November 2017
Volume 17, Issue 13
Open Access
Article  |   November 2017
Preferential processing of cardinal over oblique orientations in human vision
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Journal of Vision November 2017, Vol.17, 8. doi:10.1167/17.13.8
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      Gerald Westheimer; Preferential processing of cardinal over oblique orientations in human vision. Journal of Vision 2017;17(13):8. doi: 10.1167/17.13.8.

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Abstract

The oblique effect—poorer performance when contours are in oblique meridians—is here extended from the discrimination of line-orientation to the tilt illusion and to the detection and contextual induction of curvature. The distinction is made between a contour's susceptibility to contextual perturbation and its capacity to induce such perturbation, for which the oblique effect is only about one half. That the cardinal/oblique superiority is retained for the orientation of illusory borders and for the implicit orientation of shapes lacking explicit rectilinear delineation has implications for its neural substrate. To the extent that a geometrical-visual illusion, such as Poggendorff's or Hering's, depends on interaction in the domain of contour orientation, it manifests a corresponding orientational anisotropy. On the other hand, visual functions that govern whether and how well a boundary is visible are invariant with orientation.

Introduction
It has long been known that the discrimination of contour orientation is poorer in oblique meridians than in the horizontal and vertical (Mach, 1861), something that has since been demonstrated abundantly in a variety of situations and named the oblique effect (Appelle, 1972). A claim that the tilt illusion, where a line's apparent orientation is shifted when surrounded by inclined flanks, is absent in oblique meridians (Campbell & Maffei, 1971) failed to find confirmation: More thorough investigations (Mitchell & Muir, 1976; O'Toole & Wenderoth, 1977) showed that oblique test lines could also have their apparent orientation perturbed by contextual stimulation. But none of the extensive research in this area has tested the question whether the tilt illusion is not just present but actually more prominent in oblique meridians. Further, in spite of long-standing interest in curvature perception (Bühler, 1913; Guillery, 1899; Watt & Andrews, 1982), the possibility of an oblique effect in this attribute had remained open (Crassini & Over, 1975; Wilson, 1985) until it was settled by Fahle (1986). Yet the existence of orientational anisotropy in geometrical-optical illusions (Obonai, 1931; Robinson, 1998; Weintraub, Krantz, & Olson, 1980) suggests widespread orientation-dependent differences in contour processing. 
The present study therefore addresses cardinal/oblique differences, specifically in the spatial attributes of orientation and curvature of foveally-seen contours that are involved in the defining of form. Comparing their properties with those of detection, contrast discrimination, and edge smoothness thresholds enables the delineation of at least two hierarchical neural levels: a more peripheral boundary-generating apparatus based on local brightness, contrast, and chromaticity differences that determine whether and how well a contour is seen, and a more central stage related to shape perception and hence object recognition, in other words, how well a contour is established against how firmly the spatial attributes of a well-established boundary depend on its orientation—vertical/horizontal, i.e., cardinal versus oblique. 
The spatial grain of vision coarsens as one proceeds radially from the fovea to the retinal periphery. A radial/tangential anisotropy is a consequence of the radial gradient and can be demonstrated in a variety of spatial visual tasks (Westheimer, 2005). The orientational anisotropy in central vision that is the subject of the current study is superimposed on and more widely distributed than that due this anatomical organization. 
The distinction is here made between oblique contours on the one hand, and vertical and horizontals, lumped together and identified as cardinal orientations, on the other. This is not to deny that a further distinction between horizontals and verticals is not also in order. To the contrary, it has been understood ever since Malebranche first pointed to it in the late 18th century (Malebranche, 1997) that different measures may apply to the apparent length of otherwise identical vertical and horizontal patterns. Moreover, in the experiments here reported and others, the results on occasions differ significantly in the two cardinal meridians. But these differences are not consistent from observer to observer and in any case are minor compared with the prominent and universal oblique effects to be described. 
Methods
In psychophysical experiments, observers were shown computer-generated, high-contrast geometrical patterns in central fixation and were required after each presentation to make a yes/no response about a specific relative disposition of target components. No error feedback was provided. 
Stimuli
Stimuli were presented on the LCD screen of laptop computers at a distance, usually 70–90 cm, such that a pixel subtended 1 arcmin. Lines were 2 pixels wide, drawn with an antialiasing algorithm on a circular gray background, 4° in diameter. Test lines were white, between 40 and 150 arcmin in length depending on the experiment, and could have their orientation set with a precision of 0.2°. The contours were either straight lines or circular arcs whose curvature κ was specified in units of degrees of visual angle−1. In the study of the orientation dependency of contour smoothness, one or the other long edge of white 60′ × 12′ rectangles was shown with protrusions extending randomly 0–6 arcmin in consecutive four arcmin segments along the edge, The magnitude of the physical changes employed to investigate the underlying perceptual attributes (angle of contour orientation in degrees of orientation, curvature of arcs in the reciprocal of its radius of curvature in degrees of visual angle, extent of average deviations from smoothness in arcmin) could be set at scales small enough to probe thresholds and also at the somewhat larger ones needed to capture the range of context-induced changes. Illustrations are provided of the stimuli used in the contextually influenced changes in line orientation (Figure 1, top left) and curvature (Figure 1, top right), in the Hering (Figure 1, center and bottom left) and Poggendorff (Figure 1, center and bottom right) illusions, and in the edge corrugation (Figure 6, top). 
Figure 1
 
Configurations used in this study. Top left: Tilt illusion, orientation change induced in a test line by tilted flanking lines. Top right: Curvature induction, where a straight line appears curved when flanked on either side by curved contours. Middle left: Hering Illusion, where a pair of straight, parallel lines appears curved in the presence of a sheaf of intersecting lines. Middle right: Poggendorff illusion, in which there is an apparent misalignment of the transverse segments even when they are strictly collinear. Bottom: The Hering and Poggendorff illusions in different meridians.
Figure 1
 
