Half-Angle-to-Half-Angle (HA2HA) Methods

• The paper introduces a self-supervised denoising technique for ultrasound microvascular imaging that leverages paired complementary angles to isolate vascular signals from noise.
• HA2HA is a structural motif applied across multiple domains, including point-cloud learning, quantum signal processing, and directional statistics, each adapting the concept to specific data characteristics.
• The framework demonstrates practical benefits, achieving over 15 dB improvement in CNR and SNR for UMI, while its variant implementations highlight tradeoffs in computational cost and domain-specific limitations.

Half-Angle-to-Half-Angle (HA2HA) denotes, in its explicit contemporary usage, a self-supervised denoising framework for ultrasound microvascular imaging (UMI) that constructs paired observations from complementary angular subsets of beamformed blood-flow radio-frequency (RF) data and learns the shared vascular signal while suppressing inconsistent noise (Huang et al., 7 Jul 2025). Closely related mechanisms also appear under different names in other fields: MAP-VAE for 3D point clouds uses “half-to-half prediction” between front and back halves extracted from multiple viewing angles (Han et al., 2019); quantum signal processing (QSP) employs a recursive “halving” decomposition that splits a high-degree QSP object into two lower-degree objects of roughly half the degree (Chao et al., 2020); and directional statistics develops a half-angle principle relating wrapped Cauchy and angular central Gaussian distributions through angle halving or doubling (Kent, 2022). The common thread is not a single standardized algorithm, but a family of constructions in which complementary halves, half-angles, or half-degrees are paired so that invariant structure is retained and nuisance variation is isolated.

1. Terminology, scope, and conceptual variants

The acronym HA2HA is used literally for UMI denoising, where it abbreviates Half-Angle-to-Half-Angle and refers to a mapping between complementary angular subsets of multi-angle plane-wave acquisitions (Huang et al., 7 Jul 2025). In the point-cloud literature, the corresponding mechanism is not named HA2HA in the source paper; it is called “half-to-half prediction”, “half-to-half sequence pair”, and “half-to-half prediction … to infer a sequence of several back halves from the corresponding sequence of the complementary front halves” (Han et al., 2019). In QSP, the term is likewise not literal: the relevant procedure is a “halving” algorithm that recursively splits a degree-$d$ object into two lower-degree pieces (Chao et al., 2020). In directional statistics, the operative concept is the half-angle principle, not the acronym itself (Kent, 2022).

These usages are technically distinct. In UMI, the “half-angle” object is a subset of steering angles. In point-cloud learning, the “half” object is a front or back surface subset relative to a viewpoint. In QSP, the “half” object is a lower-degree factor in a Laurent-polynomial decomposition. In directional statistics, the “half-angle” object is literal angle halving or the reverse operation of angle doubling. This suggests that HA2HA is best understood as a structural motif rather than as a single domain-independent method.

2. HA2HA in ultrasound microvascular imaging

In UMI, HA2HA is a self-supervised deep learning framework for denoising blood-flow RF data in ultrasound microvascular imaging (Huang et al., 7 Jul 2025). It is designed for settings in which unfocused plane-wave imaging must detect very weak blood-flow echoes from red blood cells or microbubbles in deep tissue, producing very low SNR and low CNR between blood flow and background noise. The method is placed after beamforming and after clutter filtering, and before power Doppler / CDI computation.

The operational pipeline is fixed. Raw channel data are acquired with multiple steered plane waves. Beamforming is performed for each transmit angle, yielding angle-dependent RF images. The steering angles are then partitioned into two complementary subsets, and each subset is coherently compounded separately. SVD-based clutter filtering is applied to each compounded subset, producing two blood-flow RF datasets. These two datasets form HA2HA training pairs. At inference time, the trained network is applied not to half-angle data but to full-angle SVD-filtered RF data, after which envelope processing yields power Doppler and phase processing yields color Doppler imaging (CDI) (Huang et al., 7 Jul 2025).

For the pig and human data described in the source, the full set of steering angles is

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$

with $K=10$ angles from $-9^\circ$ to $9^\circ$ with $2^\circ$ steps. HA2HA splits these into

$$\Theta_{\text{odd}} = \{\theta_1,\theta_3,\theta_5,\theta_7,\theta_9\}, \qquad \Theta_{\text{even}} = \{\theta_2,\theta_4,\theta_6,\theta_8,\theta_{10}\}.$$

If $x(\theta_k,r,l,t)$ denotes beamformed RF data for angle $\theta_k$, axial index $r$, lateral index $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$0, and slow time $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$1, coherent compounding on a subset $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$2 is

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$3

After SVD clutter filtering, the two resulting blood-flow RF datasets are modeled as

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$4

where $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$5 is the unknown clean blood-flow RF and $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$6 are treated as approximately zero-mean and independent across complementary angle subsets.

