Overview

Research Questions

The Measurement Challenge

Experiment 1: Establishing Spatial Calibration

Experiment 2: Three Perceptual Measures of Cylinder Length

Experiment 3: Measuring Depth Through Action

Converting Percepts into Physical Depth

Cross-Method Comparison

Key Findings

What the Results Suggest

Research Outputs

When two non-concentric circles rotate in the image plane, observers perceive a 3D cylinder extending in depth despite the absence of physical depth cues.

Although this stereokinetic illusion has been studied for decades, a fundamental question remained unresolved: how deep is the perceived object? Competing theories made opposing predictions, with minimal deformation accounts favoring large perceived depths and slow-motion accounts favoring much shallower structures.

To investigate this question, I developed four independent measurement methods — linear perspective adjustment, binocular disparity matching, volumetric fitting, and motor reaching — to quantify perceived depth across three experiments.

While all four methods produced highly reliable measurements within observers, they yielded systematically different estimates of perceived depth. These findings suggest that perceived depth is not represented by a single quantity, but depends critically on how it is measured.

This project was designed to answer three fundamental questions.

1. Does perceived depth depend on rotational speed?
If the visual system favors the slowest plausible 3D interpretation, increasing rotational speed should alter the perceived length of the stereokinetic cylinder.

2. Is perceived depth consistent across observers?
Do different individuals perceive approximately the same 3D structure, or does perceived depth vary systematically from person to person?

3. Do different measurement methods converge?
If the stereokinetic cylinder has a stable perceived depth, independent measurement methods should produce similar estimates.

The primary challenge was that the stereokinetic cylinder has no objective physical depth.

Unlike a real 3D object, there is no ground truth against which perceptual judgments can be compared. The problem was therefore not simply measuring depth, but determining how depth should be measured in the first place.

To address this challenge, I developed four independent measurement methods that estimated perceived depth through different perceptual and behavioral systems:

• Linear Perspective Adjustment

• Wireframe Sphere Fitting

• Binocular Disparity Matching

• Motor Reaching

If these methods converged, it would suggest a common underlying representation of perceived depth. If they diverged, it would indicate that different measurement methods capture different aspects of the percept.

Before perceived depth could be compared across participants, each participant's effective viewing distance first had to be measured.

To accomplish this, I developed a virtual display apparatus using a half-silvered mirror that projected a circular stimulus into 3D space.

This calibration established a common spatial reference frame, allowing subsequent depth measurements to be compared across observers.

Participants reached toward the perceived location of the circle without touching the display. These reaches provided an individualized estimate of perceived viewing distance that served as the geometric foundation for all subsequent depth calculations.

Mean reaching distances closely matched the 30 cm ground-truth viewing distance under both monocular and binocular conditions (29.91 cm vs. 29.87 cm), with no significant difference between viewing conditions (t(6) = 0.11, p = .92). However, binocular viewing produced significantly more consistent reaches, exhibiting lower within-participant variability than monocular viewing (t(6) = 4.93, p = .002). Across both conditions, reaching variability remained small, indicating high measurement precision.

These participant-specific viewing distances became the calibration reference for every subsequent experiment. Rather than assuming a fixed viewing distance, all later depth estimates were derived using each observer's own spatial calibration, ensuring that subsequent measurements reflected individual viewing geometry.

With participant-specific viewing distance established, I measured perceived cylinder length using three independent perceptual methods. A full experimental trial is shown below, illustrating how each method estimated perceived depth from a different perceptual perspective.

Linear Perspective Adjustment measured depth through size constancy. Participants adjusted the rear circle until the rotating structure appeared to form a uniform cylinder.

Wireframe Sphere Fitting measured depth volumetrically. Participants adjusted a rotating wireframe sphere until its diameter matched the perceived length of the cylinder.

Binocular Disparity Matching measured depth stereoscopically. Participants adjusted binocular disparity until a depth probe aligned with the perceived front and back surfaces of the cylinder.

All three perceptual methods confirmed that observers consistently perceived a 3D cylinder rather than a flat pair of rotating circles. Although each method quantified perceived depth differently, they converged on the same pattern of results.

