Introduction It is often important to accurately judge the slant of surfaces in our nearby environment. Whether placing our foot on the ground when we walk or climb stairs, or our fingers on an object when we grasp it and place it elsewhere, the interaction always involves making contact with surfaces. In order to interact successfully, we need to know the orientation of these surfaces. We have many ways to judge a surface’s orientation, including ones based on texture gradients, binocular disparity and motion parallax. 1995 1994 2003 1 1950 1981 1996 2004 1998a 2003 Fig. 1 The deformation of the regular texture of a chess board provides a profound impression that the surface is slanted relative to the plane of the 2D picture  1997 1989 1979 1982 1990 1991 2001 2003 2003 1995 2002 2004 2005 2007 2004 1998b 1993 1994 2005 2005 Methods Participants Five people, four of whom were male, participated in the experiment. All participants gave their informed consent prior to their inclusion in the study. The experiment was part of an ongoing research program that was approved by the local ethics committee. All participants were right-handed, had normal or corrected-to-normal vision and good binocular vision (stereo acuity <40 arcseconds). Experimental setup 2 cp x325 1987 3 3 −2 Fig. 2 right grey ring  Fig. 3 left right left panel right panel grey ring dark grey  optotrak 3020 northern digital inc. Procedure Experiments were performed in a completely dark room. The dark environment and the low intensity of the stimuli ensured that there was no visible external reference frame. Participants were instructed to place the probe at the indicated target position on the surface. They were to start moving as soon as the target was visible. Stimuli were shown for 2.5 s. All movements were completed well within this interval. In order to avoid dark adaptation a bright lamp was turned on for 5 s immediately after each trial. During this period participants placed the probe at the starting position, 50 cm to the right of the midline of the screen. Then the light was turned off for about 5 s, during which time the experimenter adjusted the orientation of the screen in preparation for the next trial. 4 4 2 3 3 Fig. 4 continuous lines dashed lines solid disks open disks  We used four conditions in which different combinations of the available cues were presented. The choice of conditions will become clear when we describe the data analysis. We chose three conditions with which we could calculate the five parameters of our model, and one condition to test one of our assumptions. In the ‘binocular’ condition viewing was binocular and head-free in both conflict and consistent trials. In this condition all cues to slant perception were available. In the ‘monocular’ condition the conflict and consistent trials were both presented monocularly and head-free. Stimuli were viewed with the left or right eye in random order. No binocular cues were available, but all other cues were present. In the ‘biteboard’ condition the head was fixed in combination with monocular viewing. The biteboards were made individually with an impression of the participant’s teeth. The biteboard severely limits head movements, removing information from motion parallax. In the consistent trials of the ‘mixed’ condition the screen was viewed monocularly (75% of all trials), but in the conflict trials (25%) it was viewed binocularly. This condition was included to evaluate whether participants adapt their strategy at the level of a session rather than per trial. In the ‘binocular’ condition binocular information was always reliable, so participants could have learnt to use this cue. In the ‘mixed’ condition, in contrast, binocular cues were absent in the majority of trials, so participants could have learnt to use texture or motion parallax. Note that the 25% binocular, conflict trials in the ‘mixed’ condition are identical to the conflict trials in the ‘binocular condition’, whereas the 75% consistent trials are identical to the consistent trials in the ‘monocular’ condition. Thus the ‘mixed’ condition serves as a control condition to test whether the weight given to the cues stays about the same under changing viewing conditions on other trials. Analysis 5 5 Fig. 5 upper left panel curved thin line straight thick line upper right panel dark-grey lower panels  6 Fig. 6 left panel right panel  1995 i 1 \documentclass[12pt]{minimal}
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$$ {\text{S }} = {\text{ }}\frac{{{\sum\limits_i {w_{i} {\text{ s}}_{i} {\text{ }}} }}} {{{\sum\limits_i {w_{i} {\text{ }}} }}};\quad w_{i} {\text{ }} \propto \frac{1} {{\sigma ^{2}_{i} }}, $$\end{document} w i i b, m, t, r, p w i w i w i 2 \documentclass[12pt]{minimal}