The self-supervised formulation follows the Noise2Noise setting. With a denoiser $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$7, the paper first gives

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$8

together with

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$9

HA2HA extends this to a dual-path objective. With

$K=10$0

the HA2HA loss is

$K=10$1

where $K=10$2 weights a consistency term. The final objective adds $K=10$3 regularization on the weights,

$K=10$4

with $K=10$5 and $K=10$6 in the reported experiments. The key statistical assumption is that vascular structure is nearly identical between complementary angular subsets, whereas electronic noise, thermal noise, angle-dependent sidelobes, and residual clutter variations differ enough between subsets to serve as independent nuisance components.

3. Architecture, training protocol, and reported UMI performance

The denoiser is a 2D U-Net with 4 resolution levels, using two $K=10$7 convolutions, batch normalization, and LeakyReLU at each encoder stage, max pooling for downsampling, bilinear interpolation for upsampling, and skip connections between encoder and decoder (Huang et al., 7 Jul 2025). Inputs are 2D RF patches of size $K=10$8, and outputs are denoised RF patches of the same size. Training uses a dual-path configuration with shared weights: $K=10$9 is compared with $-9^\circ$0, $-9^\circ$1 is compared with $-9^\circ$2, and the two outputs are constrained to be similar.

The reported training data comprise 3 pigs, each with 300 frames of in-vivo contrast-free kidney data. 30 frames are sampled per pig, one every 10 frames, for 90 frames total. The original RF frame size is $-9^\circ$3, lateral interpolation is applied to obtain isotropic resolution with pixel size $-9^\circ$4, and each frame is split into non-overlapping $-9^\circ$5 patches. With 80 paired patches per frame, the total becomes 7200 paired patches. Augmentation consists of flipping and rotation. Training uses PyTorch, AdamW with $-9^\circ$6, $-9^\circ$7, batch size 256, initial learning rate $-9^\circ$8, reduce-on-plateau scheduling with factor $-9^\circ$9 after 10 epochs of plateau, and an NVIDIA Quadro RTX 5000 GPU. Convergence is reported in approximately 2.5 hours. At inference, each full RF frame is processed independently without patch tiling; denoising a $9^\circ$0 blood-flow RF volume takes approximately 67 s.

The reported quantitative evaluation on final power Doppler images uses

$9^\circ$1

$9^\circ$2

$9^\circ$3

Dataset	Conventional CNR / SNR / BNP	HA2HA CNR / SNR / BNP
Pig kidney, contrast-free	$9^\circ$4 dB	$9^\circ$5 dB
Pig kidney, contrast-enhanced	$9^\circ$6 dB	$9^\circ$7 dB
Human liver, contrast-free	$9^\circ$8 dB	$9^\circ$9 dB
Human kidney (CKD), contrast-free	$2^\circ$0 dB	$2^\circ$1 dB

