Perception, 1998, volume 27, pages 87-103

Depth interactions between inclined and slanted surfaces in vertical and horizontal orientations

Byron J Pierce Air Force Research Laboratory, 6001 S Power Road, Building 558, Mesa, AZ 85206-0904, USA; e-mail: pierce@alhra.af.mil

Ian P Howard Centre for Vision Research, 103 Farquharson Building, York University, Toronto, Ontario M3J 1P3, Canada; e-mail: ihoward@hpl.ists.ca

Catina Feresin Department of Psychology, Universita di Trieste, via dell'Universita 7, 34123 Trieste, Italy Received 22 July 1996, in revised form 18 September 1997

Abstract. Depth interactions between a frontal test surface and an adjacent induction surface were measured as a function of the type of disparity in the induction surface and of the vertical/horizontal orientation of the boundary between the surfaces. The types of disparity were 4° horizontal-shear disparity, 4° vertical-shear disparity, and 4° rotation disparity; 4% horizontal-size disparity, 4% vertical-size disparity, and 4% overall-size disparity. Depth contrast in a frontal surface was produced by surfaces containing horizontal-size disparity but not by those contain­ing horizontal-shear disparity. Vertical-shear and vertical-size disparities produced induced effects in both the induction and the test surface, which is here explained in terms of deformation-disparity processing. Effects of rotation disparity on the test surface can be accounted for in terms of cyclovergence, deformation disparity, and perhaps also depth contrast. The fact that horizontal-size disparity produced more depth contrast than horizontal-shear disparity is due to an anisotropy of disparity processing rather than the relative orientation of the surfaces. Ground surfaces appeared more slanted than ceiling surfaces. Surfaces containing horizontal disparities produced a sharp boundary with the test surface because horizontal disparities are processed locally. Surfaces with vertical disparities produced a gradual boundary with the test surface because vertical disparities are processed over a wider area.

1 Introduction This paper is about depth contrast and other depth-interaction effects produced in a test surface with zero disparity adjacent to a textured surface with the following types of shear or size disparity. (i) Horizontal-shear disparity, which occurs in a surface inclined in depth about a horizontal axis. (ii) Vertical-shear disparity, which produces the impression of a surface inclined in the opposite direction to that produced by the same sign of horizontal-shear disparity, (iii) Rotation disparity, which can be regarded as a horizontal-shear disparity plus a vertical-shear disparity. The effects of the two disparities cancel leaving the surface in the frontal plane (Howard and Kaneko 1994). (iv) Horizontal-size disparity, which occurs in the images of a surface slanted about a vertical axis. (v) Vertical-size disparity, which produces the impression of a surface slanted in the opposite direction to that produced by the same sign of horizontal-size disparity. This is Ogle's induced effect (Ogle 1938). (vi) Overall-size disparity, which can be regarded as a horizontal-size disparity plus a vertical-size disparity. Ogle showed that the effects of the two disparities tend to cancel.

There are three types of depth interaction between stimuli lying in distinct depth planes. First, movement or inclination of a surface in depth is more accurately and rapidly perceived in the presence of a second surface in another depth plane. We refer to this as

88 B J Pierce, I P Howard, C Feresin

depth enhancement. For example, a change in horizontal disparity applied to the whole of a frontal surface is not perceived as a change in depth unless a stationary reference object is present (Erkelens and Collewijn 1985; Regan et al 1986). Changes in disparity over the whole visual field (zero-order spatial disparity) are most likely due to unintended changes in vergence and are best ignored as signals for depth. Similarly, the inclination of a large isolated surface about a horizontal axis produced by a shear disparity (a first-order spatial derivative of horizontal disparity) is underestimated and takes a long time to perceive in the absence of a frontal-plane reference surface that provides a second-order depth discontinuity. The slant of a surface about a vertical axis defined by a first-order size disparity is even more difficult to perceive in the absence of a reference surface (Gogel 1965; Gillam et al 1984; van Ee and Erkelens 1996). We have previously measured depth enhancement (Pierce and Howard 1997; Howard and Pierce 1998).

The second type of depth interaction is apparent depth created in a frontal-plane display superimposed on or adjacent to an induction display containing a step or gradient of depth. After Werner (1938), we refer to this as depth contrast. There is contradictory evidence about the occurrence of depth contrast created by steps of disparity (see Howard and Rogers 1995). Anstis (1975) observed depth contrast between a disc surrounded by a surface with uncrossed disparity and a coplanar disc surrounded by a surface with crossed disparity although Brookes and Stevens (1989) could not see this effect. Graham and Rogers (1982) observed that a pair of frontal planes on either side of a horizontal step of disparity appeared inclined in depth. This effect is similar to the disparity analogue of the Chevreul illusion reported by Brookes and Stevens. Graham and Rogers also observed depth contrast in a stereoscopic frontal surface flanked above and below by inclined induction surfaces. Anstis et al (1978) reported the depth analogue of the Craik - O'Brien - Cornsweet illusion.

Julesz (1971) created the disparity analogue of the Herman grid in the form of a random-dot stereogram in which there was a disparity between a set of cyclopean squares and the grid of dots lying between the squares. In the luminance Herman grid the intersections appear darker than the remainder of the grid but Julesz saw no depth change at the intersections in the disparity analogue of the grid (see also Brookes and Stevens 1989). Similarly, it has been claimed that there is no disparity analogue of Mach bands for a surface consisting of two planes smoothly connected by a horizontal or vertical disparity ramp (Brookes and Stevens), although the depth gradients of the three regions are often misperceived. Lunn and Morgan (1995) challenged these nega­tive findings on the grounds that the displays were on too fine a spatial scale for the relatively coarse disparity system. They constructed larger random-dot displays of the Herman grid, embedded squares, and Mach bands and their subjects saw depth-contrast effects analogous to those seen in the luminance domain. Nevertheless, contrast effects between regions of constant disparity are weak at best.

Depth contrast has been reported in a frontal surface adjacent to an inclined or slanted surface. For example, Graham and Rogers (1982) devised random-dot stereo­grams in which a horizontal strip in the frontal plane appeared inclined in the opposite direction to the inclination of larger flanking induction surfaces. The contrast inclina­tion measured by a nulling procedure was between 20% and 60% of the inclination of the induction surfaces.

