Passable Aperture Paradigm: Theory & Applications

• Passable Aperture Paradigm is a framework that integrates rigorous mathematical modeling, non-linear optimization, and sensorimotor calibration to analyze and compensate for aperture discontinuities.
• It employs sequential deformable mirror techniques and adaptive interaction matrix recalibration to achieve significant contrast improvements in high-contrast imaging systems.
• The paradigm is applied across optical engineering and VR perception, enhancing system throughput and mitigating perceptual distortions with practical compensation strategies.

The Passable Aperture Paradigm encompasses a body of theoretical and applied methods for analyzing, optimizing, and interpreting the functional relationship between physical apertures and the systems or agents interacting with them. This broad concept is instantiated in fields ranging from optical engineering—where complex apertures and their aberrations must be actively controlled for high-contrast astronomical imaging—to perception science, where human affordances are quantified by tasks such as traversing or visually estimating passageways. Across domains, the paradigm identifies, compensates for, or leverages spatial discontinuities and perceptual distortions that modulate throughput, contrast, or perceived functional body size. Rigorous mathematical modeling, non-linear optimization, sensorimotor calibration, and computational corrections (such as those for depth compression) are key components of the contemporary approach.

1. Principles of Aperture Discontinuity and Compensation

Aperture discontinuities, including segment gaps, struts, and central obscurations, introduce pronounced amplitude and phase excursions in both wavefront-based physical systems (e.g., telescopes) and perception tasks involving spatial gaps. In astronomical imaging, discontinuities degrade the point spread function (PSF), limiting contrast and throughput essential for exoplanet detection. The Active Compensation of Aperture Discontinuities (ACAD) technique utilizes sequential deformable mirrors (DMs) to remap the pupil’s intensity distribution (Pueyo et al., 2012). DM1 applies a non-linear phase modulation, effectively reducing amplitude variations, while DM2 restores an on-axis output wavefront. The surface deformations required to achieve this are related by the Monge–Ampère equation:

$$(1 + Z \frac{\partial^2 H_1}{\partial X^2})(1 + Z \frac{\partial^2 H_1}{\partial Y^2}) - [Z (\frac{\partial^2 H_1}{\partial X\partial Y})]^2 = A(X,Y)^2$$

where $A(X,Y)$ is the target apodization profile. Discontinuity compensation is achieved by solving for DM shapes that induce phase-induced amplitude modulation matched to the discontinuity topology, yielding contrast floors orders of magnitude deeper than uncorrected systems.

In perception science, the passable aperture paradigm quantifies affordance by the relationship between physical gap width and perceived body size required to pass through the aperture. Experimental methodologies involve both action-based threshold determination (e.g., sidling through the gap) and perceptual estimation (minimum gap judged passable), often modeled with invariant scaling relationships.

2. Nonlinear Optimization and Interaction Matrix Recalibration

Because DM response and pupil remapping are highly nonlinear—especially when discontinuities reach λ-scale excursions—linear, small-stroke methods fail. Nonlinear strategies deployed include:

• Explicit Newton Algorithm: Linearizes the Monge–Ampère equation by decomposing the DM surface into a base guess plus small update ($v$), attacking the linearized PDE in a Fourier basis with damping to ensure convergence (Pueyo et al., 2012).
• Semi-Implicit Algorithm: Reformulates the problem as a Poisson-like equation isolating the Laplacian, solved iteratively.

For image-plane wavefront correction, the ACAD-OSM framework introduces adaptive interaction matrix recalibration (Mazoyer et al., 2017). When DM strokes approach the nonlinear regime, the interaction matrix—linking actuator perturbations to focal-plane electric field—is periodically rebuilt around the current DM configuration. Stroke minimization proceeds via adaptive gain control:

$$C_{\mathrm{target}}[k+1] = (1 - \gamma) \cdot C_{\mathrm{target}}[k]$$

Convergence is monitored and recalibration is triggered when progress stalls under fixed gain and iteration counts. This enables contrast improvement from the $10^{-5}$–$10^{-7}$ range (ACAD-ROS) to the $10^{-10}$ threshold required for Earth-like exoplanet detection.

