Cepheid: Decoupling Scale from Structure for Cross-Family Motion Completion

Chih-Hai Su; Tsung-Wei Ke; Yu-Lun Liu

cs.CVOct 13, 2026

Cepheid: Decoupling Scale from Structure for Cross-Family Motion Completion

Authors: Chih-Hai Su, Tsung-Wei Ke, Yu-Lun Liu

Organizations: National Yang Ming Chiao Tung University · Department of Computer Science and Information Engineering, National Taiwan University

Abstract

Sparse-to-dense motion completion recovers dense mesh motion from sparse vertex displacements. Residual priors trained on one geometry family frequently fail to beat parameter-free harmonic extension off-family. Moreover, cross-family training meets a scaling barrier: residual amplitude varies by more than an order of magnitude across geometry families and collapses as observations become denser. Since this scale depends on unknown target motion, it cannot be measured directly at test time. We decouple residual structure from physical scale by learning a dimensionless prior, estimating amplitude from observables and a lightweight correction fitted only on training sources. Our unified model surpasses harmonic extension across all evaluated families, including unseen cloth and legged bodies, without skeletons, registration, or cross-mesh correspondence. The same prior applies without retraining to point-tracked animal video and a robot grasping a soft body. Project page: https://su-terry.github.io/cepheid-website/

Figures & tables

Figure 2: Cepheid constructs dense motion by decoupling residual structure from physical scale. Given a mesh $\mathcal{M}$ with coordinates $X$ and sparse observed displacements $O$ on vertex subset $\Omega$ (A), harmonic extension first yields a closed-form base field $H$ (B). Amplitude recovery then estimates physical scale $\hat{s}$ using frozen fitted coefficients (C; Algorithm 1 ). Conditioned on $H$ and $O$ , the flow-matching network learns the normalized target $R/\hat{s}$ with $R=F-H$ during training, while mapping $k$ Gaussian noise draws to samples averaged into $\bar{R}_{\theta}$ at inference (D); it is the sole component trained by gradient descent (flame icon). Finally, the completed field is assembled as $\hat{F}=H+\hat{s}\cdot\bar{R}_{\theta}$ (E).

	Observed vertices
err $\div$ mean $\|F\|$	0.5%	2%	5%	10%	50%
harmonic	0.6255	0.3533	0.2257	0.1598	0.0754
NSDP ( Tang et al., 2022 ) , retrained	1.0406	0.5789	0.4018	0.3275	0.2683
DOMA ( Zhang et al., 2024b )	0.5750	0.2370	0.1204	0.0720	0.0246
ours (unified)	0.3533	0.1116	0.0485	0.0300	0.0180

Table 1: Baseline comparison on DT4D: NSDP and DOMA against ours, on one split. NSDP is above the no-learning base at every density; DOMA and ours are below it, and ours is lowest at all five. Seven held-out identities, five densities (2,100 conditions), error divided by that density’s mean motion magnitude. DOMA’s row is its best iteration budget and reading at each density, chosen after scoring.

Model	DT4D $\dagger$	AMASS	Franka	cloth $\ddagger$	legged $\ddagger$
	Evaluated on
harmonic	0.2685	0.0693	0.1011	0.4377	0.2474
ours (DT4D)	0.1097	0.0698	0.0782	0.4256	0.2238
ours (AMASS)	0.2701	0.0182	0.0459	0.5493	0.2482
ours (Franka)	0.2910	0.0797	0.0155	0.4717	0.2426
ours (unified)	0.1243	0.0214	0.0201	0.3692	0.2225

Table 2: Transfer matrix: three single-family priors and one trained on all three, each scored on five families. Seven of the twelve off-diagonal cells sit at or above their column’s base, while the unified checkpoint is below it in every column. 3,000 conditions per column, 3,840 on cloth; cells are completion error divided by that column’s mean motion magnitude, with harmonic the no-learning base.

\dagger

The DT4D column’s denominator is in mAU, that dataset’s own artist units, not a length.

\ddagger

No arm trains on cloth or on the legged bodies.

