arxiv: 2604.24415 · v2 · submitted 2026-04-27 · 💻 cs.SI · cs.CV· eess.SP

Recognition: unknown

Phase-Separated Complex Hilbert PCA on Markerless 3D Pose Estimation Data: A Global Phase Network and Its Extension to a Continuous Field on the Body Surface

Hiromitsu Goto , Tao Tao , Zheng-Lin Chia

Authors on Pith no claims yet

Pith reviewed 2026-05-07 17:43 UTC · model grok-4.3

classification 💻 cs.SI cs.CVeess.SP

keywords Complex Hilbert PCAmarkerless 3D pose estimationwhole-body coordinationphase analysishammer strikingglobal phase networkcontinuous phase fieldsports biomechanics

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The pith

Phase-separated Complex Hilbert PCA extracts a trunk-anchored global phase pattern from markerless 3D pose data in hammer striking.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies Complex Hilbert Principal Component Analysis separately to the backswing and downswing phases of hammer-striking motions captured by markerless 3D pose estimation. It extracts the dominant whole-body coordination pattern as a single complex eigenvector for each phase and extends the representation to a continuous phase field across 1,079 body-surface mesh vertices. This approach moves beyond pairwise joint analysis to reveal a unified kinematic chain anchored at the trunk, with measurable differences in mode contribution and consistency between preparation and execution. A reader would care because the method works without force plates or lab setups and links phase structure to kinetic energy mobilization variance.

Core claim

Applying Complex Hilbert Principal Component Analysis separately to backswing and downswing phases on markerless 3D pose estimation data extracts the dominant whole-body phase pattern as a single complex eigenvector. On 14 hammer-striking trials the analysis shows a trunk-anchored global phase architecture, a functional asymmetry quantified by Mode-1 contribution of 45.5 percent in backswing versus 70.5 percent in downswing and inter-trial Spearman consistency of 0.38 versus 0.58, and consistent reorganization across both skeletal joints and mesh vertices at p less than 10 to the minus 10. Pairwise phase differences from the Mode-1 eigenvector correlate with continuous relative phase on all

What carries the argument

Phase-separated Complex Hilbert Principal Component Analysis (CHPCA) that extracts one complex eigenvector per phase to represent the dominant whole-body phase pattern, extended from skeletal joints to a continuous field on 1,079 mesh vertices.

If this is right

Whole-body coordination can be summarized by a single complex eigenvector per phase instead of separate adjacent-joint calculations.
Preparation and execution phases exhibit different degrees of mode dominance and trial-to-trial stability.
The same phase reorganization appears on both joint skeletons and body-surface mesh vertices.
Mode-1 amplitude correlates with kinetic-energy mobilization variance only during the downswing.
The pipeline requires no force plates or inertial parameters and runs on outdoor markerless recordings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The trunk-anchored pattern suggests that interventions targeting trunk timing could alter overall coordination more efficiently than distal-joint corrections.
Extending the continuous mesh field to multiple subjects would allow statistical comparison of phase networks across athletes or skill levels.
Because the method links phase structure to kinetic variance without direct force measurement, it could serve as a surrogate monitor in field settings.
Automatic segmentation without strike-count priors opens the possibility of applying the same pipeline to other cyclic or repetitive sports motions.

Load-bearing premise

The fully automatic signal-based phase segmentation accurately partitions each trial into backswing and downswing without introducing artifacts from the markerless pose estimator.

What would settle it

Recomputing the Mode-1 eigenvectors and their contribution percentages on the same trials but with manually verified phase boundaries or with data from an independent pose-estimation system would falsify the claim if the reported asymmetry and trunk-anchored pattern disappear.

Figures

Figures reproduced from arXiv: 2604.24415 by Hiromitsu Goto, Tao Tao, Zheng-Lin Chia.