3. Impact on High-Contrast Imaging Through Arbitrary Apertures

ACAD and ACAD-OSM uniquely extend the achievable contrast for telescopes with complex apertures. Simulations demonstrate:

• ACAD: $10^{-7}$ contrast for JWST-like geometries, up to $10^{-8}$ for ELT or HST-like apertures, across 4–30 $\lambda/D$ angular separations (Pueyo et al., 2012).
• ACAD-OSM: $10^{-10}$ contrast is attainable over at least a 10% spectral bandwidth using iterative recalibration, far surpassing plateaued performance from static correction (Mazoyer et al., 2017).

Throughput metrics highlight that amplitude compensation prior to the coronagraph improves the dark hole quality and off-axis throughput, with large-stroke solutions avoiding high-frequency ringing or diffraction artifacts. Adaptive compensation also dampens near-field diffraction effects and can be further refined by quasi-linear wavefront control after the nonlinear solution is established.

4. Applications and Design Guidelines

The paradigm can be implemented across a range of optical and perception tasks:

Domain	Paradigm Application	Contrast/Accuracy Gain
Segmented Telescopes	ACAD for JWST-like, ELT-like pupils	$10^{-7}$–$10^{-8}$ contrast
On-axis Telescopes	Mitigation of central obscurations/spiders	1–2 orders higher suppression
Perception Science	VR-based aperture tasks, sensorimotor scaling	Recovered invariance after correction
Satellite Flood Mapping	CNN+Fusion for road passability from imagery	$5\%$ F1 score improvement

Design recommendations include favoring many thin spiders over few thick ones for ELTs, leveraging DM hardware already common in mission designs (e.g., LUVOIR, HabEx), and integrating real-time algorithms to adaptively correct for dynamic aberrations such as segment dephasing or misalignments.

5. Extensions to Perception and VR Environments

The passable aperture paradigm is applied in immersive VR to paper perceptual distortions and sensorimotor uncertainty. In VR, the affordance ratio (perceptual threshold/action threshold) is systematically elevated due to depth compression from the vergence-accommodation conflict (VAC). Mathematical modeling links visual angle (p) and offset ($\beta$) through:

$p' = p + \beta$

$l = (\mathrm{IPD}/2) \cdot \tan(p + \beta)$

$W = |A_L - A_R|$

$W' \propto W \cdot (D/d)$

Upon correcting for the VAC-induced distortion, the affordance ratios in VR realign with those of unmediated reality (UR), establishing a recovered invariant geometrical scaling. The shift persists as an aftereffect post-VR exposure (Wang et al., 1 Oct 2025), suggesting that perceptual recalibration can be both beneficial and transiently biasing.

6. Future Prospects and Computational Directions

Continued development focuses on:

• Enhanced numerical solvers for diffractive propagation through arbitrary remapping functions.
• Hybrid correction strategies, combining static apodization (for high spatial frequency control) with adaptive DM correction.
• Real-time adaptive calibration for dynamic aberrations and evolving aperture topologies.
• Perceptual adaptation models in VR to mitigate depth compression and sensorimotor uncertainty via predictive geometrical corrections.

For perception-action research, incorporating dynamic corrections for VAC and accounting for short- and long-term recalibration effects promises improved transfer and more accurate assessment of embodied spatial cognition in immersive environments.

7. Comparative Advantages and Limitations

ACAD and its extensions provide robust, broadband, and high-throughput solutions for systems with challenging aperture geometry, outperforming static apodization or single-DM architectures in terms of both contrast and throughput metrics. Limitations include actuator count constraints (limiting maximum spatial frequency correction) and increased computational overhead in iterative matrix recalibrations. In VR, while geometrical corrections restore invariance, lingering aftereffects necessitate consideration for sensorimotor rehabilitation protocols.

In summary, the Passable Aperture Paradigm integrates non-linear physical compensation, advanced computational optimization, and rigorous geometrical modeling to address the multifaceted impact of aperture discontinuities and perceptual distortions across physical, biological, and perceptual systems.