	Observed vertices
cloth, err $\div$ mean $\|F\|$	0.5%	1%	2%	10%	50%
Target-informed oracle	0.6283	0.5175	0.3917	0.1492	0.0529
harmonic	0.6888	0.5925	0.4953	0.2824	0.1295
Raw observed scale (no correction)	0.8073	0.7326	0.7650	1.1599	1.3197
Borrowed lookup table (best source)	0.6817	0.5530	0.4330	0.2056	0.0652
Amplitude recovery (ours)	0.6790	0.5524	0.4087	0.1513	0.0548

Table 3: Amplitude rules on cloth: one set of frozen network predictions, scaled four ways, against the no-learning base. Cloth is held out of training; each cell is error

\div

mean

|F|

. Target-informed oracle : for each condition, the single scalar that minimizes error along the residual the network predicted, chosen after seeing the target and not computable at inference. harmonic : the base field

H

alone, with no learned residual. Raw observed scale :

\hat{s}=\sigma_{\mathrm{obs}}

, the dispersion of the observed displacements, without the fitted correction. Borrowed lookup table : a per-density amplitude table fitted on one training source, taking at each density whichever of the three gives the lowest error on cloth. Amplitude recovery :

\sigma_{\mathrm{obs}}

times the correction fitted on the training sources (Algorithm 1 ), within 1–8% of the oracle at every density.

harmonic						0.438	0.247
	harmonic	mean of $k$ samples	flow matching	conditioned on harmonic	amplitude recovery	cloth $\ddagger$	legged $\ddagger$
Cepheid (ours)	-	$\checkmark$	$\checkmark$	-	$\checkmark$	1.047	0.585
	$\checkmark$	-	$\checkmark$	$\checkmark$	$\checkmark$	0.416	0.240
	$\checkmark$	$\checkmark$	-	$\checkmark$	$\checkmark$	0.415	0.226
	$\checkmark$	$\checkmark$	$\checkmark$	-	$\checkmark$	0.400	0.244
	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	-	0.391	0.409

Table 4: Component ablation on the two held-out families: each row removes one choice from Cepheid . Two removals can turn learning into negative transfer; three cannot. A check marks a component the row keeps; in the amplitude column a dash is

\hat{s}=1

, and in the samples column a check is eight samples and a dash is one. The three training columns are Table 8 .

Figure 3: Demo: five independent completions on a multi-body point set in Isaac Lab ( Mittal et al., 2025 ) . The arm and grasped body form a face-free point set coupled by a 12-NN graph. From eight proprioceptive points (red; zero on object; markers can overlap at this viewpoint), the model predicts plausible coupled motion.

Figure 4: Demo: densifying video point tracks. Across frames 4–39, the model observes the same 12 tracked vertices in every frame. (a) Twelve observed points and 120 completed points. (b) 120 camera-opposed far-side points (0 of 2,164 ever observed). Without ground truth, this shows the checkpoint connects to a tracker and completions exist on the far side, not that either is accurate. Overlays degrade away from fitted frame 18 (Appendix D ).

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

column	sampled over	masks	densities	motion norm.
DT4D	250 pairs	4	0.5 / 10 / 50%	extent
AMASS	250 pairs	4	0.5 / 10 / 50%	extent
Franka	250 pairs	4	0.5 / 10 / 50%	none
cloth	24 trajectories $\times$ 8 frames	4	0.5 / 1 / 2 / 10 / 50%	extent
legged	250 pairs	4	0.5 / 10 / 50%	none

Appendix

Table 5: What each evaluation column samples. Each source is sampled on its own density grid. Its data are organized either by frame pair or by frame across continuous trajectories. Masks are drawn from the random-mask family throughout, four per pair and density.