Stimulus duration is an important factor in depth contrast. Kumar and Glaser (1993) used a pair of test dots within a surrounding frame and found that the largest contrast occurred with the shortest exposure of 10 ms, as Werner (1937) had originally noted. With exposures of several seconds, depth contrast almost completely disappeared. In the present experiments we were interested in the steady-state properties of contrast for various types of disparity and therefore allowed subjects to inspect the stimulus for many seconds before making their judgments.

Depth contrast for inclined and slanted surfaces 89

We have previously reported evidence of depth contrast in a frontal surface from a surface slanted in depth about a vertical axis but not from a surface inclined in depth about a horizontal axis (Pierce and Howard 1997; Howard and Pierce 1998). An anisotropy between slant and inclination has been noted by several investigators. For example, inclination, involving horizontal-shear disparity, is more rapidly and accurately perceived than slant, involving horizontal-size disparity (Gillam et al 1984; Mitchison and McKee 1990; Cagenello and Rogers 1993). Also, the depth analogue of the Craik-O'Brien-Cornsweet illusion is considerably larger when the edge of the disparity discontinuity is oriented vertically (involving size disparity) than when the edge is oriented horizontally (involving shear disparity) (Rogers and Graham 1983).

Depth enhancement and depth contrast may be two aspects of the same processes, namely, a tendency to underestimate the slant or inclination of a surface relative to the frontal plane (in a headcentric coordinate system) coupled with accurate registration of the relative disparity between two surfaces. According to this theory, the more depth is underestimated in an induction surface, the greater should be the depth contrast in a frontal surface. Our results support this prediction.

The third type of interaction between surfaces arises from the fact that the perceived inclination or slant of a surface is derived from deformation disparity, or the difference between local horizontal disparity and global vertical disparity. Whereas the horizontal disparities in adjacent surfaces are detected independently, the vertical disparities are averaged across the boundary and this causes one surface to affect the other. The details of this type of interaction are described in section 4.

In this paper we investigate depth contrast and induced effects produced by textured surfaces with various types of shear and size disparity as a function of (i) the sign of disparity, (ii) the orientation of the boundary between the induction and test surface, and (iii) whether the region judged is near the boundary between the surfaces or inside one or the other surface.

2 Experiment 1: Shear-disparity displays In this experiment we examined depth contrast induced into a zero-disparity test surface as a function of three types of shear disparity in an adjacent induction surface— horizontal shear, vertical shear, and rotation (equal horizontal and vertical shear)—and of the orientation of the boundary between the surfaces.

2.1 Method 2.1.1 Stimuli and apparatus. The stimuli consisted of textured patterns constructed with MacDraw Pro in a Macintosh computer and projected onto two screens of a Wheatstone stereoscope. For each eye the induction pattern filled half the 60 deg by 60 deg display and the zero-disparity test pattern filled the other half. The two abutting patterns were either in the upper and lower halves of the display with a horizontal boundary or in the left and right halves of the display with a vertical boundary. Each induction pattern contained 362 elements irregularly placed in a forty-column by twenty-row or a twenty-column by forty-row rectangle. The test pattern was similar except that it was a thirty-nine-column by twenty-row or a twenty-column by thirty-nine-row rectangle. The elements were crosses, horizontal and vertical lines, and open and filled squares and circles. The pattern is illustrated in figure 1. The test pattern abutted the induction pattern with its centre column or row between the centre pair of columns or rows of the induction pattern.

Plus or minus 2° of shear was introduced into each eye's image of the 60 deg by 30 deg induction pattern, with the centre of shear along the mid-vertical axis (vertical shear) or mid-horizontal axis (horizontal shear) of the combined 60 deg by 60 deg display. The shear disparities are illustrated in figure 2. Stereoscopic combination of two

90 B J Pierce, I P Howard, C Feresin

• a • + • O

+ I +

• D + i I +

+ I I

I o

+ • •

a • o • <

+ i + I

o < • •

D • + I

+ ' a o

a a •

• + o D •

I -

o o

+

• I -

I o

I

—

» + o •

• +

+ • o + 1

— • I + 0

+ I • • • +

a • +

0 I D + +

+ -

0 -

• 1 + + • o

o -• o

0 • 1 -

+ +

+ -I - +

I + + +

I • + • O I

I

a o ,

> + i + •

+

I • a o i

+ • o

• a I o • I - • + • I o • a + + a o o I • o • O D O -

o • - • • o l - o o • a + • o • o l

0 • O • O l • D O B I D o - a o + - • a

" ° • + - • a + - o • a « I - o I + I •

° o # I + • • - • o • _ O - • + O l D D +

I— OH • — • • • 0 + | + • • I I •

• D • D + - • • • 1 - • o + - a

w ' • — # o n • • a • o ° a - o o + o - B •

• o a + a • - o • • o • - + • I • a

o + - • • o l • a • ° Q O I - + + - I • n

• - + 1 • - o • 7 • o + i - • a I

• ' • a • • + I + - • a — o l • I — • o •

1 a o • o o I + o • I - I - I 0 a • | o • a o - I •

• - + + o o - a • • • a I o • I - • +

• + | o - B O D o o o • • o I • + •

• a o • • • + — o • - • + - a • a •

D a - o o + — I • o ' a — + o • I a a

a a o - • o • + o o o + - - o - I -

• a • • D O + + 0 0 • + O B a a +

Figure 1. The basic stimulus pattern, which subtended 60 deg. In this example, an induction stimulus 60 deg high by 30 deg wide containing a disparity is on the right and the test stimulus is on the left. In the actual displays the texture elements were white on a black background, and all elements were the same brightness.

oppositely sheared patterns produced a stereoscopic surface with 4° of shear disparity. Theoretically, 4° of horizontal-shear disparity at a distance of 89 cm corresponds to an inclination of 43.8° from the vertical. Because of limitations of the graphics program, individual elements were not sheared. In a positive shear disparity the right-eye pattern is sheared clockwise and the left-eye pattern counterclockwise. A positive horizontal-shear disparity produces a surface inclined top away. In a negative shear disparity the right-eye pattern is sheared counterclockwise and the left-eye pattern clockwise. Surfaces with 4° of vertical-shear disparity were created by a 90° rotation of the patterns with

m left-eye image right-eye image

• ii; f II i \ i

BfW

f If

i»-0 « «*

l)m t'xmm m ! i • • f I M M

0

V m ± : m

m + £

" i t" ^ • m

m § t

; ; : : | : : : : : r #

%

* mi

m *

•r • s

y

(a) (b) (c)

Figure 2. Types of shear disparity: (a) horizontal-shear disparity, (b) vertical-shear disparity, and (c) rotation disparity. Black elements represent images in one eye and grey elements images in the other eye. Positive shear and rotation are defined as clockwise in the right-eye image and counterclockwise in the left-eye image.

