Updated, October 1st, 2015
CONTRIBUTIONS TO GEOMETRIC VISUAL ILLUSIONS
Jacques NINIO
Included in the web site http://www.lps.ens.fr/~ninio
Illustration supplement to this chapter
Illustrated pdf version of this chapter
Recent exhaustive review (Frontiers in Human Neurosciences, 2014) on geometrical illusions
TOPICS
DISCUSSED HERE 1.
Overview 2.
Theoretical work on geometrical illusions 3.
Unpublished data 4.
Abstracts of published work 5.
Cited references SEE
IN OTHER SECTIONS Zöllner
and other illusions in stereo (in Chapter on stereoscopic vision) The
“depth from illusory disparities” issue (in Chapter on
stereoscopic vision) Shape
perception (to be developed in a forthcoming chapter) ==============================================
1.
OVERVIEW Most
of what I have to say on geometrical visual illusions is now in a
review article [1] or in books [2,3]. This chapter of the web site is
a biographical account of my itinerary in the field. My position in
this field is very strange, to say the least. It is a domain to which
I devoted intense thinking in 1975-1976, and in which I performed
rather systematic psychophysical work, from 1995 to 2004, a great
deal of which is still in my drawers. I also designed original
stimuli showing counter-intuitive effects (for instance, a diamond
looking too small compared to a square – see Fig. 3 in the
Illustration supplement and [2]). I kicked off a well-known illusion
(the so-called flattening of short arcs), showing that it is
definitely not an illusion (Appendix A in [4] or Figures 9a-9c in
[1]), I announced a paradoxical result which should have interested
neuro-philosophers: two trapeziums, one above the other can never be
made perceptually equal (when the wide bases look equal, the small
bases don’t, etc.[ 1, 2, 3]). Technically, I made a number of
strong statements, based upon psychophysical work, about which types
of deformations account best for the illusions: For instance, the
Zöllner illusion, considered with all its variants, is best
described in terms of an expansion at right angles to the oblique
segments [5, 6], while the Poggendorff and related illusions are best
described by misangulation biases [7]. The Müller-Lyer illusion, and
a large number of illusions which are not usually found in its
company are best described with a “convexity rule” (for the
layman, take it provisionally as a law of contrast: “small looks
smaller, large looks larger”) [1, 4], which is quite different from
the often invoked assimilation effect. I also established a number of
other points, which are detailed below. All this has been largely
unnoticed. Perhaps I failed to make my findings intelligible to my
colleagues, and perhaps putting it all together in this web chapter
will make my work more obvious.
It
is through visual illusions that I started to work thoroughly on
visual perception. In 1975, I was a full-time molecular biologist
having developed a quantitative understanding of how errors could
arise in molecular processes [8, 9]. I contemplated the possibility
of extending such analyses to other domains. I bought two books which
played an important role: The Intelligent Eye by Richard Gregory
[10], that shaped my way of thinking about perception and The
Psychology of Visual illusion by J.O. Robinson [11], that presented
the geometrical illusions in an exhaustive manner, and provided a
critical account of all the scientific literature on the subject. It
seemed to me, reading the two books, that so far, geometrical visual
illusions had not been treated in a geometrically insightful way. Some
very elementary geometric truths may be counter-intuitive. A good
example is the fact that if you take planar sections of a circular
cone, you get a circle, an ellipse, a parabola or a hyperbola
depending upon the orientation of the plane with respect to the axis
of the cone. Therefore, any of these curves can be thought of as the
projection of any other of these curves, and this knowledge has
practical value, for it allows a geometer to predict some properties
of hyperbolas, knowing proven properties of ellipses.
This
is not to say that an ellipse or a hyperbola should be perceptually
equivalent. However, it may be the case that some strange aspects of
2d shape perception, as revealed by geometric visual illusions are
just “legal” but counter-intuitive outcomes of the internal rules
used by the brain for performing geometry. I spent several months
scrutinizing the geometrical illusions in J.O. Robinson’s book,
trying to extract their gist. In Robinson's book, the illusions were
grouped into chapters. This amounted to a formal classification of
the illusions. What if some illusions had been misplaced? I
considered that a good “objective” classification of the
illusions was crucial. The analogy I had in mind was Darwin’s
theory of evolution, that was greatly facilitated by Linnaeus’
systematic classification work. We learnt from Linnaeus that the most
obvious characters are not necessarily the most relevant ones to
understand the relatedness between living organisms. Some hidden
subtle characters may turn out to be far more relevant. Thus, a
dolphin is much closer to land mammals than to fishes, and a bird is
closer to a penguin or even a snake than to a flying insect. A very
common attribution error in geometric illusions is related to the
horizontal-vertical illusion, often illustrated with a letter T,
having two segments of equal lengths, the horizontal one looking
shorter than the vertical one. Here, most of the illusory effect
comes from the fact that the horizontal segment is split in two due
to its intersection with the vertical segment (bisection illusion –
I have some psychophysical data showing this). The
more I was looking at illusions, the more they reduced to simple
inequalities, such as “if there are two segments a and b in a
figure with a larger than b, then the a/b ratio is perceptually
increased”. Psychologists would call this a law of contrast
enhancement. Curiously, I had to invoke a law of this kind mostly in
the situations that were previously explained in terms of
assimilation, which is nearly the opposite principle. I derived a few
principles, that were “pulling in different directions”, yet
could work together, and could be embodied into a coherent framework.
