Cepheid splits residual structure from physical scale: 15.6% and 10.1% lower completion error on unseen cloth and legged bodies than harmonic extension
Synopsis
In sparse-to-dense mesh motion completion, residual priors trained on one geometry family often fail to beat parameter-free harmonic extension, because residual amplitude varies by more than an order of magnitude across families and densities while the target motion is unknown at test time. Cepheid learns only a dimensionless residual structure and recovers physical amplitude from observables plus a lightweight correction fitted on training sources. The unified model stays below harmonic extension on all five evaluated families, cutting error by 15.6% on an unseen cloth sheet and 10.1% on procedural legged bodies, without skeletons, registration, or cross-mesh correspondence.
Interpretation
Dense mesh motion completion can now rest on the harmonic extension implied by mesh connectivity and add a residual prior shared across geometry families, without skeletons, registration, or cross-mesh correspondence. Residual priors trained on a single geometry family did not transfer reliably: across three single-family models plus a unified model, seven of twelve single-family transfer cells failed to improve on harmonic extension, and on cloth the AMASS and Franka priors were 25.5% and 7.8% worse than that base. Under a protocol of five densities, seven held-out identities, thirty pairs each and two masks per pair, 2,100 conditions in total, the unified model's pooled error stays below harmonic extension on all five evaluated families, reducing it by 15.6% on unseen cloth and 10.1% on procedural legged bodies.
The construction separates the residual into a dimensionless structure and a physical amplitude, and the network models only the former. Harmonic extension solves the discrete Dirichlet problem on the mesh graph with observed vertices as boundary conditions, giving a dense base field in one sparse linear solve, so learning only has to model what that base leaves behind. Amplitude recovery multiplies a dimensionless correction by a dimensional gauge taken from observed displacement dispersion, bringing cloth completion error within 1.4%–8.1% of the target-informed scalar optimum that requires ground truth and is not deployable.
Perspective
The quantitative evaluation covers the three training sources DT4D, AMASS and Franka plus fully held-out cloth and nine procedural legged bodies, at observation densities from 0.5% to 50%, and assumes exact vertex displacements free of sensor noise, lost tracks, depth distortion or scanned-geometry error. Application demonstrations include a Franka arm gripping and lifting a soft bar in Isaac Lab, twelve vertex tracks from CoTracker3 on a DAVIS sequence, and directional validation on one recorded teleoperation episode each from three commercial manipulators in RH20T.
The quantitative evaluation assumes exact vertex displacements; only AMASS and the RH20T episodes in the appendix are recorded data, DT4D is artist-animated, Franka comes from forward kinematics on random joint trajectories, and cloth and legged robots are simulated. Training uses random masks only; the prior transfers zero-shot to a fixed gripper placement, but fixed three-marker layouts gain substantially from fine-tuning. Validation on noisy inputs and training across mask families are left to future work.