Depth contrast for inclined and slanted surfaces 91

horizontal shear. Surfaces with plus or minus 4° of rotation disparity were created by rotating each eye's induction pattern, including the texture elements, 2° about the centre of the 60 deg by 60 deg display.

The stimuli were printed with a 600 dpi laser printer and 2-inch slides were made with reverse-negative film. When projected onto rear-projection screens, the stimuli were white elements on a dark background. The elements had an average luminance of 1.3 cd m~2 at the eyepoint; the background luminance was 0.3 cd m -2.

The images on the two screens were carefully aligned to create a stereoscopic display straight ahead of the subject at a distance of 89 cm with the centre at eye level. Subjects viewed the stimuli in a dark room and all surrounding objects were painted matte black so that nothing but the stimulus was visible.

2.1.2 Procedure. A block of 12 trials consisted of [three types of disparity (vertical shear, horizontal shear, and rotation) x two disparity directions (positive and negative) x two boundary orientations (horizontal and vertical)]. Each subject performed four blocks of trials over two sessions. All combinations of disparity type, direction of disparity, and boundary orientations were sorted for each block and presented in random order. When the boundary between the induction and test surfaces was horizontal, the upper/ lower position of the induction surface was counterbalanced between subjects by using an AABB sequence; when the boundary was vertical, the left/right position of the induction surface was counterbalanced by using an ABAB sequence.

Subjects were instructed to look at the region they were judging and adjust an unseen paddle to match the perceived inclination of each surface in the boundary region where the two surfaces met and in the region away from the boundary. The paddle was a disc of 7 inches diameter pivoted about its mid-horizontal axis and placed in the midline just in front of the subject at waist height. Paddle settings were meas­ured by a potentiometer and recorded for later analysis.

Measurement order (induction or test surface) was counterbalanced between subjects by using an ABBA sequence. Next, subjects were asked to maintain their gaze on the boundary between the surfaces and report the number of columns of the vertical-shear and rotation-disparity patterns they could fuse, or the number of rows of the horizontal-shear patterns they could fuse. Last, subjects were asked whether there was a distinct or gradual transition of inclination from one surface to the other.

In a separate control procedure, subjects set the unseen paddle to match the inclina­tion of a 48 cm2 board covered with a pattern similar to those used in the experiment. The board was inclined randomly at 10° intervals between plus and minus 80° and had a full range of binocular and monocular depth cues. For each angle, subjects provided two sets of settings with four settings in each set. The results were fitted with a third-order polynomial for each subject and were used to calibrate the manual inclination settings in the main experiment.

Four subjects took part in the experiment, three males and one female, between the ages of 20 and 67 years. All subjects had normal or corrected-to-normal acuity and normal stereoscopic vision. All had participated in previous experiments involving stereoscopic surfaces.

2.2 Results 2.2.1 Horizontal-shear disparity. The mean adjusted inclinations of both the induction and the test surface for the four subjects for the horizontal-shear conditions are shown in figure 3. For each surface, results are shown for the horizontal and vertical orienta­tion of the boundary between the surfaces, as indicated on the abscissa, and also for when judgments were derived from the boundary region and from the region away from the boundary, as indicated at the top of the figure. The two directions of disparity are represented by the cross-hatched and black bars. The ordinate on the left side is

92 B J Pierce, I P Howard, C Feresin

> 40-o

eg 3 c/j

= 3

*° 20 a _o "5 g 1 io-

Induction surface outer region boundary

i Test surface

boundary outer region

1

h40g*

h30 !

20?

o 10 2

fZ23 positive disparity direction warn negative disparity direction

f-o

-10 hor vert hor vert nor vert hor vert

Boundary orientation

Figure 3. Perceived inclination of the surface with horizontal-shear disparity and of adjacent test surface with zero disparity, near the boundary and away from the boundary. The boundary was either horizontal (hor) or vertical (vert). Disparity was positive (ground surface) or negative (ceiling surface). Theoretically, the disparity corresponded to an inclination of 43.8°. Any apparent inclination of the test surface was in the opposite direction. Means and standard errors for four subjects. For details see text.

the unsigned mean of the transformed inclinations of the induction surface and that on the right is the mean of the transformed inclinations of the test surface relative to the induction surface. The sign of the perceived inclination of the test surface indicates whether it was in the same (positive) or the opposite (negative) direction to that of the induction surface.

Means of adjusted inclinations were computed for each subject by condition and submitted to a three-way analysis of variance (ANOVA) with region (induction outer, induction boundary, test boundary, and test outer), disparity direction (positive or negative), and boundary orientation (horizontal or vertical) serving as within-subject independent variables.(1) The statistical model for all ANOVAs in experiments 1 and 2 was a randomized blocks design with subjects as blocks. F-ratios were computed by using the subject by factor interaction as the error term (Keppel 1973, pages 457-462). The ANOVA for horizontal-shear conditions revealed a significant two-way interaction between region and direction of disparity (F3^9 = 5.05, p = 0.025). The main effects of both these factors were significant (Fh9 = 69.07, p = 0.001; and F^3 = 25.27, p = 0.015, respectively). The main effect of boundary orientation and all other inter­actions were not significant. Analysis of simple main effects and interactions for the induction-surface-only conditions revealed a significant interaction between region and boundary orientation (FU9 = 5.81, p = 0.039), and significant main effects of disparity direction (Flf3 = 40.78, p = 0.008) and boundary orientation (F1?3 = 10.75, p = 0.047). The simple main effect of region, and the three-way and the remaining two-way interactions were not significant.

(1) All ANOVAs in experiment 1 used both positive and negative perceived inclinations. Perceived top-farther inclinations for positive shear and rotation were signed positive and for negative shear and rotation were signed negative. Perceived top-nearer inclinations for positive shear and rotation were signed negative and for negative shear and rotation were signed positive. Thus, the dependent variable was perceived inclination adjusted for the direction of shear and rotation disparities.

Depth contrast for inclined and slanted surfaces 93

In all cases the induction surface appeared inclined in the direction expected from the sign of the disparity and apparent inclination was greater for positive disparities (ground surfaces) than for negative disparities (ceiling surfaces). Minimal inclination was perceived for zero-disparity test surfaces. Planned comparisons revealed a significant change in perceived inclination across the boundary between the two surfaces (.Flj9 = 128.69, p < 0.001). There were no significant changes in perceived slant over different regions of the induction or test surface.

