Angle Distribution Refinement Overview

• Angle distribution refinement is a set of methods that enhance coarse, noisy angular data into detailed representations by preserving local geometry and physical constraints.
• It encompasses deterministic point corrections, hierarchical coarse-to-fine regression, and continuous orientation refinement applied in imaging, astronomy, and engineering.
• Practical outcomes include improved deblurring, higher object detection precision, and optimized control maps in motor design through sample-efficient design.

Angle distribution refinement denotes a family of operations in which angular information that is initially coarse, projected, noisy, or globally summarized is replaced by a representation with finer structural content. In the literature surveyed here, the refined object may be a noisy scalar blur angle, a coarse angular sector, a trajectory of projection angles, an observable projection of a three-dimensional spin–orbit law, a spacing law for Frobenius angles, a pitch-angle distribution, an opening-angle ensemble, a commutation-angle map, or a topological configuration indexed by $z=\rho e^{i\theta}$. The common thread is not a single standardized algorithm, but a shift from a weak angular description to one that preserves local geometry, physical constraints, dynamical consistency, or higher-order statistics.

1. Principal modes of refinement

A first mode is deterministic correction of a point estimate. In semi-blind rotational deblurring, CAR-Net assumes a single noisy scalar angle,

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$

uses it in a physics-based inversion, and optionally predicts

$$\theta_\mathrm{corrected}=\mathrm{AngleDetector}(f^{-1},\theta_\mathrm{initial}),$$

before applying image residual-refinement stages. The paper is explicit that this is not an angle-distribution method: there is no posterior over angles, no confidence interval output, and no iterative refinement of an angle law across stages (Lai et al., 30 Nov 2025).

A second mode is hierarchical coarse-to-fine refinement. In arbitrary-oriented object detection, MGAR partitions $[0^\circ,180^\circ)$ into coarse bins of width

$\omega=\frac{180^\circ}{C_\theta},$

predicts a coarse class $k=\lfloor \theta_{gt}/\omega \rfloor$, and then refines within that interval by local regression, reconstructing

$$\theta'=\omega\cdot \operatorname{Argmax}\!\big(\operatorname{Sigmoid}(t'_{\theta_{\text{class}}})\big)+\big(t'_{\theta_{\text{reg.}}}\big)^2.$$

Here the refined object is a local residual conditioned on a coarse angular mode (Wang et al., 2022).

A third mode refines the statistical object itself. In the refined Sato–Tate setting, the one-point density

$\rho(\theta)=\frac{2}{\pi}\sin^2\theta$

is not treated as the end of the story; after unfolding, the ordered angles are conjectured to obey

$p_k(s)=\frac{s^k e^{-s}}{k!},$

so refinement means moving from marginal equidistribution to local spacing statistics (Kimura, 2021).

A fourth mode is projection/inversion of angle laws. For hot Jupiters, the intrinsic spin–orbit distribution $w(\Psi)$ is not directly observable; the observable quantity is the projected angle $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$0, with

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$1

Here refinement means replacing a raw projected histogram by a geometrically correct relation between latent and observed angular distributions (Crida et al., 2014).

Taken together, these examples suggest a useful distinction between point-estimate refinement, coarse-to-fine decomposition, local-statistics refinement, and projection-aware distributional inference. A plausible implication is that the phrase “angle distribution refinement” is best read contextually: in some fields it names an explicit law over angles, whereas in others it names a structured correction procedure applied to angular variables.

2. Deterministic angle correction in vision systems

In CAR-Net, rotational motion blur is modeled in Cartesian coordinates by

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$2

and, after Cartesian-to-polar transformation, by

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$3

The coarse inversion stage is

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$4

with $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$5 as a numerical regularizer. In the full CAR-Net-AD model, one inversion with $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$6 produces $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$7, an angle detector regresses $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$8, a second inversion is recomputed with that corrected angle, and image refinement proceeds through $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$9 residual stages

$$\theta_\mathrm{corrected}=\mathrm{AngleDetector}(f^{-1},\theta_\mathrm{initial}),$$0

The paper’s main angle-related conclusion is narrow but precise: the method performs learned point-estimate angle correction inside a cascaded deblurring pipeline, not explicit probabilistic angle-distribution refinement (Lai et al., 30 Nov 2025).

