We study whether bisection in visual grasp space (the region over which eye and hand can work together to grasp or touch objects) depends on fixation or on the method of judgment employed (the task). We determined observer bias and sensitivity for bisection judgments (in a fronto-parallel plane as well as along contours slanted in depth). Significant biases were found that varied across observers both qualitatively and quantitatively. These biases were stable for a given individual (across a year between data collection intervals) and across tasks (method of adjustment vs. forced-choice). When observers maintained fixation (on an endpoint or in the neighborhood of the bisection point), fixation location had a small but significant effect on bias, although those effects were small compared with bisection uncertainty. We conclude that bisection judgments differ significantly between fixations, but that the effect of fixation location on bisection is not large enough to be detected reliably by the observer moving his or her eyes during a judgment.

*visual grasp space*, errors in visual or motor estimates can be most punishing and, at the same time, the visual and motor systems can each be used to calibrate the other (Berkeley, 1709).

*relative*magnitudes of the failures of the various models under consideration and their consequences for biological vision. We return to this point in the discussion, after we have estimated these magnitudes.

*x*(left-right),

*y*(front-back), and

*z*(up-down). Pressing one key of the pair moved the point one way along the axis, pressing the other moved it in the opposite direction. A seventh button controlled the speed at which the point moved. At the start of a trial, the control program permitted “quick” movement of the point — each key press displaced the point by approximately 0.5 mm. When the observer judged that the adjustable point was near the bisection point, they pressed a seventh button, the speed toggle, which allowed them to move the point with greater precision (at the limit of resolution of the apparatus) until they were satisfied with their setting. A final press of an eighth button recorded the observer’s setting and triggered the next trial. Observers were encouraged to move their eyes across the vertex pair before completing their judgment.

*xy*plane (i.e., seen from above). The variance ellipses indicate ±2 SD of the setting. Table 1 shows the mean deviations (±1 SEM) of the settings from the Euclidean bisection point in the

*x, y*, and

*z*directions, as well as the Weber fraction (as a percentage) for the three-dimensional adjustment. Negative mean deviations from the Euclidean bisection point indicate biases leftward (

*x*), away from the observer (

*y*), and downward (

*z*).

Observer | Vertex pair | Δx (mm) (left /right) | Δy (mm) (depth) | Δz (mm) (up / down) | Weber fraction |
---|---|---|---|---|---|

HB | Left | −5.07±0.43 | −5.17±1.28 | 0.05±0.04 | 3.5% |

Back | 3.17±0.52 | 1.23±0.74 | −0.05±0.02 | 2.4% | |

Right | 5.87±0.35 | −4.64±1.05 | 0.09±0.03 | 2.9% | |

IO | Left | −1.12±0.18 | −2.70±0.80 | 0.25±0.06 | 2.1% |

Back | 0.00±0.26 | −5.48±1.00 | 0.07±0.09 | 2.7% | |

Right | 1.82±0.22 | −7.63±0.99 | 0.16±0.76 | 3.3% | |

JT | Left | 1.74±0.15 | −7.59±0.38 | 0.66±0.05 | 1.1% |

Back | 3.90±0.20 | −3.24±0.31 | 1.40±0.09 | 1.0% | |

Right | 2.15±0.18 | −10.20±0.38 | 0.78±0.05 | 1.1% | |

KB | Left | 0.06±0.21 | −3.60±0.59 | 0.16±0.04 | 1.6% |

Back | 0.16±0.19 | 1.02±0.61 | 0.26±0.05 | 1.7% | |

Right | 0.80±0.35 | −1.22±0.57 | 0.06±0.04 | 1.8% |

Deviations of mean settings from the Euclidean bisection points for the three vertex pairs and each observer. Deviations are shown for the *x* direction (negative values indicate leftward biases), *y* direction (negative values indicate backward biases), and *z* direction (negative values indicate downward biases). Data are reported as mean ±1 SEM (40 data points per vertex pair). The Weber fraction (as a percentage) for the three-dimensional adjustment was computed by dividing the averaged SD (average of the *x, y*, and *z* SDs) by the length of the configuration (140 mm) and multiplying by 100.

*t*test,

*p*< .05). The patterns of deviations differ from observer to observer and are reproducible from observer to observer (see also Experiment 2, which was performed on the same observers two weeks later, and Experiment 3, which JT performed 11 months after Experiment 1). Deviations were symmetric with respect to the stimulus vertex pair in the

*xy*plane, except for observer JT whose deviations exhibited an overall rightward bias. In the vertical direction, the bisection settings were biased upward by approximately 0.1–0.6 mm (Table 1).