In all horizontal-shear conditions, subjects could fuse all the elements of both surfaces and reported that the induction and test surfaces appeared as separate surfaces with a distinct boundary.

2.2.2 Vertical-shear disparity. The mean results for the four subjects for the vertical-shear-disparity conditions are shown in figure 4. All surfaces with vertical-shear disparity appeared inclined in a direction opposite to that of the horizontal-shear surfaces (induced effect). Thus positive disparities produced a ceiling surface and negative disparities a ground surface. A three-way ANOVA revealed a significant three-way interaction between region, direction of disparity, and boundary orientation (F3 9 = 28.59, p < 0.001). The main effect of region was significant (F3?9 = 29.69, p = 0.001), as were the interactions between region and direction (F3i9 = 22.11, p = 0.001), and region and orientation (F3f9 = 11.85, p = 0.002). Planned comparisons revealed a significant difference in perceived inclination between the boundary and outer regions of the induction surface CFlj9 = 48.67, p < 0.001) but no significant differences of inclination within the test surface or across the boundary between the two surfaces.

40-

5 30-

o 20-

10-1

0-4

Induction surface outer region boundary

Test surface boundary outer region

40 g-

h30 H

.~s*5i..J-0

20

o io 2

hor vert hor vert hor vert Boundary orientation

hor vert -10

[ZZ2 positive disparity direction ^m negative disparity direction

Figure 4. Perceived inclination of the surface with vertical-shear disparity and of adjacent test surface with zero disparity, near the boundary and away from the boundary. The boundary was either horizontal (hor) or vertical (vert). Positive vertical-shear disparity created a ceiling surface and negative shear a ground surface. Any apparent inclination of the test surface was in the same direction as the disparity surface. Means and standard errors for four subjects.

It can be seen from figure 4 that the perceived inclination of the induction surface tended to increase away from the boundary. This effect was moderated by both the direction of disparity and the orientation of the boundary. The induction and test surfaces appeared to have the same inclination in the boundary region regardless of

94 B J Pierce, I P Howard, C Feresin

boundary orientation or the direction of disparity. While the change was not significant, the perceived inclination of the test surface fell to near zero away from the boundary. This was true for all combinations of disparity direction and border orientation.

Subjects fused between one half and two thirds of the columns of the induction surface when the boundary was vertical but only about one third of the rows when the boundary was horizontal. Subjects perceived no clear boundary between the induction and test surfaces.

2.2.3 Rotation disparity. Figure 5 shows the inclination means for the rotation-disparity conditions. A three-way ANOVA revealed a significant two-way interaction between region and direction of disparity (i^?9 = 6.17, p = 0.014). Main effects of both these factors were significant (F3i9 = 30.23, p = 0.001; and i*|j3 = 30.32, p = 0.012, respec­tively). The main effect of boundary orientation was not significant, nor were interactions which included the orientation factor. Planned comparisons revealed significant changes in perceived inclination within the induction surface (F^9 = 42.69, p < 0.001) and across the boundary between the surfaces CF1?9 = 79.25, p < 0.001).

It can be seen from figure 5 that at the boundary region the induction surface appeared inclined in the direction of its horizontal-shear component but appeared frontal away from the boundary. The apparent inclination of the induction surface was greater for positive than for negative rotation disparity. The zero-disparity test surface appeared inclined in the opposite direction to the induction surface but only in the boundary region. Negative rotation of the induction surface induced more inclination of the zero-disparity test surface than positive rotation.

Subjects fused between one half and two thirds of the columns of the induction surface when the boundary was vertical but only about one third of the rows when the boundary was horizontal. In all rotation-disparity conditions, the induction and test surfaces appeared to lie in separate planes with a distinct boundary.

.40

30 H

20

10

o-f

Induction surface outer region boundary

Test surface boundary outer region

h40

30

20

*f 1

*

- 4 - 0 «»

-10 8

hor vert hor vert hor vert Boundary orientation

hor vert -20

EZ3 positive disparity direction ^m negative disparity direction

Figure 5. Perceived inclination of the surface with rotation disparity and of the adjacent test surface with zero disparity, near the boundary and away from the boundary. The boundary was either horizontal (hor) or vertical (vert). Rotation disparity was positive or negative. The disparity surface appeared inclined in the direction of the horizontal component of the rotation disparity and to a greater extent for a ground plane than a ceiling plane. Any apparent inclination of the test surface was in the opposite direction. Means and standard errors for four subjects.

Depth contrast for inclined and slanted surfaces 95

3 Experiment 2: Size-disparity displays In this experiment we examined depth contrast produced by induction surfaces with horizontal-size disparity, vertical-size disparity, and overall-size disparity in an adjacent zero-disparity test surface.

3.1 Method 3.1.1 Stimuli and apparatus. The apparatus was the same as in experiment 1, as were the texture, sizes, boundary orientations, luminance values, and viewing distances. A 4% horizontal, vertical, or overall-size disparity was introduced into the induction surface by a corresponding expansion of the image presented to either the left or the right eye. The centre of the expansion was the vertical or horizontal midline of the total 60 deg by 60 deg display area. A 4% horizontal-size disparity corresponds to a slant of 28.2° with respect to the frontal plane.

3.1.2 Procedure. A block of 12 trials consisted of [three types of disparity (vertical-size disparity, horizontal-size disparity, and overall-size disparity) x two disparity directions (positive, right eye magnified; negative, left eye magnified) x two boundary orienta­tions (horizontal and vertical)]. Each subject performed four blocks of trials over two sessions. Counterbalancing controls were the same as in experiment 1.

Four subjects took part in the experiment, three males and one female, between the ages of 22 and 34 years. None of the subjects took part in experiment 1, but all had participated in previous experiments involving slant perception. All subjects had normal or corrected-to-normal acuity and normal stereoscopic vision. Subjects were instructed to adjust the unseen paddle to match the slant of each surface, first in the boundary region and then in the centre of each surface away from the boundary. Subjects looked at the region they were judging. Next, subjects maintained their gaze on the boundary between the induction and test patterns and reported the number of columns of the horizontal-size-disparity patterns they could fuse or the number of rows of the vertical and overall-size-disparity patterns they could fuse. Last, subjects indicated whether there was a distinct or gradual transition of slant from one surface to the other.

