Motion-Blur Illusions
Frédéric Gosselin
[gosselif@psy.gla.ac.uk]
University of Glasgow
and
Claude Lamontagne [clamonta@uottawa.ca]
University of Ottawa
Abstract
The Still-Radii-Illusion (Cobbold 1881), the Figure-of-Eight Illusion (MacKay
1958), the Band-of-Heightened-Intensity Illusion (Smith 1964) and the
Dark-Blurred-Concentric-Circles Illusion (Wade 1972) have remained, until now,
isolated relatively ill-expl ained phenomena. A single algorithmic model is
proposed which explains these four visual illusions. In fact, this model
predicts phenomena produced by motion of any gray shaded patterns relative to
the eyes (termed "Motion-Blur Illusions"). Results of a computer
simulation of the model are presented. A novel instance of the proposed class
of illusions, which can be readily experienced by the reader is introduced to
illustrate the generality of the model.
1. The Still-Radii Illusion
Charles S. W. Cobbold (1881) was the first to give account of the Still-Radii
Illusion (named after Junge's 1963a Moving Radii Illusion.). To elicit it one
must move linearly in the fronto-parallel plane a set of black concentric
circles drawn on a white background (see figure
1a) relative to one's eyes. This is best achieved by eye tracking the tip
of a pencil moving at uniform speed across the pattern. The Still-Radii
Illusion is characterized by a fan-like clear struc ture (perceived lying against
a smeared background) orthogonal to the direction of motion [Footnote 1: What
we call the Still-Radii Illusion must be distinguished from the
Revolving-Propeller Effect (e.g., Wade 1978). In fact, Wade (1978) made this
dual ity quite explicit: "An analogue [italics added] to this
[Revolving-Propeller Effect]" he wrote "can be demonstrated by moving
the patterns relative to the eye [italics added]: if the concentric circles are
moved vertically up and down, two clear horizo ntal fans ... are seen, with the
rest of the pattern slightly blurred." (p. 33) The Still-Radii Illusion
is, precisely, this "analogue" illusion. For one thing, while the
Revolving-Propeller Effect seems to be more highly correlated with the lens
curvat ure fluctuation than with the eye movements (e.g., Campbell & Robson
1958), the Still-Radii Illusion is perfectly correlated with eye movements
(e.g., Cobbold 1881), and it is unaffected by lens paralysis (anonymous
referee). The distinction is, also, mi rrored in the distinct histories of the
two illusions (although they may have a somewhat "blurred" origin in
Purkinje's 1832 work). The Revolving-Propeller Effect was studied by Helmholtz
(1856), Campbell & Robson (1958), Pritchard (1958), Evans & Marsde n
(1966), Millodot (1968), Wade (1978 and 1982), etc. And, the Still-Radii
Illusion was studied by Cobbold (1881), Bowditch & Hall (1882), Pi_ron
(1901), and Junge (1963a and 1963b).]. Cobbold also provided an explanation of
the Still-Radii Illusion bas ed on visual persistence:
During vertical motion, the images of the more or less horizontal black and
white bands are constantly replacing one another upon [the same regions of] the
retina, each becoming confused with the impression immediately preceding it,
and thus producing the blurred appearance noticed.... The reverse is the case
... in which the black and white bands are practically vertical, and coincide
with the direction of motion; their images ... remain clearly defined and
unaffected by the vertical movement.... The clearly-defined narrow sector [of
the concentric circles] will then be seen extending horizontally ... while the
upper and lower portions of the disc appear somewhat ... blurred. (p. 77)
Cobbold's account of the Still-Radii Illusion implies two processing stages: first,
the continuous projection of the pattern upon the retina is transformed into
discrete "impressions", and, then, two successive
"impressions" are "confused" into an "appea rance
". To an observer, these "appearances" should not be
differentiable from the pattern represented in figure 1a when set in linear
motion in the fronto-parallel plane.
Cobbold added nothing about the first stage of processing implied by his
explanation. It therefore remains very intuitive, no detail being given
concerning the crucial "confusing" process of the second stage of
processing (affecting the successive " impressions"). In fact, it
only allows predicting that the more parallel a line in the pattern represented
in figure 1a is to the movement of the eyes, the less blurred it will appear.