*x*direction. A leftward bisection bias, often observed in line bisection experiments in the frontal plane, did not occur for any of the observers with this vertex pair. However, in contrast to standard line bisection experiments in which observers are instructed to bisect a line using a one-dimensional judgment (along the line), in our experiment observers bisected the frontal vertex pair using a full

*three-dimensional*adjustment. Being able to place the bisecting point anywhere in space, and not restricted to a position along the line, three observers chose settings significantly deviating from the bisecting line, either in depth (observers IO and JT), or in the vertical direction (observers JT and KB).

*k*, both positive (indicating a spherical geometry) and negative (indicating a hyperbolic geometry). Curvature estimates have even been found to vary within individual observers if measured at different locations in space (Koenderink et al., 2000) or when context objects are added to the scene (Cuijpers et al., 2001; Schoumans et al., 2000, 2002). In agreement with these previous results (which involved distances beyond grasp space), our results contradict a geometric interpretation of visual space, both qualitatively and quantitatively. Were the biases found in our study a result of a non-Euclidean geometry of constant curvature, we would expect all the bisection settings to be outside the triangle (for a positive curvature) or all inside (for a negative curvature). This is not true of most of our subjects, even in the small region of visual grasp space immediately in front of the observer. Second, given an empirically estimated value of curvature from a previous study, one can calculate the bias expected in the present experiment, which uses an inter-point distance that is far smaller. The predicted biases are much smaller than those we found. We conclude that our results cannot be attributed to an underlying Riemannian geometry of constant curvature (compare Cuijpers, Kappers, & Koenderink, 2001, 2002; Koenderink, van Doorn, & Lappin, 2000, 2003; Todd, Oomes, Koenderink, & Kappers, 2001).

*MOA bisection point*) is preferred over the Euclidean bisection point when presented as a forced choice.

*xy*plane. Because the MOA bisection point could have differed from the Euclidean bisection point in the

*z*coordinate, the sequence of points may also have included increments in the

*z*direction. See Table 2 for details.

Subject | Spacing (SD units) | Spacing (mm) | Number of points |
---|---|---|---|

HB | 0.70 | 1.35 | 10 |

IO | 0.85 | 1.96 | 7 |

JT | 0.65 | 0.98 | 11 |

KB | 0.62 | 1.20 | 7 |

A sequence of equally spaced points lying on a line connecting the Euclidean bisection point and the mean MOA setting from Experiment 1 was used as stimuli in Experiment 2; spacing between points is given in units of the MOA SD of Experiment 1 (Figure 4).

*p*

_{ij}] based on the actual observer’s data set (i.e., the 10 or 20 repetitions per point pair

*ij*). The preference probability

*p*

_{ij}indicates the proportion of occasions in which this observer chose point

*i*over point

*j*in a pairwise comparison.

*p*

_{ij}values. In each simulation, the modal value was chosen to represent the most preferred point. A distribution of preferred points was generated by performing 5,000 simulation runs. Figure 5 displays the bootstrap estimates of the mean and SD of the simulated 5,000 modal values, as well as the mean and SE of the MOA judgments from Experiment 1 (see also Table 3).

Subject | SE_{MOA} (mm) | SD_{2AFC} (mm) | P value |
---|---|---|---|

HB | 0.29 | 0.88 | .478 |

IO | 0.37 | 1.60 | .437 |

JT | 0.24 | 0.99 | .511 |

KB | 0.33 | 0.64 | .887 |

Results of the statistical analysis of comparing the MOA setting (Experiment 1) with the most preferred point in the 2AFC task (Experiment 2); see text and Figure 5 for details.

*z*value was computed by dividing the difference of mean MOA setting and mean preferred bisection point in the 2AFC task by the combined variance estimate of the MOA setting and the bootstrap estimate of the variance in the 2AFC task. (To reduce the probability of a type-II error, the statistical test was performed based on a

*z*value instead of testing a

*t*value corresponding to the finite number of degrees of freedom in the MOA judgment.) Table 3 contains the resulting

*p*values and indicates that preferred bisection points in the 2AFC task did not differ significantly from the MOA settings. We therefore conclude that observers select a bisection point independent of the method of judgment.

*x*direction), the point was either to the left or right of the MOA bisection point. Observers indicated whether the point was left or right of the bisection point. Back-front (

*y*direction) and up-down (

*z*direction) displacement conditions were run as well.

*x*: left vs. right,

*y*: behind vs. in front of, and

*z*: above vs. below) in separate blocks of trials.

*x*,

*y*, and

*z*directions, separately for each observer and vertex pair. Then, a series of 9 points was chosen in each Cartesian direction containing the MOA subjective bisection point plus 8 points whose distances from the MOA bisection point were chosen based on the MOA SD in steps of ±0.5 SD, ±1 SD, ±2 SD, and ±4 SD. As the SD in the

*z*direction was very small (< 0.3 mm), grid points were placed with a spacing of ±1 SD, ±2 SD, ±3 SD, and ±4 SD in the

*z*direction.