The quantitative behavior reflects that design choice. Under Gaussian angle noise with $$\theta_\mathrm{corrected}=\mathrm{AngleDetector}(f^{-1},\theta_\mathrm{initial}),$$1, CAR-Net-AD reports $$\theta_\mathrm{corrected}=\mathrm{AngleDetector}(f^{-1},\theta_\mathrm{initial}),$$2 dB / $$\theta_\mathrm{corrected}=\mathrm{AngleDetector}(f^{-1},\theta_\mathrm{initial}),$$3, matching its $$\theta_\mathrm{corrected}=\mathrm{AngleDetector}(f^{-1},\theta_\mathrm{initial}),$$4 performance, whereas CAR-Net-Base drops from $$\theta_\mathrm{corrected}=\mathrm{AngleDetector}(f^{-1},\theta_\mathrm{initial}),$$5 dB / $$\theta_\mathrm{corrected}=\mathrm{AngleDetector}(f^{-1},\theta_\mathrm{initial}),$$6 to $$\theta_\mathrm{corrected}=\mathrm{AngleDetector}(f^{-1},\theta_\mathrm{initial}),$$7 dB / $$\theta_\mathrm{corrected}=\mathrm{AngleDetector}(f^{-1},\theta_\mathrm{initial}),$$8. Module ablation shows that most of the gain comes from image refinement rather than angle correction alone: inversion only gives $$\theta_\mathrm{corrected}=\mathrm{AngleDetector}(f^{-1},\theta_\mathrm{initial}),$$9 dB / $[0^\circ,180^\circ)$0, refinement only $[0^\circ,180^\circ)$1 dB / $[0^\circ,180^\circ)$2, and the full model $[0^\circ,180^\circ)$3 dB / $[0^\circ,180^\circ)$4. This suggests that, in this architecture, angle correction functions chiefly as improved initialization for residual image refinement rather than as a full uncertainty model.

MGAR uses a more explicitly hierarchical angular representation. Coarse-Grained Angle Classification encodes

$[0^\circ,180^\circ)$5

and Fine-Grained Angle Regression predicts the local residual

$[0^\circ,180^\circ)$6

Training uses the square-root transform

$[0^\circ,180^\circ)$7

and inference decodes

$[0^\circ,180^\circ)$8

The angle loss is refined further by the IoU-aware FAR-Loss,

$[0^\circ,180^\circ)$9

which increases supervision when angle error strongly harms overlap (Wang et al., 2022).

The empirical gains are concentrated where angular precision matters most. On HRSC2016, baseline regression yields mAP$\omega=\frac{180^\circ}{C_\theta},$0, baseline+CSL $\omega=\frac{180^\circ}{C_\theta},$1, baseline+DCL(gray) $\omega=\frac{180^\circ}{C_\theta},$2, and baseline+MGAR $\omega=\frac{180^\circ}{C_\theta},$3. With $\omega=\frac{180^\circ}{C_\theta},$4, the square fitting function outperforms Linear, Sigmoid, and Exp, reaching mAP$\omega=\frac{180^\circ}{C_\theta},$5. Replacing MSE with IFL improves mAP$\omega=\frac{180^\circ}{C_\theta},$6 from $\omega=\frac{180^\circ}{C_\theta},$7 to $\omega=\frac{180^\circ}{C_\theta},$8. The method also reduces prediction-head thickness relative to fine angular classification: with $\omega=\frac{180^\circ}{C_\theta},$9, baseline+CSL has thickness $k=\lfloor \theta_{gt}/\omega \rfloor$0, baseline+DCL $k=\lfloor \theta_{gt}/\omega \rfloor$1, and baseline+MGAR $k=\lfloor \theta_{gt}/\omega \rfloor$2. In this setting, refinement means reducing global ambiguity by coarse discretization and then recovering precision by local regression.

3. Continuous orientation and trajectory refinement

In single-particle cryo-EM, angle refinement appears as a continuous optimization problem over projection orientations rather than a coarse-to-fine classifier. The stated contribution of joint angular refinement and reconstruction is to refine projection angles on the continuum, jointly with the density map: the density map is updated using an alternating-direction method of multipliers, while orientations are updated through semi-coordinate-wise gradient descent, eliminating fine discretization of orientation space and the classical template-matching step (Zehni et al., 2020). The supplied text does not provide the full body derivation, but the methodological contrast is clear: the refined object is a continuous orientation field, not a histogram over angle bins.