Figure 5: Demo: soft-bar manipulation in Newton. Five instants of a Franka arm carrying a soft bar simulated in Newton 1.5.2 ( The Newton Contributors, 2025 ) as a face-free 12-NN point set. Given eight proprioceptive points (red) and none on the sagging bar, the frozen checkpoint completes dense motion, closely matching harmonic extension on the unobserved payload.

err $\div$ mean $\|F\|$ , cloth	0.5%	1%	2%	10%	50%
harmonic	0.6888	0.5925	0.4953	0.2824	0.1295
DT4D	0.8131	0.6479	0.4578	0.1550	0.0544
AMASS	0.9511	0.8393	0.6440	0.2360	0.0765
Franka	0.7594	0.6451	0.5268	0.2926	0.1346
Cepheid	0.6790	0.5524	0.4087	0.1513	0.0548

Appendix

Table 6: Cloth by density: the four priors and the base at each rung. Every specialist crosses the no-learning base and the unified model never does. 3,840 conditions, 768 per density; Table 2 pools all five rungs.

err $\div$ mean $\|F\|$ , legged	0.5%	1%	2%	10%	50%
harmonic	0.6033	0.4228	0.2640	0.0954	0.0435
DT4D	0.6227	0.4181	0.2274	0.0383	0.0102
AMASS	0.6955	0.4293	0.2127	0.0308	0.0183
Franka	0.6626	0.4195	0.2268	0.0478	0.0175
Cepheid	0.6222	0.3990	0.2067	0.0355	0.0096

Appendix

Table 7: Legged by density: the four priors and the base at each rung. At the sparsest rung every arm is above the no-learning base; at 1% only the AMASS prior still is, and from 2% up none is. Table 2 pools three of these five rungs: 0.5, 10 and 50%.

Figure 6: Qualitative comparison on cloth, a family no model trained on. Completion from 32 observed vertices of the 1,577 free vertices, shown as geometry above and per-vertex error below on a shared ramp; the observed points are marked on the ground truth. Ours is the unified checkpoint and NSDP is retrained on our DT4D split. NSDP tears the sheet and DOMA loses its shape and self-intersects, while our completion keeps the flag intact and its error field dark. Ours is the only one of the three below harmonic extension here.

harmonic						0.269	0.069	0.101
	harmonic	mean of $k$ samples	flow matching	conditioned on harmonic	amplitude recovery	DT4D	AMASS	Franka
Cepheid (ours)	-	$\checkmark$	$\checkmark$	-	$\checkmark$	0.752	0.341	0.457
	$\checkmark$	-	$\checkmark$	$\checkmark$	$\checkmark$	0.152	0.025	0.024
	$\checkmark$	$\checkmark$	-	$\checkmark$	$\checkmark$	0.126	0.021	0.028
	$\checkmark$	$\checkmark$	$\checkmark$	-	$\checkmark$	0.166	0.027	0.032
	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	-	0.152	0.038	0.209

Appendix

Table 8: The same component ablation on the three training families. The arms, conditions and scoring are those of Table 4 , which carries the two held-out columns; together they are one five-column ablation, with rows in that table’s order (cloth, worst first). A check marks a component the row keeps; in the amplitude column a dash is

\hat{s}=1

, and in the samples column a check is eight samples and a dash is one.

Figure 7: The predicted amplitude recovers the residual; a borrowed one does not. Cloth at 10% observation, one shared arrow scale; red dots are the observed vertices. The first panel is the true residual; the three that follow show the same frozen network output, differing only in the scale applied to it. The network’s raw output carries no physical scale, and AMASS’s table restores it at a third of the predicted scale.

Feature	Mathematical Definition
log_frac	$\log(M/N)$
log_n_obs	$\log M$
log_n_vert	$\log N$
log_hop_mean	$\log(\bar{h}+1)$ , with $\bar{h}$ the mean graph hops to $\Omega$ over unobserved vertices that can reach it
log_hop_max	$\log(h_{\max}+1)$ , with $h_{\max}$ the maximum graph hops to $\Omega$ over the same vertices
unreach_frac	Share of the unobserved vertices with no path to $\Omega$

Appendix

Table 9: The eleven amplitude-recovery features, as the code computes them. Together they form

\varphi\in\mathbb{R}^{11}

feature	$w$
log_frac	$-0.202$
log_n_obs	$-0.237$
log_n_vert	$-0.173$
log_hop_mean	$+0.171$
log_hop_max	$+0.189$
unreach_frac	$+0.075$

