Straightness as the Main Factor of the Hermann Grid Illusion

Abstract

The generally accepted explanation of the Hermann grid illusion is Baumgartner's hypothesis that the illusory effect is generated by the response of retinal ganglion cells with concentric ON-OFF or OFF-ON receptive fields. To challenge this explanation, we have introduced some simple distortions to the grid lines which make the illusion disappear totally, while all preconditions of Baumgartner's hypothesis remain unchanged. To analyse the behaviour of the new versions of the grid, we carried out psychophysical experiments, in which we measured the distortion tolerance: the level of distortion at which the illusion disappears at a given type of distortion for a given subject. Statistical analysis has shown that the distortion tolerance is independent of grid-line width within a wide range, and of the type of distortion, except when one side of each line remains straight. We conclude that the main cause of the Hermann grid illusion is the straightness of the edges of the grid lines, and we propose a theory which explains why the illusory spots occur in the original Hermann grid and why they disappear in curved grids.

Full text

1 Introduction
The Hermann grid illusion is a well known classical illusion in which illusory sp ots
are visible at the intersections of the grid lines (see figure 2). It is very robust, since the
occur renc e of the illusory spots is tolerant to a wide range of geometrical parameter
changes, such as line wi dth, line spa cing, direction, angle of the lines, etc (for a review
see Spillmann 1994).
The generally accepted explanation, which may be considered as the official text-
book account of the Hermann grid illusion, is Baumgartner's (1960) hypothes is,
according to which the illusory spots are the product of centre/surround antagonism
within the
ON
^
OFF
or
OFF
^
ON
receptive fields of the retinal ganglion cells. The
generally used computer simulation of Baumgartner's explanatory pri nciple is based
on the `Mexican-hat' weighting function. This computational model is also referred to
as the difference-of-Gaussi ans (DOGs) model (Marr 1982, page 63).
Although Baumgartner's explanatory principle is a local one, some authors have
criticised it, showing that in addition to local factors, global ones also play a role in
the explanation of the illusi on (Wolfe 1984; fo r a review see Ninio and Stevens 200 0).
To prove its robustness, the purpose of these authors was to modify the Her mann gr i d
while the illusion persisted.
Our a im is the oppos ite: we have i ntroduc e d some minor distortions to the grid
lines that make the il lusory spots disappear totally. We then demonstrate that the
predictions based on Baumgartner's hypothes is contradict perception; this discrepancy
forms the basis of our paper. On the grounds of our psychophysical experiment based
on the concept of distortion tolerance we have introduced earlier (Geier et al 2005),
we demonstrate that the straightness of the black ^ white edges is the main cause of the
illusory spots. Finally, we present an alternative theory, which accounts for the appear-
ance of spots in the Hermann grid as well as for their disappearance in the distorted
grids.
Straightness as the main factor of the Hermann grid illu s ion
Perception advance online publication
Ja
¨
nos Geier, La
¨
szlo
è
Berna
¨
thô, Mariann Huda
¨
kô, La
¨
szlo
è
Se
¨
ra#
Ster eo Vision Ltd, Na
¨
dasdy Ka
¨
lma
¨
n utca 34, H 1048 Budapest, Hungary; ô Institute of Psychology,
Eo
«
tvo
«
sLora
¨
nd University, Izabella utca 46, H 1064 Budapest, Hungary; # Department of Psychology,
Kodola
¨
nyi Ja
¨
nos University College, Fu
«
rdo
«
u. 1, H 8000 Sze
¨
kesfehe
¨
rva
¨
r, Hungary;
e-mail: janos@geier.hu
Received 5 April 2006, in revised form 4 July 2007; published online 1 May 2008
Abstract. The generally accepted explanation of the Hermann grid illusion is B aumgartner's
hypothesis that the illusory effect is generated by the response of retinal ganglion cells with con-
centric
ON
^
OFF
or
OFF
^
ON
receptive fields. To challenge this explanation, we have introduced
some simple distortions to the grid lines which make the illusion disappear totally, while all pre-
conditions of Baumgartner's hypothesis re main unchanged. To analyse the behaviour of the new
versions of the grid, we carried out psychophysical experiments, in which we measured the distor-
tion tolerance: the level of distortion at which the illusion disappears at a given type of distortion
for a given subject. Statistical analysis has shown that the distortion tolerance is independent of
grid- line width withi n a wide range, and of the type of distortion, except when one side of each
line remains straight. We conclude that the main cause of the Hermann grid illusion is the straight-
ness of the edges of the grid lines, and we propose a theory which explains why the i llusory
spots occur in the or iginal Hermann grid and why they disappear in curved gri ds.
doi:10.1068/p5622