Microtomography provides a complementary case in which the refined quantity is an entire angular trajectory. TomoSLAM assumes fixed source and detector, object rotation about a stationary axis, zero axial displacement, and only rotational error. The image model for a tracked feature is

$k=\lfloor \theta_{gt}/\omega \rfloor$3

and triplets of projections yield relative-angle constraints that are filtered by RANSAC and fused with encoder increments in a one-dimensional factor graph. The MAP estimate is written as

$k=\lfloor \theta_{gt}/\omega \rfloor$4

which becomes a weighted least-squares problem over angular residuals. In this framework, refinement means re-estimating the full set of projection angles $k=\lfloor \theta_{gt}/\omega \rfloor$5 so that they are jointly consistent with motor-control signals and image-derived relative-angle measurements (Griguletskii et al., 2021).

The numerical outcome is operational rather than purely geometric. TomoSLAM reports trajectory-RMSE reduction of up to $k=\lfloor \theta_{gt}/\omega \rfloor$6 versus the pure stepper-motor trajectory, and the full $k=\lfloor \theta_{gt}/\omega \rfloor$7 optimization takes $k=\lfloor \theta_{gt}/\omega \rfloor$8 s on an Intel Core i7-10850H in Python on 12 cores. The result is a refined angular distribution over projections that is no longer the nominal $k=\lfloor \theta_{gt}/\omega \rfloor$9 sequence, but the trajectory most probable under both encoder and visual constraints.

4. Angle-resolved diffraction as a refinement target

In digital large-angle convergent beam electron diffraction, the refined object is not a single angular parameter but a large angular intensity distribution. D-LACBED is produced by collecting hundreds to thousands of CBED patterns at many beam tilts and assembling them into reflection-resolved angular maps extending beyond $$\theta'=\omega\cdot \operatorname{Argmax}\!\big(\operatorname{Sigmoid}(t'_{\theta_{\text{class}}})\big)+\big(t'_{\theta_{\text{reg.}}}\big)^2.$$0. The forward model uses Bloch-wave dynamical diffraction with a neutral, spherical independent atom model and Fourier components of the crystal potential

$$\theta'=\omega\cdot \operatorname{Argmax}\!\big(\operatorname{Sigmoid}(t'_{\theta_{\text{class}}})\big)+\big(t'_{\theta_{\text{reg.}}}\big)^2.$$1

with absorptive terms added through calculated imaginary scattering factors. Refinement minimizes a global fit index built from zero-mean normalized cross-correlation,

$$\theta'=\omega\cdot \operatorname{Argmax}\!\big(\operatorname{Sigmoid}(t'_{\theta_{\text{class}}})\big)+\big(t'_{\theta_{\text{reg.}}}\big)^2.$$2

so the comparison is patternwise and angle resolved rather than based on a small set of integrated intensities (Hubert et al., 2018).

This angular overdetermination changes what can be refined. In Cu, isotropic Debye–Waller factors are accurate and agree well with X-ray and Mössbauer benchmarks over $$\theta'=\omega\cdot \operatorname{Argmax}\!\big(\operatorname{Sigmoid}(t'_{\theta_{\text{class}}})\big)+\big(t'_{\theta_{\text{reg.}}}\big)^2.$$3 K to $$\theta'=\omega\cdot \operatorname{Argmax}\!\big(\operatorname{Sigmoid}(t'_{\theta_{\text{class}}})\big)+\big(t'_{\theta_{\text{reg.}}}\big)^2.$$4 K. In $$\theta'=\omega\cdot \operatorname{Argmax}\!\big(\operatorname{Sigmoid}(t'_{\theta_{\text{class}}})\big)+\big(t'_{\theta_{\text{reg.}}}\big)^2.$$5-Al$$\theta'=\omega\cdot \operatorname{Argmax}\!\big(\operatorname{Sigmoid}(t'_{\theta_{\text{class}}})\big)+\big(t'_{\theta_{\text{reg.}}}\big)^2.$$6O$$\theta'=\omega\cdot \operatorname{Argmax}\!\big(\operatorname{Sigmoid}(t'_{\theta_{\text{class}}})\big)+\big(t'_{\theta_{\text{reg.}}}\big)^2.$$7, the internal coordinates refine to

