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REVIEW 2 major objections 6 minor 59 references

Where Is The Ball: 3D Ball Trajectory Estimation From 2D Monocular Tracking

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read From one 2D ball track, an LSTM pipeline recovers the 3D bouncing trajectory — bounces and hits included — and does so from simulation-only training: 0.63 m landing error on broadcast tennis, 0.03 NRMSE on single-launch tests.

desk verdict Solid geometric contribution, but the SOTA claim rests on a TrackNet comparison that is not yet apples-to-apples; the method itself deserves careful review. read the letter →

arxiv 2506.05763 v1 pith:ZMCQKIJ4 submitted 2025-06-06 cs.CV

Puntawat Ponglertnapakorn , Supasorn Suwajanakorn This is my paper
classification cs.CV
keywords 3Dballtrajectoryestimationmonocularvideoheightpredictionplane-pointrayparameterizationsim-to-realgeneralizationLSTMsportsanalysisbouncingphysics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Recovering where a ball is in 3D from a single fixed camera is ambiguous: many 3D paths project onto the same 2D track. The paper's claim is that the ambiguity can be resolved when each 2D point is re-expressed as a camera-independent pair of plane intersections, the sequence starts and ends on the ground, and the network is trained on simulated bouncing footage. With those ingredients, a simple LSTM-based pipeline handles multiple continuous trajectories — several bounces and hits in a rally — without segmenting the motion into projectile arcs, and it transfers from simulation to real tennis, soccer, and ping-pong footage. The headline numbers are 0.63 m average landing error on the TrackNet tennis benchmark versus 3.58 m for SynthNet, and 0.03 normalized root-mean-square error (NRMSE) on single-launch trajectories versus 0.11 for a physics-based baseline and 1.02 for a learning baseline. If the claim holds, broadcast monocular video becomes a workable source of 3D ball motion for sports analysis and virtual replay.

What carries the argument

The load-bearing object is the plane-point ray parameterization $P = (p_{\text{ground}}, p_{\text{vertical}})$: each 2D track point is back-projected into a 3D viewing ray and represented by the ray's intersections with the ground plane ($y=0$) and a vertical plane such as the court's net plane, yielding a four-vector that is invariant to camera location, orientation, and focal length, so one network serves many viewpoints. This representation converts the ill-posed 3D-from-2D regression into per-frame height prediction: choosing the height $h$ along a ray uniquely determines the 3D point, so reprojection consistency is guaranteed by construction. Heights are produced by two unidirectional LSTMs that accumulate temporal height differences in opposite directions, both initialized to zero because the sequence begins and ends on the ground, then blended by the ramp sum $h_t = (1-w_t)h_t^f + w_t h_t^b$ with $w_t = (t-1)/(N-1)$, which confines forward integration error to the start and backward error to the end. An end-of-trajectory LSTM estimates the probability $\varepsilon_t$ that the current motion ends at frame $t$, used as a soft boundary signal rather than a hard segmentation, and a residual-style refinement LSTM predicts deltas to the lifted 3D coordinates so that real-world 2D tracking noise is not baked into the trajectory.

What would settle it

Take an untrimmed real broadcast clip that starts mid-rally, with the first frame's true height known to be roughly one to two meters, and ends before any ground bounce, then run the pipeline with all settings unchanged; if the reported 0.63 m landing error depends on the zero-height anchors, the predicted heights and landing positions should deteriorate measurably as the true initial height grows, and measuring that error increase as a function of initial height would settle whether the ground-endpoint assumption rather than the learned motion prior carries the reconstruction.