2 Theoretical problems raised by Baumgartner's hypothesis
First, let us point out some deficiencies of Baumgartner's hypothesis. It raises at least
one unresolved question and one discrepancy.
2. 1 The unresolved q uestion
According to empirical data, the illusion occurs for a wide range of grid-line widths,
while concentr ic receptive fields of several different sizes can be discerned. Simple
geometric considerations and computer simulation (see figure 1) demonstrate that the
compu tational mod el of concentric receptive fields works only in the cas e of an `opti-
mally' set diameter of the Mexican-hat weighting function at any given gr id-line width.
Figures 1b, 1c, and 1d were generated by a computer program. The input in all cases
was the same bitmap image of the Hermann grid (figure 1a). The program generated
(a)
(b)
(c) (d)
Figu re 1. An undistorted Hermann grid (a) and its fil tered outputs generated by computer sim-
ulation, using the Mexican-hat weighting function of different s parameters: (b) s  4,(c)s  8,
(d) s  16. It is clear that illusory spots occur only in the case of one optimally set diameter,
figure (c), while in the case of a smaller diameter, no spots are produced. Moreover, spots
enscon ce themselves in the grid- line sections and the intersections tur n light when a s large r than
the optimal one i s used [see (d)] and the two diagrams in which the grey scale cross-sections
of the centre lines of the images are illu strated. Several unwanted byeffects occur even in the
case of the optimal diam eter, such as small balls at the corners of the squares, or blurred grid line
edges. In addition, the interior of the squares also turns lighter. We have placed the cross-section and
the plan of the Mexican hat on the filtered images. In the plan diagram, the light-gr ey level stands
for the inhibition, whereas the dark one represents the stimulation. Image sizes  6006600 pixels;
grid line width  24 pixels; distance of the lines  144 pixels.
2JGeier,LBerna
¨
th, M Huda
¨
k,LSe
¨
ra

filtered versions of figure 1a by replacing each of its points with the weighted means of
their environments, where the weighting function was a Mexican hat with s deviation.
The deviations in figures 1b, 1c, and 1d were s  4, s  8,ands  16, respectively.
Obviously, it is only figure 1c which produ c es an effect somewhat similar to human
perception. In figure 1b, the spots cannot be circumsc ribed and the centres of the
grid lines becom e dark. The spots are away from the intersectio ns in figu re 1d, and
the entries to the grid lines appear darker than the intersections. Hence, the following
question arises: for a given grid-line width, why are only tho se receptive fields a ctive
whose diameter is `optimally set', while all the others are not? Or, in computational te rms:
what would be the algorithm for d etermining the optimal diameter of the Mexican-hat
weighting function for a given grid-line width?
Baumgartner's hypothesis provides no answe r to th ese questions. Theoretically, it
would be possible to devise an algorithm to select the opti mal receptive field diameter,
but such models are not available in the literature.
2.2 The di screpa ncy
When gazing at a classical Hermann gr id, one observes homogeneous black squares,
sharp edges, and a homogeneous white border surrounding the Hermann grid. Illusion
(ie the difference between reality and perception) occurs on ly at the intersections.
Nonetheless, what one can see in figure 1 is that the sharp edges and corners of the
squares are blurred, and sm all balls have been generated at the corners by the com-
puter simulati on of Baumgartne r's a ccount. While the computer simulation pre dicts
illusory spots at the intersections almost correctly, there are several other predictions
that disagree with human perc eption.
Furthermore, as shown in figure 1, i f the s parameter of the Mexican hat is optimal
fo r generating illusory spots (s  8), the corners of the black squares become rounded,
and vice versa: if s is small er than the optimal value, the rounding is negligible, but
no illusory spots are predicted by the simulation.
Schiller and Carvey (2005) propose an alternative account of the Hermann grid
illusion. In fact, their hypothe sis is a modification of Baumgartner's hypothesis, i n
which the illusory effect is l argely attributed to the relative activity of neurons driven
by the
ON
^
OFF
systems. They assign principal role to the orientation-selective
simple cells (S1) in the primary visual cortex (V1), whose receptive f ields are elongated
along their axis of orientation. T heir main suggestion is th at ``illusory smudges are
the result of the relative degree of activity of the
ON
and
OFF
S1 cells at th e inters-
ections, as compared with activity at non-i ntersecting locations'' (Schiller and Carvey
2005 , page 1 389) .
This hypothesis raises two objections. First, it i s obvious that there is only one
specific size and orientation of the S1 receptive field, which is optimal for eliciting the
illusory effect. But what about S1 cells whose receptive fields are at the same location
but have different axes of orientation? Here the authors provide no explanation. If
these neurons are inactive, the mechanism activating solely those cells that are optimal
fo r eliciting the illusion should be specified in their argume nt. On the other hand, if
these cells are assumed to be active, the process in which the responses of cells sharing
the same location are combined to produce th e output response of the particular
location should be described in detail. These two questions are, in fact, identical to those
we have raised during the analysis of Baumgartner's acc ount. However, the authors leave
both questions unanswered in their pape r.
Second, in their figu re 11 (page 1387), Schiller and Carvey place the axes of the
elongated receptive fields precisely on the edges of the grid lines; therefore the inner
areas of the intersections are left empty, and none of the elongated receptive fields is
indicated as being located there. Thus, no spot is pred icted to appear in the middle of
Str aightness as the main factor of the Hermann grid illusion 3