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$$ s_{P} = c $$\end{document} 3 \documentclass[12pt]{minimal}
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$$ s_{i} = s $$\end{document} 4 4 \documentclass[12pt]{minimal}
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$$ s_{T} = 10^\circ - s $$\end{document} 5 \documentclass[12pt]{minimal}
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$$ S_{{{\text{consistent}}}} = \frac{{c\;w_{P} + s\;w_{T} + s{\sum\limits_{i \ne (P,T)} {w_{i} } }}} {{{\sum\limits_i {w_{i} } }}} $$\end{document} 6 \documentclass[12pt]{minimal}
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$$ S_{{{\text{conflict}}}} = \frac{{c\;w_{P} + (10^\circ - s)\;w_{T} + s{\sum\limits_{i \ne (P,T)} {w_{i} } }}} {{{\sum\limits_i {w_{i} } }}} $$\end{document} 5 6 6 7 \documentclass[12pt]{minimal}
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$$ \beta _{{{\text{consistent}}}} = \frac{{\partial S_{{{\text{consistent}}}} }} {{\partial s}} = \frac{{\;w_{T} + {\sum\limits_{i \ne (P,T)} {w_{i} } }}} {{{\sum\limits_i {w_{i} } }}} $$\end{document} 8 \documentclass[12pt]{minimal}
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$$ \beta _{{{\text{conflict}}}} = \frac{{\partial S_{{{\text{conflict}}}} }} {{\partial s}} = \frac{{\; - w_{T} + {\sum\limits_{i \ne (P,T)} {w_{i} } }}} {{{\sum\limits_i {w_{i} } }}} $$\end{document} 7 8 9 \documentclass[12pt]{minimal}
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$$ \frac{{\; - w_{T} + {\sum\limits_{i \ne (P,T)} {w_{i} } }}} {{w_{T} + {\sum\limits_{i \ne (P,T)} {w_{i} } }}} = \frac{{\beta _{{{\text{conflict}}}} }} {{\beta _{{{\text{consistent}}}} }} $$\end{document} 10 \documentclass[12pt]{minimal}
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$$ \frac{{w_{P} }} {{w_{T} }} = \frac{{2\beta _{{{\text{consistent}}}} - 2}} {{\beta _{{{\text{conflict}}}} - \beta _{{{\text{consistent}}}} }} $$\end{document} 9 10 11 \documentclass[12pt]{minimal}
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$$ {\left( {\frac{{\beta _{{{\text{conflict}}}} }} {{\beta _{{{\text{consistent}}}} }}} \right)}_{{{\text{binocular}}}} = \frac{{\;w_{B} + w_{M} + w_{R} - w_{T} }} {{w_{B} + w_{M} + w_{R} + w_{T} }} $$\end{document} w B 12 \documentclass[12pt]{minimal}
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$$ {\left( {\frac{{\beta _{{{\text{conflict}}}} }} {{\beta _{{{\text{consistent}}}} }}} \right)}_{{{\text{monocular}}}} = \frac{{\;w_{M} + w_{R} - w_{T} }} {{w_{M} + w_{R} + w_{T} }} $$\end{document} 13 \documentclass[12pt]{minimal}
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$$ {\left( {\frac{{\beta _{{{\text{conflict}}}} }} {{\beta _{{{\text{consistent}}}} }}} \right)}_{{{\text{biteboard}}}} = \frac{{\;w_{R} - w_{T} }} {{w_{R} + w_{T} }} $$\end{document} 14 \documentclass[12pt]{minimal}
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$$ w_{B} + w_{T} + w_{M} + w_{R} + w_{P} = 1 $$\end{document} w P w T 10 w P w T 11 14 w B w T w M w R conflict consistent Results methods Conditions 7 P 7 P 7 P 10 14 Fig. 7 open symbols closed symbols error bars grey lines black lines lower right corner Asterisks P P  consistent P consistent conflict P conflict Cue weights 8 Fig. 8 horizontal axis 10 14  Head movements Head movements were measured in the monocular viewing condition for three of the participants. Participants move their head considerably when placing the cylinder: EB moved on average 103 mm, JG 53 mm and DdG 37 mm in the lateral direction. Interestingly, the head movement only started just before the arm movement. Shortly before (100 ms) the onset of arm movement (when the probe was 10 mm from the starting position), the head had only moved 10 mm (EB), 4 mm (JG) or 6 mm (DdG). Thus the information from motion parallax is mainly picked up during the arm movement. None of the participants were aware of having made head movements. Discussion We used a physically rotatable screen as a surface. The projected stimulus was viewed in a completely dark environment within a space in which objects are normally manipulated. In our analysis systematic deformations that affect a single cue (like depth compression resulting from an erroneous depth estimate) were not considered. Moreover, we assume that the cue weights are the same in all conditions, so that their contributions to the percept only depend on which cues are available. We determined the contributions of binocular disparity, texture cues, motion parallax, a rest category and a prior. Under these conditions and based on these assumptions we conclude that participants mainly relied on binocular information (between 50 and 90%). Texture cues contributed between 2 and 18% to the estimated slant. Motion parallax contributed up to 9%. The prior contributed between 6 and 23%. Residual cues may account for up to 9%. 8 7 1997 2005 7 2004 2000 1995 2002 2007 2003 1998 1999 2003