Configurations used in this study. Top left: Tilt illusion, orientation change induced in a test line by tilted flanking lines. Top right: Curvature induction, where a straight line appears curved when flanked on either side by curved contours. Middle left: Hering Illusion, where a pair of straight, parallel lines appears curved in the presence of a sheaf of intersecting lines. Middle right: Poggendorff illusion, in which there is an apparent misalignment of the transverse segments even when they are strictly collinear. Bottom: The Hering and Poggendorff illusions in different meridians.
Procedure
Two psychophysical procedures were employed. For the determination of thresholds and contextual changes in the orientation or rectilinearity of lines, it was the method of constant stimuli where each trial in a run of several hundred featured the presentation of a pattern in one of a set of seven possible configurations, equally spaced in the domain of the test parameter. The geometrical-optical illusions of Poggendorff and Hering were estimated quantitatively by nulling them out using the adjustment psychophysical method. 
Thresholds
To set the stage for the subsequent experiments, the smallest detectable deviation from a standard was determined for the orientation of a short foveal line lying in the horizontal, 45° vertical, and 135° meridians. In addition a companion experiment was performed to determine, for a similar set of line-orientations, the smallest detectable deviation from rectilinearity, implemented by finding the least curvature whose direction, concave or convex, could be correctly identified on 75% of presentations. 
Context-induced changes
Orientation—the tilt illusion
Experiments were performed to measure the change in the perceived orientation of a briefly-presented line, as influenced by a lateral inclined flank on each side. To provide a standard for comparison purposes, the test line was first shown by itself for 400 ms and, after a 400-ms pause, also for 400 ms, as part of the full line triplet (Figure 2, top), i.e., the test line accompanied by its flanking pair. These were at a fixed separation and with an orientation difference whose value was the same for the whole run but whose direction, clockwise or counter-clockwise, varied randomly on each presentation (Figure 2, bottom). The central line's orientation was always fixed when it was part of the triplet, but when it was first shown by itself for comparison purposes, it was at an orientation that differed randomly by −3°, −2°, −1°, 0°, 1°, 2° or 3°, or multiples thereof. The observer responded by a button press whether the central line when part of the triplet appeared tilted clockwise or counterclockwise with respect to the previously shown comparison. Three hundred presentations, with data accumulated separately according to whether the flank tilt was clockwise or counterclockwise, constituted a run, and two runs each were conducted for flank inclinations of 22° with the whole stimulus pattern vertical, horizontal, and in each of the two 45° oblique meridians. Frequency-of-seeing curves for the concurrently accumulated responses to clockwise- and counterclockwise-inclined flanks were analyzed separately to yield the tilt induced for that flank inclination and pattern orientation. After each run, the curve of the frequency of “yes” responses versus the seven stimulus stations was subjected to probit analysis, yielding a mean and slope, each with a standard deviation. Half the difference in their 50% points constitutes the measure of the shift induced by flanks of that particular configuration (Figure 2, bottom). It represents the orientation change, in degrees, that has to be imparted to a test line in order to null out the tilt induced in it by the pair of lateral flanks of the particular orientation difference. 
Figure 2
 
Top: Sequence of presentations during each trial in the tilt illusion experiment. The test line is first shown for 400 ms by itself with an orientation that was randomly perturbed by one of seven amounts from the orientation it had during the subsequent presentation (after a 400-ms pause) of the full configuration, which included a flank at either side, with a tilt whose amount was fixed during the run. In long runs of trials, the direction of flank tilt was randomly either clockwise or counterclockwise. The observer signaled whether the test line, when part of the line triplet, appeared tilted clockwise or counterclockwise, using the test line's orientation when first shown for comparison purposes as a reference. Bottom: Example of data analysis in tilt illusion experiment. Proportion of responses “Test lined appears tilted clockwise” in runs totaling 600 trials in which accompanying flanks were shown tilted randomly either clockwise (squares, blue probit fit) or counterclockwise (circles, red probit fit). Analyzed separately, they showed that for the test line to appear vertical (i.e., seen tilted clockwise and counterclockwise with equal probability), it needs to be shown at a 87.8° orientation when accompanied by counterclockwise flanks, and at 92.3° when accompanied by clockwise flanks. This expresses the repulsive tilt induced by the flanks and evaluates numerically the oppositely-directed tilt needed to make the test line appear vertical when flanked. The arrow indicates the magnitude of the total induction caused by the two oppositely directed flanks; halving it yields a bias-free estimate (here 2.25°) of the apparent tilt induced in a vertical test line by a set of 22° flanks.
Figure 2
 