the intersections on the basis of the working principle of S1 cells; it is only at the
entrance of the gr id lines where illusory darkening is expected to occur in accordance
with this idea. Consequently, the foregoing line of reasoning does not provide sufficient
explanation for the m ost essential asp ect of the Herman n grid illusion, namely that
illusory spots are manifestly darkest at the centre and tend to lighten towards the
pe riphery, as has also been reported by our subjects. As for the central area of the inter-
se ction, Schiller and Carvey merely give the following hint: ``Presumably, the percep tion
of lightness and darkness in those regions of the fig u re that contain no edges is
produced largely by responses elicited in unor iented cells in V1 that receive either
ON
or
OFF
inputs'' (page 1389). Nevertheless, the fundamental question here is: why
does one observe illusory spots in the middle of the inter sections? Their explanation
leaves this question unanswered, too.
On the basis of this analysis, it is cle ar that neither Baumgartne r's, nor S chiller's
propositions provide sufficient explanation for the phenomena occurring in the classical,
unmodified Hermann grid.
3 Distorted grids
Although we have demonstrated two shortcomings of Baumgartner's explanatory princi-
ple, this alone is not sufficient for rejecting that account totally. One possible respons e
to our argu ment could be, as noted by Sekuler and Blake in an answer to Wolfe's
(1984) criticism, that, although criticism is justifiable, ``the explanation offered here is
probably basically correct'' (Sekuler and Blake 1994, page 98).
Here, we shall present some distortions which will prove the untenability of
Baumgartner's explanation. All these distortions are of the kind in which Baumgartner's
hypothesis would still predict illusory spots; however, the illusory spots totally disappear
in all cases. The co mmon properties of the distorted grids are that the intersections
remain right- angled, and only the straightnes s and/or the directions of the edges of
the gr id lines are varied. The m ost effective distortion is to substitute sinusoid curves
for the straight lines. The result is shown in figure 2
ö
the illusory spots have totally
disappeared.
It is evident that the applied distortion has no impact on the response of conc entric
receptive fields (ie on the relationship between the image and the Mexican hat); there-
fo re Baumgartner's explanation predicts the same illusory spots in the distorted g rid
as in the undistorted Herman n gr id. However, as observers do not see illusory spots
in the distorted grids, the prediction of Baumgartner's model is certain ly discrep ant from
perception.
Now, by simple logic, we shall demonstrate that Baumgartner's explanatory prin-
ciple on its own does not account for the illusory spots perceived in the Her mann grid
(at least in the versions that have been origi nal ly formulated, cited, and freque ntly
described). To specify the conditions of the o riginal explanatory hyp othe sis, let us
consider the citation below:
``The subjective brightening and darkening effects at the crossing of a g rid may be explained
by relative di fference of lateral inhibition and activation. A neurone of the B-system
`looking' at an intersection receives more light in its receptive field surround and produces
more lateral inhibition than when it is stimulated by a bar.'' (Jung 1972, page 223)
This citation implies that the increment in the sur round stimulation of the receptive
field is a sufficient condition for the illusion if the central stimulation is constant.
Accordingly, no other conditions are required by Baumgartner's explanati on than the
different ratio of centre/surround stimulation. Hence, the straightness, continuity, and
homogeneity of the lines are not required by h is hypothesis, n or i s collin earity of
the intersections demand ed. But, although we have not changed any conditions of the
4JGeier,LBerna
¨
th, M Huda
¨
k,LSe
¨
ra

original explanation, the spots have disappeared in the sinusoid grid (figure 3). This
line of evidence undermi nes Baumgartner's hypothesis as an ac count of the illusory
spots in the Hermann grid. Thus, Baumgartner's hypothesis should be either supple -
mented by furth er conditions, or declined as a whole.
To demonstrate the foregoing, let us have a look at figure 4, in which the output
image of the sinusoid grid is displayed, filtered by the Mexican-hat weighting func-
tion. The lin e widths and distances of th is sinus oid gri d are identical to those of
figure 1a. The size of Mexican-hat function applied here is s  8, which is the same as
the `optimal' one shown in figure 1c. It is obvious that the Mexican-hat simulation i s
not sensitive to the cur vature of the grid lines: the output spots are exactly like those
in figure 1c, which correspo nds to the prediction of Baumgartner's explanatory princ iple.
This entirely contradicts perception.
Figure 2. The classical Her mann grid (above) and the sinusoid grid (below). In the case of
straight grid lines, illusor y spots are seen in the intersection s, but they totally disappear in the
case of the curved grid lines. The ampl itude of sine curve is less than 10% of its wavelength.
Str aightness as the main factor of the Hermann grid illusion 5

(a) (b)
Figure 3. (a) A usual explanatory image of Baumgartner's hypothesis [a similar one is in Spillmann
(1994), page 693, figure 1]. According to this account, to observe illusory spots, it is sufficient to
inhibit the lateral ring of the receptive fields located at the intersection twice as much as those
located in the grid-line sections, while the stimulation of the centres remains constant. Though
these conditions are totally fulfilled by the sinusoid grid (b), no illusory spots are perceived at line
intersections. If double lateral inhibition were sufficient in itself to elicit the illusory effect, spots
would appear in both images. However, they do not. Thus, the question arises whether Baumgartner's
hypothesis could be extended to be a satisfactory account of the illusion by including the straight-
ness of the lines as a condition, or should it be rejected totally.
(a) (b)
Figure 4. Sinusoid gr id (a) and its output image by the Mexican-hat computer simulation at
s  8 (b). It is noticeable that the simulation predi cts the same spots at the interse ctions as in (a).
The curvature has no effect. This result correspo nds to Baumgartner's conception; however,
it contradicts human perce ption.
6JGeier,LBerna
¨
th, M Huda
¨
k,LSe
¨
ra