In no way does it allow predicting the absolute degree of blurring o f a line
in the pattern, nor does it allow predicting the various extents of the blur
for the various lines of the pattern.
2. Figure of Eight Illusion
Donald M. MacKay (1958) was the first to report that when a ray figure (see figure
1b) is moved linearly in the fronto-parallel plane relative to the eyes,
blurred concentric "figures of eight" perpendicular to the motion's
direction are perceived against a clear background. Again, the best way to
produce the illusion is to eye track the tip of a pencil moving at uniform
speed across the pattern. MacKay (1958, 1961) proposed an explanation of the
Figure-of-Eight Illusion which he generalized to phenomena (Moir_ Effects, as
he called them) produced by motion of all periodical black and white figures in
the fronto-parallel plane. Since the pattern represented in figure 1a is a
periodical black and white pattern, MacKay's m odel should also account for the
Still-Radii Illusion.
Although MacKay never refered to Cobbold, his explanation can be understood as
an elucidation of Cobbold's. Like Cobbold, MacKay proposed two stages of
processing, which involve highly similar processes: first, the continuous
projection of a black and white periodical pattern upon the retina is
transformed into discrete "images", and, then, the "images"
are "superposed", causing a result which should not be
distinguishable, to an observer, from the moving stimulus. As opposed to
Cobbold, however, who provided no further detail about the first stage of
processing, MacKay stated clearly that the " [images] are displaced
replicas of the [physical] pattern" (1958, p. 362). The difference between
the "images" results from the movement of the pattern r elative to
the eyes. In other words, the "images" are instantaneous snapshots of
the visual world taken at different loci "as the eyes move" (MacKay
1961).
The second stage of processing in MacKay's theory is based on the combination
of two "successive images" or "impressions" as is
Cobbold's, but a much clearer description of how these two "successive
images" are combined is now provided: the two "suc cessive
images" are "superposed". MacKay appeared to indicate that the
"superposition of [two] successive images" is equivalent to the physical
superposition of an "[image] with the transparency of itself" (1958,
p. 362). Thus, a white point (which is t ransparent on a transparency)
superposed on a white point appears white, a white point superposed on a black
point appears black, whilst a black point superposed on a white point appears
black, and a black point superposed on a black point appears black [ Footnote
2: This type of superposition mimics the behavior of a logical "and"
with white corresponding to "true" and black corresponding to
"false".]. Whenever a periodical pattern (like the pattern
represented in figure 1a or that represented in figure 1b, for instance) is
superposed to a copy of itself in that fashion, moiré patterns are produced
(e.g., Oster & Nishijima 1963). This is the reason why MacKay chose to call
this class of phenomena "Moiré Effects".
The effects produced by the superposition of the patterns represented in figure
1a or figure 1b with transparencies of themselves are quite reminiscent of the
Still-Radii Illusion and the Figure-of-Eight Illusion (e.g., Spillmann 1993;
Gregory 1990) , respectively. However, MacKay's model falls short of accounting
for the "blurred appearance" noticed by Cobbold in the Still-Radii
Illusion (Moiré patterns are black or white.).
3. The Band-of-Heightened-Intensity and the Dark-Blurred-Concentric-Circles
Illusions
Babington Smith (1964) was the first to report that when a grating (see figure
1c) is rotated slowly relative to the eyes (between 0.175 and 1.047 rad per
s) in the fronto-parallel plane, a "band of heightened intensity" (l
ying against a smeared background) centered on the origin of the rotation is
perceived at an angle with the grating's lines. In fact, the so-called
"band of heightened intensity" is neither of "decreased"
intensity nor of "heightened" intensity; it is s imply simply more
contrasted (Wade 1974). The speed of rotation is crucial to the production of
the phenomenon. By rotating the pattern represented in figure 1c faster (beyond
10_ rad per s) in the fronto-parallel plane, Wade observed (1972) a completel y
different effect featuring "dark blurred concentric circles" or, more
specifically, alternatively light and dark concentric circles that become less
and less contrasted as their radii increase.
Since the pattern represented in figure 1c is a black and white periodical
pattern, MacKay's model of Moiré Effects should account for the
Band-of-Heightened-Intensity and the Dark-Blurred-Concentric-Circles Illusions
(following Wade's 1972 de scription of the illusion).