For the contrast-free pig kidney case, AP reports $2^\circ$2 dB CNR, $2^\circ$3 dB SNR, and $2^\circ$4 dB BNP; ST-NLM reports $2^\circ$5 dB CNR, $2^\circ$6 dB SNR, and $2^\circ$7 dB BNP. For the contrast-enhanced pig kidney case, AP reports $2^\circ$8 dB CNR, $2^\circ$9 dB SNR, and $$\Theta_{\text{odd}} = \{\theta_1,\theta_3,\theta_5,\theta_7,\theta_9\}, \qquad \Theta_{\text{even}} = \{\theta_2,\theta_4,\theta_6,\theta_8,\theta_{10}\}.$$0 dB BNP, whereas ST-NLM reports $$\Theta_{\text{odd}} = \{\theta_1,\theta_3,\theta_5,\theta_7,\theta_9\}, \qquad \Theta_{\text{even}} = \{\theta_2,\theta_4,\theta_6,\theta_8,\theta_{10}\}.$$1 dB CNR, $$\Theta_{\text{odd}} = \{\theta_1,\theta_3,\theta_5,\theta_7,\theta_9\}, \qquad \Theta_{\text{even}} = \{\theta_2,\theta_4,\theta_6,\theta_8,\theta_{10}\}.$$2 dB SNR, and $$\Theta_{\text{odd}} = \{\theta_1,\theta_3,\theta_5,\theta_7,\theta_9\}, \qquad \Theta_{\text{even}} = \{\theta_2,\theta_4,\theta_6,\theta_8,\theta_{10}\}.$$3 dB BNP. For the human liver, AP reports $$\Theta_{\text{odd}} = \{\theta_1,\theta_3,\theta_5,\theta_7,\theta_9\}, \qquad \Theta_{\text{even}} = \{\theta_2,\theta_4,\theta_6,\theta_8,\theta_{10}\}.$$4 dB CNR, $$\Theta_{\text{odd}} = \{\theta_1,\theta_3,\theta_5,\theta_7,\theta_9\}, \qquad \Theta_{\text{even}} = \{\theta_2,\theta_4,\theta_6,\theta_8,\theta_{10}\}.$$5 dB SNR, and $$\Theta_{\text{odd}} = \{\theta_1,\theta_3,\theta_5,\theta_7,\theta_9\}, \qquad \Theta_{\text{even}} = \{\theta_2,\theta_4,\theta_6,\theta_8,\theta_{10}\}.$$6 dB BNP, whereas ST-NLM reports $$\Theta_{\text{odd}} = \{\theta_1,\theta_3,\theta_5,\theta_7,\theta_9\}, \qquad \Theta_{\text{even}} = \{\theta_2,\theta_4,\theta_6,\theta_8,\theta_{10}\}.$$7 dB CNR, $$\Theta_{\text{odd}} = \{\theta_1,\theta_3,\theta_5,\theta_7,\theta_9\}, \qquad \Theta_{\text{even}} = \{\theta_2,\theta_4,\theta_6,\theta_8,\theta_{10}\}.$$8 dB SNR, and $$\Theta_{\text{odd}} = \{\theta_1,\theta_3,\theta_5,\theta_7,\theta_9\}, \qquad \Theta_{\text{even}} = \{\theta_2,\theta_4,\theta_6,\theta_8,\theta_{10}\}.$$9 dB BNP. For the CKD kidney, AP reports $x(\theta_k,r,l,t)$0 dB CNR, $x(\theta_k,r,l,t)$1 dB SNR, and $x(\theta_k,r,l,t)$2 dB BNP, whereas ST-NLM reports $x(\theta_k,r,l,t)$3 dB CNR, $x(\theta_k,r,l,t)$4 dB SNR, and $x(\theta_k,r,l,t)$5 dB BNP. The source summarizes the aggregate outcome as “an improvement exceeding 15 dB in both CNR and SNR” across datasets (Huang et al., 7 Jul 2025).

Qualitatively, the reported findings are that HA2HA suppresses noisy background between microvascular branches and in pure background regions, reveals fine vessels that are not visible or are only weakly visible with conventional processing, and improves CDI by reducing spurious high velocities in non-vascular regions. In line profiles across vessels, the denoised CDI preserves a parabolic profile consistent with laminar flow. The method is also evaluated under a variable-duty-cycle SNR study. For DC $x(\theta_k,r,l,t)$6, HA2HA yields the highest CNR, highest SNR, and lowest BNP; at DC $x(\theta_k,r,l,t)$7, AP marginally outperforms HA2HA. The source therefore identifies extremely low SNR as a failure regime, and also reports that training with Single-Angle-to-Single-Angle (SA2SA) pairs significantly degrades performance relative to HA2HA, indicating that training-pair SNR matters.

4. Half-to-half prediction in multi-angle point-cloud learning

In the point-cloud literature, the closest HA2HA analogue is MAP-VAE’s “half-to-half prediction”, defined on 3D point clouds observed from multiple viewing angles (Han et al., 2019). Let $x(\theta_k,r,l,t)$8 be a point cloud with 2048 points. The method chooses $x(\theta_k,r,l,t)$9 viewpoints evenly spaced on a circle around the shape in the horizontal plane, with $\theta_k$0 in the experiments. For each viewpoint $\theta_k$1, the point cloud is split into a front half $\theta_k$2 and a back half $\theta_k$3. A naive Euclidean-distance split was found to produce non-contiguous, non-semantic regions, so the paper uses geodesic splitting: the Euclidean nearest point $\theta_k$4 to the viewpoint is found, geodesic distances from $\theta_k$5 to all points are computed using the algorithm of Crane et al. [62], and the front half is defined as $\theta_k$6 plus its nearest $\theta_k$7 points in geodesic distance while the back half consists of the farthest $\theta_k$8 points. In the experiments, each half has $\theta_k$9 points.