Top: Sequence of presentations during each trial in the tilt illusion experiment. The test line is first shown for 400 ms by itself with an orientation that was randomly perturbed by one of seven amounts from the orientation it had during the subsequent presentation (after a 400-ms pause) of the full configuration, which included a flank at either side, with a tilt whose amount was fixed during the run. In long runs of trials, the direction of flank tilt was randomly either clockwise or counterclockwise. The observer signaled whether the test line, when part of the line triplet, appeared tilted clockwise or counterclockwise, using the test line's orientation when first shown for comparison purposes as a reference. Bottom: Example of data analysis in tilt illusion experiment. Proportion of responses “Test lined appears tilted clockwise” in runs totaling 600 trials in which accompanying flanks were shown tilted randomly either clockwise (squares, blue probit fit) or counterclockwise (circles, red probit fit). Analyzed separately, they showed that for the test line to appear vertical (i.e., seen tilted clockwise and counterclockwise with equal probability), it needs to be shown at a 87.8° orientation when accompanied by counterclockwise flanks, and at 92.3° when accompanied by clockwise flanks. This expresses the repulsive tilt induced by the flanks and evaluates numerically the oppositely-directed tilt needed to make the test line appear vertical when flanked. The arrow indicates the magnitude of the total induction caused by the two oppositely directed flanks; halving it yields a bias-free estimate (here 2.25°) of the apparent tilt induced in a vertical test line by a set of 22° flanks.
The possibility of response bias in favor of the veridical always exists. Because, as distinct from the oblique, observers have an excellent intrinsic representation of cardinal orientations, all pattern components, comparison and stimulus, were given a common orientation jitter in the range ±2° in each trial. In this way the perception of any line as strictly horizontal or vertical provided no reliable clue to veridicality, and the observers were forced to concentrate on the direction of change, if any, of its orientation from when the line was shown by itself to when it was part of the configuration. The jitter was small enough not to have any substantial impact on the identity of the meridian to which the run of trials was devoted. 
Curvature induction
A second experiment was performed, similar except that the line attribute was now curvature instead of orientation, and the observer's task was to judge the apparent curvature of a test line, 1.6° in length. The inducing pattern consisted of a pair of arcs, curvature 0.3° visual angle−1, laterally separated from the test line by distances 40 and 60 arcmin (Figure 1, top right). The central test line could assume at random one of a set of seven curvatures, three with progressively more curvature concave to one side of the line, straightness, and three with progressively more curvature concave to the other side. In a single run, presentations with right and left curvature flanks were randomly interdigitated. The measure of curvature induction would be the curvature that has to be imparted to the test line to null out the apparent curvature induced in it when surrounded by the curved arcs, and this is given by half the lateral shift needed to superimpose two ogives like the ones in Figure 2, bottom. It is expressed in units of curvature, with dimension degree of visual angle−1, that is, the reciprocal of the radius of curvature of the osculating circle at the center of the test line when it appears straight in the presence of the flanking arcs. 
Edge quality
At each presentation, the observer saw a white rectangle, 1° × 20 arcmin, against the gray circular background. The smoothness of one of the long edges was compromised at, randomly, one of seven levels of degradation, and the observer responded by a button press, if necessary by guessing, as to which of the two edges appeared less smooth. Probit analysis of runs of 150 responses allowed the calculation of the threshold, i.e., the level of degradation at which the compromised edge was correctly identified on 75% of occasions (50% would represent pure guesses). Results of two such runs, at each of four edge orientations (horizontal, 45°, vertical, 135°) were averaged to allow the determination whether there is an oblique effect for this contour property. 
Poggendorff illusion
The procedure to measure the Poggendorff illusion, i.e., the apparent lack of alignment of the two segments of a traversing line that is interrupted by a pair of parallel lines (Figure 1, middle right), was to find the displacement of one of the segments needed to null out the illusion. It was implemented by having the observer, using computer button presses, move one of the segment of the transverse back and forth along its parallel until the two transverse segments appeared collinear, and then registering the position value. This was repeated 15 times in a session, each time with a starting position that had been randomly displaced from actual collinearity. The experiment was performed with the parallel lines vertical (Figure 3, middle right), horizontal, and in the 45 and 135° meridians. In each case the transverse line had an inclination of 22° with respect to the parallels. The experiment was repeated on another day and the results of the two runs averaged, yielding a mean and its standard error based on a total of 32 settings. Because the emphasis in this study is on line orientation, and a widely-held view of the origin of the Poggendorff illusion is an expansion of the acute angle between the parallels and the transverse line, the measure employed for a quantitative estimate of the illusion was the orientation difference (in degrees) between the transverse lines and the virtual line joining the junction points where the parallels meet the transverse segments when the illusion has been nulled out. The orientation of the configuration could also be manipulated to test propositions concerning possible differences between the susceptibility of lines in oblique orientations to contextual perturbation and their potency in exerting such perturbations. 
Figure 3
 
From top down: (A) Threshold for discrimination of the orientation of a foveal line for four line orientations in three observers. (B) Threshold for detection of curvature in a centrally-fixated line, 1.6° in length as a function of orientation in three observers. Curvature in units of 0.0069° of visual angle−1. (C) Induced tilt in a foveal line that is flanked on each side by a line 40 arcmin away and inclined by 22°, as a function of orientation in four observers. (D) Curvature induced in a centrally fixated line when flanked on each side by two curved arcs separated from it by distances of 40 and 60 arcmin as a function of orientation of the whole configuration.
 
The obliquity index, (45 +135)/(H+V), averaged for all observers is 1.92., 1.89, 2.47, and 1.80 in the four experiments, and close to 2 overall for all observers and conditions.
Figure 3
 
From top down: (A) Threshold for discrimination of the orientation of a foveal line for four line orientations in three observers. (B) Threshold for detection of curvature in a centrally-fixated line, 1.6° in length as a function of orientation in three observers. Curvature in units of 0.0069° of visual angle−1. (C) Induced tilt in a foveal line that is flanked on each side by a line 40 arcmin away and inclined by 22°, as a function of orientation in four observers. (D) Curvature induced in a centrally fixated line when flanked on each side by two curved arcs separated from it by distances of 40 and 60 arcmin as a function of orientation of the whole configuration.
 
The obliquity index, (45 +135)/(H+V), averaged for all observers is 1.92., 1.89, 2.47, and 1.80 in the four experiments, and close to 2 overall for all observers and conditions.
Hering illusion
Hering first pointed out (Hering, 1861) that a pair of parallel straight lines appears curved when superimposed on a sheaf of intersecting lines (Figure 1, middle left). The effect can be estimated numerically by imparting such curvature to the lines as is needed to make them appear straight and parallel, the nulling procedure. This was carried out, as described under Poggendorff above, with the parallel line pair in the vertical, horizontal, 45° and 135° meridians. The measure of the illusion was the curvature that had to be imparted to each of the parallel lines, in degrees of visual angle−1, to make them appear straight and parallel. 
Oblique index
In accord with previously-used terminology (Li & Westheimer, 1997), the oblique index  
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has been computed as a measure of preferred performance in the cardinal meridians. Its calculation is based throughout on the actual data from each observer in each condition, all contained in Figures 38, utilizing psychometric curve parameters each based on at least 300 responses when the method of constant stimuli was used, or the means of 36 separate null settings for the illusions. Values of the oblique index are meant as general indicators of the data trend, good at most to the first decimal place.  
Figure 4
 
