaus figures (small and big
context circles with a fixed target size and target-context distance;
Bressan and Kramer, 2013), and reports of simple reaction time
studies (Sperandio et al., 2010). Whereas Bressan and Kramer
found that individuals with a longer response time tended to
show less illusion effects (Bressan and Kramer, 2013), others
reported that the reaction time was shorter when the target
appeared bigger/longer (Sperandio et al., 2010; Ponzo illusion:
Plewan et al., 2012). We found that strong illusion effects went
hand in hand with long response times. Thus, rather than being
scaled according to the perceived target size, we found that
the response time scaled with the (absolute) illusion magnitude.
It may be that, at least to some extent, these discrepancies
are due to methodological differences: in the reaction time
studies quickness of response was stressed and the illusions
were presented briefly only (ranging from 10 to 250 ms), unlike
our study. Regardless, the question remains what the origin
of the increase in response time is, and how response time
and illusion magnitude causally relate (if so). Given the widely
accepted view that response time somehow reflects the cognitive
processes involved in a given performance, and the more easily
comprehendible effects relative to the control condition and the
moment of assessing it (i.e., baseline vs. the area of uncertainty),
we believe that response time, which is typically discarded in
studies using visual illusion as a means to investigate the ventral-
dorsal visual pathway distinction, may provide an interesting
novel entry point to its effects. We will return to this issue in the
section below.
Models Describing the Ebbinghaus Illusion
Until now it has not been possible to predict the illusion
magnitude given a certain set of parameters. Massaro and
Anderson (1971) and Roberts et al. (2005) described a simple
model of the Ebbinghaus illusion. The model developed by
Massaro and Anderson (1971), to which they refer as judgmental
model, is based on the idea that the Ebbinghaus figure works as
a simple size-contrast illusion with a fixed number of context
circles. They did not take into account that the completeness
of the surroundings would influence the illusion magnitude as
previously shown (Massaro and Anderson, 1971; Roberts et al.,
2005). Nemati (2009) extended the hypothesis of Massaro and
Anderson (1971) with the idea that the area of empty space
influences the magnitude and direction of the illusion effect.
By controlling for the completeness of the surroundings, as
in Roberts et al. (2005), we controlled for differences in the
empty space. The sole remaining explaining factor, thus, would
accordingly be the size-contrast effect. As said above, this was
not the case. Roberts et al. (2005) proposed a model according to
which the illusion magnitude scales exponentially with inducer
distance. Their model could not explain our data in 78% of all
factor combinations. We incorporated three times the number
of participants, and should therefore have shown an exponential
decaying trend if the model would have been correct.
In that regard, a potential shortcoming of existing models
is that they do not allow for non-linear effects like hysteresis,
multi-stability, etc. Dynamical systems are described in the space
spanned by its state variables. If one stable solution exists in
that space (an attractor), the system will invariantly evolve
toward it. If multiple stable solutions exist (multi-stability), it
will evolve toward one of the attractors, dependent on the
initial conditions. In a bifurcation, the number and/or nature
of the system’s solution changes when the so-called bifurcation
parameter is (gradually) scaled. Hysteresis only occurs in multi-
stable systems, and refers to the phenomenon that when changing
a bifurcation parameter the system’s history determines to which
stable attractor the system will evolve. Such effects are the
hallmark of non-linear systems, and evidence that behavioral,
perceptual, and cognitive systems belong to that class of non-
linear systems abounds (e.g., Haken et al., 1985; Tuller et al.,
1994; van Gelder, 1998; see also Kelso, 1995 for a review). Our
present results only hint at the existence of non-linear effects
(note that the experiment was not designed so as to reveal them).
The response time data and, in particular their exponential decay
as a function of the size of the area of uncertainty (Figure 4), may
provide indications that are consistent with non-linear effects.
Within the borders of the area of uncertainty, the responses are
at chance level. Outside this area of uncertainty the participants
perceive a clear difference between the target and the probe. This
observation is open to interpretation in terms of the existence of
two distinct “states” or regimes (multi-stability). In this sense, the
borders of the area of uncertainty are linked to the bistability
regime of the coexistent two distinct states (see Figure 5).
They are, however, not synonymous therewith. Intuitively, it
makes sense to assume that response time scales with the
degree of (perceptual) uncertainty. Consequently, the shorter an
observer’s distance to the area of uncertainty, the slower her/his
response. In the present staircase procedure, the participants’
initial conditions were the same, but the size of their area of
uncertainty varied. In other words, their distance to the area of
uncertainty, which scales inversely with its size, was different
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Knol et al. Quantifying the Ebbinghaus figure effect
FIGURE 5 | Cartoon illustration of the Ebbinghaus figure parameter
space. PP1and PP2contain the area of uncertainty for two different
participants; the black (D01,D11 ) and gray arrows (D02,D12) represent the
corresponding distances to the area of uncertainty from the start of the two
staircases (SCup,SClow ), respectively. The non-shaded vs. shaded areas
(PP1, PP2) may represent distinct regimes in parameter space in which
perceptual decision-making is deterministic vs. random, respectively.