In a control procedure similar to that used in experiment 1, subjects set the unseen paddle to match the slant of a 48 cm2 board covered with a pattern similar to those used in the experiment. The board was slanted randomly at 10° intervals between plus and minus 60° and had a full range of binocular and monocular depth cues. For each angle, subjects provided two sets of settings with four settings in each set. The results were fitted with a third-order polynomial function for each subject and were used to calibrate the manual slant settings in the main experiment.

3.2 Results 3.2.1 Horizontal-size disparity. The mean perceived slants for the four subjects for the horizontal-size disparity conditions are shown in figure 6. Labels at the top indicate the region of the induction or test surface being judged. Labels beneath the figure indicate whether the boundary between the induction and test surfaces was vertical or horizontal. The direction of disparity is indicated by cross-hatched and black bars. The ordinate on the left is the unsigned mean of the transformed perceived slant of the induction surface and that on the right side is the perceived slant of the test surface relative to the induction surface, signed positive when the test surface appeared to slant in the same direction as the induction surface.

96 B J Pierce, I P Howard, C Feresin

40

30

20 H

S io

Induction surface outer region boundary

Test surface boundary outer region

V77\ positive disparity direction mm negative disparity direction

40

h30

20

o 10 *

h - 1 0

hor vert hor vert hor vert Boundary orientation

hor vert

Figure 6. Perceived slant of the surface with horizontal-size disparity and of adjacent test surface with zero disparity, near the boundary and away from the boundary. The boundary was either horizontal (hor) or vertical (vert). Disparity was positive (right-eye magnified) or negative (left-eye magnified). The test surface appeared to slant in the opposite direction to the disparity surface. Means and standard errors for four subjects.

Means of adjusted perceived slants for each subject were submitted to a three-way ANOVA(2) with region (induction outer, induction boundary, test boundary, and test outer), direction of disparity (positive and negative), and boundary orientation (horizontal or vertical) serving as within-subject independent variables. There was a significant three-way interaction (F3?9 = 6.11, p = 0.015). The main effect of region and the region by boundary orientation interaction were significant (F3?9 = 91.61, p = 0.001; and F^9 = 5.78, p = 0.017, respectively). The main effects of disparity direction and boundary orientation, and the three-way and all other two-way interactions were not significant.

All surfaces with horizontal-size disparity appeared to slant in the direction appro­priate to the sign of the disparity. It can be seen from figure 6 that the perceived slant of the test surface was opposite to that of the induction surface. Planned comparisons revealed significant differences between the perceived slants of the induction and test surfaces at the boundary region (FU9 = 167.6, p < 0.001). This is a slant-contrast effect. There were no significant changes in perceived slant over either the induction or the test surface. The induction surface appeared to slant more when the boundary was horizontal than when it was vertical. This effect was moderated by both direction of disparity and region.

Subjects fused all columns and rows of both surfaces in all horizontal-size-disparity conditions. The induction and test surfaces appeared on distinct planes with a well-defined boundary.

(2)A11 ANOVAs in experiment 2 used both positive and negative perceived slants. Perceived right-side-father slants for positive size disparity were signed positive and for negative dispar­ity were signed negative. Conversely, perceived right-side-nearer slants for positive size disparity were signed negative, and for negative disparity were signed positive. Thus, the dependent vari­able was perceived slant adjusted for the direction of size disparity.

Depth contrast for inclined and slanted surfaces 97

40-1

30

•a 20 H

V. 10

o-4

Induction surface outer region boundary

Test surface boundary outer region

T Z r " " . cJ=|_. |yJ-» •

U77\ positive disparity direction M negative disparity direction

40 p

30

20 ~

io a

J-o

hor vert hor vert hor vert Boundary orientation

hor vert -10

Figure 7. Perceived slant of the surface with vertical-size disparity and of the adjacent test surface with zero disparity, near the boundary and away from the boundary The boundary was either horizontal (hor) or vertical (vert). Disparity was positive or negative. Means and standard errors for four subjects.

3.2.2 Vertical-size disparity. The mean perceived slants for vertical-size-disparity con­ditions are shown in figure 7. A three-way ANOVA revealed significant main effects of region (F^9 = 13.11, p = 0.001) and boundary orientation (FU3 = 40.4, p = 0.008). The main effect of direction of disparity and all interaction effects were not significant.

All surfaces with vertical-size disparity appeared to slant in a direction opposite to a surface with the same sign of horizontal-size disparity (induced effect). Generally, slants were greater when the boundary was vertical rather than horizontal. In all conditions subjects reported minimal slant of the test surface that was generally in the same direction as that of the induction surface. Planned comparisons revealed significant differences between the perceived slants of the induction and test surfaces near the boundary (Fly9 = 30.93, p < 0.001) but the perceived slant of neither surface changed away from the boundary.

Subjects fused all rows of the induction surface when the boundary was horizontal and all but three of four rows when the boundary was vertical. Subjects perceived no clear boundary between the induction and test surfaces.

3.2.3 Overall-size disparity. Figure 8 shows the mean perceived slants for overall-size-disparity conditions. A three-way ANOVA revealed a significant two-way interaction between region and boundary orientation (F3^9 = 16.20, p = 0.001). The main effect of region was significant (F3i9 = 32.72, p = 0.001). The main effects of boundary orientation and direction of disparity and all other interactions were not significant.

All surfaces with overall-size disparity appeared to slant in the direction of the horizontal component of the disparity. This slant of the induction surface was reduced away from the boundary, especially when the boundary was vertical. The zero-disparity test surface appeared to slant in the opposite direction. Planned com­parisons revealed significant changes of slant between the induction and test surfaces near the boundary (FU9 = 78.82, p < 0.001), between the boundary and outer regions of the induction surface (F^9 = 12.09, p = 0.007), but not between regions of the test surface.

98 B J Pierce, I P Howard, C Feresin

40

30

a 20

* 10

.£ 0-

Induction surface outer region boundary

Test surface boundary outer region

o H

h-io

hor vert hor vert hor vert Boundary orientation

hor vert -20

T77i\ positive disparity direction ^m negative disparity direction

Figure 8. Perceived slant of the surface with overall-size disparity and of adjacent test surface with zero disparity, near the boundary and away from the boundary. The boundary was either horizontal (hor) or vertical (vert). The direction of disparity was positive or negative. The disparity surface appeared to slant in the direction of the horizontal component of the overall-size disparity. The test surface appeared to slant in the opposite direction. Means and standard errors for four subjects.

Subjects fused all rows of the induction surface when the boundary was horizontal and all but two or three rows when the boundary was vertical. The induction and test surfaces appeared to lie in different planes with a sharp boundary.