Magnitude of the induced tilt in a foveal line, 2° in length, when flanked on either side by a similar line separated from it by 20 arc min, for four inducing tilts of the flanks with the configuration oriented vertically, at 135° horizontally, and 45°. Each bar represents measurement based on 600 responses.
Figure 4
 
Magnitude of the induced tilt in a foveal line, 2° in length, when flanked on either side by a similar line separated from it by 20 arc min, for four inducing tilts of the flanks with the configuration oriented vertically, at 135° horizontally, and 45°. Each bar represents measurement based on 600 responses.
Figure 5
 
Top: Tilt induced in a test line in the 22° meridian when accompanied by a pair of flanking lines in a cardinal meridian and (lower) in the 45° meridian. The ratio of tilt-inducing capacity is on average half as much as the oblique index for the same observer and the same flank inclination.
Figure 5
 
Top: Tilt induced in a test line in the 22° meridian when accompanied by a pair of flanking lines in a cardinal meridian and (lower) in the 45° meridian. The ratio of tilt-inducing capacity is on average half as much as the oblique index for the same observer and the same flank inclination.
Figure 6
 
Top: Screen shot showing stimulus, consisting of a white rectangle on a gray circular background when randomly one or the other long edge had an irregular contour, the parameter being the average depth of the irregularity in arcmin. Bottom: Thresholds values for this parameter, with their standard errors, when the observer correctly identified on 75% of occasions which edge was corrugated for three observers, as a function of the orientation of the configuration.
Figure 6
 
Top: Screen shot showing stimulus, consisting of a white rectangle on a gray circular background when randomly one or the other long edge had an irregular contour, the parameter being the average depth of the irregularity in arcmin. Bottom: Thresholds values for this parameter, with their standard errors, when the observer correctly identified on 75% of occasions which edge was corrugated for three observers, as a function of the orientation of the configuration.
Figure 7
 
Curvature needed to null out Hering illusion, that is to give the line pair the appearance of being straight and parallel in the presence of the intersecting sheaf, as a function of orientation of the whole configuration in three observers. When the overall pattern size was 1.3° × 2.5°, the curvature was in units of 0.0069° of visual angle−1.
Figure 7
 
Curvature needed to null out Hering illusion, that is to give the line pair the appearance of being straight and parallel in the presence of the intersecting sheaf, as a function of orientation of the whole configuration in three observers. When the overall pattern size was 1.3° × 2.5°, the curvature was in units of 0.0069° of visual angle−1.
Figure 8
 
Top: Measurement of the Poggendorff illusion, in which the observer moves one of the transverse line segments until it appears aligned with the other, in four configurations. Numerical values signify the difference in orientation of the transverse line segments and the covert line joining their intersection points with the parallels when the illusion has been nulled out, a measure taken to be the apparent expansion of the acute angle, here always 22°, between the transverse and parallel lines. In conditions I and IV the transverse line was at an inclination of 22° to the vertical, and the last column gives the ratio of the inducing strengths of cardinal to oblique parallels. The complementary data III/II show the oblique/cardinal ratio in the susceptibility to having perturbation induced by a given set of parallels.
Figure 8
 