3.1 Possible solutions
Theoretically, there are two possible ways to resolve the strong contradictions between
human perception and the predictions of Baumgartner's hypothesis. The first is to
find an extension that provides an answer to the question: if it is the concentr ic
rec eptive field s that produce the illusory spots in the unmodified Hermann grid,
what is the reason for their disappearance in the distorted grids? The second one is to
reject B aumgartner's hypothesis as an explanation of the Hermann spots.
Taking into consideration the unresolved questions and problem s raised by
Baumgartner's classical explanation, our opinion is that it should b e totally rejected:
it is not the
ON
and
OFF
centred rec eptive fields that underlie the phenome non of
illusory spots in the Hermann grid. However, as long as Baumgartner's hypothesis is
not supplemented by further conditions to explain the differenc e b etween the cases of
the classical and the dis torted Hermann grids, the grounds of this explanatory principle
seem to b e problematic.
In saying this, we do not deny the role of
ON
and
OFF
centred receptive fields in
the visual process in general, but we consider the explanation to be more complex
than that proposed in Baumgartner's hypothesis. Perhaps
ON
and
OFF
systems play an
important role in the backg round, but their effect does not manifest itself directly in
the p erc eptual sensation.
4 Experiment
Here the question emerges what the `right' explanation could be. In order to develop
a new the ory, one should get to know the principles of the appearance and disap-
pearance of the illusion. Our exp er iment was planned in o rder to reveal the effects
of distortions on the illusion, which might provide a vi able basis of a new model.
The experiment here was a tool to provide an answer to the basic question: what is the
rationale for the illusory spots in the Hermann grid?
Earlier experiments concerning the Hermann grid illusio n were bas ed on asking
the subjects how intensively they pe rceived the illusory spots (eg Wolfe 1984). However,
since we can eliminate the illusion completely, it seems more plausible to pose a much
more objective question: when does th e illusion disapp ear? D etermining the limit when
the spots are no longer visible is an even more reliable method than rating the subjec-
tive intensity on a scale.
We define the term `distortion tolerance' as the degree of distortion at which the
illusory spots disappear. Potentially, the distortion tolerance may dep end on the distortion
type, grid-line width, or on the individual subject. Our aim was to reveal, by means
of empirical data, what distortion tolerance depended on, and whether it would differ
significantly in the case of different distortion types.
4. 1 Selection of the applied distortion types
In addition to the sine curves, further distortion types were used (f igure 5).
4. 1 .1 Wavy grid is introduced for testing the role of the collinearity of the intersections.
If collinearity plays a role in the presence of spots, there should be significant differences
in the means of the distortion tolerance of the sinusoid grid and wavy grid.
4. 1 .2 Knotted g rid is introduced for testing the role of the (white and black) area sizes,
which is a crucial point of Baumgartner's model. Increa sing the light intensity only at
the
OFF
surrounds of the receptive fields, while leaving the stimulation of the centre
constant, allowed us to examine the role of the proportion of white and black areas.
4. 1 .3 Half-sided humped grid and an asymmetrical humped grid were introduced in order
to verify whether straight gr id-l ine edges play an essential role. The half-sided humped
grid was generated from the humped grid by replacing the lower curve of the latter
Str aightness as the main factor of the Hermann grid illusion 7

by a straight edge segment. The asymmetrical humped grid was also derived from the
humped grid, by flipping its lower curve vertically along the edge of th e grid line,
so that the lower curves of the line edges were identical; the only difference was the
proportion of the black and white areas.
4.2 Modifying g rid-line widths
In the light of our earlier experiments (Geier et al 2004, 2005), we assumed that grid-
line width does not significantly affect the degree of distortion tolerance. This assumption
corresponds to the well-known fact that the illusion is independent of grid-line width
within a wide range. However, by introducing the concept of distortion tolerance a more
reliable experimental method has been devised.
(a) (b)
(c) (d)
(e) (f)
Figu re 5. The six distortion types used in our experiment: (a) sinus, (b) wave, (c) kn ot, (d) hump,
(e) half-sided hump, (f) asymmetrical hump. The magnitude of the distortion was set by the
subject by pressing the arrow buttons of the computer keyboard until the illusion disappeared.
These images were presented with three different lin e widths (11, 17, and 23 pixels; the resolution
of the monitor was 10246768 pixels), but the distance of the lines was identical in all cases (102
pixels between the left sides of th e lines). In this figure, only the images of 17 p ixel line width
are presented.
8JGeier,LBerna
¨
th, M Huda
¨
k,LSe
¨
ra

