# On the Aperture Problem of Binocular 3D Motion Perception

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## Abstract

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**Dataset:**osf.io/2j6sq.

## 1. Introduction

#### 1.1. The Inverse Problem and Geometric Defaults

#### 1.2. Vector Normal and Cyclopean Average

#### 1.3. Bayesian Inference

#### 1.4. Spherical Motion Prior

#### 1.5. Bayesian Vector Normal

## 2. General Methods

#### 2.1. Participants

#### 2.2. Apparatus

#### 2.3. Stimulus

#### 2.4. Procedure

## 3. Results

#### 3.1. Results of Experiment 1

#### 3.2. Results of Experiment 2

## 4. Discussion

#### Limitations and Implications

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Viewing Geometry

#### Appendix A.2. Velocity Constraints

#### Appendix A.3. Brightness and Velocity Constraints

#### Appendix A.4. Bayesian Vector Normal Model

#### Appendix A.5. Model Fit and Model Comparison

**Table A1.**Noise ratios (%), goodness-of-fit (${\chi}^{2}$), and model comparison for Bayesian vector normal model with one (BVN1) and two noise parameters (BVN2) in Exp. 1 (slanted lines) and Exp. 2 (tilted and slanted lines). Model comparison in terms of likelihood ratio LR and Bayes factor $BF$.

Obs. | BVN1 | BVN2 | Mod. Comp. | ||||
---|---|---|---|---|---|---|---|

Exp. 1 | $\sigma $:${\sigma}_{0}$ | ${\chi}_{12}^{2}$ | ${\sigma}_{m}$:${\sigma}_{0}$ | ${\sigma}_{d}$:${\sigma}_{0}$ | ${\chi}_{11}^{2}$ | LR | $BF$ |

S1 | 7.03 | 93.0 | 8.97 | 6.85 | 21.7 | 1.89 | 0.69 |

S2 | 6.42 | 227.4 | 14.48 | 6.03 | 4.32 | 8.27 | 16.67 |

S3 | 6.68 | 182.2 | 13.53 | 6.35 | 12.0 | 5.08 | 3.39 |

S4 | 6.20 | 54.7 | 9.55 | 6.00 | 7.01 | 3.28 | 1.38 |

S5 | 6.19 | 24.96 | 10.22 | 5.95 | 7.63 | 3.37 | 1.44 |

Exp. 2 | $\sigma $:${\sigma}_{0}$ | ${\chi}_{24}^{2}$ | ${\sigma}_{m}$:${\sigma}_{0}$ | ${\sigma}_{d}$:${\sigma}_{0}$ | ${\chi}_{23}^{2}$ | LR | $BF$ |

S1 | 6.41 | 486.3 | 12.71 | 5.31 | 15.2 | 6.65 | 7.43 |

S2 | 6.18 | 570.8 | 13.77 | 5.00 | 29.3 | 5.81 | 4.89 |

S3 | 5.58 | 859.1 | 11.20 | 4.48 | 24.9.0 | 6.62 | 7.33 |

S4 | 4.67 | 632.9 | 11.01 | 3.36 | 27.8 | 6.06 | 5.54 |

S5 | 5.59 | 1741 | 13.99 | 4.21 | 46.6 | 6.92 | 8.52 |

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**Figure 1.**Illustration of velocity constraints and the 3D aperture problem. The left eye (LE) and right eye (RE), with nodal points separated by interpupillary distance i (exaggerated here), fixate point F on an image plane (screen) at distance D. Stereoscopic rendering of a tilted moving line on the image plane (grey lines) suggests two image-motion gradients (grey arrows) originating from point F. Inverse projection of each image gradient through the nodal points of each eye do not intersect but the intersection of velocity constraint planes (shaded triangles) describe all possible motion directions in a single line (ICP). Different 3D velocity estimates are shown in colour (blue—VN: Vector Normal; red—CA: Cyclopean Average; light purple – intermediate directions, see text for explanation).

**Figure 2.**Illustration of the Bayesian vector normal (BVN) model. Inverse projection of the line stimulus into the left and right eye are illustrated by 2D velocity constraints on an image plane (grey lines), resulting in two 3D velocity constraint planes (shaded triangles), and their intersection (ICP). The ICP describes an oriented line moving from the start point F to the left and away from the observer. The translucent sphere centred on fixation point F denotes a weak prior for slow motion in 3D. If the constraint planes have very little noise and are combined with the spherical prior in a Bayesian inference then the resulting 3D velocity estimate $\widehat{\mathbf{v}}$ approximates the vector normal (blue arrow).

**Figure 3.**Illustration of the stimulus display and adjustment method. White hairlines indicate the axes (x, y, and z) and angles (longitude, latitude) in 3D and were not visible in the display. In each trial an oriented line stimulus moved back and forth on a 3D trajectory behind a circular aperture. The three grey lines illustrate a line stimulus at its far (left), start (centre), and near (right) position. The observer adjusted a string of five red dots until they matched the perceived 3D motion direction of the moving line stimulus. Motion direction was expressed in longitude and latitude angle, measured in degrees from the midpoint of the depth probe. See text for details.