Top: Measurement of the Poggendorff illusion, in which the observer moves one of the transverse line segments until it appears aligned with the other, in four configurations. Numerical values signify the difference in orientation of the transverse line segments and the covert line joining their intersection points with the parallels when the illusion has been nulled out, a measure taken to be the apparent expansion of the acute angle, here always 22°, between the transverse and parallel lines. In conditions I and IV the transverse line was at an inclination of 22° to the vertical, and the last column gives the ratio of the inducing strengths of cardinal to oblique parallels. The complementary data III/II show the oblique/cardinal ratio in the susceptibility to having perturbation induced by a given set of parallels.
Observers
Participating observers included five undergraduate biology students and in addition one senior experienced researcher. Because of the randomizing process in the stimulus generation and the random orientation jitter, no individual trial contained reliable clues as to the veridicality of the shown pattern, so in a sense all observers were naïve. Nevertheless, all findings were confirmed in at least three participants lacking specific knowledge of the concept underlying the question being asked. The optometric status of all subjects was unexceptional for the purpose of these experiments. The experiments were approved by the Institutional Review Board and were in conformity with the principles of the Helsinki Declaration. 
Results
Orientation of straight line contours
Thresholds
Since these investigations were directed at cardinal versus oblique differences, in each instance it was first ascertained that the observers manifested the typical oblique effect in their orientation discrimination thresholds in the setting of the current experiments. Figure 3, top shows that this is indeed the case. Orientation discrimination thresholds are higher by a factor of about two for lines in the 45° and 135° meridians than for horizontals and verticals. 
Susceptibility to context-induced tilt
The next set of measurements involved the apparent tilt induced in a test line by a pair of laterally placed flanks inclined to it by 22°. The interaction is repulsive; that is, when flanked, the test line's orientation appears inclined in a direction opposite to the flanks'. As is seen in Figure 3, this is also more pronounced for configurations in the 45° and 135° meridians. The obliquity index, (45 +135)/(H + V), has about the same value for this measure of a line's susceptibility to tilt induction as it has for its orientation discrimination threshold. 
A survey in one observer of a more extensive range of inducing tilts (Figure 4) suggests that this phenomenon is regular. 
Tilt-inducing potency
Given that lines lying in oblique meridians have a less secure orientation attribute—oblique orientations have poorer orientation discrimination threshold and higher susceptibility to tilt induction than those in cardinal meridians, the question arises whether their apparently weaker orientation signal makes them also less potent in inducing contextual tilt. A comparison test was therefore designed in which a test line with a moderate degree of obliquity (22°) is shown flanked by inclined lines with equal angular disposition with respect to the test line, which means that one flank orientation was in a cardinal orientation and the other in the 45° oblique (Figure 5). The ratio of the tilt induced in the moderately oblique test line by the fully oblique inducing flanks to that by the flanks in a cardinal orientation, i.e., the ratio of their strength in inducing contextual perturbation, can be compared with the susceptibility to such induction. The data (Figure 5) show that the poorer orientational sturdiness of oblique lines is also mirrored in a reduced capacity to exert contextual disturbance, but only to about half the extent. A somewhat different approach to this question via the Poggendorff illusion (see below) confirms this conclusion. 
Contour curvature
Investigated so far has been the orientation of a smooth straight line segment. The simplest deviation from that shape is to impart curvature to it, that is, to render it as an arc of a circle whose center of curvature lies on the normal to the line segment at its midpoint. So long as that condition remains intact, the line's orientation, defined as the tangent at its midpoint, remains invariant. Curvature is expressed as the reciprocal of the radius of curvature as the measure, being zero for a straight line, and its sign, defined by the side on which the center of curvature is located. 
Curvature threshold
The minimum detectable deviation from straightness that has to be imparted to line for the detection that it is no longer straight, that is, the minimum curvature whose direction (concave to one or the other side of the line segment) can be correctly identified on 75% of presentations, was determined as a function of orientation in three observers and is shown in Figure 3, middle. Here too there is a prominent oblique effect. 
Curvature induction
The procedure for the measurement of curvature induction was in all essential details the same as that described above for the tilt illusion, except that the line attribute was curvature rather than orientation. Just as for line orientation, the interaction is repulsive: When flanked by prominent curved arcs, a line assumes the appearance of small oppositely directed curvature, and more so when the configuration's orientation lies in the 45° or 135° than in the horizontal or vertical meridians (Figure 3, bottom). 
Contour edge properties
Edge irregularity
Straight edges were generated whose edges could be varied in smoothness from a being strictly straight to featuring small, irregular edge corrugations. Sequential 4 arcmin segments of a long high-contrast edge were given random deviations from the edge's mean location of between 0 and 6 arcmin (Figure 6, top). The smallest average deviation in arcmin that could be distinguished from smoothness on 75% of presentations was designated as the smoothness or irregularity threshold. It was determined as a function of edge orientation in three observer (Figure 6, bottom), revealing no oblique effect for this variable. 
Geometrical-optical illusions
Here considered are configurations consisting of high-contrast lines, where simple geometrical properties (e.g., straightness, parallelism, collinearity) differ in observers' reports from those objectively verified. Of the several that are known to vary in magnitude with orientation, the orientation anisotropy is here studied, using the method of adjustment, in the Hering illusion of apparent curvature and in the Poggendorff illusion of apparent misalignment. 
Hering illusion
The curvature induced in the pair of parallel straight lines by the sheaf of intersecting lines was nulled out by making the straight lines into circular arcs of opposing curvature. The means of 16 observer settings that satisfied this criterion, duplicated on a second day, in four overall orientations of the whole configuration, are plotted in Figure 7 for three observers. The obliquity index is in agreement with that for curvature detection threshold and for curvature induction due to curved contextual flanks. 
Poggendorff illusion
Orientation anisotropy in the Poggendorff illusion had been thoroughly documented, quite early and remarkably clearly by Obonai (1931) and more recently in the Weintraub and Hotopf laboratories (Hotopf & Hibbard, 1989; Weintraub et al., 1980), and does not, therefore, require further elaboration here. Although the manifestation is that of positional misalignment, the illusion is usually interpreted as due to expansion of the apparent angle between the parallels and the transverse line segments, an example of contextually induced line-orientation changes that has been demonstrated above to be more pronounced when the affected line is oblique (Figures 3 and 4). To obtain a quantitative estimate of the illusion, observers were instructed to null it out, that is to move the lower transverse segment of the display until the two transverse segments appeared to be in alignment. The average of 16 such settings each on two separate days was converted into the angular difference between the inclinations of the overt transverse segments and the covert line joining their junction points with the parallels. This conforms with the approach leading to the data in Figure 3 and is the induced orientation change due the particular context. But, as compared with that pattern in which all the components are the same, here there is a relative weight difference between the elements of the configuration. This asymmetry allows concentration on orientation differences in the relative potency of the parallels to act as inducers as well as on the susceptibility of the short transverse line segments to interacting influences exerted by the long double parallel lines. 
This was implemented as follows (Figure 8): First, the Poggendorff illusion was measured for the parallels at a 22° obliquity with respect to the vertical, with two orientations of the transverse, both at an inclination of 22°, which made one of them fully oblique and the other vertical (Figure 8, II and III). Since the same inducing element is involved, any difference must be ascribed to the cardinal/oblique difference in the susceptibility of the transverse line segment to orientation perturbation (Figure 8, III/II). 
Second, the situation was now reversed: The transverse segment remained at a 22° tilt with respect to a cardinal throughout, and the Poggendorff illusion measured for orientations of the parallels 22° either side, one in the cardinal (Figure 8, I) and the other in an fully oblique (Figure 8, IV) meridian. Since the same accepting element is involved, any difference must be ascribed to an oblique/cardinal difference (Figure 8, I/IV) in the capacity of the parallels to exert contextual induction. 
In the interpretation of the data in Figure 8, a possible complication needs discussion. If the segments of the interrupted transverse line manifested a standing offset from collinearity, in the manner of a vernier displacement, then this would be a component in the angles which constitute the data in the table. Fortunately, earlier results (figure 5i of Westheimer & Wehrhahn, 1997) speak to that. The magnitude of the Poggendorff illusion for oblique transverse lines varies depending on the nature of the inducing contours, including virtual borders, but the illusion disappears in the total absence of inducing contours. Hence the data in Figure 8 give a definitive answer to the question whether the obliques' capacity to induce interaction is as potent as their susceptibility to induction. In agreement with the data in Figure 5, it can be concluded that whereas vertical and horizontal contours are more resistant to contextual perturbation than oblique ones, the difference in their ability to generate such an influence is only about half as much. 
Table
 
Summary data across all observers and relevant conditions of the performance superiority of horizontal and vertical over oblique contours.
Table
 