**Figure 1.** Figure 1: Diagnostic plot for the automatic phase segmentation. Top: speed norm of the right view at source ↗

**Figure 2.** Figure 2: Scree plots of the phase-separated CHPCA on the skeleton. Blue: observed eigen view at source ↗

**Figure 3.** Figure 3: Phase-separated Hodge potentials of the 20 joints (Approach A). Blue: backswing. view at source ↗

**Figure 4.** Figure 4: Network representations of the kinematic chain. Left column: conventional CRP (bone view at source ↗

**Figure 5.** Figure 5: Scatter plot of pairwise phase differences between CRP and CHPCA Mode 1 (all 190 view at source ↗

**Figure 6.** Figure 6: Scatter plots of CHPCA Mode-1 amplitude Ai against the energy-mobilisation variance proxy Var(s 2 i ) on the 20-joint skeleton. Left: backswing. Middle: downswing. Right: full duration pooled. Horizontal axis is log-scaled. During the downswing the two quantities are strongly aligned, whereas during the backswing no correspondence is observed. The same asymmetry holds on the 1,079-vertex mesh ( view at source ↗

**Figure 7.** Figure 7: Pairwise phase-difference comparison between the Hilbert-transform energy phase view at source ↗

**Figure 8.** Figure 8: Per-joint Mode-1 phase from the speed-norm CHPCA ( view at source ↗

read the original abstract

Quantitative analysis of the kinematic chain in sports motion is essential for performance evaluation and injury prevention. Conventional methods such as the kinematic-sequence (KS) and continuous relative phase (CRP) are confined to adjacent joint pairs and lack a unified framework for whole-body coordination, while segmental power-flow analysis requires force plates and inertial parameters that restrict it to laboratory environments. We apply Complex Hilbert Principal Component Analysis (CHPCA) separately to each motion phase (backswing and downswing) on markerless 3D pose estimation data, extracting the dominant whole-body phase pattern as a single complex eigenvector. The pipeline further includes a fully automatic signal-based phase segmentation (no priors on strike count or rest location) and an extension to 1,079 body-surface mesh vertices, so that the kinematic chain is represented as a continuous phase field across the body. On 14 hammer-striking trials of a single subject, the framework reveals (i) a trunk-anchored global phase architecture, (ii) a functional asymmetry between preparation and execution phases quantified by Mode-1 contribution (45.5% vs. 70.5%) and inter-trial Spearman consistency (0.38 vs. 0.58), and (iii) a consistent reorganisation across both skeletal joints and mesh vertices ($p < 10^{-10}$ on 1,079 vertices). As a methodological consistency check, pairwise phase differences from the Mode-1 eigenvector are compared against CRP on all 190 joint pairs by a permutation test ($\rho = 0.473$, $p = 0.0005$). A correspondence analysis between Mode-1 amplitude and kinetic-energy mobilisation variance further shows a strong positive correlation in the downswing ($\rho \approx 0.71$ on both skeleton and mesh) and no correlation in the backswing, indicating that the proposed framework bridges kinematic and kinetic descriptions of coordination through phase structure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a workable pipeline for phase-separated CHPCA on markerless pose data with mesh extension, but single-subject results and unvalidated automatic segmentation limit how far the reported asymmetries can be trusted.