(see Figure 5). Consistent with our present argument, the results
in Figure 4 indicate an exponential relation between the size of
the area of uncertainty and response time. For the argument to
hold, however, a similar trend should exist for the participants
individually. We tested this in two ways: first, for each participant
we calculated the distance to the area of uncertainty for the upper
and lower staircase and the corresponding response times for
the second response. For most of the participants (11 out of 12),
the distance was larger for the upper staircase and the response
times were shorter. Both effects were significant (paired t-test;
both p<0.001). Second, we linearly regressed each participant’s
response times against the distance to the area of uncertainty.
Unfortunately, due to the high variability, only three out of 12
regressions were significant at α= 0.05. Their average slope was
–1.23. Regardless, all regressions had a negative slope; the mean
slope of the non-significant regressions was –0.49. That is, across
participants the response time tended to decrease as the area
of uncertainty increased. In combination, these results argue in
favor of a relation between the distance to the area of uncertainty
and response time, and are suggestive of the existence of distinct
regimes of operation. Clearly, however, future efforts will be
needed to either falsify or reject this idea.
Illusion Effects in Motor Tasks
The combination of an increased illusion magnitude,
standard deviation (as suggested by visual inspection of
Figures 2B,C), and the increase in response time as target
size and, concomitantly, the area of uncertainty decreased,
might indicate that strong illusion effect evoking parameters
induce instability in the participants’ decision-making. But
which processes underlie this change in stability is uncertain. As
discussed in Section Models Describing the Ebbinghaus Illusion,
possibly the area of uncertainty and the longer response time
hint at hysteresis. If hysteresis indeed exists, then the mechanism
underlying the change of strength of the illusion effect is linked
to multistability and transitions from one state to another. The
parameter space in Figure 2 offers a starting point to develop
experimental paradigms, in which the Ebbinghaus illusion is
used to drive parametrically coordination behavior through a
“perceived” parameter such as size in contrast to the “physical”
parameter, the actual size. It should be noted, however, that it
should not be naively assumed that the parameter space of the
illusion effects in Figure 2 is the same, when the Ebbinghaus
illusion is used in sensorimotor coordination experiments.
This assumption holds, from the dynamical system perspective,
only for weak coupling of the perception-action system. Weak
coupling means that two dynamic systems, when coupled,
display the same qualitative dynamics as in absence of coupling,
and undergo changes that can be regarded as small perturbations.
For instance, two systems that display oscillations in absence
of coupling can realize arbitrary relative phase relations; when
weakly coupled, they still display oscillations, but now only
certain relative phase relations are stable, others unstable.
For strong coupling, the intrinsic oscillation may disintegrate
and different behaviors may occur that cannot be understood
anymore through the notion of relative phase. This limitation
should be kept in mind when developing applications of the
Ebbinghaus illusion parameter space, in which the perception-
action coupling, if strong, may alter the system dynamics
significantly.
How visual illusion figures affect perception and action has
previously been shown to be a complex puzzle, and highly
depending on the research method and selected parameters
(Bruno and Franz, 2009). In the present perceptual study,
the effect sizes and direction of the effects resulting from the
perception of the Ebbinghaus figure appeared to be highly
dependent on the selected parameters. Observed illusion effects,
if present, up to (exceptionally) 10% of the target size might
explain why illusion effects in motor tasks have sometimes failed
to materialize.