The effect produced by the superposition of the pattern represented in figure
1c with a transparency of itself is, in fact, somewhat reminiscent of the
Band-of-Heightened-Intensity Illusion. However, MacKay's theory cannot explain
the Dark-Blurred-C oncentric-Circles Illusion. Covering the pattern represented
in figure 1c with a transparency of itself and rotating it through all the
possible moiré patterns convincingly shows that none possess the slightest
resemblance with the Dark-Blurred-Co ncentric-Circles Illusion.
Smith (1964) proposed that the Band-of-Heightened-Intensity Illusion is the
result of "the simultaneous perception of what is seen over a measurable
period of time" (p. 27A). Smith's proposition involves a single processing
stage: the continuous pr ojection of the rotating pattern upon the retina over
a certain period of time is "simultaneously perceived". The result of
this "simultaneous perception" should not be differentiable, to an
observer, from the pattern represented in figure 1c set in rota tion. How this
"simultaneous perception" is achieved is so vague in Smith's
explanation that it is difficult to see how anything could be predicted from
it.
Barbur (1980) gave a much more satisfying description of this
"simultaneous perception". According to him, the perception at a
given receptor is the average of the sampled luminance at that receptor over a
period p. Building upon this "averaging hy pothesis", Barbur
elaborated a formal model accounting for part of Smith's Band-of-Heightened-Intensity
Illusion.
Independently, Glunder (1987) showed that a time-integrating system (e.g., the
human visual system) exposed to a suitably chosen fronto-parallel rotating
spatial frequency (i.e., a sinusoidally modulated luminance grating)
approximates its "temporal transfer function" (TTF) in the frequency
domain. The TTF can be understood as a linear filter applied to a visual input
signal. If the impulse response of this filter is chosen as 1/p over the period
p, then the time-integrating system averages the inp ut signal over p. Such a
time-integrating system is, thus, equivalent to the one postulated by Barbur
(1980). Now, the Fourier transform of that particular TTF (its representation
in the frequency domain) is highly similar to the Band-of-Heightened-Inte nsity
Illusion.
4. An Algorithmic Model of Motion Blur Illusions
We propose a general model of motion blur which has a bearing on all the
phenomena produced by the motion of gray shaded patterns relative to the eyes,
and not only on the Band-of-Heightened-Intensity Illusion as in Barbur's (1980)
and Glunder's (198 7) cases. We refer to these phenomena as "Motion-Blur
Illusions". Glunder (1987) distinguishes two types of time-integrating
systems: those for which p is equal to the time of observation (e.g., a
camera), and those for which p is smaller than the obse rvation time (e.g., a
cine-camera, the human visual system). A time-integrating system of the second
type can always be reduced to finite or infinite series of time-integrating
systems of the first type (e.g., a cine-camera could be reduced to a finite s
eries of cameras). Time-integrating systems of the first type can be thought of
as building blocks of time-integrating systems of the second type. Our model
will be concerned with the building blocks of the human visual time-integrating
system and will be formulated at the "representation and algorithm"
level of understanding (Marr 1982).
The model will be presented in three steps. First (section 4.1), the problem of
computing the projection of a gray shaded image onto the retinal plane at any
single moment (i.e. no integration over time yet) will be addressed. The
problem of extendi ng this algorithm to computing the integration over time, of
the continuous projection of the stimulus pattern onto the retinal surface will
then be addressed (section 4.2), completing the presentation of the model as
such. The third step (section 4.3) will be concerned with the presentation of
the model at work, as it is tested with various gray shaded images of interest
in the present context.
4.1 Projecting the Visual World Upon the Retina
Since we wish to provide a general model of Motion-Blur Illusions we must
conceive a way of projecting the visible portion of the visual world upon the
retina which can accommodate any type of motion. Our formal visual world (W)
possesses four conti nuous dimensions: three spacial dimensions (X,Y,Z) (see
"Visual World" in figure
2) and one temporal dimension (T). We will restrict our analysis to a where all points are invisible except
for the gray shaded points lying on one plane, the image (see "Image"
in figure 2). We have associated each gray shade with a positive real number
corresponding to its relative position on a continuous gray scale (anywhere
from black to white). Our gray scale is proportional to the lumin ance scale.