In order to investigate the role of grid-line width, we displayed all the six grid
types in th ree versions of line width: the lines of the grids were 11, 17, and 23 pixels
wide. This is quite a wide range, since th e large st width is more than twice the smallest
one.
4.3 Method of the experiment
The dependent variable was the distortion tolerance; the independent variables were the
distortion type and the line width. Stimuli were shown in random order on a 15 inch
wide computer monitor: screen size  10246768 pixels; viewing distance  60 cm.
Distorted Hermann grids included 765 white lines on a black background, with
constant line spacing of 102 pixels. There were six categories of distortion type: sinusoid,
waved, knotted, humped, half-sided hu mped, and asymmetrical humped. Three catego-
rie s of line width were used: 11 pixels, 17 pixels, and 23 pixels. There were twenty-two
subjects.
The task of the subjects was to set the amplitude of the given type of curves by
pressing the arrow keys of the computer keyboard when they could no longer see the
illusory spots. The starting point of all distortions was the classical Herm ann grid.
If the subject overshot the limit at which the illusion disappeared, he could backtrack
the previous setting and could adjust it until he re-established the right amplitude.
After each grid, maski ng stimuli consisting of random squares were shown fo r 3 s. The
ty pe of distortion and the width of the lines were randomly generated by the computer
program.
4.4 Results of the experiments
The distortion tolerance (figure 6) was analysed by a 3 (line width)66 (distortion type)
repeated-measures
ANOVA
and by Bonferroni pairwise compar ison. T he
ANOVA
showed
for the main effect of grid-line width: F
242
 0:649, p 4 0:05; for the main effect of
distortion type: F
5 105
 15:708, p 5 0:01; and fo r their interaction: F
10 210
 1:163,
p 4 0:05. One can see that the only significant effect is the distortion type. The
Bonferroni pairwise comparison showed that the m e an difference between the pairs of
sinusoid/wavy, sinusoid/knotted, and wavy/knotted is not significant; the mean differ-
enc e between the pairs of hump/asymmetr ical hump is not significant; but the mean
difference between the half-sided hump an d all the others is h ighly significant.
,
, ,
8
7
6
5
4
3
Tolerance means
Line width
thin (11 pixels)
medium (17 pixels)
thick (23 pixels)
sinus knot half-sided hump
wave hump asymmetrical hump
Distortion type
Figure 6. Experimental results. The horizontal axis illustrates the six distortio n types, while the
vertical axis represents the means of the dis tortion tolerance of twenty-two subjec ts. The three
lines, representing three line widths, are nearly parallel to each othe r, which implies that the
distortion tolerance is independent of line width. The distortion tolerance of the half-sided hump
is seen to be much greater than that of the others.
Str aightness as the main factor of the Hermann grid illusion 9

The fact that the main effect of the grid-line width in the
ANOVA
is not significant
confirms our assumption that grid-line width does not play a significant role in the
perception of the Hermann grid illusion. The significant main effect of distortion type
in the
ANOVA
, in addition to the results of the Bonferroni pairwise comparison,
demonstrates th at the distortion tolerance of the half-sided hump distortion type is
an exception; this contains the essence of our results. Since it is only the half-sided
humped grid that includes straight grid-l ine edges, the conclusion is that the main
cause of the Her mann grid illusion is the straightness of th e black ^ white edge s of
the gr id lines, but their width plays no significant role; also the collinearity of the
intersec ti ons plays no sig nificant role.
5 Towards a new theory
As expected , an essential principle em e rged fro m the experiment. The straightness of
the edges of the grid l ines lies behind the perc eptio n of illusory spots. The basic idea
of the theory presented in this paper stems from these findings.
Havi ng analysed the distortion typ es and their effects, we can make an essential,
but not trivial, remark: the location of the change in the illusion is far removed from
the location of the physical change.
In all distortion types, the curving of line edge s is applied exclusively to the middle
part of each line se ction. Nonetheless, the appe arance and the disappearance of the
illusory spots take place at the intersections, where no physical change is applied. How
does the change in the middle part of the gr id-line sections get to the inter sections,
then?
Our answer is that, in fact, it does not even get there. We suggest that, instead of
the spots being darker, it is the line segments that are perceived lighter than the inter-
se ctions. On the basis of this assumption, we formulate the main hypothesis, as the
axioms of our theory.
5.1 Radiating edge hypothesis:
(a) The short segments of white ^ black edges radiate `darkness' on their dark side and
`lightness' on their light side.
(b) The straighter a continuous edge is, the stronger is the radiation of its elemental
segments.''
The direction of the radiation is perpendicul ar to the orientation of the edge s, and
it disperses at a certain `fuzzy' angle (see figure 7). The intensity of the radiation is
strongest in two opposite directions perpendicular to the edge. The radiation diffuses
all over the retinal points that are located i n heading of the radiation, thereby affect-
ing the perceived brightness at each point by producing an additive effect, whose
weighted value d ecreases with distance.
These axiom s allow u s to bridge over our question by introducing a change of
viewpoint: only at first sight does the location of change in the illusion seem distant
from the location of the physical change. In fact, it is not distant at all, since according
to point (a) the grid-line edge directly illuminates the section next to it. Consequently,
the line sections will be much lighter than the intersections, as they are irradiated by the
edge segments of the grid lines. On the other hand, the intersections are not affected
by this radiation.
The analogy of the `illuminating' or `radiating' effect of straight lines is based on
the study of electronics. It is known that the potential of electronic or magnetic dipoles
decreases with distance [the exact rule is that the potential decreases as the reciprocal
of third power ratio of the distance
ö
see eg Feynman et al (1969)]. Let us now image
the line edges of the Hermann gr i d consisting of dipoles placed next to each other
(`particles' that are radiating black in one direction, and white in the opposite one).
10 J Geier, L Berna
¨
th, M Huda
¨
k,LSe
¨
ra

