Smooth Interpolation: Methods & Applications

• Smooth interpolation is a family of constructions that recovers continuous functions, fields, or geometric objects from discrete data under explicit regularity constraints.
• It applies across domains such as geospatial analysis, geometric modeling, and graph signals by utilizing techniques like spline fitting, variational penalties, and geodesic operations.
• The methods balance exact data constraints with minimizing curvature or derivative penalties to ensure mass preservation, edgewise smoothness, and stable interpolation.

Searching arXiv for relevant papers on smooth interpolation across major subfields. Smooth interpolation denotes a family of constructions in which intermediate values, functions, fields, or geometric objects are recovered from discrete data under explicit regularity constraints. Across the literature, the interpolant may be a nonnegative $C^m$ function on $\mathbb{R}^n$, a density field over enumeration units, a graph signal, a curve or surface, a shell trajectory on a Riemannian manifold, a curve of probability measures in Wasserstein space, or a learned predictor that varies smoothly in input space (Fefferman et al., 2016, Gastner et al., 2022, Huber et al., 2017). What counts as “smooth” is correspondingly domain-specific: continuity up to a polygonal boundary, small curvature or Laplacian energy, low graph variation, non-oscillatory high-order behavior near discontinuities, or regular geodesic behavior on a nonlinear state space.

1. Problem statements and admissibility conditions

A recurring formulation starts from partial or aggregated data and asks for an interpolant satisfying exact constraints together with smoothness. In geospatial analysis, smooth pycnophylactic interpolation seeks a smooth, nonnegative density $\rho(x,y)$ such that for enumeration units $U_1,\dots,U_n$ with totals $P_i$,

$$\iint_{U_i}\rho(x,y)\,dx\,dy=P_i,\qquad i=1,\ldots,n,$$

thereby preserving unit-level aggregates exactly (Gastner et al., 2022). The same preservation principle motivates the term pycnophylactic, namely mass-preserving interpolation.

For polygonal domains, the formulation may instead be boundary-driven. Transfinite mean value interpolation considers a polygon $\Omega\subset\mathbb{R}^2$ with continuous boundary data $f:\partial\Omega\to\mathbb{R}$, and constructs an interior interpolant $g$ from edge integrals weighted by angular geometry. The key guarantee is continuity up to the boundary,

$$g\in C(\overline{\Omega}),\qquad g|_{\partial\Omega}=f,$$

including at both edge interiors and vertices (Floater et al., 2019).

In Whitney-type interpolation with inequality constraints, the question becomes whether prescribed values $\mathbb{R}^n$0 on a finite or arbitrary set $\mathbb{R}^n$1 can be extended by a global nonnegative function $\mathbb{R}^n$2 or $\mathbb{R}^n$3. The decisive result is a finiteness principle: existence of such an interpolant is determined by testing all subsets of uniformly bounded cardinality $\mathbb{R}^n$4 (Fefferman et al., 2016). This places smooth interpolation within the broader Whitney-extension framework, but with the nontrivial additional constraint $\mathbb{R}^n$5.

2. Smoothness as an objective, penalty, or exact inequality

In many settings, smoothness is formalized by variational penalties. For polynomial and spline-like interpolation, one classical criterion is integrated curvature. In one dimension,

$\mathbb{R}^n$6

and in higher dimensions,

$\mathbb{R}^n$7

with $\mathbb{R}^n$8 the Hessian. Smooth supersaturated models enlarge the polynomial basis and use the additional degrees of freedom to minimize this criterion under exact interpolation constraints, producing spline-like behavior while remaining algebraically polynomial (Bates et al., 2008).

An analogous principle appears in pycnophylactic interpolation. Tobler’s 1979 cellular-automaton method is described as minimizing

$\mathbb{R}^n$9

subject to unit-preserving and nonnegativity conditions (Gastner et al., 2022). Here smoothness is curvature suppression of a density surface rather than fidelity to point samples.

Other formulations characterize smooth interpolation through exact inequalities rather than an optimization functional. In smooth strongly convex interpolation, a finite set of triples $\rho(x,y)$0 is $\rho(x,y)$1-interpolable if and only if a closed-form inequality holds for every pair $\rho(x,y)$2. In the smooth convex special case $\rho(x,y)$3, the criterion reduces to

$\rho(x,y)$4

which yields a finite representation of the function class and underlies exact semidefinite performance estimation for first-order methods (Taylor et al., 2015).

A related extremal viewpoint appears for convex scattered data in $\rho(x,y)$5. There the admissible interpolants are smooth edge-convex curve networks, and the objective is to minimize the $\rho(x,y)$6-norm of the second derivative along the edges of a triangulation. For the previously open case $\rho(x,y)$7, the problem always has a solution; on each edge, the minimizer is a quadratic spline with at most one knot (Vlachkova, 2022). Smooth interpolation is therefore not merely about continuity class, but about minimizing edgewise curvature in a prescribed norm.

3. Geometric design, geometric continuity, and manifold-valued interpolation

In geometric modeling, smooth interpolation is frequently governed by geometric continuity rather than scalar Sobolev-type penalties. For Pythagorean normal vector surfaces, a dual construction maps point-normal Hermite data into the isotropic model of dual space, constructs bicubic Coons patches there, and pulls them back to $\rho(x,y)$8. The result is a local patch construction with global $\rho(x,y)$9 continuity and rational offsets (1602.01224). The smoothness notion is geometric continuity of tangent-plane fields, not simply coordinatewise $U_1,\dots,U_n$0 regularity.