Fitts’ law predicts the time required to rapidly move between
two targets as a ratio of the width of the target and the distance
to the target (Fitts, 1954). The Ebbinghaus (like) figure has
been implemented in a Fitts’ task to test whether a perceptual
illusion would affect the motor behavior (van Donkelaar, 1999;
Fischer, 2001; Ellenbürger et al., 2012). van Donkelaar (1999) and
Ellenbürger et al. (2012) found that movement was affected by
the illusion (in terms of movement time (van Donkelaar, 1999);
van Donkelaar, dwell time, and harmonicity Ellenbürger et al.,
2012). However, Van Donkelaar and Ellenbürger and colleagues
did not quantify the illusion magnitude of their Ebbinghaus
figures. In contrast, Fischer (2001) found an effect of context
size and context-target distance on perception but no effect
on movement (at least, in the absence of stimulus-movement
delays). The perceptual effects, while significant, were rather
small; they ranged from –0.3 to 0.2 mm, that is, about an order
of magnitude smaller than the range reported here. It remains to
be seen, however, to which degree the method used by Fischer
to quantify the illusion effect on perception, namely scaling a
probe until it matches the perceived target size, provides robust
results (see also Introduction). In fact, we found no illusion
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Knol et al. Quantifying the Ebbinghaus figure effect
effect in 33% of the parameter combinations for a similar target
size as in Fischer’s study (1 vs. 1.2 cm, respectively). Thus, if
the reported perceptual results fail to be robust, the results of
Fischer’s movement study might simply be due to the lack of
illusion effects. Furthermore, since the illusion magnitude was
often found to be relatively small, it might be that the measures
used for motor studies were too coarse to capture small effects
of the illusion. In conclusion, studies like these hamper drawing
firm conclusions on how perceptual and motor effects relate,
and to what degree the ventral and dorsal stream operate in a
functionally distinct manner.
Methodological Concerns
Both the Ebbinghaus figure and the staircase procedure can be
adapted by changing numerous parameters such as parameters
a, b, c in the Ebbinghaus figure (Figure 1A), and the (adaptive)
stepsize, procedure, starting point, and number of reversals
for the staircase procedure. Due to the contradictory results
of various methods to quantify the illusion effect, and due to
the large number of Ebbinghaus figure configurations tested
in this study, the widely studied and applied two-up, one-
down staircase procedure was chosen, which is a two alternative
forced choice method (2AFC). Several previous studies also
applied the staircase procedure to study different features of the
Ebbinghaus figure (Roberts et al., 2005; Im and Chong, 2009;
McCarthy et al., 2013). Another version of the 2AFC method
to study perception is the method of constant stimuli, in which
a fixed number of combinations of (Ebbinghaus) figures are
shown a certain number of times in a random order. In this
case, the sampling is random and every possible stimulus-pair
combination is presented equally often. This method allows
for the full sampling of a so-called psychometric function. The
slope and the horizontal shift of this psychometric function (i.e.,
a cumulative probability distribution) and the X50 value (also
called the Point of Subjective Equality) then specify the illusion
effect. A big area of uncertainty might be linked to a shallow
slope of the psychometric function, and the PT should be equal
to the point of subjective equality. McCarthy et al. (2013) have
performed 4 experiments with using both the staircase procedure
(experiment 2) and the method of constant stimuli (experiments
1, 3, and 4) showing that both methods result in similar points
of subjective equality. Considering the long history of staircase
procedures in the field of psychophysics (García-Pérez, 1998),
and the magnitude of the illusion effect being in a similar range as
in the similar study of Roberts et al. (2005) the staircase procedure
opens new doors in order to quantify the Ebbinghaus illusion
effect in a systematic way.
Clearly, this method has its own limitations and assumptions.
For example, to which percent-larger responses (referred to
as percent-correct responses in visual contrast and luminance
studies) the staircases converge using different protocols, is still
under debate (for a review see García-Pérez, 1998). At chance
level, a two-up, one-down procedure will bias responses in the
“up” direction rather than the “down” direction. However, a
two-up, one-down procedure also assures a better precision
than a one-up, one-down procedure (García-Pérez, 1998). Two
staircases that start from positions bigger and smaller than
the actual target size, assure a fully symmetrical procedure,
and a bias in both the “up” and “down” direction, for the
upper and lower staircase, respectively. Anyhow, regardless these
limitations, visual illusion research, be it in the context of the
visual stream dissociation or otherwise, may benefit from these
(and potentially other) more or less standardized and in-depth
investigated methods.
CONCLUSION
Concluding, since the Ebbinghaus figure is widely used but
no clear rule is set, inter-comparison of the broad range of
parameters remains difficult. We haven shown that the illusion
magnitude highly depends on an interplay of target size,
context size and target-context distance, and that a third of the
parameter combinations here used did not evoke an illusion
effect. Importantly, however, even if the group-averaged illusion
magnitude can be predicted by a set of stimulus configuration
parameters (or established rules), the predictive value for
individual performances would likely be limited given the
marked inter-individual variability. Thus, the implementation of
the Ebbinghaus figure in various fields of research needs to be
handled with care and quantified per study.
ACKNOWLEDGMENTS
The research reported herein was supported by the Conseil
regional Provence-Alpes-Côte d’Azur, and the Brain Network
Recovery Group through the James S. McDonnell Foundation.
The authors acknowledge Dr. Bruno Berberian and Dr. Andreas
Spiegler for the fruitful discussions.
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Frontiers in Psychology | www.frontiersin.org 11 November 2015 | Volume 6 | Article 1679