Watch

Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that 3D ball trajectory estimation from a 2D monocular track reduces to height prediction along known viewing rays. Each 2D position is back-projected to a 3D ray and reparameterized as the ray's intersections with the ground plane ($y=0$) and a vertical plane such as the net plane, giving a four-vector $P = (p_{\text{ground}}, p_{\text{vertical}}) \in \mathbb{R}^4$ that is independent of the camera's location, orientation, and focal length. The height network integrates temporal height differences with two LSTMs running forward and backward, both seeded at zero because the first and last frames are on the ground, and merges them with the ramp sum $h_t = (1-w_t)h_t^f + w_t h_t^b$, where $w_t = (t-1)/(N-1)$; because a height on a ray uniquely fixes the 3D point, every predicted point reprojects exactly to its input pixel. An end-of-trajectory network supplies soft boundary probabilities instead of hard segmentation, and a refinement network predicts coordinate corrections $(\delta x_t, \delta y_t, \delta z_t)$ so that noisy real-world 2D tracking is not forced into the reconstructed path. Trained purely on the PhysX physics engine used for simulation, the method is reported to generalize to real footage and to exceed prior work on every comparison: on the TrackNet benchmark its average landing error is 0.63 m against 3.58 m for SynthNet, and on synthetic single-launch trajectories its NRMSE is 0.03 against 0.11 for Shen et al. and 1.02 for Mocanu et al.

Load-bearing premise

The load-bearing premise is that the input sequence begins and ends with the ball on the ground plane ($y=0$), which seeds both height accumulators at zero; if the first or last frame is airborne, the integrated height error propagates through the entire sequence, and the paper notes that real clips may need trimming, or a learned initial height in future work, to satisfy it.

Editorial extensions

If this is right

  • Broadcast monocular footage of tennis and soccer becomes a viable source of 3D ball motion: with camera calibration and sequences trimmed to ground contacts, no shadow cues, ball-size estimates, multi-camera triangulation, or projectile segmentation are required.
  • A full rally with several bounces and hits can be reconstructed in one pass, whereas physics-based baselines that fit separate projectile arcs break down when segmentation is noisy or when motion includes lateral drift and spin-like curvature.
  • Because the input representation is invariant to camera parameters, a single network trained once can be applied across multiple viewpoints and camera setups without per-camera retraining.
  • The pipeline tolerates imperfect 2D tracking: across tested noise levels up to 25 pixels, its reported NRMSE rises only from 0.03 to 0.14 on single-launch sequences, whereas the physics-based baseline degrades from 0.11 to 0.64.
  • Training entirely on synthetic data suffices for real-world generalization, and fine-tuning on a small real motion-capture set improves the reported real-data NRMSE from 0.17 to 0.08.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: nothing in the pipeline is tennis-specific, so the same ground/vertical-plane hierarchy and ramp-sum height accumulation should apply to other ground-anchored bouncing objects, such as basketballs, volleyballs, and table-tennis balls, provided sequences can be trimmed to contacts and the camera is calibrated.
  • Editorial inference: the paper closes by suggesting a separate initial-height predictor as the fix for its trimming requirement; a concrete extension would train the forward and backward accumulators to start from a learned height prior instead of zero, letting unedited mid-rally clips run as-is.
  • Editorial inference: because the reported fine-tuned error of 0.08 NRMSE is roughly half the simulation-only error of 0.17, a natural test is to measure how few real sequences close the sim-to-real gap, which would quantify how much of the result is carried by the learned motion prior rather than the synthetic training distribution.
  • Editorial inference: the plane-point representation assumes no viewing ray is near-parallel to the ground or the vertical plane, which ties the method to high, downward-looking cameras; low, level sideline shots would violate the intersection scheme and constitute an untested failure mode.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper presents a learning-based pipeline for estimating the 3D trajectory of a bouncing ball from a monocular 2D tracking sequence, assuming known camera parameters and that the sequence begins and ends with the ball on the ground plane. The pipeline converts each 2D point to a viewing ray parameterized by its intersections with the ground plane and a vertical plane, predicts end-of-trajectory probabilities with an LSTM, then estimates per-frame heights via forward and backward LSTM accumulators anchored to zero at both endpoints, followed by a refinement LSTM that outputs 3D coordinate corrections. The networks are trained jointly on simulated PhysX data using reconstruction, below-ground, and EoT losses, and are evaluated on four synthetic and three real datasets. The central claims are that the canonical plane-point representation enables generalization across viewpoints and that, despite training only on simulation, the method achieves state-of-the-art accuracy, in particular a 0.63 m average landing error versus 3.58 m for SynthNet on Real TrackNet.

