Radial Compensation: Methods & Applications

• Radial Compensation (RC) is a collection of techniques for correcting or decoupling radial artifacts by leveraging the link between radial displacements and underlying geometries.
• It is applied in quantum ion trapping, computational imaging, and generative modeling to enhance precision, reduce distortions, and improve statistical interpretability.
• RC methods utilize tailored measurements, parametric modeling, and information-theoretic frameworks to achieve robust calibration and deliver enhanced system performance.

Radial Compensation (RC) spans multiple research domains as a collection of techniques and frameworks for correcting, nullifying, or decoupling radial artifacts, distortions, or parameter entanglements. Methodologies described as "radial compensation" arise in quantum ion trapping (for micromotion minimization), geometric deep generative modeling (for curvature-invariant radial semantics), computational imaging (for lens and trajectory correction), and video analysis (for fisheye distortion and motion compensation). Despite the diversity of application, each instantiation leverages a mathematical link between "radial" displacement or parameters and an underlying geometry or artifact, and applies performant correction by dedicated analysis and measurement, parametric modeling, or information-theoretic construction.

1. Information-Geometric RC on Riemannian Manifolds

Radial Compensation in generative modeling formalizes an information-geometric construction for disentangling radial parameters from chart curvature and numerical artifacts on Riemannian manifolds. Given a complete manifold $M$ of constant curvature $\kappa$ and geodesic pole $p$, chart-based manifold models typically sample in $T_p M \cong \mathbb{R}^n$ and map to $M$ via a scalar-Jacobian azimuthal chart $T(x)$. The classical exponential map $\exp_p$ precisely preserves geodesics but induces a radius-dependent Jacobian factor, namely

$$\left( \frac{s_\kappa(R)}{R} \right)^{n-1}, \quad s_\kappa(R) = \begin{cases} R_c \sin(R / R_c), & \kappa>0 \ R_c \sinh(R / R_c), & \kappa<0 \end{cases}$$

which entangles radial prior statistics with curvature and dimension, impeding statistical interpretability and inflating gradient variance in flows.

RC addresses this by selecting the tangent-space base density in polar form as

$$f^{\mathrm{base}}_\theta(r, \omega) = \frac{1}{Z(\theta)} \varphi_\theta(r) J_T(r), \quad \omega \sim \mathrm{Unif}(S^{n-1}),$$

where $J_T(r)$ is the chart Jacobian and $\varphi_\theta(r)$ is the target (semantically meaningful) radial law. The crucial property is that the resulting manifold density depends only on the geodesic radius: $\rho_\theta(q) = \varphi_\theta(d(p, q))$ with all chart- and curvature-induced volume factors and Jacobians cancelled out. The Fisher information and Kullback-Leibler divergence in $\theta$ match exactly those of the target 1D Euclidean problem: $$I_{M}(\theta) = I_{\mathbb{R}}(\theta), \quad \mathrm{KL}(\rho_\theta \Vert \rho_\eta) = \mathrm{KL}(\varphi_\theta \Vert \varphi_\eta)$$ making RC the unique information-geometric construction that ensures geodesic-radial, Fisher-invariant modeling. The Balanced-Exponential (bExp) family of charts interpolates between volume-preserving and geodesic-preserving charts, permitting optimal tuning of numerical properties (gradient variance, function evaluations) without altering model likelihood or Fisher information. Empirical results demonstrate RC stabilizes generative modeling, prevents radius blow-ups in latent flows, and yields interpretable learned curvature and improved likelihoods across manifold types and datasets (Papamichals et al., 18 Nov 2025).

2. Radial Compensation in Quantum Ion Trapping

RC methods in ion trapping focus on minimizing excess micromotion induced by stray radial electric fields in linear Paul traps. The fundamental principle is the proportionality between radial displacement and the local stray field. The ion’s equilibrium shift under a static stray field $\mathbf E_{\rm stray}$ is given by

$$r_{0,i} = \frac{Q E_{\rm stray, i}}{m \omega_i^2}, \quad i \in \{x', y'\}$$

where $Q$ is the ion charge, $m$ its mass, and $\omega_i$ the secular (radial) frequency.

The canonical two-frequency displacement procedure comprises:

• Measuring the ion equilibrium at two distinct radial confinements ($\omega_i$, $\omega_i'$), producing position change

$$\Delta r_i = r_{0,i} - r_{0,i}' = \frac{Q E_{\rm stray,i}}{m} \left( \omega_i^{-2} - {\omega'_i}^{-2} \right)$$

• Executing a 2D scan over compensation voltages applied to calibrated DC electrodes, and fitting the resultant $\|\Delta \mathbf r\|$ surface to locate the voltage pair that nulls the displacement.
• Achieving residual stray field uncertainties as low as $3.5 \, \text{V/m}$ in optimized directions.

This methodology is entirely self-calibrating and is agnostic to laser k-vector orientation or detuning. It is broadly suitable for planar, stylus, and miniaturized traps (Saito et al., 2021).

A complementary approach leverages structural sensitivity in multi-ion crystals, notably near the zig-zag phase transition in bright–dark–bright ion triplets. Here, stray fields induce dramatic axial deformations as the soft mode frequency approaches zero; thus, RC via spatial monitoring can achieve full 2D field nulling below $0.2 \,\text{V/m}$. Single bright-ion variants exploit non-degenerate radial modes and RF amplitude switching, enabling axis-resolved compensation with a 1D camera readout (Barnea et al., 16 Mar 2025).