$$\theta'=\omega\cdot \operatorname{Argmax}\!\big(\operatorname{Sigmoid}(t'_{\theta_{\text{class}}})\big)+\big(t'_{\theta_{\text{reg.}}}\big)^2.$$8

differing from high-quality X-ray values by only $$\theta'=\omega\cdot \operatorname{Argmax}\!\big(\operatorname{Sigmoid}(t'_{\theta_{\text{class}}})\big)+\big(t'_{\theta_{\text{reg.}}}\big)^2.$$9 and $\rho(\theta)=\frac{2}{\pi}\sin^2\theta$0. By contrast, the IAM DWFs in GaAs and $\rho(\theta)=\frac{2}{\pi}\sin^2\theta$1-Al$\rho(\theta)=\frac{2}{\pi}\sin^2\theta$2O$\rho(\theta)=\frac{2}{\pi}\sin^2\theta$3 disagree strongly with X-ray values, indicating that bonding models are needed. In this literature, refinement of the angular diffraction distribution serves as a high-information route to structural refinement, and the failures are themselves diagnostic of missing physics rather than lack of angular sensitivity.

5. Explicit angle laws, local statistics, and asymptotic structure

In analytic number theory, angle distribution refinement can mean replacing one-point equidistribution by local-statistical laws. For a non-CM elliptic curve, the classical Sato–Tate theorem gives

$\rho(\theta)=\frac{2}{\pi}\sin^2\theta$4

The refinement unfolds the angles via

$\rho(\theta)=\frac{2}{\pi}\sin^2\theta$5

and conjectures Poisson $\rho(\theta)=\frac{2}{\pi}\sin^2\theta$6-th neighbor spacing laws

$\rho(\theta)=\frac{2}{\pi}\sin^2\theta$7

The same section of the literature also asks whether angles are log-integrable random variables in multiplicative ergodic theory. In dimension $\rho(\theta)=\frac{2}{\pi}\sin^2\theta$8, the angle between Oseledets directions of an i.i.d. $\rho(\theta)=\frac{2}{\pi}\sin^2\theta$9 cocycle need not be log-integrable under finite first moment, but it is always log-integrable under finite second moment; outside the i.i.d. regime, the joint distribution of the Oseledets spaces may be chosen arbitrarily (Kimura, 2021, Bochi et al., 6 Mar 2025).

In geometric measure theory and combinatorial geometry, refinement takes the form of measure and multiplicity control. If $p_k(s)=\frac{s^k e^{-s}}{k!},$0 is compact with

$p_k(s)=\frac{s^k e^{-s}}{k!},$1

then every angle is equitably represented in the sense that

$p_k(s)=\frac{s^k e^{-s}}{k!},$2

and the set of determined angles has positive Lebesgue measure. For $p_k(s)=\frac{s^k e^{-s}}{k!},$3 points in convex position in the plane and not all on a circle, the distinct-angle count is bounded below by

$p_k(s)=\frac{s^k e^{-s}}{k!},$4

with the structural dichotomy that if the set determines $p_k(s)=\frac{s^k e^{-s}}{k!},$5 distinct angles, then either $p_k(s)=\frac{s^k e^{-s}}{k!},$6 or there are $p_k(s)=\frac{s^k e^{-s}}{k!},$7 co-circular points [(Iosevich et al., 2011); (Konyagin et al., 2024)].

Topological and meshing contexts supply two more technically different refinements. For an angle-valued map $p_k(s)=\frac{s^k e^{-s}}{k!},$8, the paper on Novikov invariants assigns a finite configuration

$p_k(s)=\frac{s^k e^{-s}}{k!},$9

refining the Novikov–Betti number, together with free $w(\Psi)$0-modules $w(\Psi)$1 refining Novikov homology. In Delaunay mesh refinement, angle quality is controlled algorithmically: with desired minimum angle $w(\Psi)$2 and

$w(\Psi)$3

the method yields size-optimal meshes away from small input angles, while across a small PSLG angle $w(\Psi)$4 the minimum angle tends to

$w(\Psi)$5

and the maximum angle tends to

$w(\Psi)$6

These cases show that refinement may mean either localization of homological information over $w(\Psi)$7 or asymptotic control of the full angle spectrum of a mesh (Burghelea, 2015, Sastry, 2018).

