Draughtsmen at Work

Abstract

To obtain the representation of a contour, the visual system integrates fragments of a pattern. One of the 'binding rules' governing this process requires that a path of conjunction in which contrast polarity is preserved be followed. Here we show that this rule has a corollary: where two alternative paths compete to emerge in opposite directions, the one with greater contrast luminance is likely to prevail.

Full text

Abstract. To obtain the representation of a contour, the visual system integrates fragments of a
pattern. One of the `binding rules' governing this process requires that a path of conjunction
in which contrast polarity is preserved be followed. Here we show that this rule has a corollary:
where two alternative paths compete to emerge in opposite directions, the one with greater contrast
luminance is likely to prevail.
Figure 1a shows a checkerboard with four columns of draughtsmen, arranged away
from the rectangle centres to partly cover one horizontal edge and tangential to a
vertical dividing line. Though geometrically parallel, the columns appear to diverge/
converge. The illusory effect is strengthened if each draughtsman is flanked by a twin
isoluminant disk as in figure 1b.
A small displacement of the disks helps to locate the source of the illusory effect.
In figure 1c each pair of draughtsmen has been slid apart: they now appear as they
actually are, arranged in vertical parallel columns. This vanishing of the illusory tilt
suggests that it originates within a particular pattern formed by the two side-by-side
disks at the convergence of four surfaces (forming a so-called X-junction, visible after
the disks have been separated). Within this configuration, the contour completion can
take two different paths. Each rectangle side can continue straight along the collinear
edge or else deviate by 908, going round the rectangular perimeter; these two options
are shown in figure 2a as two dotted lines which demonstrate that, whichever the
solution, one rule is obeyed and the other is violated. When the rectangle side con-
tinues along the collinear side, the Gestalt law of `good continuity' is obeyed, but
the sign of the contrast (or contrast polarity, CP) inverts. When the side turns up the
perpendicular side (going round the rectangle perimeter), the abrupt change in direction
violates the `good continuity' law, but the CP is preserved. The checkerboard therefore
presents a condition in which the law of `good continuity' is challenged by a competing
binding rule
ö
already documented by Roncato and Casco (2003)
ö
which states that
of two directions of contour completion, the one preserving the CP will perceptually
prevail. It is this competition that can play a role in the occurrence of misperception
of linearity on a checkerboard when the X-junctions are erased. As can be seen in the
striking demonstrations of Pierce (1898) and Kitaoka et al (2004), these illusory distor-
tions document contour completions deviating from linearity in which the CP rule is
to some extent responsible. A simple stratagem can turn this conflict into an alliance.
Figure 2b illustrates how this may happen. A portion of the X-junction is hidden by
a dark circle that is tangential to two rectangle sides. As a result, a new direction of
completion opens up because the CP is preserved at a cost of a small smooth deviation
from straightness (a straight side extending into an arc
ö
see figure caption). Direct
manifestation of this binding effect is not found in figure 1b because we do not perceive
LAST BUT NOT LEAST
Draughtsmen at work
Perception, 2010, volume 39, pages 255 ^ 2 59
Oronzo Parlangeli
Dipartimento di Scienze della Comunicazione, Universita
©di Siena, Banchi di Sotto 55, 53100 Siena,
Italy; e-mail: parlangeli@unisi.it
Sergio Roncato
Dipartimento di Psicologia Generale, Universita
©di Padova, via Venezia 8, 3 5131 Padua, Italy;
e-mail: sergio.roncato@unipd.it
Received 24 June 2009, in revised form 7 January 2010
doi:10.1068/p6500

(a)
(b)
(c)
Figure 1. Checkerboards with draughtsmen: the disks are arranged in vertical columns but in (a)
and (b) appear to tilt.
256 Last but not least

a straight curved border distinct from the others. We can only infer its occurrence
on the basis of the assumption that this local deviation from straightness generates
a cue of local slant, and this, together with similar collinear effects, gives rise to an
overall effect for the whole column of disks. The configurations in figure 1 move us a
step forward in understanding the role of luminance contrast in contour binding,
providing a demonstration of the local effects of slanting as well as the role of contrast
polarity and magnitude. A check of this double role of contrast is essential, in view of
the recent challenge by van Lier and Csatho
¨(2006), who found an exception to the CP
rule. In particular, we aim to demonstrate that the effects of contour binding arising
between edges of the same CP are stronger when the contrast magnitude is higher.
(a) (b) (c)
(d)
(e)
Figure 2. CPs (indicates the brighter side of an edge) and direction of edge-merging (dotted
curves) for various luminance conditions on a checkerboard. (a) Two directions of contour bind-
ing within an X-junction (dotted lines). The straight direction obeys the `good continuity' rule
but generates a contour of inverted CP. The angled path (white dotted line) preserves the CP.
In (b) ^ (e) a dotted line indi cates the path formed by two edges that bind through having the
same CP. (b) Illustration of the luminance conditions in figure 1a. A conjunction path forms that
is tilted clockwise from the vertical. (c) Illustration of the luminance conditions in figure 1b
(dark disks). Both the upper and lower straight edges bind with the adjacent circular edge of
same CP. Two local slant cues arise. (d) and (e) Illustration of the luminance conditions in figure 1b
(dark and light disks). The path of invariant CP binds a vertical edge with an arched edge on the
upper right and lower left. The local slant effects are more pronounced than in (b).
Last but not least 257