Summary data across all observers and relevant conditions of the performance superiority of horizontal and vertical over oblique contours.
Explicit versus virtual contours
There is convincing evidence that such properties as the oblique effect and contextual interaction are present in configurations whose contours are only sketched in by tokens, often called virtual borders (Westheimer, 1990), or when what matter is the overall orientation of the whole configuration rather than its constituent borders, often called implicit orientations (Li & Westheimer, 1997). In this spirit, some of the findings that context-induced illusions of tilt and of curvature are more prominent when patterns lie in oblique rather than the cardinal meridians, were replicated using 50% dashed lines (lines consisting only of 10 arcmin segments separated by 10 arcmin) instead of the continuous contours used so far. The data confirmed that contours do not need to be rendered explicitly for the effects to be exhibited. Figure 9 allows an informal glimpse of this. The literature on virtual contours is, however, voluminous and the strength of virtual, illusory, implicit and sketched-in borders varies greatly, depending on many factors. This report is, therefore, intended merely to decouple the findings from the requirement that lines and borders be continuous and explicit. The demonstration that all oblique borders, no matter how tenuous, have weaker spatial properties than equivalent horizontal and vertical ones, which may very well be true, is beyond the scope of this study. 
Figure 9
 
Illustration that the Hering and Poggendorff illusions survive the substitution of virtual lines for explicitly-drawn contours.
Figure 9
 
Illustration that the Hering and Poggendorff illusions survive the substitution of virtual lines for explicitly-drawn contours.
Discussion
The findings in this paper extend the oblique effect that had so far been described for orientation discrimination also to the tilt illusion and to curvature detection and induction. Further, the distinction is developed between a contour's susceptibility to allow a perturbation to be induced in it and its ability to generate a context-induced perturbation. The search for the neural substrate properly begins in the orientation-selective neurons in the primate primary visual cortex, because the simpler border attributes of detection and contrast discrimination, many of retinal origin, don't manifest an oblique effect (Westheimer & Beard, 1998). The context-induced changes studies here are doubtless the results of antagonistic interaction between neighboring orientations that was proposed as the basis of perceived changes in orientation of lines (Blakemore, Carpenter, & Georgeson, 1970) and has since been well documented (Kapadia, Westheimer, & Gilbert, 2000). There is some evidence that horizontal and vertical retinal meridians are favored in the distribution of V1 orientation-selective cells, and, apart from numerosity, there may also be differences in tuning width. Until reliable data for the primate, perhaps even human, are available, reticence is in order in attempts to design detailed neural models. Cortical circuitry is complex with much local as well as reentrant and top-down interconnectivity, which in the behaving organism is significantly plastic, depending on such factors as alertness, attention, memory, and expectation. So it is apparent that the receptive field and connectivity properties of cortical units, or even ensembles of such units, would be limited to the stimulus patterns and states for which they had been characterized. For example, in even as narrow a span as the V1-V2 interface, it is difficult to ascertain specificity of the orientation feedback signal (Stettler, Das, Bennett, & Gilbert, 2002). Formulation of models must now also accommodate the novel observation here reported of a conspicuous quantitative difference between a contour's strength in exerting a contextual influence on a neighboring contour, and the contour's own susceptibility to such influences. 
It has often been pointed out that there is more to contour-orientation sensitivity than the traditional concept of filtering retinal images through the receptive fields of simple cortical cells. At a minimum, recourse has to be taken to the collector cell proposal (Morgan & Hotopf, 1989; Brincat & Westheimer, 2000). The data presented here can be complemented by the observation of oblique effects in configurations, such as three rings several degrees apart, whose overall orientation is clearly not directly dependent on the stimulation of orientation-selective neurons in the primary visual cortex (Westheimer, 2003). That relatively simple geometrical relationships and illusions survive being rendered by contours that are virtual or even implicit further serves to decouple their perceptual origin from obligatory dependence on receptive field shape of neurons early in the visual stream. 
That behaviorally there is an oblique effect even in configurations that cannot immediately be traced to the functioning of V1 orientation-selective units draws attention to the demonstration of orientation anisotropy by imaging techniques in other visual cortical areas: middle-temporal lobe (Xu, Collins, Khaytin, Kaas, & Casagrande, 2006), para hippocampal place area (PPA; Nasr & Tootell, 2012). But before attempting to relate them to human psychophysical findings, one needs to be sensitive to the preparation from which neurophysiological data were acquired. For example, the distribution of preferred orientations in V1 may have been gathered from recordings in the anesthetized cat (Li, Peterson, & Freeman, 2003) or paradoxical oblique effects may be based on oscillatory potentials in the infratemporal lobe on stimulation in only one oblique orientation in the retinal periphery (Koelewijn, Dumont, Muthukumaraswamy, Rich, & Singh, 2011). 
Though contour curvature seems a perceptual attribute sui generis, its neural substrate may nevertheless reside in the analysis of the orientation difference in sequential segments of the contour. Formally, curvature is defined by the rate of change of the tangent to the contour's osculating circle, a definition that would suffice also for the contour's local orientation. The curvatures and arc lengths in the present experiments were such that the included orientations were contained in a range within which no significant change in the oblique effect might be expected, but this would no longer be the case for more highly curved arcs. A small control experiment, which revealed no oblique effect in curvature difference thresholds of prominently curved arcs which encompassed a 45° range of orientations, supports this contention. Of interest in the question of the neural locus of the oblique effect is the fact that the neural analysis of curvature has been demonstrated in the primate to be located in cortical area V4 (Yau, Pasupathy, Brincat, & Connor, 2013), that is well central to the first orientation-selective filtering of visual signals in V1. 
The results here reported on the oblique effect in the Poggendorff illusion accord with an established viewpoint about predominance of cardinal over oblique orientations in this phenomenon. Weintraub, following thorough documentation of most of the important aspects of the Poggendorff illusion, speaks of “increased error susceptibility at the visual obliques” (Weintraub et al., 1980, p. 724). Referring to the same phenomenon, Hotopf and Hibbard, claiming agreement with almost a dozen earlier studies, wrote about “greater resistance … on the part of vertical and horizontal lines than ones that are in other orientations” (Hotopf & Hibbard, 1989, p. 358). A quantitative estimate is here also given for the Hering illusion, where it manifests itself in induced curvature in straight lines. By giving a measure to the illusions, it is possible to make a comparison of their oblique effect with that in more direct estimates of context-induced perturbation of orientation and curvature. In each case the illusory geometrical distortion within the whole configuration are larger than when the concerned components are examined in isolation, and so also is the oblique effect. (Table 1 and Figure 3C vs. Figure 8 and Figure 3D vs. Figure 7). It has been the experience in this laboratory on several previous occasions that it is more difficult to obtain reliable quantitative measures of geometrical-optical illusions than more routine spatial psychophysical visual thresholds. There is more day-to-day scatter and inter-subject variability (Morgan & Dillenburger, 2016, Westheimer & Wehrhahn, 1997) in the average of even a large number of settings, very likely as a result of the more variegated judgment involved in whether an illusion has been nulled out. But the larger oblique effect in the illusions seen here, prominent and uniform across several experiments, could be interpreted as a more pronounced cardinal/oblique dominance at perceptually more demanding levels where obliquity might be more of a handicap than earlier on in the visual stream. 
Contour orientation and curvature are here called “form-defining” to distinguish them from the simple orientation-invariant properties of detection and discrimination of contrast and small-scale irregularities. As early as 1894, Guillery (1894) postulated a processing step that, while utilizing the first elaboration of the demarcation of spatial nonuniformities in the visual field, presumably with signatures of location, brightness, and chromaticity contrasts, is beyond them. It is tempting, therefore, to view the oblique effect as entering at that stage. Yet it is by no means evident that it is a universal manifestation of form perception, for it is absent in, for example, line length discrimination (Westheimer, 2001). In target movement perception there are also attributes, such as the discrimination of the direction of streaming random dots that exhibit an oblique effect, and of their velocity, which does not (Matthews & Qian, 1999). 
But the prominent superiority of cardinal orientations found so consistently in the present study raises the question of the extent to which this is a more general feature of these higher visual stages. 
Acknowledgments
Commercial relationships: none. 
Corresponding author: Gerald Westheimer. 
Address: Division of Neurobiology, University of California, Berkeley, Berkeley, CA, USA. 
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Figure 1
 