6. Physical angle distributions in radiation, jets, and fission

In synchrotron radiation, refinement of the pitch-angle distribution changes the emitted spectrum itself. Instead of isotropic pitch angles, the electron distribution is taken to be Gaussian in $w(\Psi)$8 with width $w(\Psi)$9. When

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$00

the low-frequency spectrum reaches

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$01

whereas for

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$02

the spectrum becomes broken,

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$03

with

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$04

Here refinement means replacing isotropic angular averaging by a narrower angular law that preserves the single-electron $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$05 asymptote (Yang et al., 2018).

In strongly coupled plasma, angle distribution refinement appears at the ensemble level rather than at the level of single objects. In the holographic jet model, each individual jet broadens as it propagates, but the opening-angle distribution for jets selected in a fixed final-energy bin can move toward smaller angles because wide jets lose more energy and the initial spectrum

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$06

suppresses replenishment from higher energies. The net result is a final distribution with fewer narrow and wide jets, even though every jet in the ensemble broadens (Rajagopal et al., 2016).

The fission-fragment spin problem supplies a quantum version of the same theme. The final state is expanded as

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$07

and the opening angle is extracted from coupled-spin channels through

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$08

Classically, the strict $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$09 planar case gives a flat opening-angle distribution, whereas isotropic $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$10D gives

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$11

Quantum mechanically, however, the distribution is depleted near $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$12 and $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$13; the smallest attainable angle in a spin space truncated at $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$14 is

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$15

The realistic TDFFT/TDDFT opening-angle distribution is intermediate between the 2D and isotropic 3D cases (Scamps, 2023).

Across these physical systems, a common pattern emerges: refining the angular law alters not only a descriptive statistic but the observable response itself. Spectral slopes, ensemble jet shapes, and fragment-spin opening-angle endpoints all depend on how sharply or broadly angle space is populated.

7. Projected observables and engineering angle maps

For transiting hot Jupiters, the central refinement problem is that the observable is not the true spin–orbit angle $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$16 but its sky-plane projection $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$17. In the exact transiting limit,

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$18

which yields the conditional kernel

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$19

on its allowed support, and hence

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$20

The paper analyzes a sample of $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$21 transiting hot Jupiters, of which only $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$22 have $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$23, and concludes that disk-torquing can reproduce the excess of aligned systems if torquing is not always efficient, whereas scattering and Kozai-cycle-with-tidal-friction models cannot by themselves account for all hot Jupiters (Crida et al., 2014).

In interior permanent-magnet synchronous motors, the refined angular object is a control map rather than a probability law. The commutation angle $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$24 is defined as the angle between the fundamental of the phase current and the fundamental of the back-EMF, with

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$25

Because the torque law

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$26

contains a reluctance component, the optimum $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$27 must be mapped over the torque-speed plane under both current and voltage limits. The paper formulates

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$28

for MTPA and

$$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$29

for voltage-constrained operation, and uses a multi-criteria local Latin hypercube refinement system to optimize rotor geometry and angle maps jointly (Asef et al., 5 Mar 2025).

The reported gains are explicitly angle-map related. Relative to a standard LHS-NSGA-II workflow, the MLHR-NSGA-II system reaches the global optimum between the $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$30th and $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$31th generations rather than the $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$32th to $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$33th, reduces computation time from $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$34 min to $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$35 min, and finds the best solution at the $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$36th generation rather than the $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$37th. For the optimized M.V2 rotor, the overall torque-per-commutation-angle factor is $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$38 (reported also as $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$39), versus $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$40 for the Toyota Prius reference. The same design reduces magnet mass by $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$41, raises torque density to $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$42, and gives the lowest overall commutation angle over the operating envelope. In this engineering setting, angle distribution refinement is the construction of a control-optimal $$\theta_\mathrm{initial}=\theta_\mathrm{GT}+\epsilon,$$43-map whose geometry is itself improved by sample-efficient design optimization.