Repeated-measures ANOVAs revealed a robust main effect of inter-center distance (ICD) across all three perceptual methods (F = 44.23 – 98.85, all p < .001). As the distance between the circles increased, participants consistently perceived longer cylinders. In contrast, doubling rotational speed produced no reliable change in perceived depth, providing little support for slow-motion accounts of stereokinetic perception.

As inter-center distance increased, Linear perspective adjustment produced smaller back-to-front size ratios, wireframe sphere fitting yielded larger sphere diameters, and binocular disparity matching produced greater disparity differences. All three methods indicated that inter-center distance, rather than rotational speed, was the primary determinant of perceived depth.

While Experiment 2 estimated perceived depth using perceptual methods, Experiment 3 measured the same percept through motor reaching. Participants first adjusted the stimulus until it appeared as a uniform cylinder and then reached toward its apparent front and back surfaces.

These reaching movements provided an independent behavioral estimate of perceived cylinder length that could be directly compared with the three perceptual methods from Experiment 2.

Repeated-measures ANOVAs revealed robust effects of inter-center distance (ICD) on both reaching distance and reaching variability (F values ≥ 72.19, all p < .001). As in Experiment 2, increasing ICD produced significantly longer perceived cylinders.

Interestingly, the front surface remained closely aligned with the viewing distance of the display, while the back surface progressively shifted farther away as ICD increased. Reaching movements toward the back surface also became more variable, suggesting greater uncertainty when estimating depth farther from the observer.

Each measurement method produced responses in different units, including size ratios, sphere diameters, binocular disparities, and reaching distances. To compare them directly, I converted each measure into centimeters using each participant’s calibrated viewing distance, allowing different measurement methods to be evaluated on a common physical scale.

The full geometric derivations, uncertainty analyses, and statistical methods are provided in the accompanying publication.

Within-subject permutation tests revealed significant pairwise differences between methods across both stimulus conditions (p ≤ .019), confirming that the methods did not produce interchangeable estimates of perceived depth. At the same time, between-subject analyses found no consistent group-level differences (p ≥ .074), indicating that these measurement differences varied systematically across individuals rather than reflecting a single population-wide pattern.

Despite these differences, the methods converged on several important findings. All four showed that increasing inter-center distance produced longer perceived cylinders, while rotational speed had little influence on perceived depth. Across repeated trials, within-subject variability remained substantially smaller than between-subject variability, demonstrating high measurement precision despite pronounced individual differences.

The strongest agreement emerged between linear perspective adjustment and motor reaching, which were highly correlated across participants (r = .97, p < .001). In contrast, binocular disparity matching and wireframe sphere fitting often produced different absolute depth estimates, suggesting that each method captured a different aspect of the underlying percept.

Taken together, these findings suggest that perceived depth is not represented by a single, method-independent quantity. Instead, different perceptual and behavioral measurements appear to access partially distinct representations of the same 3D experience while remaining internally precise and highly reproducible.

Stimulus geometry dominates perceived depth. Across all four measurement methods, increasing inter-center distance consistently produced longer perceived cylinders, whereas rotational speed had little measurable influence.

Perceived depth is highly reproducible within individuals. Despite differences between methods, repeated measurements exhibited strong within-subject precision across perceptual and behavioral tasks.

Measurement method influences the estimated percept. Independent perceptual and behavioral methods produced systematically different estimates of cylinder length, even after conversion to a common physical scale.

Together, these findings suggest that minimal deformation is an important constraint on stereokinetic depth perception, but not its sole determinant. Rather than relying on a single cue, the visual system appears to integrate geometric relationships, motion, and viewing geometry to produce a stable, finite 3D percept.

More broadly, this work demonstrates that different measurement methods need not produce identical estimates to reveal a common underlying phenomenon. Although observers consistently perceived the same coherent 3D object, linear perspective, binocular disparity, volumetric fitting, and motor reaching each captured different aspects of that percept. The choice of measurement method therefore shapes the conclusions that can be drawn about perceptual experience.

• Four independent methods for quantifying perceived stereokinetic depth across perceptual and behavioral tasks

• Participant-specific geometric calibration framework enabling direct comparison across measurement methods

• Common physical measurement scale for comparing size judgments, binocular disparity, volumetric fitting, and motor reaching

• Evidence of high within-subject precision despite systematic differences between measurement methods

• Demonstration that measurement method influences estimated perceived depth, even for a stable and coherent 3D percept

• Published in Vision Research (2026)