read the letter

The main takeaway is a method that splits hammer-strike motion into backswing and downswing using only the signal itself, runs complex Hilbert PCA on each phase separately, pulls out the dominant whole-body eigenvector, and maps it onto a 1079-vertex surface mesh. On 14 trials from one subject they find a trunk-anchored pattern that changes between phases, with Mode-1 explaining more variance and showing higher trial-to-trial consistency in the downswing, plus a reorganization that reaches p < 10^{-10} on the mesh vertices. They also run a permutation check against continuous relative phase on all joint pairs and report a moderate correlation with kinetic-energy variance in the downswing only. That pipeline and the mesh extension are the genuinely new pieces; CHPCA itself is not new, but applying it this way to markerless whole-body data with automatic phasing is a fresh combination. The CRP permutation test and the kinetic correlation are useful internal checks that give the eigenvectors some external grounding. The soft spots are straightforward. Everything rests on one person and 14 trials, so the specific percentages (45.5 % vs 70.5 %) and Spearman values are descriptive rather than general. The automatic phase segmentation has no reported comparison to manual labels or marker-based ground truth, and markerless pose estimators are known to introduce several-centimeter errors during rapid motion; those errors can shift phase boundaries and get amplified by the Hilbert transform. The stress-test concern about boundary artifacts therefore lands. The skeleton-to-mesh consistency is interesting but still depends on the same processing steps. This is for biomechanists and sports scientists who already work with pose data and want a global phase view instead of pairwise CRP. A reader who wants to adapt the code or try the mesh extension on their own striking or throwing data will get concrete value from the description. It deserves peer review because the method is clear enough to be tested and the authors have included at least one external consistency check; referees can push for multi-subject replication and segmentation validation without starting from zero.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces Complex Hilbert Principal Component Analysis (CHPCA) applied separately to automatically segmented backswing and downswing phases of hammer-striking motions from markerless 3D pose estimation data. It extracts a dominant Mode-1 complex eigenvector as a global phase pattern (trunk-anchored), reports functional asymmetry via Mode-1 variance (45.5% backswing vs 70.5% downswing) and inter-trial Spearman consistency (0.38 vs 0.58), demonstrates consistent reorganisation on skeletal joints and 1,079 mesh vertices (p < 10^{-10}), and validates pairwise phase differences against continuous relative phase (CRP) via permutation test (ρ = 0.473, p = 0.0005). It further correlates Mode-1 amplitude with kinetic-energy mobilisation variance (ρ ≈ 0.71 in downswing).

Significance. If the automatic segmentation and pose data prove robust, the framework offers a unified whole-body phase representation that extends beyond pairwise CRP or kinematic-sequence methods, with the mesh extension providing a continuous surface field and the kinetic correlation bridging kinematic-kinetic descriptions. The CRP permutation check is a positive methodological strength for eigenvector fidelity.

major comments (3)

[Methods] Methods (automatic phase segmentation): The fully automatic signal-based segmentation (no priors on strike count or rest location) is load-bearing for all phase-specific claims, yet the manuscript provides no validation against manual labels, marker-based ground truth, or sensitivity analysis to boundary jitter. The Hilbert transform is known to be sensitive to such boundaries, and markerless estimators can introduce several-cm errors in rapid motions; this directly risks artifacting the reported Mode-1 asymmetry (45.5% vs 70.5%) and consistency values.
[Results] Results (single-subject design): All central findings—the trunk-anchored architecture, phase asymmetry, p < 10^{-10} reorganisation on 1,079 vertices, and CRP correspondence—are based on 14 trials from one subject. While inter-trial Spearman consistency is quantified, the absence of multi-subject replication or cross-validation leaves open whether these patterns reflect general coordination structure or subject- and estimator-specific features.
[Data and Methods] Data acquisition and preprocessing: No quantitative metrics are reported for the markerless 3D pose estimator's accuracy during hammer strikes (e.g., joint RMSE or temporal jitter). Because CHPCA eigenvectors and subsequent permutation tests depend on clean phase time series, uncharacterised reconstruction noise could inflate or distort the observed skeleton-to-mesh consistency and the ρ = 0.473 CRP match.

minor comments (2)

[Abstract] Abstract and Results: The reported percentages (45.5%, 70.5%) and correlations lack error bars or confidence intervals; adding these (or bootstrap estimates) would strengthen interpretability of the asymmetry claims.
[Results] The kinetic-energy correspondence analysis is mentioned but lacks explicit description of how kinetic energy mobilisation variance was computed from the pose data (e.g., velocity derivatives, mass assumptions); a brief methods paragraph or equation would aid reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the scope and limitations of our work. We respond point-by-point to the major comments below, indicating where revisions will be made.

read point-by-point responses

Referee: [Methods] Methods (automatic phase segmentation): The fully automatic signal-based segmentation (no priors on strike count or rest location) is load-bearing for all phase-specific claims, yet the manuscript provides no validation against manual labels, marker-based ground truth, or sensitivity analysis to boundary jitter. The Hilbert transform is known to be sensitive to such boundaries, and markerless estimators can introduce several-cm errors in rapid motions; this directly risks artifacting the reported Mode-1 asymmetry (45.5% vs 70.5%) and consistency values.