Significance. If the claims are substantiated, this is a valuable contribution to monocular sports analysis: it avoids explicit projectile segmentation, handles multiple bounces and hits in a single sequence, and provides a viewpoint-invariant input representation. The height-on-ray parameterization is an elegant way to enforce projection consistency at the coarse stage, and the ablations in Tables 1 and 2 systematically justify the main design choices. The planned release of the real mocap dataset and source code would be useful to the community. However, the headline state-of-the-art claim rests on a comparison with SynthNet whose evaluation protocol is not fully specified; until that comparison is placed on a common footing, the empirical significance cannot be fully assessed.

major comments (2)
  1. [Section 4.2.1 / Table 4 / Appendix C.7] The state-of-the-art claim against SynthNet is not verifiable from the information provided. The Ours column of Table 4 is produced with the authors' own PnP camera calibration, the authors' preprocessing (including any trimming needed to satisfy the ground-start/ground-end assumption), and contact-point ground truth generated by their own ray-tracing from TrackNet's labels, while the SynthNet column is taken verbatim from [11]. The manuscript does not identify the TrackNet clips used, the trimming rule, the number of rallies per game, or the exact contact-point annotation frames, and Appendix C.7 describes the Real TrackNet evaluation as 'qualitative' on 118 trajectories from 13 clips in one match, which is difficult to reconcile with Table 4's per-game numbers for 10 games. Moreover, SynthNet's published numbers may have been obtained under a different calibration or a different set of 2D tracks. Please specify the exact evaluation protocol and, ideally, rerun SynthNet's released code on the same preprocessed inputs; otherwise the reported 5.7x improvement may be a test-set artifact.
  2. [Section 3.1.3 / Eq. (3) / Section 5] The height prediction relies on forward and backward accumulators that are anchored to zero at the sequence endpoints (hf_0 = 0, hb_N = 0), and the ramp-sum in Eq. (3) forces the output height at the last frame to be zero by construction. This makes the assumption that both endpoints lie on the ground plane load-bearing: if a real clip starts or ends mid-rally, the forced-zero endpoints introduce a systematic error that propagates through the whole trajectory. The paper correctly lists this as a limitation in Section 5, but it does not report how many of the real TrackNet or IPL sequences had to be trimmed to satisfy the assumption, nor the sensitivity to a violation of the assumption. To assess the practical scope of the claimed generalization to real-world scenarios, please quantify the trimming rate and provide a small experiment with a nonzero initial height or with untrimmed sequences.
minor comments (6)
  1. [Section 3.1.3] The recurrence for the forward accumulator is ambiguous: the text writes hf_t = hf_{t-1} + Δhf_{t-1} with hf_0 = 0, but it is not clear whether Δhf_0 is defined and how the first output height is produced; please state the indexing convention explicitly.
  2. [Abstract] The phrase 'Please visit our page' is not standard in an abstract and should be moved to a footnote or the introduction.
  3. [Appendix C.7] The sentence 'We qualitatively evaluate our performance on 118 trajectories from 13 clips in one match' is in tension with the quantitative per-game results in Table 4; please reword to clarify that this appendix presents a separate qualitative subset and not the full evaluation underlying Table 4.
  4. [Section 4.1 / Real IPL] The fact that only 9 of the IPL sequences were successfully calibrated is a significant limitation for real-world evaluation; this should be stated in the main text rather than only in the appendix.
  5. [Table 3] In the row for Mocanu et al., the standard error at ±15 pixels is 0.38, which is an order of magnitude larger than the neighboring entries; please verify that this is not a typo.
  6. [Figure 2] The mapping from the raw input (ut, vt) to the plane-point representation P = (pground, pvertical) is not clearly shown in the pipeline diagram; adding explicit labels to the arrows would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the height prediction is trained on simulated 3D ground truth and verified against independent real-world measurements; the Table 4 SynthNet comparison raises benchmark-fairness concerns that are not circular reductions.