Appendix

Table 10: The deployed standardized amplitude-recovery weights

w

and intercept

b=-2.439

. The rule is the leave-cloth-out fold, fitted on standardized features from DT4D, AMASS and Franka with the mesh group dropped; individual weights are not interpretable because log_frac

=

log_n_obs

-

log_n_vert and log_base_rough_hop

=

log_base_rough

+\log\bar{d}

exactly (Table 9 ).

cloth, err $\div$ mean $\|F\|$	0.5%	1%	2%	10%	50%
harmonic	0.6888	0.5925	0.4953	0.2824	0.1295
Amplitude recovery (ours)	0.6790	0.5524	0.4087	0.1513	0.0548
Oracle table ( $\alpha^{*}$ )	0.6720	0.5529	0.4205	0.1670	0.0614
Relative difference vs $\alpha^{*}$	$+1.04\%$	$-0.09\%$	$-2.81\%$	$-9.40\%$	$-10.77\%$

Appendix

Table 11: Oracle table comparison on held-out cloth. Conditions match Table 3 , normalized by mean motion magnitude. Relative to the ground-truth-fitted oracle table (

\alpha^{*}

), amplitude recovery differs by

+1.04\%

at 0.5%,

-0.09\%

at 1%, and reduces error by 2.81% to 10.77% from 2% upward.

DT4D held-out, err $\div$ mean $\|F\|$	0.5%	2%	5%	10%	50%	pooled
MGN-T (ours)	0.3997	0.1963	0.1224	0.0898	0.0548	0.1837
Message-passing control	0.4321	0.2012	0.1249	0.0929	0.0562	0.1933
Relative reduction	$7.5\%$	$2.4\%$	$2.0\%$	$3.3\%$	$2.3\%$	$4.9\%$

Appendix

Table 12: Attention ablation on DT4D. Comparison of MGN-T against a matched-budget message-passing control across seven held-out identities at five densities (0.5, 2, 5, 10, 50%). MGN-T achieves a 4.9% pooled relative reduction and yields lower error on all seven identities.

$k$	DT4D	AMASS	Franka	cloth $\ddagger$	legged $\ddagger$
harmonic	0.2685	0.0693	0.1011	0.4377	0.2474
1	0.1519	0.0254	0.0244	0.4154	0.2404
2	0.1372	0.0232	0.0221	0.3905	0.2313
4	0.1287	0.0220	0.0208	0.3765	0.2258
8	0.1241	0.0214	0.0202	0.3690	0.2226
16	0.1216	0.0210	0.0198	0.3651	0.2209

Appendix

Table 13: Averaging more samples at inference: the same checkpoint read at five values of

k

. Little is left by

k=8

, with

k=8

sitting only 0.8–2.1% above

k=16

. Every row is read off the same sixteen draws per condition, so its

k=1

and

k=8

are not the cells of Tables 4 and 8 , and no row can sit above the one before it.

\ddagger

Held out at source level.

	input	0.5%	1%	2%	10%	50%
Laplacian	mesh	53.26	41.73	30.08	13.61	6.42
TPS	points	46.98	31.42	20.24	6.92	2.63
DOMA	points	48.96	33.94	20.18	6.13	2.10
Laplacian + GNN (ours)	mesh	37.47	26.52	17.64	8.73	5.29
TPS + kNN (ours)	points	43.16	26.82	15.71	4.63	1.91

Appendix

Table 14: Completion without mesh connectivity: harmonic extension on mesh edges replaced by a thin-plate spline on a

k

-NN graph. It stays viable, at a cost concentrated at sparse observation. The DT4D density ladder, in mAU rather than the err

\div

mean

|F|

of the other tables. The two (ours) rows are 200k-step models on their own matched pair. DOMA’s cells are each its best over seven iteration budgets and two readings of its prediction, chosen per density after scoring.

	$k=3$
harmonic	2.652
zero-shot	2.283
fine-tuned	0.991

Appendix

Table 15: Fine-tuning the pretrained prior to a fixed sensor placement on a Franka gripper. It more than halves the error against the same checkpoint zero-shot. 12 held-out episodes, 36 frame pairs, error over the free arm links 1–7 in mm;

k=3

is the number of observed markers.

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