Authors: We agree that validating the automatic segmentation is important given the sensitivity of the Hilbert transform. Manual labels and marker-based ground truth are not available for this dataset, but we will add a sensitivity analysis in the revised Methods by perturbing segmentation boundaries by small amounts (e.g., shifting by 3-7 frames) and re-running CHPCA to assess stability of the Mode-1 variance (45.5% vs 70.5%) and Spearman consistency values. This will directly test for boundary-induced artifacts. revision: partial
Referee: [Results] Results (single-subject design): All central findings—the trunk-anchored architecture, phase asymmetry, p < 10^{-10} reorganisation on 1,079 vertices, and CRP correspondence—are based on 14 trials from one subject. While inter-trial Spearman consistency is quantified, the absence of multi-subject replication or cross-validation leaves open whether these patterns reflect general coordination structure or subject- and estimator-specific features.

Authors: This is a single-subject study designed to characterize intra-individual phase architecture and trial-to-trial consistency in detail, as quantified by the reported Spearman values. We will revise the Discussion to explicitly frame the single-subject design as a limitation and to state that the trunk-anchored pattern and phase asymmetry require multi-subject replication to establish generality. The internal statistical evidence (p < 10^{-10} reorganisation and CRP permutation test) supports reliability within the subject studied. revision: partial
Referee: [Data and Methods] Data acquisition and preprocessing: No quantitative metrics are reported for the markerless 3D pose estimator's accuracy during hammer strikes (e.g., joint RMSE or temporal jitter). Because CHPCA eigenvectors and subsequent permutation tests depend on clean phase time series, uncharacterised reconstruction noise could inflate or distort the observed skeleton-to-mesh consistency and the ρ = 0.473 CRP match.

Authors: We will expand the Methods and Discussion sections to include published accuracy benchmarks for the markerless estimator in dynamic upper-body tasks and to acknowledge that hammer-strike-specific RMSE was not quantified here. We note that the observed inter-trial consistency, the highly significant mesh reorganisation, and the CRP correspondence provide indirect support that noise did not dominate the extracted modes, but we will add this as an explicit limitation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; pipeline applies standard CHPCA to independently segmented phases with external CRP validation

full rationale

The paper's core chain—automatic signal-based phase segmentation (no priors), separate CHPCA per phase to extract Mode-1 complex eigenvectors, statistical reporting of variance contributions and Spearman consistencies, plus permutation test of pairwise phases against CRP—does not reduce any claimed result to its inputs by construction. Mode-1 contribution percentages and inter-trial correlations are direct empirical outputs from the data after segmentation and decomposition; they are not tautological. The CRP permutation test (ρ=0.473) supplies an independent external check rather than a self-referential loop. No self-citations, uniqueness theorems, or ansatzes are invoked as load-bearing justifications for the trunk-anchored architecture or skeleton-to-mesh reorganisation claims. The derivation remains self-contained against the stated benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard assumptions of Hilbert transform for instantaneous phase and PCA for dominant mode extraction; no explicit free parameters are fitted beyond data-driven eigenvector selection, and the global phase network is an output rather than an invented entity with independent evidence.

axioms (2)

domain assumption Complex Hilbert PCA extracts meaningful dominant phase relationships from multivariate kinematic time series
Invoked when treating the complex eigenvector as the global phase architecture.
domain assumption Automatic signal-based segmentation can reliably separate backswing from downswing without external priors
Required for the phase-separated analysis described in the abstract.

pith-pipeline@v0.9.0 · 5673 in / 1445 out tokens · 63117 ms · 2026-05-07T17:43:44.435301+00:00 · methodology

discussion (0)

Reference graph

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