3. RC for Radial Distortion in Imaging and Computer Vision

Radial Compensation in imaging addresses nonlinear radial distortion imposed by lens optics, notably in wide field-of-view or sports-broadcast cameras. In this context:

• The observed (distorted) radius is related to the undistorted (ideal) radius via a polynomial model:

$$r_{d} = r_{u} \left( 1 + k_1 r_{u}^2 + k_2 r_{u}^4 \right)$$

with $k_1, k_2$ encoding barrel or pincushion distortion.

• RC is learned via a convolutional neural network regressor, trained on a synthetic dataset of rectified/cropped images and corresponding distortion parameters. The network predicts "apparent" distortion coefficients $\tilde{k_1}, \tilde{k_2}$ normalized by image scale.
• Correction consists of inverting the distortion mapping for each pixel using a polynomial approximation (e.g., 4-term Drap–Lefèvre inversion), rescaling coordinates, and remapping via standard interpolation.
• The approach enables real-time undistortion (25+ fps on commodity CPUs) and meets requirements for calibration-free, per-frame compensation under dynamic optical parameters (Janos et al., 2021).

4. RC in Fisheye and Motion Compensation Video Pipelines

For fisheye video, RC describes transformations and algorithms that compensate both distortion and parallax-induced motion artifacts:

• The fisheye projection is modeled, e.g., equisolid-angle: $r_f(\theta) = 2f \sin(\theta/2)$.
• Each image pixel is mapped to the unit sphere, rotated according to a viewport axis (front/back, top/bottom, left/right), then projected onto a local perspective plane.
• The proposed virtual image plane compensation (VIPC) applies corrections for points residing "behind" the perspective camera by inverting motion vectors and transforming angles, guaranteeing exact geometrical fidelity in correspondence.
• Experimental results demonstrate substantial quantitative gains (e.g., +2.40 dB average PSNR over previous state-of-the-art) and qualitative improvements in block motion estimation (Regensky et al., 2022).

5. RC in MRI: Trajectory Correction in Radial Imaging

Radial Compensation in magnetic resonance imaging targets phase errors and trajectory deviations in non-Cartesian (radial) k-space acquisitions:

• The measured signal in k-space segments is modeled as

$$\widehat I_i(\mathbf k) = \widehat\rho(\mathbf k + \Delta \mathbf k_i)$$

where $\Delta \mathbf k_i$ is a segment-specific shift.

• Data from segments with shared phase errors are arranged into a multi-block Hankel matrix, which is low-rank under smooth phase variations.
• Recovery exploits this structure by nuclear-norm minimization, subject to parallel-coil consistency and measurement constraints:

$$\min_{\{I_i\}, x, \{\phi_i\}} \sum_{i,c} \left\| \mathcal{F}_i^{\text{nu}}\{ S_c I_i \} - d_{c,i} \right\|_2^2 + \lambda \| H(\{\widehat{I}_i\}) \|_*$$

where $H$ is the block-Hankel operator.

• The algorithm robustly reconstructs artifact-free images universally across partial Fourier, golden-angle, and conventional radial acquisitions, matching or outperforming calibration-based competitors (e.g., TrACR), with no explicit measurement of the underlying trajectory or phase errors required (Mani et al., 2018).

6. Empirical Performance and Limits

Across all domains, performance gains are reported in terms of reduced uncertainty, greater measurement precision, improved statistical interpretation, or direct figures of merit (PSNR, likelihood, residual artifact signal, etc.). Table 1 summarizes representative RC outcomes.

Domain	Key Metric	RC Performance
Generative Models (manifold)	ELBO/test likelihood	$\sim$7 nats ELBO gain, stable variance
Ion traps (displacement)	Residual field uncertainty	$\lesssim 3.5$–$10$ V/m (2D)
Ion traps (multi/bright-dark)	Micromotion amplitude	$\lesssim$10 nm or $<0.2$ V/m
Imaging (sports vision)	Frame rectification error	$\lesssim$2$$px at 1080p</td> </tr> <tr> <td>Fisheye video</td> <td>PSNR gain</td> <td>+2.40 dB over prior work</td> </tr> <tr> <td>MRI (radial imaging)</td> <td>NRMSE, <a href="https://www.emergentmind.com/topics/general-speech-restoration-gsr" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">GSR</a></td> <td>NRMSE$$0.031$, GSR$0.025$

Empirical evidence consistently supports RC’s ability to isolate and correct for radial entanglements, with the limitation that ultimate performance is bounded by the resolution and sensitivity of the measurement system, numerical error, or—for learning-based RC—the training distribution coverage.

7. Theoretical Guarantees and Uniqueness Properties

For information-geometric RC, uniqueness is formally proven: within isotropic base families and scalar-Jacobian charts on constant curvature manifolds, RC is the only construction yielding geodesic-radial, Fisher-invariant likelihoods. In other domains, RC frameworks systematically exploit the underlying physics or signal structure (e.g., low-rankness under phase errors, mode softening in ion chains) to guarantee compensation up to the noise floor or system limits.

A general implication is that, across applications, RC techniques provide a systematic scheme to decouple or null radial deviations, regardless of whether such artifactual coupling arises from geometry, instrument imperfections, or algorithmic chart choice.