This theoretical work was ultimately published in 1979 [4]. There
were a number of quite good insights in this work, that led me to
fruitful developments later, and there were a few attribution errors,
now corrected in [1]. One
published result attracted my attention. Bela Julesz had shown that
most geometrical illusions are maintained when they are presented as
random-dot stereograms, the Zöllner illusion being the only notable
exception [12]. Seymour Papert had shown earlier that the Müller-Lyer
illusion was maintained in camouflaged stereograms [13].Thus, it
seemed, stereoscopic vision provided an objective way to classify the
illusions into two groups. A first group containing most illusions
would occur late in visual processing, after the stage in which the
streams from the two eyes are usually combined to provide
stereoscopic interpretation. The shape giving rise to the geometric
illusion does not exist prior to this stage, since it is, by design,
monocularly undetectable. Contradistinctively the Zöllner illusion
is destroyed when camouflaged in the form of random-dot stereograms.
So, it seemed, it arises rather early in visual processing, before
the stage at which the streams from the two eyes are normally
combined. (I proved much later that this reasoning was based upon an
incorrect hidden assumption - [14]). In any event it seemed to me at
that time that stereoscopic vision provided a criterion for an
objective classification of geometric illusions, and I thus started
to think about the geometric problems of stereoscopic vision (see the
web chapter on stereo vision). After
moving to Ecole Normale Supérieure, I started doing some
psychophysical measurements on geometrical illusions. I knew
Robinson’s book almost by heart, and followed whatever was
published in Gregory’s scientific journal “Perception”, but
was quite ignorant of what was published elsewhere on the topic. My
first work was on Poggendorff illusion. I attempted to measure the
effect on several variants at several orientations, not knowing that
Weintraub had already done an excellent job [15]. I found that the
misalignment effect was essentially that which could be observed in
the absence of the parallels. Fortunately, I did not publish this
result, it was an artefact: I had used rather long collinear
segments, and in this case they are minimally deviated when they
intersect the parallels of the Poggendorff figure, so what remains is
the “pure misalignment” Zehender illusion, that is observed in
the absence of parallels [16].
A
few years later, I happened to have frequent discussions with Kevin
O’Regan, and we decided to do some psychophysical work on the
Zöllner illusion. We constructed several rather unusual variants
(people could have described them as “new” illusions) , and
studied the strength of the illusion at eight different orientations.
We concluded, rather firmly [5] that (i) the illusion occurred at the
level of a single stack of oblique segments (ii) the distortion was
not an apparent rotation of the stack, but could be a shear, or an
expansion at right angles to the obliques (iii) that the way the
segments ended was not of primary importance.
At
that time, both Kevin and I were also working on memory, but
independently. Kevin published his seminal paper on change blindness
[17]. I was involved in precise determinations of the capacity of
visual memory [18, 19]. We found nevertheless the time to pursue the
work on geometrical illusions. This time we performed extensive work
on the Poggendorff illusion and several of its variants, including
the corner-Poggendorff variant [20]. We did not invent any new
pattern, but performed rather systematic measurements, that allowed
us to separate the measured illusory effect into two components [7]:
a minor one, the pure misalignment illusion, and a major one, that
was clearly a misangulation effect (contrary to my theory in [4]). We
could also make sense of why the illusion was weak in some patterns
(the corner-Poggendorff), and rather strong in others (a Weintraub
variant in which a segment is not collinear with another segment, but
with a dot). Kevin
then started taking responsibilities as director of the Laboratoire
de Psychologie Expérimentale, and I continued the work alone. (Why,
within the French context, I could not have students working with me
is another story). I designed and carried out to the end several
series of psychophysical measurements, along the same lines as the
Poggendorff work: many related stimuli, studied at 8 or 16 different
orientations, 10 subjects, each subject going through a whole series
ten times. I thus obtained results on the horizontal-vertical
illusions, the square-diamond illusions, the trapezium illusions, the
Müller-Lyer illusions. I also studied hybrids between Poggendorff
and Zöllner. I did not attempt yet to publish the results (but see
the “unpublished results” section at the end of this chapter).
However, I made some salient conclusions leak in various places. In
2002, I attended the 25th ECVP in Glasgow, organized by Pascal
Mamassian. I became acquainted there with Baingio Pinna, with whom I
immediately sympathized. He showed me a collection of perhaps 50 or
100 of his yet unpublished illusory patterns, and I went through them
one by one, trying to form an opinion on what could be reduced to
previously known effects, and what was really puzzling. Among the
geometrical effects, I was struck by two drawings. One was a
marvellous variant of the Müller-Lyer illusion, that flatly
contradicted all published theories of the illusion, except mine: the
illusion of the diagonal (Fig. 14-17 in [21] also shown in [1],
Figure 6a). Another one represented squares, that, in the
neighbourhood of oblique segments, appeared as trapeziums. I also
had, among my unpublished patterns, a distorted square illusion. In
both cases, two families of inducing segments or inducing lines at
nearly orthogonal orientations were simultaneously present, and
seemed to have additive rather than subtractive effects. We put our
distorted square patterns in common, and discussed them in the
context of the tilt illusions. Looking at the whole set, plus some
additional variants, designed on this occasion, we proposed that
there was a common theme to most of these illusions, namely
“orthogonal expansion” – a tendency for sets of parallel or
nearly parallel lines to expand at a right angle to the lines [6].
