Physics-Based 3D Ball Trajectory Reconstruction from Monocular Soccer Video: A Multi-Model Benchmark

Łukasz Grad^1,2,*, Krzysztof M. Czajkowski^2,*, Aliaksandr Varashylau^2,*

¹University of Warsaw ²ReSpo.Vision
^*Equal contribution
Contact: l.grad@mimuw.edu.pl
CVPR CVsports 2026

Monocular 3D reconstruction from our best monocular model across four trajectory segments (Legia-Piast broadcast and static cameras, Ekstraklasa-Broadcast). In each panel, the predicted 3D trajectory (red) and its pitch-plane shadow (blue) track the triangulated 3D ground truth (green) and its shadow (dark green); per-clip 3D RMSE is shown in the bottom-right corner.

Abstract

We study physics-based 3D ball trajectory reconstruction from monocular videos. Our pipeline segments the trajectory at contact events and fits each segment by optimizing a forward-simulated flight model with a reprojection-based objective. We implement seven arc models of increasing complexity, ranging from gravity-only parabolas to MuJoCo simulations with drag, spin, and fluid forces. All seven are evaluated under end-to-end monocular reconstruction, with four additionally evaluated under multi-view 3D reconstruction.

The two protocols yield reversed model rankings: a model with fitted gravitational acceleration achieves the best monocular accuracy on all soccer evaluation settings, yet a spin-decomposition model leads on four of five datasets when fitted to 3D ground truth. These results suggest that observation noise and single-view geometric ambiguity, rather than model expressiveness, are the primary limiting factors for monocular soccer reconstruction.

We evaluate across five datasets spanning nearly 6,000 trajectory segments and publicly release two new soccer datasets with triangulated 3D ground truth along with segment-level annotations for APIDIS and ISSIA-3D.

Method Overview

Overview of the 3D Trajectory Reconstruction Pipeline. A monocular input video, ball and player features and camera calibration (K, [R|t]) are given. Pivot points and playtime from manual annotation segment the sequence, which is fitted in an optimization loop: a physics engine simulates and reprojects the 3D path to iteratively improve trajectory parameters. Finally, the pipeline corrects high pivots, re-fits the affected segments, and rejects invalid segments.

Physics Models

Seven arc models of increasing complexity. A check (✓) marks an enabled force or feature; #Params is the number of free parameters optimized per arc segment.

Model	MuJoCo	Fitted Gravity	Drag	Magnus	Spin (ω₀)	#Params
parabola	—	—	—	—	—	6
kinetic	—	—	✓	—	—	7
fitg	—	✓	✓	—	—	8
angular	—	—	✓	✓	—	9
MuJoCo kinetic	✓	—	✓	—	—	6
MuJoCo angular	✓	—	✓	—	✓	9
MuJoCo ellipsoid	✓	—	✓	✓	✓	12

Monocular 3D Reconstruction

Balanced mAP (mAP_bal) and arc-specific mAP (mAP_arc); higher is better. Best per column in bold.

Model	LP-static		LP-broadcast		EB		SW		ISSIA-3D		APIDIS
Model	bal	arc	bal	arc	bal	arc	bal	arc	bal	arc	bal	arc
parabola	.325	.274	.493	.330	.533	.357	.421	.351	.260	.159	.577	.577
kinetic	.332	.289	.508	.359	.532	.359	.414	.355	.253	.153	.476	.476
fitg	.343	.307	.530	.397	.557	.412	.459	.371	.262	.163	.507	.507
angular	.336	.296	.517	.377	.544	.383	.412	.353	.255	.157	.463	.463
MuJoCo kinetic	.329	.282	.496	.337	.525	.341	.422	.352	.260	.157	.552	.552
MuJoCo angular	.330	.283	.499	.341	.527	.345	.422	.351	.260	.158	.543	.543
MuJoCo ellipsoid	.324	.271	.494	.334	.522	.339	.437	.367	.261	.160	.547	.547

Multi-view 3D Reconstruction

Frame-weighted mean 3D error (dm) and arc-specific error (dm); lower is better. Per-frame errors capped at 25 dm. Best per column in bold.

Model	LP		EB		SW		ISSIA		APIDIS
Model	ē	ē_a	ē	ē_a	ē	ē_a	ē	ē_a	ē	ē_a
parabola	3.16	4.68	3.80	5.88	1.92	7.24	5.90	3.64	4.55	4.55
kinetic	3.20	4.74	3.68	5.26	1.86	6.05	5.54	2.80	3.68	3.68
fitg	2.75	3.12	3.48	4.14	1.78	5.89	5.34	2.36	3.12	3.12
angular	2.24	1.83	3.28	3.32	1.83	5.77	5.22	2.08	2.56	2.56

Static vs Broadcast Camera

Mean 3D error (m) by camera type on LP for the two best-performing models.

Model	LP-static			LP-broadcast
Model	Full	Str	Arc	Full	Str	Arc
fitg	0.93	0.74	1.44	1.24	0.85	2.59
angular	0.89	0.67	1.48	1.20	0.80	2.62

Key Finding

The ranking reversal between protocols reflects an observation-noise bottleneck: monocular geometric ambiguity prevents spin-decomposition models from exploiting their extra parameters, so the single-parameter fitg model suffices under 2D fitting, whereas 3D inputs unlock the richer angular model. This shows that—among the models tested—observation noise and the inherent geometric ambiguity of monocular reconstruction, rather than model expressiveness, are the current bottleneck.

Qualitative Results

Qualitative model comparison on the Legia-Piast dataset (static and broadcast cameras). Each column corresponds to one physics model. Red dots show the predicted 3D trajectory projected onto the video frame; green dots show the oracle 3D ground truth; blue and dark-green dots show the corresponding pitch-plane (ground) projections of the prediction and ground truth; yellow markers indicate segment endpoints (pivot points). Per-panel 3D RMSE in meters is overlaid in the bottom-right corner.

Qualitative model comparison on the Ekstraklasa-Broadcast and APIDIS datasets

Qualitative model comparison on the Ekstraklasa-Broadcast (five matches) and APIDIS (basketball) datasets. Each column corresponds to one physics model. Red dots show the predicted 3D trajectory projected onto the video frame; green dots show the oracle 3D ground truth; blue and dark-green dots show the corresponding pitch-plane (ground) projections of the prediction and ground truth; yellow markers indicate segment endpoints (pivot points). Per-panel 3D RMSE in meters is overlaid in the bottom-right corner.

Poster

BibTeX

@InProceedings{Grad_2026_CVPR,
    author    = {Grad, {\L}ukasz and Czajkowski, Krzysztof M. and Varashylau, Aliaksandr},
    title     = {Physics-Based 3D Ball Trajectory Reconstruction from Monocular Soccer Video: A Multi-Model Benchmark},
    booktitle = {Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops},
    month     = {June},
    year      = {2026},
    pages     = {9940-9948}
}