Chebyshev Differential Flows for Shape Matching and Interpolation with Endpoint Guidance

Estimating relationships between 3D shapes requires both dense endpoint correspondence and a deformation path connecting the source to the target. These tasks are coupled: the correspondence constrains the interpolation endpoint, while a plausible trajectory can help disambiguate the map. However, when only endpoint shapes are observed, the interior trajectory is under-constrained. Existing joint frameworks typically model interpolation through sampled per-vertex displacements regularized by local rigidity or temporal smoothness, which can indirectly restrict the stretch and shear needed for non-isometric or poorly aligned shape pairs. We propose Chebyshev Differential Flows (Chef), an unsupervised framework that models motion as a continuous time-varying field of per-face Jacobians rather than vertex trajectories. Each Jacobian trajectory is represented by a low-order shifted Chebyshev expansion, yielding a compact arbitrary-time deformation model whose odd and even modes separate endpoint-visible deformation from endpoint-invisible interior control. A spectral functional-map branch estimates the endpoint correspondence, while a differentiable Poisson solver integrates the predicted Jacobian field into globally consistent intermediate shapes. We further introduce a trend regularizer that guides intermediate stretch and shear without imposing per-snapshot As-Rigid-As-Possible (ARAP) rigidity. Experiments on near-isometric and non-isometric benchmarks show competitive correspondence accuracy, improved interpolation proxy metrics, and better stability when the standard rigid pre-alignment step is omitted. Code will be publicly released for research.