4 Discussion 4.1 Anisotropy in perceived inclination The surface with positive horizontal-shear disparity appeared as a ground surface and that with a negative disparity appeared as a ceiling surface. On average, the ground surface appeared inclined from the vertical between 12° and 15° more than the ceiling surface.

Two factors could contribute this asymmetry. First, corresponding vertical meridians are relatively inclined inwards by about 2°, which causes the vertical horopter to be inclined top away by an amount that varies with viewing distance (see Howard and Rogers 1995). On the assumption that the vertical horopter is the locus of apparent vertical, vertical lines should appear inclined top forward. However, Cogan (1979) found that a line was set close to true vertical in the median plane even though it had to be inclined 30° to stimulate corresponding vertical meridians. Thus the inclination of the vertical horopter does not displace the apparent vertical. But the inclination of the vertical horopter does mean that the disparity produced by a ground surface at a given inclination has a smaller Weber fraction than a disparity produced by a ceiling surface with the same inclination. This could account for why ground surfaces are more accurately perceived. We perceive more ground surfaces than ceiling surfaces and it has been suggested that the inclination of the vertical horopter is an adaptation to this ecological fact (Cooper and Pettigrew 1979; Krekling and Blika 1983).

The second factor that could contribute to directional asymmetry in perceived inclination is the fact that crossed disparities are detected more accurately than uncrossed disparities (Mustillo 1985), combined with the fact that disparities in the

Depth contrast for inclined and slanted surfaces 99

lower visual field are more precisely perceived than those in the upper visual field (Manning et al 1992). A ground plane presents the more accurately perceived sign of disparity in the better, lower half of the visual field.

The same asymmetry shows in the inclinations produced by rotation disparity, especially near the boundary with the zero-disparity surface (figure 5). Positive rotation disparity produces stronger inclination because it contains a horizontal component which is produced by a ground surface. Negative rotation disparity produces weaker inclination because it contains horizontal disparity produced by a ceiling surface.

Surfaces with vertical-shear disparities produced the same asymmetry; ground surfaces produced by negative disparity appeared more inclined than ceiling surfaces produced by positive disparity (figure 4).

The interaction between the direction of disparity and surface region for vertical-shear and rotation disparities reflects a decrease in the directional asymmetry as perceived inclination declined, obviously reaching zero as perceived inclination fell to zero.

4.2 Depth contrast from shear disparities The ceiling surface with horizontal-shear disparity produced a hint of depth contrast in the zero-disparity surface but for neither the ceiling nor the floor surface was the effect significant (figure 3). This was true when the boundary between the surfaces was horizontal or vertical.

Each surface with vertical-shear disparity appeared inclined in the opposite direction to a surface with the same sign of horizontal-shear disparity (figure 4). This is the induced effect in the domain of vertical-shear disparity reported by Howard and Kaneko (1994). However, the induced effect was very small near the boundary between the surfaces. The induced effect can be explained if it is assumed that inclination is coded in terms of deformation disparity—the difference between horizontal-shear disparity detected locally and the mean vertical-shear disparity extracted over a large area. In the boundary region the estimate of vertical-shear disparity is derived from both surfaces. We have shown previously that an equal mixture of disparate and nondisparate texture elements does not produce an induced effect (Kaneko and Howard 1997a; Howard and Pierce 1998). It seems that greater weight is given to elements with zero disparity. The small induced effect spread into the zero-disparity surface because the regional measure of vertical disparity was applied to both surfaces. Since the two surfaces had the same horizontal-shear disparity there was no basis for perceiving distinct inclina­tions in the boundary region. Away from the boundary the zero-disparity surface appeared in the frontal plane because the mean vertical disparity in that region approached zero. The induction effect in the boundary region of the zero-disparity surface is opposite in direction to depth contrast. Note also that the greater the induced effect in the disparity surface, the greater the spread of the effect into the zero-disparity surface. This is the opposite of what happens in depth contrast but is consistent with the idea of averaging of vertical disparities over the two surfaces.

Away from the boundary the surface with rotation disparity appeared almost frontal (figure 5). In previous experiments we have shown that a large surface with rotation disparity appears to lie in the frontal plane, which is explained if one assumes that inclination is coded in terms of the difference between local horizontal-shear disparity and global vertical-shear disparity (Howard and Kaneko 1994). Near the boundary the surface with rotation disparity appeared inclined in a direction appropriate to the horizontal-shear component of its disparity. This is explained by the fact that, near the boundary, mean vertical disparity is reduced because it is derived partly from the zero-disparity surface. The locally derived horizontal disparity of the disparity surface was assessed in terms of this reduced vertical disparity, and the surface therefore appeared inclined.

100 B J Pierce, I P Howard, C Feresin

The zero-disparity surface adjacent to the surface with rotation disparity appeared inclined at the boundary region, especially when the disparity surface contained a negative rotation disparity (figure 5). This inclination of the zero-disparity surface was in the opposite direction to that of the disparity surface and, in that sense, qualifies as a contrast effect. However, evidence presented below and previously (Pierce and Howard 1997) shows that contrast effects, when they occur, are not spatially localized. Instead, at least part of the present effect can be accounted for in terms of deformation disparity. The mean vertical-shear disparity extracted from both surfaces in the boundary region is subtracted from the local zero horizontal disparity of the zero-disparity surface to produce a deformation disparity with a predominant vertical-disparity component, which accounts for the apparent inclination of the zero-disparity test surface in the opposite direction to that of the disparity surface. This theory also predicts the larger effect produced by the disparity surface with negative disparity. Away from the boundary the mean vertical disparity was reduced and the zero-disparity surface appeared frontal.

Cyclovergence probably contributes to these effects of rotation disparity. A surface with rotation disparity evokes cyclovergence which would cancel image disparity com­pletely if the response had a gain of 1. However, surfaces with rotation disparity appear frontal even for people with a low gain of cyclovergence, so we conclude that there is also a neural process which extracts deformation disparity (Howard and Kaneko 1994). Ogle (1944) claimed that Werner's inclination-contrast effect is due to cyclovergence but he measured cyclovergence by the perceived inclination of a vertical line, which is a flawed procedure (see Howard et al 1993).