So, having eliminated in a first work the “rotation” description
of the Zöllner illusion, I eliminated the “shear” description of
the illusion in subsequent work, and am now convinced that
“orthogonal expansion” is the key to these illusions. In
April or May 2007, I listened to a seminar given by Pierre Pica at
the Laboratoire de Psychologie Expérimentale in Paris. Pierre Pica
is a linguist who has been spending several months every year with an
Indian tribe of Amazonia, the Mundurucus. He made many puzzling
observations about the structure of their language, their counting
habits [22], their fascination for symmetry, their spatial sense,
their understanding of geometry [23]. I designed for him a battery of
visual tests, which he embarked in his subsequent expedition in the
land of the Mundurucus (dec. 2007 - July 2008). The tests were
designed in such a way as to make sense for the Mundurucus, at least
to the best of Pierre Pica’s knowledge. For instance, there were
tests on the geometrical illusions, that I embodied as tests on the
appreciation of symmetry with respect to an axis, of the two figures
to be compared. It seems that the subjects took pleasure in
performing these tests. Their responses to the tests were in most
cases (Zöllner, Poggendorff, trapezium, Müller-Lyer, Helmholtz)
very similar to those in our contemporary occidental culture. There
were however two differences. One difference related to the Delboeuf
illusions with circles. The two stimuli were presented on the two
sides of a symmetry axis, and the subjects were not sensitive to the
traditional effect. This was expected, by analogy with previous
observations on Titchener's circles in a Himba population [24]. It
seems that the critical cultural issue is whether the subjects focus
on the two parts of the stimuli as a single entity, or separate
entities. Another result was quite unexpected: The Mundurucus were
not subject to the square/diamond illusion. Again, the square and the
diamond were presented on the two sides of a symmetry axis, a feature
that normally reduces the illusion by one half [25]. This, perhaps
can be related to their widespread use of diamond motifs on their
body ornaments. I hope that Pierre Pica will write one day about
these experiments. Although the raw data are preserved, the notebooks
in which Pierre Pica was recording the circumstances of the tests
were lost, so it would be hard to publish the results in a regular
scientific journal. Later, my co-worker in stereo vision, Svetlana
Rychkova e.g., [26], became interested in testing geometrical
illusions on people with normal or impaired vision. She used
slightly modified variants of the programs I had written for Pierre
Pica, and her work is in progress [27]. Another
encounter turned out to be quite important. Being an assiduous reader
of the journal “Perception” I had noticed two very innovative
illusions by Vicario, the sloping step, and the rarefaction
illusions: [28, 29], see also da Pos and Zambianchi, 1996 [30] and
Vicario [31] for a collection of rarely discussed illusions. A very
rewarding correspondence was initiated between Vicario and I. I
learnt many effects from him, thanks to his unique, encyclopaedic
knowledge of geometric illusions and their history. So,
what is my theory of geometrical illusions, people will ask?
Actually, my main work is not centred on why there are illusions, its
main focus is an attempt to determine what is the nature of the
distortion in the illusion. Typically, what is the correct way to
describe the Zöllner illusion: Is it a rotation of the stacks, a
relative sliding of the parallel segments (shear distorsion), an
increased separation between the segments (orthogonal expansion)?
Once the nature of the distortion is correctly described, one can
then try to find a rationale for the distortions. The explanation may
be purely mechanistic, such as: inhibition between neurons having
receptive fields tuned to similar orientations. I do not have the
ambition to defend a particular neuronal model. It is enough to know
that, in principle, there is at least one neuronal model that may
account for the observations. Whatever it is, there is then hope that
a more appropriate one will be found later. The explanation may also
be a teleological one: there is some purpose for the observed
effects. Here, my general idea is that the brain constructs some
representation of shapes in which it incorporates geometrical
relationships derived from visual processing, and it has to satisfy a
number of constraints. The main problem, I propose, which is at the
origin of geometrical illusions is the conflict between sizes or
orientations, as may be derived when observing a scene with a fixed
eye (which then obey the laws of linear perspective) and the sizes
and orientations which can be acquired as the eyes move and explore a
scene, in which case apparent (angular) sizes dominate (which then
follow the laws of curvilinear perspective). How this may lead to
Müller-Lyer, and so many other effects is a bit technical, but
should become clear at the end. Here also, I am satisfied with the
fact that there is at least one plausible teleological theory for at
least one large class of illusions.
2.
THEORETICAL WORK ON GEOMETRICAL ILLUSIONS [1, 4] 2.1
The convexity principle In
1975, I spent months and months scrutinizing the illusions in
Robinson’s book [11], trying to extract the basic geometrical rules
behind these. The book often showed several variants of each
illusion, and this protected me from expeditous erroneous
generalizations. For
instance, people who encounter the Müller-Lyer illusion for the
first time are seized with amazement, then they say “of course,
this illusion is due to the arrows”. It was very clear, from the
examples in Robinson’s book, that the arrows in the Müller-Lyer
pattern may be replaced by squares, circles, or by almost any shape
without damage to the illusory effect.
