Overview

Research Questions

Theoretical Proposal

Experimental Conditions

Measuring Perceived Cone Height

Perceived Cone Height Increases Toward the Minor Axis

Minimal Deformation Constrains Perceived Cone Height

Key Findings

What the Results Suggest

Research Outputs

After extended viewing of a rotating ellipse containing an eccentric dot, observers perceive a 3D cone extending in depth despite the stimulus being entirely 2D. Although the stereokinetic cone has been studied for decades, little work had quantified why its perceived height changes as the dot moves across the ellipse.

This project tested whether minimal deformation could explain changes in perceived cone height. The hypothesis was that the visual system prefers the 3D interpretation requiring the least change in the cone's shape as it rotates, while remaining consistent with the observed motion. Observer depth judgments were then compared against quantitative predictions generated by the minimal deformation model across multiple stimulus configurations.

Observers' depth estimates closely matched the theoretical predictions, providing evidence that the visual system favors 3D interpretations that minimize deformation while preserving coherent structure.

This project was designed to answer two fundamental questions.

1. What factors influence the perceived height of a stereokinetic cone?
Examine how rotational speed, ellipse aspect ratio, eccentric dot position, and their interactions influence perceived cone height.

2. Does the visual system favor minimally deforming structures?
Test whether principles of rigidity and minimal deformation explain the perceived 3D structure generated by ambiguous motion by comparing observers' depth estimates with quantitative predictions from the theoretical model.

The stereokinetic cone presents a fundamental ambiguity. A rotating ellipse with an eccentric dot is compatible with many possible 3D interpretations, yet observers consistently perceive a cone of a particular height.

To explain this phenomenon, I developed a geometric model based on a preference for minimal deformation. Rather than assuming that the visual system selects an arbitrary 3D solution, the model proposes that it favors the interpretation that minimizes the relative motion between the cone's apex and its rotating circular base, thereby preserving the most stable and coherent structure.

This principle produces a clear quantitative prediction. When the eccentric dot lies near the major axis of the ellipse, a relatively shallow cone minimizes relative motion. As the dot moves progressively toward the minor axis, progressively taller cones minimize relative motion and therefore become the preferred percept.

The model therefore predicts that perceived cone height should increase systematically as the eccentric dot moves from the major axis toward the minor axis, providing a direct, testable relationship between stimulus geometry and perceived depth.

The theoretical model predicted that perceived cone height is determined by the geometric relationship between the ellipse and the eccentric dot. To test this prediction, I systematically manipulated three stimulus properties while measuring observers' perceived cone height using a psychophysical adjustment task.

• Dot location: four angular positions spanning the major axis (0°) to the minor axis (90°)

• Ellipse aspect ratio: 0.6 or 0.8

• Rotation speed: 60°/s or 90°/s

These manipulations formed a 4 × 2 × 2 within-subject design with 16 stimulus configurations, each repeated 13 times for 208 trials per participant.

To evaluate the theoretical model, participants viewed rotating ellipse-and-dot stimuli and estimated the cone's apparent height using a psychophysical adjustment task. On each trial, they adjusted the length of a rotating depth probe centered on the ellipse until it matched the perceived distance between the cone's apex and its circular base.

The final probe length provided a quantitative estimate of perceived cone height. This allowed theoretical predictions to be directly compared with human perception on the same physical scale.

A repeated-measures ANOVA evaluated the effects of dot location, ellipse aspect ratio, and rotation speed on perceived cone height. Consistent with the theoretical model, dot location emerged as the primary determinant of perceived cone height. Observers reported progressively taller cones as the eccentric dot moved from the major axis (0°) toward the minor axis (90°) (F(3,21) = 485.51, p < .001). Across both ellipse aspect ratios, mean perceived cone height increased systematically from 0.13 cm at 0° to 2.25 cm at 30°, 4.89 cm at 60°, and 6.45 cm at 90°, demonstrating a clear relationship between stimulus geometry and perceived cone height.

Ellipse aspect ratio also produced a significant main effect (F(1,7) = 194.31, p < .001), with ellipses of higher aspect ratios producing proportionally taller perceived cones. A significant dot location × aspect ratio interaction (F(3,21) = 122.88, p < .001) further showed that the influence of dot location became progressively larger as ellipse aspect ratio increased. In contrast, rotation speed produced no significant effect, indicating that the final percept was governed by geometric structure rather than the speed of image motion.

Together, these results closely matched the theoretical model's predictions, providing strong evidence that perceived cone height is primarily determined by geometric relationships that minimize deformation.

To evaluate the theoretical model, I compared its predicted cone heights against observers' judgments using multilevel regression, allowing both the slope and intercept to vary across participants. This approach tested whether the model accurately predicted perceived cone height while accounting for repeated measurements within individuals.

Across both ellipse aspect ratios, the agreement between theory and perception was strong. The regression slopes were essentially unity (0.6 aspect ratio: β = 1.00, R² = 0.83; 0.8 aspect ratio: β = 0.98, R² = 0.74), indicating that observers' depth estimates closely tracked the values predicted by the minimal-deformation model. Intercepts remained near zero, suggesting little systematic bias across participants.

Although the model accurately described overall performance, observer variability was not uniform across conditions. Judgments were most consistent when the eccentric dot lay on the major (0°) or minor (90°) axes, where the stimulus provided strong geometric reference points. Variability increased significantly at the intermediate 30° and 60° positions (F = 9.00–11.03, p < .01), where the visual system had fewer geometric constraints for resolving the cone's 3D structure.

Together, these findings provide strong evidence that minimal deformation appears to quantitatively predict human judgments of stereokinetic cone height.

Geometry governs perceived cone height. Across all stimulus conditions, moving the eccentric dot from the major axis toward the minor axis produced progressively taller perceived cones, while rotation speed had little measurable influence on the final percept.

Human perception closely follows the principle of minimal deformation. Multilevel regression showed that observers' depth judgments closely matched the predictions of the minimal deformation model, indicating that the visual system consistently favored the least deforming 3D interpretation across both ellipse aspect ratios.

Together, these findings suggest that minimal deformation is a fundamental computational constraint on stereokinetic depth perception. Rather than selecting an arbitrary 3D interpretation, the visual system appears to favor the solution that minimizes relative deformation.

• Quantitative computational model explaining perceived stereokinetic cone height through principles of minimal deformation

• Psychophysical paradigm for measuring perceived cone height across ambiguous 3D motion stimuli

• Validation of a theoretical model through controlled experiments spanning 16 stimulus configurations and 208 trials per participant

• Evidence that human judgments closely follow quantitative geometric predictions across multiple stimulus conditions

• Published in Vision Research (2018)