Configurations used in this study. Top left: Tilt illusion, orientation change induced in a test line by tilted flanking lines. Top right: Curvature induction, where a straight line appears curved when flanked on either side by curved contours. Middle left: Hering Illusion, where a pair of straight, parallel lines appears curved in the presence of a sheaf of intersecting lines. Middle right: Poggendorff illusion, in which there is an apparent misalignment of the transverse segments even when they are strictly collinear. Bottom: The Hering and Poggendorff illusions in different meridians.
Figure 1
 
Configurations used in this study. Top left: Tilt illusion, orientation change induced in a test line by tilted flanking lines. Top right: Curvature induction, where a straight line appears curved when flanked on either side by curved contours. Middle left: Hering Illusion, where a pair of straight, parallel lines appears curved in the presence of a sheaf of intersecting lines. Middle right: Poggendorff illusion, in which there is an apparent misalignment of the transverse segments even when they are strictly collinear. Bottom: The Hering and Poggendorff illusions in different meridians.
Figure 2
 
Top: Sequence of presentations during each trial in the tilt illusion experiment. The test line is first shown for 400 ms by itself with an orientation that was randomly perturbed by one of seven amounts from the orientation it had during the subsequent presentation (after a 400-ms pause) of the full configuration, which included a flank at either side, with a tilt whose amount was fixed during the run. In long runs of trials, the direction of flank tilt was randomly either clockwise or counterclockwise. The observer signaled whether the test line, when part of the line triplet, appeared tilted clockwise or counterclockwise, using the test line's orientation when first shown for comparison purposes as a reference. Bottom: Example of data analysis in tilt illusion experiment. Proportion of responses “Test lined appears tilted clockwise” in runs totaling 600 trials in which accompanying flanks were shown tilted randomly either clockwise (squares, blue probit fit) or counterclockwise (circles, red probit fit). Analyzed separately, they showed that for the test line to appear vertical (i.e., seen tilted clockwise and counterclockwise with equal probability), it needs to be shown at a 87.8° orientation when accompanied by counterclockwise flanks, and at 92.3° when accompanied by clockwise flanks. This expresses the repulsive tilt induced by the flanks and evaluates numerically the oppositely-directed tilt needed to make the test line appear vertical when flanked. The arrow indicates the magnitude of the total induction caused by the two oppositely directed flanks; halving it yields a bias-free estimate (here 2.25°) of the apparent tilt induced in a vertical test line by a set of 22° flanks.
Figure 2
 
Top: Sequence of presentations during each trial in the tilt illusion experiment. The test line is first shown for 400 ms by itself with an orientation that was randomly perturbed by one of seven amounts from the orientation it had during the subsequent presentation (after a 400-ms pause) of the full configuration, which included a flank at either side, with a tilt whose amount was fixed during the run. In long runs of trials, the direction of flank tilt was randomly either clockwise or counterclockwise. The observer signaled whether the test line, when part of the line triplet, appeared tilted clockwise or counterclockwise, using the test line's orientation when first shown for comparison purposes as a reference. Bottom: Example of data analysis in tilt illusion experiment. Proportion of responses “Test lined appears tilted clockwise” in runs totaling 600 trials in which accompanying flanks were shown tilted randomly either clockwise (squares, blue probit fit) or counterclockwise (circles, red probit fit). Analyzed separately, they showed that for the test line to appear vertical (i.e., seen tilted clockwise and counterclockwise with equal probability), it needs to be shown at a 87.8° orientation when accompanied by counterclockwise flanks, and at 92.3° when accompanied by clockwise flanks. This expresses the repulsive tilt induced by the flanks and evaluates numerically the oppositely-directed tilt needed to make the test line appear vertical when flanked. The arrow indicates the magnitude of the total induction caused by the two oppositely directed flanks; halving it yields a bias-free estimate (here 2.25°) of the apparent tilt induced in a vertical test line by a set of 22° flanks.
Figure 3
 
From top down: (A) Threshold for discrimination of the orientation of a foveal line for four line orientations in three observers. (B) Threshold for detection of curvature in a centrally-fixated line, 1.6° in length as a function of orientation in three observers. Curvature in units of 0.0069° of visual angle−1. (C) Induced tilt in a foveal line that is flanked on each side by a line 40 arcmin away and inclined by 22°, as a function of orientation in four observers. (D) Curvature induced in a centrally fixated line when flanked on each side by two curved arcs separated from it by distances of 40 and 60 arcmin as a function of orientation of the whole configuration.
 