My
main working assumption was that the brain was constructing a
representation of a scene, in the same spirit with which a geometer
constructs a geographical map: Measurements are taken from one
landmark to another, with more or less reliable instruments,
according to well-defined procedures, then the measurements are
combined according to well-defined rules. The resulting
representation may appear distorted but the distortions are not
necessarily due to carelessness. These are due to fundamental
constraints. In the case of geographical maps, there is the
constraint of representing a portion of spherical surface on a planar
one. I considered that the brain also had to construct a kind of map,
combining distance and orientation measurements taken with its
neuronal instruments. One of the instruments would provide
measurements that were biased in a systematic way, the brain would
combine the measurements and apply corrective factors. In the end, a
rather satisfactory representation would be constructed, except in a
few cases. I
focused first on extent illusions and found several geometrical
themes, which turned out to be related. Imagine that the measure of
an extent x is a function m(x). Many illusions can be accounted for
by a rule of contrast: There are two segments of lengths a and b,
with b larger than a, and the figure is perceptually distorted in
such a way that the contrast between b and a is increased, which can
be spelled out as: m(b)
/ m(a) is larger than b / a if b is larger then a (1) An
assimilation effect would be just the opposite, m(b) / m(a) would be
closer to one than b / a. If
you add the obvious property that m(0) = 0 (the measure of a null
extent is zero), then relation (1) is true for any convex function
m(x) going through the origin – a function which instead of being
linear, rises with a curvature of constant sign (See Fig. 1 in the
illustration supplement, Figure 5a in [1] or Figure 7 in [4]).
This
rule of contrast, or “convexity” applies in an obvious way to
patterns such as the “contrast illusion” with arcs shown in Fig.
2 of the illustration supplement, which is not a “best-selling”
illusions. As an application of the principle, I designed the
original pattern with four circles in Fig. 9 of the illustration
supplement. Many more illusions, discussed in [4] seem to support the
convexity principle. The
convexity principle also applies to an apparently unrelated illusion,
the bisection illusion (Fig. 12 in the illustration supplement).
Here, you can admire the power of mathematics, because if Eq. (1) is
true and m(0) = 0, then: m(a+b)
> m(a) + m(b) (2) And
in particular, m(2a) > 2m(a) (2a). Actually,
there are two sides to the bisection illusion. The known part of it
is that the subdivided segment looks smaller than the undivided one.
The hidden part of the illusion is that, paradoxically, an isolated
half segment looks smaller than half the subdivided segment (Fig. 12
in the illustration supplement and Figure 5b in [1]). According to my
analysis, the real illusion, expressed by Eq. (2a) is the fact that a
full undivided segment looks more than twice as long as an isolated
half. However, this is not perceived as an illusion. See again Fig.
12. I
have surreptitiously used here a subsidiary principle – a principle
of compromise: If measurements are in conflict, then a compromise is
made. The Müller-Lyer illusion can be analysed in terms of convexity
+ compromise principles (See Fig. 6 - 8 in the illustration
supplement).
Here
is a metaphor for the convexity principle, published in [1]:
Assume
that you are on the sea front, and you wish to represent the layout
of a number of floating targets. Your only instrument is a
chronometer. You measure the time it takes to swim from one target to
the other. When two targets are close, one can swim rapidly from one
to the other. When the targets are distant, one swims less rapidly,
and the swimming speed diminishes as the targets become more and more
distant. Thus, the measured time to connect two targets grows more
than proportionately with the distance between targets. This time,
provided by the chronometer, overestimates large distances with
respect to small ones. In psychological language, it increases the
contrast between large and small. In mathematical language, the
measure is a convex one. The relationship between an extent x and its
measured value m(x) can be represented by a parabola, or any curve
starting at the origin, and rising with a curvature of constant
positive sign.
Recently,
Baingio Pinna produced the “illusion of the diagonal”, which I
think is an excellent illustration of the convexity principle (See
Fig. 4 in the illustration supplement, and Figure 6 in [1]). By
adding an arrowhead to the diagonal, one obtains a “reversed
Müller-Lyer” effect – a figure in which a shaft with an ingoing
finn appears larger than expected. The convexity principle accounts
in one stroke for the standard Müller-Lyer and the reverse Pinna
variant, as it explained in one stroke the standard and the reverse
bisection illusion. Kennedy, Orbach and Löffler [32] recently
produced an illusion with triangles (See Fig. 11 in the illustration
suplement and Figure 5g in [1]) which they christened
“isocele/scalene triangle illusion”. In these figures, I explain
with a geometrical construction how this effect can be deduced from
the convexity principle. The “gravity lens illusion” by Naito and
Cole [33] also seems to me to be predictable by the convexity
principle (Fig. 7 in the illustration supplement and Figure 5j in
[1]). In
order to illustrate the convexity principle, I designed a paradoxical
figure with squares and diamonds, showing that the convexity effect
is stronger than the square-diamond illusion. It is reproduced here
in Fig. 3 of the illustration supplement. A simplified version is
shown in Fig. 5 of the supplement. The effect is related to Pinna's
diagonal illusion (see [1]). So,
the convexity principle seems to be at work in many apparently
unrelated illusory patterns. It allows one to construct patterns that
seem to contradict well-known effects, including the bisection
illusion, the square-diamond illusion, or the Müller-Lyer illusion. In
my early work [4], I used the convexity principle to explain the
Poggendorff illusion, but I am now convinced that the explanation was
incorrect (see section on Poggendorff below). While
a large number of illusions were described in terms of a convexity
rule, a number of other illusions seemed to go in the opposite
direction, and these required two subsidiary principles: A principle
about the effect of subdividing a figure, and a principle about the
space occupied by the figure. Both acted as corrective terms to the
convexity principle. I shall examine the two subsidiary principles in
turn.