Across several angles, the method builds a front-half sequence

$r$0

and a back-half sequence

$r$1

with elements aligned by angle. A training sample is

$r$2

and from one point cloud the procedure generates $r$3 such samples by sliding the starting angle. Experimentally, one sequence uses $r$4 angles, uniformly spaced over the circle rather than contiguous; this was reported as the best tradeoff.

The prediction is performed in feature space, not directly in xyz coordinates. A local encoder, with the same architecture as a PointNet++ encoder but operating on $r$5 points, maps $r$6 and $r$7 to local features $r$8 and $r$9. A global encoder, also PointNet++ but operating on $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$00 points, maps the full point cloud $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$01 to a global feature $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$02. These are aggregated by an RNN denoted $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$03, implemented as a GRU with hidden dimension 512, run for $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$04 steps: step 0 takes $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$05, and steps $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$06 take the front-half features $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$07 in circular order. The final hidden state is the angle-specific feature $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$08.

A second RNN, $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$09, predicts the back-half feature sequence $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$10. The paper states that at each of $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$11 steps, $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$12 predicts one back half in the same order as the ground-truth sequence, thereby learning half-to-half correspondence and spatial relationships among halves. The prediction loss is

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$13

This local self-supervision is combined with a VAE-style reconstruction branch. From $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$14, the model predicts $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$15 and $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$16, samples

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$17

and reconstructs the full point cloud using Earth Mover’s Distance plus KL regularization. The full objective is

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$18

with $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$19 for the KL weight and $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$20 in most experiments. After training, angle-specific features $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$21 are max-pooled over all $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$22 starting angles to produce the final global representation.

The ablation study reports 94.82\% classification accuracy on ModelNet10 for the full model, compared with 92.40\% for “No $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$23”, 91.63\% for “No $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$24”, 93.17\% for “No KL”, 92.29\% for “AE”, 93.28\% for “VAE”, and 93.61\% for “Eucli”. For global features, MAP-VAE reports 90.15\% on ModelNet40 and 94.82\% on ModelNet10. For local features, the reported mean IoU is 67.95\%, compared with 57.04\% and 56.28\% for two LGAN variants. The paper attributes the gains to the combination of multi-angle half-to-half prediction and global reconstruction. It also notes several limitations implicit in the design: viewpoint dependence, geodesic computation cost, a fixed single-circle view configuration, and the fact that prediction is in feature space rather than in actual point coordinates.

5. Recursive halving in quantum signal processing

In QSP, the mechanism closest in spirit to HA2HA is the halving algorithm for machine-precision angle finding (Chao et al., 2020). The setting begins with a signal unitary $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$25, its controlled version

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$26

and a sequence of single-qubit unitaries $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$27. The generic QSP circuit has the form

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$28

and in the one-dimensional case $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$29, the top-left block is a Laurent polynomial

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$30

The angle-finding problem is to determine a QSP phase sequence that realizes a target $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$31.

Earlier decomposition procedures, especially Haah’s carving method, peel off one degree-1 factor at a time. The contribution of the halving method is to split a degree-$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$32 unitary parity element $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$33 into two factors of roughly half degree instead of removing one factor sequentially. If $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$34, the method seeks a unitary $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$35 of degree $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$36 such that $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$37 has degree at most $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$38. The key algebraic result is that under the system

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$39

there is a unique solution $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$40, and $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$41 is itself unitary and remains in the Low algebra when $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$42 does. This converts factorization into a balanced binary recursion rather than a linear chain.

The source stresses that this is not based on explicit trigonometric half-angle identities such as $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$43. The “half-angle” interpretation is structural: polynomial degree, and therefore phase-sequence length, is repeatedly halved. The practical algorithm solves a block-banded Toeplitz linear system in least-squares form at each recursion level. Its preprocessing step, capitalization, adds highest- and lowest-degree terms of order $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$44 to the target Laurent polynomial so that leading coefficients are not below the floating-point noise floor. Completion then constructs $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$45 such that

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$46

after which recursive decomposition proceeds.