The shade of any particular point on the image is given by the function
I(u,v) (1)
where (u,v) is the location of that point relative to the image. All the pages
of this article can be considered portions of such I functions.
Thus, we have points located within a (U,V)-based "private" or
"objective" coordinate system (i.e. irrespective of the visual world
(X,Y,Z)-based coordinate system) and associated with a gray shade via a
function I. When such a gray shaded image is placed in the visual world , its
(U,V)-based point coordinates must be transformed into (X,Y,Z)-based point
coordinates. Allowing for any type of motion in W, this can be achieved by
using three standard motion functions. A simple linear motion in the
fronto-parallel plane as required to produce the Still-Radii Illusion and the
Figure-of-Eight Illusion, for instance, could be expressed as follows:
x=u+k_X+k_S*(t-k_T) (2)
y=v+k_Y (3)
z=k_Z (4)
where k_T corresponds to the value of t at the beginning of the computation;
k_X, k_Y and k_Z are the three coordinates of point (0,0) on the image in W at
t equal to k_T; and k_S corresponds to the constant speed of displacement along
the X axis of W.< br>
With the (u,v) coordinates of the points of the image transformed, at any
moment, into (x,y,z) coordinates, we can now address the issue of computing the
projection of that gray shaded image onto the retinal plane. Our formal eye has
a visual field of 90 deg, and, in order to respect the biconvex lens properties
of the optics of the human eye, it is composed of two invisible bounded planes,
embodied in W, and parallel to the plane defined by the X and Y axes of W (see
"Eye" in figure2). The sides of the smaller bounded plane, called
"the retina", are all of length n, with the intersection of its two
diagonals being located at (0,0,0) in W. Thus, all retinal points (receptors)
lie at z=0, and take the form (r_X,r_Y,0), with -n/2<=r_X, r_Y<=n/2. T he
sides of the larger bounded plane, called the "pseudo lens", are all
of length 2n; this larger plane being separated from the retina by a distance
of n/2, the intersection of its two diagonals is located at (0,0,n/2) in W.
The point (x,y,z) on the image projects, at any moment, onto point (r_X,r_Y,0)
on the eye's retina following a receptor line passing through point
(2r_X,2r_Y,n/2) on the pseudo lens (see "Receptor Line" in figure2).
Therefore, all we need in order t o be able to compute the projection is the
parametric equations of receptor lines, which are
x=r_X(2z/n+1) (5)
y=r_Y(2z/n+1) (6)
with -n/2<=r_X, r_Y<=n/2. Since we deal with half-lines and not with full
lines, we must put a restriction on equations 2 and 3: only the portions of
receptor lines on the positive side of the Z axis are to be considered.
All that remains to be done then, is to combine this projection process,
characterized by the above "receptor line parametric equations"
(equations 5 and 6), and the transformation process described earlier as characterized
by the "standard motion fu nctions" (and examplified by equations 2,
3 and 4). This can be done by creating two functions, F_U(r_X,r_Y,t) and
F_V(r_X,r_Y,t), which we will call the "motion intersection
functions", and which, in the case of the simple fronto-parallel
horizontal lin ear motion functions described in equations 2, 3 and 4, will
yield (by combining equations 2,3,4,5 and 6) the following equations
u=r_X(2k_Z/n+1)-k_X-k_S(t-k_T) (7)
v=r_Y(2k_Z/n+1)-k_Y. (8)
Finally, by replacing u and v in equation 1 with their respective expressions
in equations 7 and 8, we obtain the shade sampled by the receptor beginning at
the point (r_X,r_Y,0) in W at any moment.
It is important to note that integration over time has been, to this point, of
no concern whatsoever. The motion intersection functions exclusively concern
the geometry of how, at any single moment t, gray shaded points given their
motion relative to an eye, project onto this eye's retina. Let us now turn
precisely to this topic of the integration of gray shades over time, to the
heart, in fact, of the present attempt to establish a computational rationale
for motion blur illusions.
4.2 Integrating the Continuous Projection of the Visual World Upon the Retina
For integrating over time the gray shaded image projected onto the retinal
plane, we chose Barbur's "averaging hypothesis" (1980). In other
words, a gray shaded image set in motion in should not be differentiable, to an observer, from the
average of the continuous projection of that image upon the retina over a fixed
period p. Formally,
NSum[1/p*I(F_U,F_V),{t,k_T,k_T+p}] (9)
for -n/2<=r_X, r_Y<=n/2.