These dipoles quasi-irradiate the line sections, and, of course, they irradiate the squares
as well in the opposite direction by the opposite sign of radiation. Therefore, the light
side of each seg m ent radiates lightness while its dark side radiates darkness.
We take it that the radiating angle is not too large; therefore the radiation cannot
reach the intersections. On the other hand, in the case of curved line edge s, some
oblique segments may irradiate the intersections as well. Acc ording to point (b) the
segments of curved edges radiate much more weakly than the segments of straight edges.
Therefore, the line sections enclosed by curved edg es will receive much less lightne ss than
those enclosed by straight edges.
Of course, the radiating edge hypothe sis should be treated as an analogy. It is not
the light reaching the retina that radiates. Yet, we believe that photosensitive cells in the
retina and other neural structures (not only in the retina) connected to them transmit
neural signals to each other in accordance with a pattern that is largely akin to the
analogy of radiation.
5.2 A theoretical model based on the radiating edge hypothesis
The model consists of the following two-dimensional layers: sensor layer, edge-detector
layer, and diffusion layer (see figure 8).
The sensor layer comprise s photosensitive units which are directly equivalent to
the retinal photosensitive neurons.
Grey-level cross-section at y ÿ370
y  0
y  ÿ370
Figu re 7. The radiation characteristics of a horizontally oriented edge segment. The intensity of
the radiation is indicated by the grey-scale values. The value of mediu m grey is 0. Grey values
lighter than 0 represent stimulation, darker values stand for inhibition. The edge segment of the
sensor layer is a small fragment of the white ^ black edge crossing the (x, y)  (0, 0) point hor i-
zontally (not indicated here). According to point (a) of the radiating edge hypothesis, the edge
radiates `black' in the upward direction and `white' downwards. The cross-section of the radia-
tion intensity is a Gaussian curve at any y value. The deviation of the Gaussian cur ve is propor-
tional to the absolute value of y, and its magnitude is proportional to 1=y. The cross-section of
grey level at y ÿ370 is shown as an example.
Str aightness as the main factor of the Hermann grid illusion 11

The edge-detector layer allocates the direc tion of the normal vector of any e dge
crossing any point in the area it monitors. Th is should not be con fused with a recep-
tive field; instead, it can be characterised as a unit compr ising several receptive fi elds,
which are sensitive to different di rections. T he inner processes of the edge-detector
units select the receptive field which produces the most intensive respo nse; the sele cte d
or ientation will therefore be the output of the edge-detector unit. We propose that the
diameters of the edge dete ctors are very small, since it is the derivatives that need to
be computed here. In computer simulations, the so called Sobel filter of 363 pixel
size is generally used.
The diffusion laye r is a `tissue' in which signals diffuse in ac cordance with the
following rules. When a dire cti on signal reaches a given point, two diffusion processes
of opposite directions are set in motion. In the direction of the nor mal vector, the
diffusing signal will be stimulating (`light'), whereas in the opposite direction it will be
inhibitory (`dark').
The process of the radiation of edges described here is akin to the so- called
standard diffusion process (Grossberg and Todorovic
¨
1988). This diffusion model uses
a DOG filter as its first step. Thereafter, the m odel applies a heat diffusion process,
constrained w ithin the so-called `boundary contours' (see the referred paper).
In contrast, in our theoretical model, the direction of the edges is specified by the
first derivative of the edge s. Then two opposite `radiation' processes (which might be
compared to Galton's board) start from the elemental edge segments perpendicularly,
in two opposite directions. This radiation spreads through the entire image. The
radiations that meet cross each other in accordan ce with the principle of superposition
(as two waves do in water). We do not apply real or `emerge d' b oundary contours.
The theoretical radiation character istics of an elemental edge segment is indicated in
figure 7.
Point (b) suggests that the straightness of a continuous edge acts upon the radiation
intensity of its elemental segments. This assumptio n is similar to the collector unit theory
put for ward by Morgan and Hotopf (1989). According to this theory, the Hermann
grid is related to the pincushion-grid phenomenon. The basis of this explanation for
the pincushion diagonals is that the output of the local long receptive fields adjoins
one common unit, which integrates the signals of the local units connected to it.
The sti muli of th e local units are the corne r pairs positioned diagonally relative to each
other in the intersections.
In contrast, we do not postulate that such collector units would participate in the
edge-detectio n process. Instead, we assume that units close to each other send to each
other signals about orientation they have detected. The more similar these two orien-
tations are, the mo re intensive ampl ification
ö
and, as a consequenc e, the larger the
signal
ö
that both units will produce. The effect of interactio n decreases with growing
distanc e between th e two points monitored by the two units.
Be aring in mind the foregoing, it is obvious that the interaction described above
is an apt realis atio n of point (b). I f a series of edge-d etector units i s pl aced along a
straight edge, then neighbours located clos e to each other wil l ampl ify the signals of
each other. This principle entails that e ach segme nt of a straight edge will radiate
more intens ively than those of a curved edge, where interaction is much weaker.
The model accounting for the background of the radiating edge hypothesis is illus-
trated in figure 8. A possible feature of the m odel is that it is suffici ent to require
short-range interactions, as long-range effects are a consequence of the diffusion of
the short-range effect.
12 J Geier, L Berna
¨
th, M Huda
¨
k,LSe
¨
ra