*x*direction), behind or in front of (judgment in the

*y*direction), or above or below (judgment in the

*z*direction) the bisection point.

*x*,

*y*or

*z*) and by vertex pair (for observer JT) in blocks of 135 trials. Fixation conditions were randomized within each block. Observers completed 50 judgments per point and fixation, amounting to a total of 4,050 trials per vertex pair. The experiment was completed in 13 (KL) and 25 (JT) sessions of approximately 30 min each.

*x*), behind (

*y*), or below (

*z*) judgments using the maximum-likelihood method as implemented by Wichmann and Hill (2001a, 2001b). The level corresponding to 50% (the point of indifference) was taken to indicate the bisection point.

Condition | Bisection point (mm) | JND_{(25–75%)} (mm) | SD_{MOA} (mm) | |
---|---|---|---|---|

JT, Left | x, front | −0.60 ± 0.48 | 1.76 | |

x, back | 0.28 ± 0.35 | 1.57 | 0.57 | |

x, center | −0.39 ± 0.20 | 0.79 | ||

y, front | 0.83 ± 0.74 | 3.33 | ||

y, back | −1.72 ± 0.65 | 2.83 | 2.38 | |

y, center | −0.50 ± 0.44 | 1.95 | ||

z, front | −0.60 ± 0.08 | 0.31 | ||

z, back | −0.06 ± 0.08 | 0.28 | 0.26 | |

z, center | 0.11 ± 0.07 | 0.25 | ||

KL, Left: | x, front | 1.09 ± 0.62 | 0.72 | |

x, back | 2.23 ± 0.62 | 0.51 | 1.39 | |

x, center | 1.73 ± 0.28 | 0.38 | ||

y, front | −1.84 ± 0.59 | 3.59 | ||

y, back | 0.73 ± 0.84 | 4.32 | 2.22 | |

y, center | 0.21 ± 0.50 | 2.73 | ||

z, front | −1.08 ± 0.13 | 0.36 | ||

z, back | −0.99 ± 0.11 | 0.42 | 0.20 | |

z, center | −1.01 ± 0.10 | 0.31 | ||

JT, Back: | x, left | −0.77 ± 0.22 | 2.55 | |

x, right | −0.76 ± 0.20 | 2.39 | 0.77 | |

x, center | 0.07 ± 0.14 | 1.31 | ||

y, left | 1.42 ± 0.89 | 2.53 | ||

y, right | −2.35 ± | 0.91 | 2.88 | 1.84 |

y, center | −1.56 ± | 0.61 | 1.86 | |

z, left | −0.55 ± 0.09 | 0.47 | ||

z, right | −0.60 ± 0.09 | 0.40 | 0.34 | |

z, center | −0.38 ± 0.07 | 0.33 |

Bisection point relative to the Euclidean bisection point (± SD) and slope estimates resulting from the fit of the psychophysical function (see Figure 6 and text for details), displayed for each observer and configuration separately. Just noticeable difference (JND) estimated as the interval between the 25% and 75% level of the fitted psychophysical functions. SD of the MOA setting estimated based on 20 settings per observer and configuration.

*x*,

*y*, or

*z*), we calculated the maximum difference Δ between the measured bisection points based on the three fixation conditions. Next we assumed that the underlying bisection points were identical, differing only in their measurement variability (the variance of the estimated 50% point of the fit). 50,000 times, we drew three random values for each of the three bisections based on identical means and the measured variabilities, and calculated a new value of Δ. The

*p*value reported in Table 5 is based on the percentile in the distribution of Δ values corresponding to the measured value of Δ. For each subject and condition, at least one of the three axes of bisection displays a significant difference across fixation conditions.

Condition | Δ (mm) | p value | |
---|---|---|---|

JT, Left: | x | 0.88* | <.001 |

y | 2.56* | .001 | |

z | 0.71* | .003 | |

KL, Left: | x | 1.14* | <.001 |

y | 2.57 | .092 | |

z | 0.09 | .067 | |

JT, Back: | x | 0.83* | <.001 |

y | 3.77 | .064 | |

z | 0.22* | <.001 |

Maximum distance Δ between bisection points under different fixations (see also Figure 6). A bootstrap analysis was performed to test whether Δ indicated a significant difference between different fixation points. The *p* values from this analysis are shown. Significant differences, employing a Bonferroni correction for the nine tests, are indicated by an asterisk.

*z*direction in the back configuration). We therefore reject the conjectures of Haubensak (1970) and Ehrenstein (1977). Bisection performance

*does*depend on fixation. However, the effect of fixation location on bisection is not large enough to be detected reliably by the observer moving his or her eyes during a judgment, and the effect is not large enough to explain the discrepancies observed between bisection judgments and a model based on Euclidean or, more generally, a Riemannian geometry.