2.2
The subdivision rule. If
we follow the logic of the convexity principle, and partition a
figure in 2, 3, 4 or more parts, the higher the number of
subdivisions, the more it should appear contracted. Actually the
effect is clearly observed only for n = 2 (bisection illusion).
Starting with n = 4 we have clearly the opposite trend, an expansion
of subdivided figure, as observed for instance in Helmholtz squares
illusions. So, I introduced a “subdivision principle” according
to which there is an expansion effect on subdivided figures, the
magnitude of the effect depending upon the number of subdivisions.
Combining the convexity with the subdivision principle is
mathematically possible. One can get both an overall contraction
effect for n = 2 and an overall expansion effect for n > 3, as I
showed in [4]. I was recently led to re-examine this principle, in
the light of the Zöllner and other tilt illusions [6], as will be
explained later. See also Fig. 14 in the illustration supplement or
Figure 7 in [1]. 2.3
The space occupation rule. A
number of geometric illusions(including Titchener’s circles and the
Ponzo illusion) clearly could not be accounted for by a combination
of the convexity and the subdivision principles. I needed to add a
“space-occupation” rule according to which a size normalization
factor is applied to all figures. The large ones are perceptually
reduced, and the small ones are perceptually enlarged. In this way,
illusions that are classically explained by a contrast effect are
reinterpreted in terms of an almost opposite principle. Fershad
Nemati [34] was aware of that, and proposed a principle of expansion
when there was empty space around a figure. Here,
my ideas have evolved considerably. For recent, precise formulations,
see [1]. 3.
UNPUBLISHED WORK I
have several unpublished results on orientation profiles (Figures
16-23 in the illustration supplement).
In
one study, I compared the Zöllner illusion with or without explicit
axes along the stacks. The measured effect is larger in the case of
explicit axes (Figure 16 of the illustration supplement). 17 subjects
took part in the experiment, and each data point represents the
average over 170 measurements. In all other studies, reported below,
there were 10 subjects, and 10 measurements per subject for each data
point. In
another study, I determined orientation profiles in variants of the
square-diamond illusion (Figures 17-19 of the illustration
supplement). The
square-diamond illusion is usually presented with one apex of the
diamond pointing towards the square. I found that when the figures
were displayed more symmetrically the illusion was significantly
reduced. Furthermore, it is surpassed, for all subjects, by an
illusion that goes in the opposite direction, in which the diagonal
of a small diamond is underestimated with respect to the side of a
larger square. The results were presented in a talk at ECVP 2011
(Toulouse), and reported in the corresponding abstract [25]. I
also determined orientation profiles for variants of the trapezium
illusion (Figures 20-22 of the illustration supplement). The
trapezium illusion was maximal when the bases of the trapeziums were
horizontal, and minimal when they were vertical. The oblique sides,
but not the bases, were essential to the illusion, suggesting the
existence of a common component between the trapezium and the Zöllner
illusion. The study is made somewhat difficult by the fact that
figures with trapeziums often lead to interpretations in perspective
that perturb the comparison of trapeziums as flat figures.
One
philosophically important side-result of the study is that two
trapeziums in the standard configuration can never be altered in such
a way as to be seen equal! When you try to equalize (by a nulling
procedure) the two large bases and the orientations of the two sides,
the small bases look unequal, and when you try equalize the two small
bases and the orientations of the sides, then the two large bases
look unequal! It is thus impossible to draw two trapeziums, one above
the other, so that they would look identical! The
results were presented in a talk at ECVP 2011 (Toulouse), and
reported in the corresponding abstract [25]. The
most important result, in my opinion, is that obtained on hybrid
Zoellner-Poggendorff patterns (Figure 23 of the illustration
supplement). It clearly rules out the “shear” hypothesis for
Zöllner, and it is clearly favourable to the “orthogonal
expansion” interpretation.
My
experiments on orientation profiles with Müller-Lyer patterns were
frustrating. I studied stimuli containing Müller-Lyer patterns and
visually related stimuli, including the receding arrow illusion
(Figure
3h in [1]),
and Judd's bisected arrow illusion (Figure
1c in [1]).
Unfortunately, my orientation profile experiments failed to show a
relationship between the Müller-Lyer, the Judd and the receding
arrow illusions. The results with Müller-Lyer patterns were erratic.
They were strongly subject-dependent, there was no simplifying
symmetry when the patterns were turned upside down, etc. My
provisional, not too satisfactory, explanation is that a subject may
compare the lengths of the segments between the fins according to
various criteria, (for instance, forming a virtual rectangle with a
pair of segments, looking at orientations, etc.) and the criterion
he/she chooses depends upon the orientation of the stimulus.
4.