In the boundary region, the surface with rotation disparity and the zero-disparity test surface appeared to lie on distinct depth planes with a sharp boundary between them. This confirms the general rule that adjacent surfaces with distinct components of horizontal disparity produce a sharp discontinuity of depth, unlike surfaces which differ only in vertical-shear disparities, which produce a gradual boundary (Kaneko and Howard 1997a; Howard and Pierce 1998). In natural scenes, horizontal-shear disparities are produced locally by distinct inclined surfaces, and therefore need to be detected locally. Vertical-shear disparities occur as a component of global rotation disparity caused by torsional misalignment of the eyes. Global vertical-shear disparity is thus the most reliable indicator that the eyes are misaligned. It provides the effective stimulus for cyclovergence and a signal for correction of residual rotational misalignment of the images. For these purposes, a global estimate derived from all or a large part of the binocular visual field provides the most reliable measure and is all that is required.

4.3 Depth contrast from size disparities All surfaces with horizontal-size disparity appeared to slant in the direction appropriate to the disparity. In all cases, the adjacent zero-disparity test surface appeared to slant in the opposite direction by about 2° to 7°, both near and away from the boundary. This is a depth-contrast effect which cannot be explained in terms of deformation disparity in the domain of size disparity, since any effect due to that cause would be in the opposite direction to the effect we found.

All surfaces with vertical-size disparity appeared to slant in the opposite direction to a surface with the same sign of horizontal-size disparity. This is Ogle's induced effect. The effect was evident over the whole disparity surface and, although subjects did not see a clear boundary between the disparity surface and the adjacent zero-disparity surface, the induced effect did not spread to a significant degree into the zero-disparity test surface. By comparison, the induced effect produced by vertical-shear disparity was reduced near the boundary but did spread into the zero-disparity test surface. This suggests that vertical-size disparity is averaged over smaller regions than vertical-shear disparity.

Depth contrast for inclined and slanted surfaces 101

Other evidence from this laboratory supports this conclusion. Kaneko and Howard found that vertical-shear disparity is averaged over the whole binocular field (1997a), and that vertical-size disparity is averaged over regions subtending about 20° (1997b). The size-disparity induced effect can be explained if it is assumed that surface slant is coded in terms of deformation disparity—the difference between horizontal-size disparity detected locally and the mean vertical-size disparity extracted over a larger area, but not over an area as large as that over which vertical-shear disparity is averaged.

Surfaces with overall-size disparity appeared to slant in the direction of the horizontal-size-disparity component. In this respect overall-size disparity produces an effect equivalent to that produced by rotation disparity. Away from the boundary the perceived slant fell to near zero when the boundary was vertical and by about 40% when it was horizontal. In previous experiments we have shown that a large surface with an overall-size disparity of 4 deg appears to slant about 7° in the direction of the horizontal component of disparity when presented alone (Kaneko and Howard 1996; Pierce and Howard 1997). This effect is explained if one assumes that slant is coded in terms of the difference between local horizontal-size disparity and global vertical-size disparity (deformation disparity) with more weight given to horizontal disparity. The surface with overall-size disparity adjacent to the zero-disparity surface appeared slanted about 13° when the boundary was vertical and about 23° when it was horizontal. With an adjacent zero-disparity surface, space-averaged vertical-size disparity is reduced because it is derived partly from the zero-disparity test surface. This reduces deformation disparity, and the surface near the boundary therefore appears slanted more than the part away from the boundary. Away from the boundary the display with overall-size disparity appeared almost frontal when the boundary was vertical and with about 10° less slant than in the boundary region when the boundary was horizontal. Away from the boundary, vertical disparity is extracted mainly from a region containing vertical disparity. This reduces deformation disparity and the perceived slant of the surface.

The zero-disparity surface adjacent to the surface with overall-size disparity appeared slanted between 5° and 10° near the boundary region (figure 8). This slant was in the opposite direction to that of the disparity surface, and in that sense qualifies as depth contrast. The mean vertical-size disparity extracted from the boundary region when subtracted from the local zero horizontal-size disparity of the test surface produces a deformation disparity which accounts for the apparent slant of the zero-disparity surface opposite to that of the disparity surface. Away from the boundary the apparent slant was reduced, especially when the boundary was vertical. This is what one would predict from the change in deformation disparity, because away from the boundary the mean vertical disparity is reduced. Depth contrast may have contributed to these effects because horizontal-size disparity does produce depth contrast in a zero-disparity surface. A contribution of depth contrast would explain why the effects on a zero-disparity surface produced by overall-size disparity are greater than those produced by rotation disparity.

In previous papers, we also found depth contrast induced by surfaces slanted about a vertical axis but not by surfaces inclined about a horizontal axis (Pierce and Howard 1997; Howard and Pierce 1998). Rogers used random-dot stereograms and reported slant contrasts of over 70% for slant and about half this magnitude for inclination (see Howard and Rogers 1995, page 477). However, he used induction surfaces inclined at only a small angle. Graham and Rogers (1982) found that the percentage of depth contrast declined from about 40% to about 20% as the inclination of the induction surface increased from 2.5 to 10 min. Another reason for the difference in results may be that they used two flanking induction surfaces rather than a single induction surface. Another factor may be a difference in time of exposure of the stimulus. Contrast effects

102 B J Pierce, I P Howard, C Feresin

are known to decline with increasing duration of exposure (Kumar and Glaser 1993). We do not deny that depth contrast produced by inclined surfaces occurs in some circumstances. In any case, we agree with Rogers that there is a marked anisotropy of depth contrast. This anisotropy cannot be due to whether the surfaces are abutted along a vertical boundary or along a horizontal boundary because the induction effects we obtained were largely independent of boundary orientation.

We propose the following apparently paradoxical principle. Surfaces with a given disparity in which depth is not well detected produce more depth contrast than surfaces in which depth is accurately perceived. Thus the large inclination produced by horizontal-shear disparity induced little inclination contrast (figure 3) and more contrast was induced by ceiling surfaces, in which inclination was underestimated, than by ground surfaces, in which inclination was more accurately perceived. Also, the large slant produced by horizontal-size disparity induced no more contrast than the smaller slant produced by overall-size disparity (figures 6 and 8). Other lines of evidence support this conclusion. First, Werner (1938), in the first experiments on depth contrast, noticed that contrast was strongest when the depth in the induction stimulus was not perceived. Also, depth contrast is strong during the initial period of stimulus exposure when the depth of the induction stimulus is not fully apparent (Kumar and Glaser 1993). Van Ee (1995) obtained depth contrast both from shear disparity and from size disparity when the inclination or slant of the induction surface was not evident. Second, the visual system is less sensitive to slant about a vertical axis, which produces strong contrast, than to inclination about a horizontal axis, which produces weak or no contrast (Rogers and Graham 1983; Gillam et al 1984; Mitchison and McKee 1990; Cagenello and Rogers 1993).