**Figure 4.**Stereoscopic illustration of the vertical line stimulus with horizontal disparity in the top row (Exp. 1) and an oblique line stimulus with horizontal and orientation disparity in the bottom row (Exp. 2). Cross-fusing the two dots to the left and right of the middle aperture or fusing the middle and right aperture (uncrossed fusion) gives an impression of the 3D orientation of the line stimulus at a “far”, “centre”, and ”near” position. In each trial the line stimulus moved back and forth between the near and far inflection position, suggesting a motion direction in 3D.

**Figure 5.**Model predictions for line stimuli in Exp. 1. Each of the seven black hairlines corresponds to a constraint (ICP) for a motion trajectory to the left and away from the observer (−IOVD/far) and to the left and towards the observer (+IOVD/near). Red dots correspond to the cyclopean average (CA) and dark blue dots to vector normal (VN) predictions. Light blue and light purple open circles correspond to BVN1 and BVN2 predictions, respectively. See text for details.

**Figure 6.**Motion directions in Experiment 1. In two columns pairs of graphs are shown for Observer S1 to S5 with −IOVD/far on the left +IOVD/near on the right. In each graph latitude is plotted against longitude angles. Circles connected by a line indicate motion directions for the seven line stimuli slanted in depth from vertical with orientation difference $-{6}^{\xb0}$ to $+{6}^{\xb0}$ in steps of ${2}^{\xb0}$. Black open circles represent observed longitude and latitude angles, averaged across repeated trials. Horizontal and vertical error bars denote $\pm 1$ SEM. Light blue and light purple circles denote motion direction estimates of the best-fitting BVN1 and BVN2 model, respectively. The last pair illustrates geometric predictions of VN (blue circles) and CA (red circles).

**Figure 7.**Individual noise estimates of BVN2 in Exp. 1. Monocular motion (${\sigma}_{m}$) and dynamic depth (${\sigma}_{d}$) estimates for Observer S1 to S5 from bootstrapped data. Thick horizontal lines denote medians, boxes the second to third quartile with error bars (whiskers) extending 1.5 times of the interquartile range. Open circles denote outliers. All pairwise differences between individual motion and depth noise estimates were statistically significant (p < 0.002, two-tailed t-test, Bonferroni-adjusted).

**Figure 8.**Model predictions for line stimuli in Exp. 2. Each of the seven black hairlines corresponds to a constraint (ICP) for an upwards trajectory to the left and away from the observer (−IOVD/far) and to the left and towards the observer (+IOVD/near). Red circles correspond to the cyclopean average (CA) and dark blue circles to vector normal (VN) predictions. Light blue and light purple open circles correspond to BVN1 and BVN2 predictions, respectively. See text for details.

**Figure 9.**Motion directions in Exp. 2. In two columns pairs of graphs are shown for Observer S1 to S5 with −IOVD/far on the left +IOVD/near on the right. In each graph latitude is plotted against longitude angles. Circles connected by a line indicate motion directions for the seven line stimuli slanted in depth from oblique with orientation difference $-{6}^{\xb0}$ to $+{6}^{\xb0}$ in steps of ${2}^{\xb0}$. Black open circles represent observed longitude and latitude angles, averaged across repeated trials. Horizontal and vertical error bars denote $\pm 1$ SEM. Light blue and light purple circles denote angular estimates of the best-fitting BVN1 and BVN2 model, respectively. The last pair of graphs illustrates the geometric predictions of VN (blue circles) and the CA (red circles).

**Figure 10.**Individual noise estimates for BVN2 in Exp. 2. Estimates of monocular motion (${\sigma}_{m}$) and dynamic depth (${\sigma}_{d}$) for Observer S1 to S5 for bootstrapped data. Thick horizontal lines denote medians, boxes the second and third quartile with error bars (whiskers) extending 1.5 times of the interquartile range. Open circles denote outliers. All pairwise differences between individual motion and depth noise estimates were statistically significant (p < 0.0002, two-tailed t-test, Bonferroni-adjusted).

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**MDPI and ACS Style**

Lages, M.; Heron, S.
On the Aperture Problem of Binocular 3D Motion Perception. *Vision* **2019**, *3*, 64.
https://doi.org/10.3390/vision3040064

**AMA Style**

Lages M, Heron S.
On the Aperture Problem of Binocular 3D Motion Perception. *Vision*. 2019; 3(4):64.
https://doi.org/10.3390/vision3040064

**Chicago/Turabian Style**

Lages, Martin, and Suzanne Heron.
2019. "On the Aperture Problem of Binocular 3D Motion Perception" *Vision* 3, no. 4: 64.
https://doi.org/10.3390/vision3040064