ABSTRACTS OF PUBLISHED WORK. An
algorithm that generates a large number of geometric visual
illusions. Ninio,
J. (1979) Journal of Theoretical Biology 167-201. ABSTRACT An
algorithm is described which, starting with any geometrical figure,
constructs a representation in which the deviations from the model
coincide with the known perceptual distortions. First, the algorithm
specifies a measurement process: drawing a straight line across the
figure and measuring the encountered segments, assigning to every
segment of length x its measure m(x). Next, the measures taken along
a line D are corrected with a normalizing factor N(D) which is a
function of the measures made on this line. Finally, a representation
of the analysed figure is constructed, using for every segment its
normalised length n(x)=m(x).N(D), instead of its actual length. It
is first established that within this general framework, a large
number of illusions can be immediately predicted by specifying a
property of the measure or of the norm. Only four properties are
required to justify most illusions. They are (1) and (2) m(x) and
n(x) must be convex functions (3) the norm must increase with the
number of segments measured on a line (4) the norm must decrease when
the average segment on a line increases in length. It is then shown
that the four requirements, conflicting as they may be in some
circumstances, can nevertheless be condensed into one single
expression of n(x). This simple formula predicts a large number of
widely different illusions (Delboeuf, Titchener, Ponzo, trapeze,
Müller-Lyer, flattening of short arcs, etc). It permits to predict
new illusions and new effects in old illusions, but fails to predict
the Zöllner illusion, and the reversal of the Müller-Lyer illusion
when the outgoing finns are becoming very large. -------- The
half - Zöllner illusion Ninio,
J. and O’Regan, J.K. (1996) Perception 25, 77-94. ABSTRACT The
Zöllner figure contains stacks of short parallel segments oriented
obliquely to the direction of the stack. Adjacent parallel stacks of
opposite polarity seem to diverge where their top segments form an
arrowhead. To probe whether or not the opposite polarities are
necessary to the illusion, three ‘half-Zöllner’ configurations
were designed, containing stacks of a single polarity. The
‘orientation profile’ of these configurations was studied, that
is, the way the strength of the perceived illusion varies with the
orientation of the stacks. The subjects had to align two stacks or
align stacks with target segments situated at a slight distance from
them. All three half-Zöllner configurations produced errors that
could be assimilated to global-orientation misjudgments. These errors
were of opposite sign for the two types of stacks and varied with the
orientation of the stacks as in the standard Zöllner illusion.
A
further study was conducted in which the effects of several
configurational parameters was explored for a single observer. The
standard Zöllner illusion increases with the separation of the
stacks. The illusion is also increased when the orientations of the
segments in different stacks are orthogonal, independently of the
particular longitudinal orientations of the stacks.
When
the ends of the short segments are curved so that at their endpoints
they become precisely perpendicular to the axes of the stacks, the
standard and half-Zöllner illusions are reduced, but not abolished.
Therefore, they cannot be entirely accounted for by a mechanism of
alignment of illusory contours generated at these endpoints. The
results are consistent with the existence of a single common
mechanism at work in both the standard and the half-Zöllner
illusion. It is suggested that the illusion itself is not a rotation
of the stacks but either a shear deformation in which the segments of
the stack slide with respect to one another, or an expansion of the
stacks orthogonally to the segments. ---------- Characterization
of the misalignment and misangulation components in the Poggendorff
and corner-Poggendorff illusions. Ninio,
J. and O’Regan, J. K. (1999) Perception 28, 949-964. ABSTRACT
In
the Poggendorff illusion, two colinear segments abutting obliquely on
an intervening configuration (often consisting of two long parallel
lines) appear misaligned. We report here the results of a component
analysis of the illusion and several of its variants, including in
particular the "corner Poggendorff" illusion, and variants
with a single arm. Using a nulling method, we determined an
"orientation profile" of each configuration, that is, how
the illusions varied as the configuration was rotated in the plane of
the display. We were able to characterize a pure misalignment
component (having peaks and dips around the ±22.5 degree and ±67.5
degree orientations of the arms) and a pure misangulation component
of constant sign, having peaks at the ±45 orientations of the arms.
Both these components were present in both the classic and the corner
Poggendorff configurations. Thus, the misangulation component appears
clearly in the classic Poggendorff illusion, once the misalignment
component is partitioned out. Similarly, the corner Poggendorff
configuration, which essentially estimates a misangulation component,
contains a misalignment component which becomes apparent once the
misangulation is nulled. While our analysis accounts for much of the
variability in the shapes of the profiles, additional assumptions
must be made to explain the relatively small misangulation measured
in the corner-Poggendorff configuration (1.5 degrees, on average, at
peak value), and the relatively large illusion measured in the
configurations with a single arm (above 6 degrees, on average, at
peak values). We invoke the notion that parallelism and colinearity
detectors provide counteracting cues, the first class reducing
misangulation in the corner-Poggendorff configuration, and the second
class reducing the illusion in the Poggendorff configurations with
two arms.
----------- Orthogonal
expansion: a neglected factor in tilt illusions.
Ninio,
J. and Pinna, B. (2006) Psychologia 49, 23-37. ABSTRACT A
broad collection of illusions belonging to the Zöllner and the
Poggendorff families, including new variants - in particular, tilted
and tilting squares - are examined in the light of two possible
formal principles : a principle of regression to right angles (RRA)
and a principle of "orthogonal expansion", which is a
perceptual expansion of the extent perpendicularly to the inducing
lines. The domains of validity of the two principles are compared. We
propose that RRA is more pertinent when the target line is
explicitely present and makes real intersections with the inducing
lines. Orthogonal expansion can produce RRA as a side-effect. It
would be more pertinent when there are several parallel or nearly
parallel inducing lines, and it does not require the presence of a
real target. Both principles may be grounded on neurophysiological
mechanisms. Orientation detectors would influence each other in the
orientation domain, generating RRA and accounting for the illusions
of the Poggendorff family. They would also influence each other in
the extent domain, generating orthogonal expansion, and accounting
for the illusions of the Zöllner family.