5.3 The radiating edge hypothesis as a unified explanatory principle
We are now going to enumerate all the essential components of the Hermann grid and
curved grid phenomena demonstrating that the radiating edge hypothesis can readily
account for all the effects listed below.
(i) Illusory spots are perceived at the intersections of the classical, straight-edged
Hermann grid. E xplanation: the line sections are lighter than the intersections, since
the i ntensity of the radiation is high (because of the straightness), and intersections are
not subje cted to radiation.
(ii) The black squares and the white grid lines are homogeneous, and the edges of the
squares are sharp. Explanation: the opposite edges of the squares radiate black towards
each other. Although the radiation of the four edges of a square de creases with grow-
ing distance, the sum of the radiation crossing the square remains nearly invariant.
(iii) The occurrence of the illusion is independent of line width within a rather loosely
restricted range. Explanation: when the line widths are altered, the lines still remain
homogenous. The sum of the irradiation may decrease, yet the lines will necessarily b e
lighter than the intersections.
(iv) When the edges of the lin es are cur ved, the illusory spots disappear at a certain
amplitude. Explanation: according to p oint (b), the intensity of radiation of elemental
edge segments decreases i n the case of curved edges. Besides, the oblique segments
irradiate the interse c ti ons as well, sinc e the diffusion process starting from them still
disperses at a given angle, s o now the `fans' can get into the intersections. It is of no
significance whether the edges are curved inward s or outwards (see symmetrical hump
and asymmetrical hump), since the decrease of radiation i n the lines may be equal
to the increase of the radiation at the intersections, as the radiation intensity depends
merely on the curvature. This also accounts for the nearly equal distortion tolerance
of the humped and the asym metrical humped grids.
(v) The distortion toleranc e of the half-sided humped grid is significantly larger than
that of all the others. Explanation: if both line edges are curved, the de crease in the
irradiation of the lines and the increase in the irradiation of intersections are both
much greater than in the case when one of the edges remai ns straight.
Light
Perceived brightness
sensor
layer
edge-detector
layer
diffusion
layer
Figure 8. Model of the radiating edge hypothes is. The
connection of two edge- detector u nits monitoring two
points clos e to each other is shown. The more similar
the orientation of the detected edge, the more they
increase the output signal of each other. The output
signal reaches the diffu sion layer, where it diffuses
towards the appropriate direction as an excitatory
or inhibitory signal, towards the border of the image.
The perc eived brightness is additively modulated by the
diffusion at the points it reaches. Each signal channel
repres ents 2-D vector signals.
Str aightness as the main factor of the Hermann grid illusion 13