full rationale

No circular reduction appears in the derivation chain. The central output (per-frame height) is supervised by PhysX-simulated 3D ground truth (Sec. 3.2, Appendix B), not by the test trajectories being 'predicted,' and it is evaluated against genuinely external measurements: Real Mocap uses eight synchronized IR cameras (Appendix C.6) and Real IPL uses multi-view triangulation (Appendix C.5). The projection-consistency property is a geometric lifting constraint—solving r_y(s*) = h along the viewing ray (Sec. 3.1.3)—which constrains the output to the input ray but cannot by itself produce accurate heights; accuracy still comes from the learned mapping, so the constraint is not a circular target. The ground-endpoint assumption (hf_0 = 0, hb_N = 0) is an acknowledged applicability limitation (Sec. 5), not a device that forces the answer, since interior bounce phases and heights are still learned. There are no load-bearing self-citations: the only preprocessing citation, [38] (Rematas et al.), is not authored by the present authors, and no uniqueness theorem is imported. Two concerns were flagged and weighed but are correctness risks rather than circularity: (1) Table 4 compares SynthNet's numbers 'taken directly from their paper' with the authors' own PnP calibration and endpoint-required trimming, so the two columns are not known to share inputs; (2) Appendix C.7 describes the Real TrackNet evaluation as 'qualitative' on 118 trajectories from 13 clips in 'one match,' in tension with Table 4's ten-game quantitative results. Neither concern exhibits an equation-level reduction of the output to its inputs, so per the circularity rules they do not raise the score; they should be resolved by releasing clip IDs and the trimming rule.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on geometric and domain assumptions that are stated but only partially validated: known cameras, ground-anchored endpoints, and synthetic-to-real transfer. Free parameters are limited to hand-chosen training hyperparameters; no new physical entities are introduced.

free parameters (3)
  • Loss weights λε, λ3D, λB = 10, 1, 10
    Chosen by hand for the total loss in Eq. 7; the paper reports the full combination works best but does not provide sensitivity analysis.
  • EoT class-balance weight γ = not reported
    Used in the weighted binary cross-entropy in Eq. 4; the paper never gives its value.
  • Training-time Gaussian noise level = not reported
    Random Gaussian noise is added to 2D inputs during training, but the standard deviation is not specified, which matters for the reported robustness results.
assumptions (6)
  • domain assumption Both the beginning and end of the input sequence lie on the ground (y = 0).
    Section 3 states this assumption; it initializes forward and backward height accumulators to zero and is acknowledged in Section 5 as requiring sequence trimming.
  • domain assumption Camera intrinsic and extrinsic parameters are known for every input sequence.
    Section 3.1.1 computes rays from pixel coordinates using E, f, px, py; on Real TrackNet and IPL these come from calibration pipelines whose errors are not quantified.
  • domain assumption No viewing ray is parallel to the ground plane or the chosen vertical plane.
    Section 3.1.1 relies on the ray intersecting both planes so the (pground, pvertical) representation is well-defined and y/z coordinates can be dropped.
  • domain assumption PhysX simulation without spin, aerodynamics, or court-type effects is an adequate proxy for real ball motion.
    Appendix B and Section 5 state these factors are ignored; Appendix G shows failures on volleys and chest traps that are absent from the simulation.
  • domain assumption The autoregressive LSTM interpolation of missing 2D points in Real IPL is accurate enough not to distort the trajectory.
    Appendix D.1 uses a separately trained LSTM to complete missing track points; errors in this preprocessing enter the pipeline and are not separately evaluated.
  • domain assumption Ground-truth contact points on Real TrackNet, from TrackNet labels plus PnP calibration, are accurate enough for landing-error evaluation.
    Section 4.2.1 generates 3D contact ground truth by ray tracing annotated contact frames; annotation and calibration noise is not quantified.

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Cite this review

Pith. "Pith review of Where Is The Ball: 3D Ball Trajectory Estimation From 2D Monocular Tracking." pith.science (2026). https://pith.science/paper/ZMCQKIJ4

@misc{pith2026250605763,
  author       = {Pith},
  title        = {Pith review of: Where Is The Ball: 3D Ball Trajectory Estimation From 2D Monocular Tracking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZMCQKIJ4}},
  note         = {Machine review of arXiv:2506.05763}
}
read the original abstract

We present a method for 3D ball trajectory estimation from a 2D tracking sequence. To overcome the ambiguity in 3D from 2D estimation, we design an LSTM-based pipeline that utilizes a novel canonical 3D representation that is independent of the camera's location to handle arbitrary views and a series of intermediate representations that encourage crucial invariance and reprojection consistency. We evaluated our method on four synthetic and three real datasets and conducted extensive ablation studies on our design choices. Despite training solely on simulated data, our method achieves state-of-the-art performance and can generalize to real-world scenarios with multiple trajectories, opening up a range of applications in sport analysis and virtual replay. Please visit our page: https://where-is-the-ball.github.io.