The reported empirical outcome is that the algorithm finds more than 3000 angles within 5 minutes in standard double precision arithmetic. For Hamiltonian simulation with $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$47 and $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$48, the paper reports simulation up to $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$49, corresponding to a Laurent polynomial of degree 3261, in under 5 minutes on a laptop. The runtime is described as cubic in degree, and the achievable region in $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$50 space is larger than that of carving. The limitations are also explicit: there is no rigorous global stability proof, the completion step depends on a root pairing strategy that is currently randomized, pathological instances exist for small degrees, and the present implementation remains $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$51. The paper further points to a possible hybrid direction with the optimization-based QSP angle-finding method of Dong et al. (Dong et al., 2020).

6. The half-angle principle in directional statistics

In directional statistics, the half-angle principle formalizes literal angle halving and doubling on the circle and on higher-dimensional spheres (Kent, 2022). On the circle $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$52, an angle $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$53 is represented by

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$54

The half-angle variable is

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$55

and the paper imposes

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$56

Angle doubling is implemented by the squaring map

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$57

so that if $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$58, then $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$59. In complex notation this is $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$60, while angle halving corresponds set-theoretically to $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$61.

The core probabilistic identification is between the wrapped Cauchy distribution and the angular central Gaussian distribution on the circle. The wrapped Cauchy density is

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$62

The circular ACG density with $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$63 is

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$64

Their parameters match under

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$65

If $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$66 and $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$67, then $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$68 with this parameter mapping. The paper also derives the fundamental diagonal $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$69 identity

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$70

where $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$71 is a rescaled diagonal linear transformation and $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$72 is a diagonal Möbius-type transformation. This identity intertwines linear transformation at the half-angle level with Möbius transformation after angle doubling.

The higher-dimensional extension fixes a north pole $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$73 on $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$74 and writes

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$75

The gnomonic projection uses $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$76, whereas the stereographic projection uses $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$77. Under angle doubling, $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$78, so the two projections coincide at the Euclidean-coordinate level. This yields a geometric interpretation of stereographic projection through angle halving.

The distributional behavior becomes more intricate beyond the circle. The general ACG density on $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$79 is

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$80

and its gnomonic projection is a multivariate Cauchy distribution in $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$81. The spherical Cauchy density is

$\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$82

and its stereographic projection is a multivariate $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$83 distribution with $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$84. The paper explicitly states that when $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$85, the SC distribution can never be identified with the ACG distribution under angle doubling. The clean wrapped-Cauchy–ACG equivalence is therefore specific to the circle.

7. Comparative interpretation, misconceptions, and limits of the term

Several misconceptions are ruled out by the source material. First, HA2HA is not a universally standardized acronym across all four literatures. It is explicit in UMI, while the point-cloud, QSP, and directional-distribution papers use the terms half-to-half prediction, halving, and half-angle principle, respectively (Huang et al., 7 Jul 2025). Second, in UMI, training uses half-angle pairs but inference uses full-angle SVD-filtered RF data, so the deployment pathway is not a half-angle-to-half-angle mapping at test time. Third, in MAP-VAE, the prediction target is feature space rather than xyz coordinates, so the mechanism is not direct geometric completion of back-half point sets. Fourth, in QSP, the relevant operation is degree halving, not a trigonometric half-angle identity. Fifth, in directional statistics, the wrapped-Cauchy–ACG identification is a circle-specific phenomenon and does not extend unchanged to $\Theta = \{\theta_1,\theta_2,\dots,\theta_K\},$86.

The limitations are similarly domain-specific. UMI HA2HA degrades at extremely low SNR and is currently 2D, denoising-only, without temporal convolutions, super-resolution, or deconvolution. MAP-VAE’s half-to-half formulation depends on a chosen viewpoint circle, incurs geodesic-splitting cost, and uses a fixed number of views. QSP halving has strong algebraic uniqueness guarantees but lacks a full finite-precision condition-number theory and remains cubic in degree. The half-angle principle in directional distributions extends geometrically to higher dimensions, but the elegant probabilistic equivalence present on the circle is lost.

Taken together, these usages suggest a recurrent methodological pattern: complementary halves are constructed so that the desired structure is shared across them, while noise, hidden geometry, or algebraic complexity varies. In UMI, this supports label-free RF denoising; in point-cloud learning, local self-supervision across views; in QSP, stable phase extraction by balanced recursive factorization; and in directional statistics, exact parameter matching under angle halving or doubling. The term therefore names a specific denoising framework in one field and, more broadly, a recognizable family of half-structured constructions across several technically unrelated areas.