4.3 Computer simulations of the model
We performed a computer simulation of the proposed model using the motion
intersection functions of the simple horizontal linear motion in the
fronto-parallel plane described above and the I functions or rather, the
portions of I functions represente d in figures 1a and 1b. The program produced
the patterns represented in figures
3a and 3b, respectively. These patterns are quite similar to the
Still-Radii Illusion and to the Figure-of-Eight Illusion. The proposed model pr
edicts that they should not be differentiable, to an observer, from the
patterns represented in figures 1a and 1b filling approximatively 90 deg of his
visual field set in a fronto-parallel linear motion of about 3.6 deg per p
relative to his eyes. If p was equal to 100 ms (e.g., Patterson 1990), that
speed would be equal to 36 deg per s approximatively [Footnote 3: The proposed
model is not based on involuntary eye movements. If the human visual system
always works as stated by the proposed model then any type of motion of the eye
relative to gray shaded images would generate some blurring. Now, under the
heading "involuntary eye movements", one, usually, includes:
microsaccades, drifts, and tremors (Yarbus 1967). All of these eye movements
are lin ear, hence, they cannot account for the Band-of-Heightened-Intensity
and the Dark-Blurred-Concentric-Circles Illusions. The retina undergoes linear
displacement as large as 40 minutes of angle during microsaccades. This is
about 10 times more than durin g the average drift (over p, of course), and
more than 60 times more than during tremors. Nonetheless, it is not enough for
the proposed model to produce any noticeable "radii" in the
concentric circles represented in figure 1a filling 90 deg of the visu al
field.]. From now on, we will use this conservative estimate of p.
Equations 10 and 11 are the motion intersection functions of a simple rotation
in the fronto-parallel plane about the central receptor of the retina
u=(2k_Z/n){r_X*cos[k_S(t-k_T)]-r_Y*sin[k_S(t-k_T)]}-k_X (10)
v=(2k_Z/n){r_X*sin[k_S(t-k_T)]+r_Y*cos[k_S(t-k_T)]}-k_X (11)
where k_S corresponds to the constant speed of the rotation. We performed
another computer simulation of the proposed model using these motion
intersection functions and the portion of I function represented in figure 1c,
and the program generated th e patterns represented in figures
4a and 4b. These patterns are very similar to the
Band-of-Heightened-Intensity Illusion and to the
Dark-Blurred-Concentric-Circles Illusion, respectively [Footnote 4: The program
used to simul ate the algorithmic model trims the boundaries of the predicted
patterns. This trimming -- especially noticeable in the pattern represented in
figure 4b -- is not a prediction of the model.]. The proposed model predicts
that the patterns represented in figures 4a and 4b should not be
differentiable, to an observer, from the pattern represented in figure 1c
filling approximatively 90 deg of his visual field, and set in a
fronto-parallel rotation of about 0.6 rad per s and 20_ rad per s,
respectively.
Equations 12 and 13 are the motion intersection functions of a simple back or
forth motion perpendicular to the fronto-parallel plane
u=r_X{2/n[k_S(t-k_T)+k_Z]+1}-k_X (12)
v=r_Y{2/n[k_S(t-k_T)+k_Z]+1}-k_Y (13)
where k_S corresponds to the constant speed of the linear motion. We performed
a final computer simulation of the proposed model using these motion
intersection functions and the portion of I functions represented in figure 1a,
and we obtained the pa ttern represented in figure
5. The shape of the blurred portion of the image is quite reminiscent of
that of a ghost. As far as we know, this is a new prediction. One can easily
test it by moving back (fronto-perpendicularly) relative to one's eyes at a
speed of about 0.4 m per s a corner of the pattern represented in figure 1a
encompassing 90 deg of one's visual field at 0.5 m.
Acknowledgments
We wish to thank N J Wade and an anonymous referee for their insightful
reviews. We are indebted also to this anonymous referee for having performed a
critical experiment showing that the Still Radii Illusion is unaffected by lens
paralysis. The first a uthor was supported by a scholarship from the Natural
Sciences and Engineering Research Council of Canada (NSERC) during this
research.
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