----------- Orientation
profiles of the trapezium and the square-diamond geometrical
illusions. Ninio, J. (2011) Perception 40 supplement, 45 ABSTRACT
of talk presented at ECVP Toulouse, 2011 In
previous work, “orientation profiles” (describing how the
strength of an illusion varies with its orientation in the plane)
were determined for several variants of the Zöllner and the
Poggendorff illusions (e.g., Ninio and O'Regan, 1999, Perception,
28(8), 949-964). The study is extended here to two other classical
illusions. Illusion strengths were determined for 10 subjects at 16
orientations on 4 variants of the trapezium illusion and 8 variants
of the square-diamond illusion. The trapezium illusion was maximal
when the bases of the trapeziums were horizontal, and minimal when
they were vertical. The oblique sides, but not the bases, were
essential to the illusion, suggesting the existence of a common
component between the trapezium and the Zöllner illusion. The
square-diamond illusion is usually presented with one apex of the
diamond pointing towards the square. I found that when the figures
were displayed more symmetrically, the illusion was reduced by one
half. Furthermore, it is surpassed, for all subjects, by an illusion
that goes in the opposite direction, in which the diagonal of a small
diamond is underestimated with respect to the side of a larger
square. --------------- Geometrical
illusions are not always where you think they are: a review of some
classical and less clasical illusions, and ways to describe them.
Ninio,
J. (2014) Frontiers in Human Neurosciences volume 8, article 856, 21
pages. Geometrical
illusions are known through a small core of classical illusions that
were discovered in the second half of the 19th century. Most
experimental studies and most theoretical discussions revolve around
this core of illusions, as though all other illusions were obvious
variants of these. Yet, many illusions, mostly described by German
authors at the same time or at the beginning of the 20th century have
been forgotten and are awaiting their rehabilitation. Recently,
several new illusions were discovered, mainly by Italian authors, and
they do not seem to take place into any current classification.
Among
the principles that are invoked to explain the illusions, there are
principles relating to the metric aspects (contrast, assimilation,
shrinkage, expansion, attraction of parallels) principles relating to
orientations (regression to right angles, orthogonal expansion) or,
more recently, to gestalt effects.
Here,
metric effects are discussed within a measurement framework, in which
the geometric illusions are the outcome of a measurement process.
There would be a main “convexity” bias in the measures: the
measured value m(x) of an extant x would grow more than
proportionally with x. This convexity principle, completed by a
principle of compromise for conflicting measures can replace, for a
large number of patterns, both the assimilation and the contrast
effects.
We
know from evolutionary theory that the most pertinent classification
criteria may not be the most salient ones (e.g., a dolphin is not a
fish). In order to obtain an objective classification of illusions, I
initiated with Kevin O’Regan systematic work on “orientation
profiles” (describing how the strength of an illusion varies with
its orientation in the plane). We showed first that the Zöllner
illusion already exists at the level of single stacks, and that it
does not amount to a rotation of the stacks. Later work suggested
that it is best described by an ‘orthogonal expansion’ — an
expansion of the stacks applied orthogonally to the oblique segments
of the stacks, generating an apparent rotation effect. We showed that
the Poggendorff illusion was mainly a misangulation effect. We
explained the hierarchy of the illusion magnitudes found among
variants of the Poggendorff illusion by the existence of control
devices that counteract the loss of parallelism or the loss of
collinearity produced by the biased measurements. I then studied the
trapezium illusion. The oblique sides, but not the bases, were
essential to the trapezium illusion, suggesting the existence of a
common component between the trapezium and the Zöllner illusion.
Unexpectedly, the trapeziums sometimes appeared as twisted surfaces
in 3d. It also appeared impossible, using a nulling procedure, to
make all corresponding sides of two trapeziums simultaneously equal.
The square-diamond illusion is usually presented with one apex of the
diamond pointing towards the square. I found that when the figures
were displayed more symmetrically, the illusion was significantly
reduced. Furthermore, it is surpassed, for all subjects, by an
illusion that goes in the opposite direction, in which the diagonal
of a small diamond is underestimated with respect to the side of a
larger square. In general, the experimental work generated many
unexpected results. Each illusory stimulus was compared to a number
of control variants, and often, I measured larger distortions in a
variant than in the standard stimulus. In
the Discussion, I will stress what I think are the main ordering
principle in the metric and the orientation domains for illusory
patterns. The convexity bias principle and the orthogonal expansion
principles help to establish unsuspected links between apparently
unrelated stimuli, and reduce their apparently extreme heterogeneity.
However, a number of illusions (e.g., those of the twisted cord
family, or the Poggendorff illusions) remain unpredicted by the above
principles. Finally, I will develop the idea that the brain is
constructing several representations, and the one that is commonly
used for the purpose of shape perception generates distortions
inasmuch as it must satisfy a number of conflicting constraints, such
as the constraint of producing a stable shape despite the changing
perspectives produced by eye movements.
5.
CITED REFERENCES [1]
Ninio, J. (2014) Geometrical illusions are not always where you think
they are: a review of some classical and less classical illusions,
and ways to describe them. Frontiers in Human Neusosciences volume 8,
article 856, 1-21. doi: 10.3389/fnhum.2014.00856 [2]
Ninio, J. (1998) La science des illusions. Odile Jacob, Paris. German
translation Nacht Schwarz Schlank? Gustav Kiepenhauer Verlag, Leipzig
(1999) and Büchergilde Gutenberg, Frankfurt (2000). Greek
translation E epistimi ton psevdaithiseon, Katoptro, Athenes. (2001).