We propose the following explanation of the above principle. Relative-disparity signals are processed even though the absolute inclinations or slants of surfaces in terms of bodycentric coordinates are weakly registered. A relative-disparity signal indicates only that two surfaces have different inclinations or slants and, in the absence of information to the contrary, one tends to partition the relative disparity between neighbouring surfaces, so that an inclined or slanted surface appears more frontal than it is and a frontal surface appears inclined or slanted.

The way depth is partitioned between two surfaces probably depends on their relative sizes. If one of the surfaces is larger than the other, as in Werner's original displays and in the display used by Graham and Rogers (1982), this surface serves as a frame of reference and is referred to the frontal plane and the slant of the smaller surface is judged relative to this frame. By analogy, in induced visual motion, a large moving display appears stationary and a small stationary display appears to move in the opposite direction (Duncker 1929). The actual dispositions of two neighbouring surfaces are appreciated only after one has had time to assess them in a headcentric coordinate system. Some depth contrast persists with a slanted surface so that, for some reason, the independent slants of neighbouring surfaces about a vertical axis must be most difficult to assess than the independent inclinations of neighbouring surfaces about a horizontal axis. For control of posture and the perception of the vertical, accuracy in the perception of inclination about a horizontal axis is more important than accuracy in the perception of slant about a vertical axis. Some depth contrast also persists when one surface is larger than the other because of a frame-of-reference effect, but that question needs further study.

Acknowledgement. This work was part of DCIEM contract W7711-4-7217/01-XSE.

References Anstis S M, 1975 "What does visual perception tell us about visual coding", in Handbook of

Psychobiology Eds C Blakemore, M S Gazzaniga (New York: Academic Press) pp 269 - 323 Anstis S M, Howard I P, Rogers B, 1978 "A Craik-Cornsweet illusion for visual depth" Vision

Research 18 213 -217

Depth contrast for inclined and slanted surfaces 103

Brookes A, Stevens K A, 1989 "The analogy between stereo depth and brightness" Perception 18 601-614

Cagenello R, Rogers B R, 1993 "Anisotropics in the perception of stereoscopic surfaces: the role of orientation disparity" Vision Research 33 2189-2201

Cogan A I, 1979 "The relationship between the apparent vertical and the vertical horopter" Vision Research 19 655-665

Cooper M L, Pettigrew J D, 1979 "A neurophysiological determination of the vertical horopter in the cat and owl" Journal of Comparative Neurology 184 1-26

Duncker K, 1929 "Uber induzierte Bewegung" Psychologische Forschung 22 180-259 Ee R van, 1995 Stability of Binocular Depth Perception PhD thesis, Helmholtz Institute, University

of Utrecht, Utrecht, Netherlands Ee R van, Erkelens C J, 1996 "Temporal aspects of binocular slant perception" Vision Research

36 45-51 Erkelens C J, Collewijn H, 1985 "Motion perception during dichoptic viewing of moving random-

dot stereograms" Vision Research 25 583 - 588 Gillam B, Flagg T, Finlay D, 1984 "Evidence for disparity change as the primary stimulus for

stereoscopic processing" Perception & Psychophysics 36 559 - 564 Gogel W C, 1965 "Equidistance tendency and its consequences" Psychological Bulletin 64 153 -163 Graham M E, Rogers B J, 1982 "Simultaneous and successive contrast effects in the perception

of depth from motion-parallax and stereoscopic information" Perception 11 247-262 Howard I P, Kaneko H, 1994 "Relative shear disparities and the perception of surface inclination"

Vision Research 34 2505-2517 Howard I P, Ohmi M, Sun L, 1993 "Cyclovergence: a comparison of objective and psychophysical

measurements" Experimental Brain Research 97 349 - 355 Howard I P, Pierce B, 1998 "Types of shear disparity and the perception of surface inclination"

Perception 27 (in press) Howard I P, Rogers B J, 1995 Binocular Vision and Stereopsis (New York: Oxford University Press) Julesz B, 1971 Foundations of Cyclopean Perception (Chicago, IL: University of Chicago Press) Kaneko H, Howard I P, 1996 "Relative size disparities and the perception of surface slant" Vision

Research 36 1919-1930 Kaneko H, Howard I P, 1997a "Spatial properties of shear disparity processing" Vision Research

37 315-324 Kaneko H, Howard IP, 1997b "Spatial limitation of vertical-size disparity processing" Vision Research

37 2871-2878 Keppel G K, 1973 Design and Analysis: a Researcher's Handbook (Englewood Cliffs, NJ: Prentice-

Hall) Krekling S, Blika S, 1983 "Development of the tilted vertical horopter" Perception & Psychophysics

34 491-493 Kumar T, Glaser D A, 1993 "Temporal aspects of depth contrast" Vision Research 33 947 - 957 Lunn P D, Morgan M J, 1995 " The analogy between stereo depth and brightness': a reexamination"

Perception 24 901 - 904 Manning M L, Finlay D C, Dewis S A M , Dunlop D B, 1992 "Detection duration thresholds and

evoked potential measures of stereosensitivity" Documenta Ophthalmologica 79 161 -175 Mitchison G J, McKee S P, 1990 "Mechanisms underlying the anisotropy of stereoscopic tilt

perception" Vision Research 30 1781 -1791 Mustillo P, 1985 "Binocular mechanisms mediating crossed and uncrossed stereopsis" Psychological

Bulletin 97 187-201 Ogle K N, 1938 "Induced size effect I. A new phenomenon in binocular space-perception asso­

ciated with the relative sizes of the images of the two eyes" Archives of Ophthalmology 20 604-623

Ogle K N, 1944 "The binocular depth contrast phenomenon" American Journal of Psychology 59 111-123

Pierce B, Howard I P, 1997 "Types of size disparity and the perception of surface slant" Perception 26 1503-1518

Regan D, Erkelens C J, Collewijn H, 1986 "Necessary conditions for the perception of motion in depth" Investigative Ophthalmology & Visual Science 27 584 - 597

Rogers B J, Graham M E, 1983 "Anisotropics in the perception of three-dimensional surfaces" Science 221 1409-1411

Werner H, 1937 "Dynamics in binocular depth perception" Psychological Monographs 49 1 - 120 Werner H, 1938 "Binocular depth contrast and the conditions of the binocular field" American

Journal of Psychology 51 489-497