In figure 3 the draughtsmen are of the same colour. There is no effect of orientation
alteration, but the line separating the pairs of disks seems to wave. Concentrating on a
single vertical side gives the impression that it bends at its extremities, following round
the disk perimeter. This is simply the slant effect predicted by the hypothesis that the
rectangle side binds with the curved edge of same CP (see caption to figure 3).
Figure 4 challenges the CP rule. The grey shades chosen for the checkerboard and
draughtsmen are combined in such a way that all the binding paths between the rectangle
sides and arcs have invariant CP (see enlargement).
Figure 3. Disks of the same colour. In the enlargement, the dotted lines indicate straight edge
binding preserving the CP. Locally the bending effect generates the impression of a rounding
of the rectangle side. An overall effect arises of the column axis waving.
Figure 4. The circles have different colours, but grey shades intermediate between the rectangle
luminances. A curved path of same CP can continue along the same edge at both its extremities and
in opposite directions. Dotted arcs indicate paths ofedge completion with low contrast (0.04 and 0.27
of Michelson contrast on CRT screen); continuous lines indicate higher contrast paths (0.60 and
0.85).
258 Last but not least

In this condition local slants are not expected to occur because the rectangle sides
do not have a privileged direction to extend into. This prediction is disconfirmed by
the illusory effects visible in figure 4: they have the same direction as in figure 1b,
albeit less vivid.
What is the origin of this illusion? One important clue comes from close inspection
of all the paths of edge conjunction in the enlargement in figure 4. Of the two bind-
ing directions at each extremity, one has a high luminance contrast, the other a low
luminance contrast. Since the direction of the former coincides with the overall tilt
direction, our hypothesis is that the binding effects
ö
both local and overall
ö
are the
same as in figure 1b, with the difference that in this case it is the contrast magni-
tude that determines what local slant will emerge. In other words, where there are two
alternative directions of binding preserving CP, the one with the higher luminance
contrast will prevail. According to this hypothesis, the illusion will vanish as the differ-
ence in contrast magnitude between the two arched edges falls.
To test this hypothesis we set up an experiment with patterns similar to those in
figure 4. A basic configuration with disks very different in luminance (see figure 4 cap-
tion) served as an initial stimulus from which to generate other stimuli by gradually
lowering the contrast magnitudes. This was accomplished by gradually darkening the
light disks and lightening the dark ones until a final condition was reached of all
disks isoluminant. Five subjects were presented with the series of stimuli following
the method of limits. They were asked to judge whether they perceived parallel or
converging/diverging columns, making their judgments twice, starting with stimulus
disks (i) highly contrasted, (ii) isoluminant. The average luminances were calculated
that signal the transition in percept (impression) from converging/diverging to parallel;
these correspond to 9.3 cd m
ÿ2
(SD 0:54 cd m
ÿ2
) for dark disks and 16.6 cd m
ÿ2
(SD 1:1cdm
ÿ2
) for light disks. The Michelson contrasts calculated at the border of
these disks were found to be 0.50 and 0.63, respectively. For the luminance conditions
of figure 4, these can be considered the minimal contrast magnitudes necessary for
the binding path to prevail and the tilt illusion to arise.
This result demonstrates that to be complete the CP-preserved path-conjunction
binding rule should have the corollary: ``if there are two contour completions that
preserve the CP, the one with the higher contrast will prevail''. This addendum may
contribute to a better understanding of the illusory effects generated by Kitaoka et al
(2001) and van Lier and Csatho
¨(2006).
References
Kitaoka A, Pinna B, Brelstaff G, 2001 ``New variations of the spiral illusion'' Percepti on 30
637 ^ 646
Kitaoka A, Pinna B, Brelstaff G, 2004 ``Contrast polarities determine the direction of Cafe
¨Wall
tilts'' Perce ptio n 33 11 ^ 20
Lier R van, Csatho
¨A
è, 2006 ``Dancing shapes: A comparison of luminance-induced distortions''
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Roncato S, Casco C, 2003 ``The influence of contrast and spatial factors in the perceived shape
of boundaries'' Perception & Psychophysics 65 1252 ^ 1272
Pierce A H, 1898 ``The illusion of the kindergarten patterns'' Psychological Review 5233 ^ 253
ß 2010 a Pion publication
Last but not least 259

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