English translation The science of illusions. Cornell University
Press, Ithaca (2002). Japanese translation, Shin'Yosha, Tokyo (2004). [3]
Ninio, J. (2011, last edition) L'empreinte des sens. Odile Jacob,
Paris.
[4]
Ninio, J. (1979) An
algorithm that generates a large number of geometric visual
illusions. Journal of Theoretical Biology 167-201. [5]
Ninio, J., and O’Regan, J. K. (1996) The half - Zöllner illusion.
Perception 25, 77-94. [6]
Ninio, J. and Pinna, B. (2006) Orthogonal expansion: a neglected
factctor in visual illusions. Psychologia 49, 23-37. [7]
Ninio, J. and O'Regan, J.K. (1999) Characterization of the
misalignment and misangulation components in the Poggendorff and
corner-Poggendorff illusions. Perception 28, 949-964. [8]
Ninio, J. (1974) . A semi-quantitative treatment of missense and
nonsense suppression in the strA and ram ribosomal mutants of
Escherichia coli . Evaluation of some molecular parameters of
translation in vivo. Journal of Molecular Biology 84, 297-313. [9]
Ninio, J. (1975) Kinetic amplification of enzyme discrimination.
Biochimie 57, 587-595. [10]
Gregory, R. (1970) The Intelligent Eye. Weidenfeld and Nicholson,
London. [11]
Robinson, J.O. (1972) The psychology of visual illusion. Hutchinson
University Library, London. [12]
Julesz, B. (1972) Foundations of Cyclopean Perception. Chicago
University Press, Chicago, Illinois, USA. [13]
Papert, S. (1961) Centrally produced geometrical illusions. Nature
191, 733. [14]
Ninio, J. and Herbomel, P. (1990) Stereoscopic study of the Zöllner
illusion. Perception 19, [ECVP abstract issue], 362.
[15]
Weintraub, D. J., Krantz, D. H., and Olson, T. P. (1980) The
Poggendorff illusion: consider all the angles. Journal of
Experimental Psychology- Human Perception and Performance 6, 718-725 [16]
Zehender, W. V. (1899) Über geometrisch-optische Taüschungen.
Zeitschrift für die Psychologie 20, 65-117. [17]
Rensink, R.A., O'Regan, J.K. and Clark, J.J. (1997) To see or not to
see: The need for attention to perceive changes in scenes.
Psychological Science 8, 368-373. [18]
Brunel, N. and Ninio, J. (1997) Time to detect the difference between
two images presented side by side. Cognitive Brain Research 5,
273-282. [19]
Ninio, J. (1998) Acquisition of shape information in working memory,
as a function of viewing time and number of consecutive images :
evidence for a succession of discrete storage classes. Cognitive
Brain Research 7, 57-69. [20]
Greene, E. (1988) The corner Poggendorff. Perception 17, 65-70. [21]
Pinna, B. (2003) Riflessione fenomenologiche sulla percezione della
qualità emergenti: verso una riconsiderazione critica della Teoria
della Pregnanza. Annali della Facoltà di Lingue e Litterature
Straniere, 3, 211-256 (Fig. 14-16).
[22] Pica, P., Lemer, C.,
Izard, V. and Dehaene, S. (2004) Exact and approximate arithmetic in
an Amazonian indigene group 306, 499-503.
[23] Dehaene, S., Izard,
V., Pica, P. and Spelke, E. (2006) Core knowledge of geometry in an
amazonian indigene group. Science 311, 381-384.
[24]
de Fockert, J., Davidoff, J., Fagot, J., Parron, C. and Goldstein, J.
(2007) Accurate size contrast judgements in the Ebbinghaus illusion
by a remote culture. Journal of Experimental Psychology: Human
Perception and Performance 33, 738-742. [25]
Ninio, J. (2011)
Orientation profiles of the trapezium and the square-diamond
geometrical illusions. Perception 40 Supplement, 45.
[26]
Rychkova, S., and Ninio, J. (2011) Alternation frequency thresholds
for stereopsis as a technique for exploring stereoscopic
difficulties. i-Perception 2, 50-68. [27]
Rychkova, S., and Bolshakov, A. (2013) Zollner and Poggendorff
illusions in children with ophthalmology. Perception 42 supplement,
70. [28]
Vicario, G.B. (1972) Phenomenal rarefaction and visual acuity under
'illusory' conditions. Perception 1, 475-482. [29]
Vicario, G.B. (1978) Another optical-geometrical illusion. Perception
7, 225-228. [30]
Da Pos, O. and Zambianchi, E, eds. (1996) Visual illusions and
effects - A collection. Guerini Studio, Milan. [31]
Vicario, G. B. (2011) Illusioni ottico-geometriche. Una rassegna di
problemi. Venezia: Istituto veneto di scienze, lettere ed arti. [32]
Kennedy, G.J., Orbach, H.S. and Löffler, G. (2008) Global shape
versus local feature: An angle illusion. Vision Research 48,
1281-1289. [33]
Naito, S. and Cole, J.B. (1994) The gravity lens illusion and its
mathematical model. In Contributions to Mathematical Psychology,
Psychometrics and Methodology (G.H. Fischer abd D. Laming, Eds).
Springer-Verlag, New-York, pp. 39-50. [34]
Nemati, F. (2009) Size and direction of distortion in
geometric-optical illusions: conciliation between the Müller-Lyer
and Titchener configuratons. Perception 38, 1585-1600.