(vi) The di stortion tolerances of the sinusoid and wavy versions are nearly equal;
collinearity plays no sig nificant role in it. Explanation: the decrease of the radiation effect
depends exclusively o n the curvature of the edge, where the direction plays no role.
(vii) In the case of inverse grids (ie black lines and white squares), the illusory spots
are lighter at the intersections. Explanation: points (i) ^ (vi) are all valid even in this case,
if the word `black' is substituted for `white' in those statements.
6 Discussion
In conclusion, it is now clear that the radiating edge hypothesis is a unifying principle,
since it has ac c ounted for the presence of spots in the classical Hermann grid, and
also fo r their disappearance in th e curved grids. It also provides an explanation for
the higher distortion toleranc e of the half-sided humped grid, in which one of the line
sides remains straight.
Our model also manages to solve the old problem that the Hermann grid is very
sensitive to the size of the rec eptive fields in modeling used generally, but scale-
invariant in human perception. Our solution is that we do not use DOG filters of large
diameters; instead, we use small edge dete ctors only. The cause of the spots is that less
radiation gets into the intersections.
6.1 Perspectives, further questions
An alter native solution may perhaps be provided by applying multiscale models of
other lightness/brig htness illusions, eg Morrone's local energy model (Morrone and
Burr 1988), which may b e suitable for modeling Herm ann spots independently of line
widths. Unfortunately, neither the above paper nor others show the applicability of a
local energy model to modeli ng the Hermann spots. There is also some doubt whether
the spots would disappear in the curved grids on applying this model.
At present, no such explicit alternative models are found in the literature that
would serve specifically as the exp lanatory prin ciple of the Hermann spots; it was only
the DO G model that focused on this spe cific issue. A s a matter of fact, in the paper
of Grossberg and Todorovic
¨
( 1988) , there is a Hermann spot simulation. However, its
first step involves the application of the DOG filter, s o it seems that the spots are
caused by that filter, and not by the diffusion. For this reason, if Grossberg's diffusion
model were applied to the curved grids, the illusory spots would not disappear.
Though the Herman n gri d i s generally k n own for its illusory spots, little research
has been devoted to another illusion inherent in it: when the Herman n grid is rotated
by 458, most people see di agonal lines of the same polarity as the squares that pass
through the corners of the squares. Morgan and Hotopf (1989) consider this phenom-
enon to be related to the pincushion grid illusion.
The reason that l ittle attention has been paid by researchers to the illusory diagonals
may be due to the fact that only approximately 5% ^ 10% of viewers see diagonal lines in
a horizontally positioned Herm ann grid (and this applies to most Hermann grids that
have been drawn so far). In the 458 grid, howeve r, almost everyone reports seeing the
diagonals. In the case of 458 curved grids, the number of p eople seeing such diagonals
strongly d ecreases. Future research is needed here.
Our paper was aiming at Marr's 1st level: we have developed a theoretical computa-
tional theory on the basis of our empi rical results. The computational algorithm (Marr's
2nd level) and its correspondi ng neural background (implementation
ö
Marr's 3rd level)
are to be elicited by future computer programming and physiological me asurements. All
the same, one can predict that if the underlying n eural background processes of the radiat-
ing edge hypothesis are successfully derived in the future, the explanation will not be
based on local neurons working independently of each other. Instead, the working principle
here wi l l probably reflect the information processing characteristics of the visu al system.
14 J Geier, L Berna
¨
th, M Huda
¨
k,LSe
¨
ra

Acknowledgments . We are grateful to Ilona Kova
¨
cs and Esther Stocker who have made essential
contributions to the publication of our idea at EVCP2004, Budapest. We wish to thank George N
Bernath for his assistance in translating an earlier version of this paper.
Refer ences
Baumgartner G, 1960 ``Indirekte Gro
«
benbestimmung der rezeptiven Felder der Retina beim
Menschen mittels der Hermannschen Gitterta
«
uschung'' Pflu
«
gers Archiv fu
«
r di e gesamte Psycho-
logie 272 21 ^ 22 (abstract)
Feynman R P, Leighton R B, Sands M, 1969 Feynman Lectures on Physics volume 2, chapter 6
(Readings, MA: Addison-Wesley)
Geier J, Se
¨
ra L, Berna
¨
th L, 2004 ``Stopping the Hermann grid illusi on by si mple sine distortion''
Perception 33 Supplement, 53
Geier J, Se
¨
ra L, Berna
¨
th L, 2005 ``Distortion toleranc e of the Hermann grid'' Perception 34
Supplement, 103
Grossberg S, Todo rovic
¨
D, 1988 ``Neural dynamics of 1-D and 2-D brightness perception: A unified
model of classical and recent pheno mena'' Perception & Psychophysics 43 241 ^ 277
Jung R, 1972 ``Neurophysiological and psychophysical correlates in vision research'', in Brain a nd
Human Behavior Eds A G Karczmar, J C Eccles (New York: Springer) pp 209 ^ 258
Marr D, 1982 Vision (San Fran cisco, CA: W H Freeman)
Morgan M J, Hotopf N A, 1989 ``Perceived diagonals in grids and lattices'' Vis ion Research 29
1005 ^ 1015
Morrone M C, Burr D C, 1988 ``Feature detection in human visio n: a phase dependent energy
model'' Proceedings of the Royal Society of London, Series B 235 221 ^ 245
Ninio J, Stevens K A, 2000 ``Variations on Hermann gr id: an extinction illusion'' Pe rception 29
1209 ^ 1217
Schi ller P H, Carvey C E, 2005 ``The Hermann grid illusion revisited'' Perception 34 1375 ^ 1397
Sekuler R, Blake R, 1994 Perception 3rd ed ition (New York: McGraw-Hill)
Spill mann L, 1994 ``The Hermann grid illusion: a tool for studying hu man perceptive f ield organi-
zation'' Perception 23 691 ^ 708
Wolfe J, 1984 ``Global factors in the Hermann grid illusion'' Perception 13 33 ^ 40
ß 2008 a Pion publication
Str aightness as the main factor of the Hermann grid illusion 15

ISSN 0301-0066 (print)
This article is an advance online publication. It will not change in conten t under normal circumstances
but will be given full volume, issue, and page numbers in the final PDF version, which will be made
available shortly before production of the printed version.
Conditions of use. This article may be downloaded from the Perception website for personal research
by members of subscribing organisations. Authors are entitled to distribute their own article (in printed
fo rm or by e-mail) to up to 50 people. This PDF may not be placed on any website (or other onli ne
distribution system) without permission of the publisher.
www.perceptionweb.com
ISSN 1468-4233 (electronic)