Figures

Figures reproduced from arXiv: 2506.05763 by the authors.

Figure 1
Figure 1. Given a 2D ball tracking sequence, we estimate the ball’s 3D motion, which includes multiple bounces and hits. Abstract We present a method for 3D ball trajectory estimation from a 2D tracking sequence. To overcome the ambiguity in 3D from 2D estimation, we design an LSTM-based pipeline that utilizes a novel canonical 3D representation that is inde￾pendent of the camera’s location to handle arbitrary views and a ser… view at source ↗
Figure 2
Figure 2. Method overview. Given a 2D ball tracking sequence (ut, vt), we first convert each tracked point to our novel 3D plane points parameterization P = (pground, pvertical) and then predict the 3D ball coordinates (xt, yt, zt). Our pipeline consists of 3 main components. 1) EoT network takes in the plane point temporal differences and predicts the end-of-trajectory (EoT) probability. 2) Height network takes in the EoT pr… view at source ↗
Figure 3
Figure 3. Ray parameterization. We represent a 2D track point as the associated 3D viewing ray, parameterized as two intersec￾tion points pground, pvertical of the ray with the ground plane (y=0) and a vertical plane (e.g., z=0). on real-world bouncing scenarios across different sports. Each input sequence may contain multiple trajectories, each beginning when a force is applied to the ball (e.g., a soc￾cer player’s kick) and… view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Different input/output parameterization types [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: LSTM ablation study. The predictions are in blue and ground truth in red. (y-axis points up, each block is 50×50 cm2 ) portant, and without them the errors significantly increase across all datasets. Without the end-of-trajectory flags from LSTMε , the negative effect …
Figure 6
Figure 6. Figure 6: From the 2D tracking of the tennis ball on the left, our method can successfully predict multiple consecutive 3D trajectories [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Comparison with learning [30] and physics-based [46] methods. We add ±25-pixel noise to the 2D input in the bottom row. Blue: ours. Yellow: prior work. Red: ground truth. 5. Limitations & Discussion Our method assumes the first and last frames are on the ground, which …
Figure 8
Figure 8. Figure 8: Unity Game Engine for Mocap, IPL and Simpler [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: Unity Game Engine for Tennis. B.3. Synthetic Single-Launch Trajectory Dataset For comparison with Mocanu et al. [30] and Shen et al.[46], we matched their input assumptions (single force / sin￾gle trajectory) and simulated single-trajectory sequences by 11 [PITH_FULL_…
Figure 10
Figure 10. Figure 10: Motion capture studio. The top left is a ping-pong ball attached with IR reflective materials. The right image is our motion capture studio used to collect data. The bottom left is one of the eight IR cameras used in the studio. C.6. Real Mocap This dataset captures t…
Figure 11
Figure 11. Figure 11: Qualitative results on synthetic datasets. [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: Qualitative results on Real Mocap dataset. [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: Qualitative results on Real IPL dataset. [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: Qualitative results on Real Tracknet (Tennis) dataset. [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: State-of-the-art comparison with a learning-based approach Mocanu et al.[30] and a physics-based approach Shen et al.[46] on a simplified test trajectory that matches their requirements. Each row uses a different noise level. Our predictions are shown in blue, prior w…
Figure 16
Figure 16. Figure 16: Failure cases on Real Tracknet(Tennis) dataset. [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]
Figure 17
Figure 17. Figure 17: Failure cases on Real Mocap dataset. Blue: our predictions. Red: ground truth. Each checkerboard block is 75×75 cm2 [PITH_FULL_IMAGE:figures/full_fig_p022_17.png]
Figure 18
Figure 18. Figure 18: Failure cases on Real IPL(soccer) dataset. [PITH_FULL_IMAGE:figures/full_fig_p022_18.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.