The obliquity index, (45 +135)/(H+V), averaged for all observers is 1.92., 1.89, 2.47, and 1.80 in the four experiments, and close to 2 overall for all observers and conditions.
Figure 3
 
From top down: (A) Threshold for discrimination of the orientation of a foveal line for four line orientations in three observers. (B) Threshold for detection of curvature in a centrally-fixated line, 1.6° in length as a function of orientation in three observers. Curvature in units of 0.0069° of visual angle−1. (C) Induced tilt in a foveal line that is flanked on each side by a line 40 arcmin away and inclined by 22°, as a function of orientation in four observers. (D) Curvature induced in a centrally fixated line when flanked on each side by two curved arcs separated from it by distances of 40 and 60 arcmin as a function of orientation of the whole configuration.
 
The obliquity index, (45 +135)/(H+V), averaged for all observers is 1.92., 1.89, 2.47, and 1.80 in the four experiments, and close to 2 overall for all observers and conditions.
Figure 4
 
Magnitude of the induced tilt in a foveal line, 2° in length, when flanked on either side by a similar line separated from it by 20 arc min, for four inducing tilts of the flanks with the configuration oriented vertically, at 135° horizontally, and 45°. Each bar represents measurement based on 600 responses.
Figure 4
 
Magnitude of the induced tilt in a foveal line, 2° in length, when flanked on either side by a similar line separated from it by 20 arc min, for four inducing tilts of the flanks with the configuration oriented vertically, at 135° horizontally, and 45°. Each bar represents measurement based on 600 responses.
Figure 5
 
Top: Tilt induced in a test line in the 22° meridian when accompanied by a pair of flanking lines in a cardinal meridian and (lower) in the 45° meridian. The ratio of tilt-inducing capacity is on average half as much as the oblique index for the same observer and the same flank inclination.
Figure 5
 
Top: Tilt induced in a test line in the 22° meridian when accompanied by a pair of flanking lines in a cardinal meridian and (lower) in the 45° meridian. The ratio of tilt-inducing capacity is on average half as much as the oblique index for the same observer and the same flank inclination.
Figure 6
 
Top: Screen shot showing stimulus, consisting of a white rectangle on a gray circular background when randomly one or the other long edge had an irregular contour, the parameter being the average depth of the irregularity in arcmin. Bottom: Thresholds values for this parameter, with their standard errors, when the observer correctly identified on 75% of occasions which edge was corrugated for three observers, as a function of the orientation of the configuration.
Figure 6
 
Top: Screen shot showing stimulus, consisting of a white rectangle on a gray circular background when randomly one or the other long edge had an irregular contour, the parameter being the average depth of the irregularity in arcmin. Bottom: Thresholds values for this parameter, with their standard errors, when the observer correctly identified on 75% of occasions which edge was corrugated for three observers, as a function of the orientation of the configuration.
Figure 7
 
Curvature needed to null out Hering illusion, that is to give the line pair the appearance of being straight and parallel in the presence of the intersecting sheaf, as a function of orientation of the whole configuration in three observers. When the overall pattern size was 1.3° × 2.5°, the curvature was in units of 0.0069° of visual angle−1.
Figure 7
 
Curvature needed to null out Hering illusion, that is to give the line pair the appearance of being straight and parallel in the presence of the intersecting sheaf, as a function of orientation of the whole configuration in three observers. When the overall pattern size was 1.3° × 2.5°, the curvature was in units of 0.0069° of visual angle−1.
Figure 8
 
Top: Measurement of the Poggendorff illusion, in which the observer moves one of the transverse line segments until it appears aligned with the other, in four configurations. Numerical values signify the difference in orientation of the transverse line segments and the covert line joining their intersection points with the parallels when the illusion has been nulled out, a measure taken to be the apparent expansion of the acute angle, here always 22°, between the transverse and parallel lines. In conditions I and IV the transverse line was at an inclination of 22° to the vertical, and the last column gives the ratio of the inducing strengths of cardinal to oblique parallels. The complementary data III/II show the oblique/cardinal ratio in the susceptibility to having perturbation induced by a given set of parallels.
Figure 8
 
Top: Measurement of the Poggendorff illusion, in which the observer moves one of the transverse line segments until it appears aligned with the other, in four configurations. Numerical values signify the difference in orientation of the transverse line segments and the covert line joining their intersection points with the parallels when the illusion has been nulled out, a measure taken to be the apparent expansion of the acute angle, here always 22°, between the transverse and parallel lines. In conditions I and IV the transverse line was at an inclination of 22° to the vertical, and the last column gives the ratio of the inducing strengths of cardinal to oblique parallels. The complementary data III/II show the oblique/cardinal ratio in the susceptibility to having perturbation induced by a given set of parallels.
Figure 9
 
Illustration that the Hering and Poggendorff illusions survive the substitution of virtual lines for explicitly-drawn contours.
Figure 9
 
Illustration that the Hering and Poggendorff illusions survive the substitution of virtual lines for explicitly-drawn contours.
Table
 
Summary data across all observers and relevant conditions of the performance superiority of horizontal and vertical over oblique contours.
Table
 
Summary data across all observers and relevant conditions of the performance superiority of horizontal and vertical over oblique contours.
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