Compactly supported interpolatory generators extend this design perspective. By taking linear combinations of half-integer shifts of exponential B-splines, one obtains basis functions that are interpolatory, compactly supported, smooth to order $U_1,\dots,U_n$1, refinable, and capable of reproducing polynomial, trigonometric, or hyperbolic primitives (Schmitter et al., 2017). This construction makes control points lie on the curve or surface itself, while preserving local shape control and multi-resolution refinement.

On nonlinear configuration spaces, Euclidean averaging is replaced by geodesic operations. In the Riemannian shell space of Loop subdivision surfaces, interpolation between key-frame shells is built from discrete geodesic interpolation, a discrete logarithm, a discrete exponential map, and discrete parallel transport. These ingredients support discrete Bezier curves, Riemannian cardinal splines, and interpolatory subdivision schemes for shell animation (Huber et al., 2017). The shell metric measures both membrane distortion and bending distortion, so the interpolated motion is tied to elastic shell geometry.

A further generalization constructs interpolating space curves in $U_1,\dots,U_n$2 from four modules: local functions, redistribution functions, blending functions, and gluing functions. With $U_1,\dots,U_n$3-blending functions whose derivatives vanish at the interval endpoints, the resulting curve is $U_1,\dots,U_n$4, local, cusp-free, and adaptable to corner preservation, convexity preservation, and sphere preservation (Hu et al., 2024). This makes explicit that smooth interpolation of curves is often a local-to-global assembly procedure rather than a single global fit.

4. Structured discrete data: graphs, streaming signals, and discontinuous samples

When the domain itself is discrete or only sequentially observed, smooth interpolation is reformulated in terms of the ambient structure. For graph signals, the Markov variation

$U_1,\dots,U_n$5

measures deviation from a local Markov averaging process. Interpolation is then posed as recovery of a full signal from sampled nodes while favoring an approximately sparse spectrum in the eigenbasis of the Markov transition matrix $U_1,\dots,U_n$6 (Heimowitz et al., 2018). Smoothness here has a probabilistic interpretation: a signal is smooth if each node value is well predicted by the probability-weighted average of its neighbors.

For streamed observations with zero delay, smooth interpolation becomes a sequential decision problem. A spline segment

$U_1,\dots,U_n$7

must be emitted immediately when a new sample arrives, while continuity constraints propagate through the derivative state $U_1,\dots,U_n$8. The one-step cost is a sum of residual error and a derivative-based smoothness penalty,

$U_1,\dots,U_n$9

and the proposed method augments the myopic spline rule with an RNN and a differentiable convex optimization layer to reduce cumulative cost on average (Ruiz-Moreno et al., 2022). This suggests that, in streaming settings, smooth interpolation is partly a prediction problem about future constraints.

High-order interpolation near discontinuities requires a different balance. A fourth-order WENO interpolant combines two quadratic substencil interpolants on a four-point stencil with nonlinear weights determined by interpolation-specific smoothness indicators. The method achieves fourth-order accuracy in smooth regions, remains non-oscillatory near discontinuities, and, unlike Bézier and monotonic high-order Hermite interpolations, does not degenerate to linear interpolation near smooth extrema (Janett et al., 2021). Smooth interpolation in this context is compatible with shock-like behavior precisely because smoothness is enforced adaptively, not globally.

5. Transport, mass preservation, and interpolation on non-Euclidean state spaces

For probability measures, smooth interpolation can be defined through the geometry of optimal transport. A practical construction on Wasserstein space first couples the measures at the interpolation knots via successive Monge maps and then fits ordinary Euclidean natural cubic splines to the coupled particle trajectories. This yields a curve of measures with an explicit particle-flow interpretation and, in the Gaussian/Bures setting, a second-order approximation guarantee

$P_i$0

under the stated regularity and mesh assumptions (Chewi et al., 2020). The method reduces measure-valued interpolation to classical spline fitting after a transport-based coupling step.

A related but distinct use of transport appears in algorithmic fairness. Extended Total Repair constructs a smooth extension of a discrete optimal-transport repair map from observed training points to unseen points. Starting from repaired pairs $P_i$1, a convex potential

$P_i$2

is regularized by the Moreau–Yosida envelope

$P_i$3

and the repaired point is $P_i$4 (Diego et al., 19 Mar 2025). The resulting map is smooth, cyclically monotone, and avoids recomputing the full OT problem for each incoming batch.

In reciprocal-space electronic-structure calculations, smooth interpolation takes yet another form. Within the optimally smooth subspace defined by maximally localized Wannier functions, matrix elements $P_i$5 vary smoothly in $P_i$6. Direct Wannier Interpolation replaces real-space Fourier reconstruction by direct multidimensional interpolation in the OSS, which applies to both periodic and non-periodic observables and avoids explicit use of Wannier-center positions (Volpato et al., 2024). Here smoothness is a gauge property of the chosen Bloch subspace.

6. Learning-based, physical, and robotic interpretations

Recent work has also treated smooth interpolation as an empirical property of learned models. In deep overparameterized networks, smooth interpolation is studied by examining the loss $P_i$7 as a function of the input near training samples, using the input-gradient norm and input-Hessian norm as pointwise sharpness measures and geodesic Monte Carlo integration for finite-volume sharpness. The central empirical finding is that sharpness exhibits both model-wise and epoch-wise double descent, and that sufficiently overparameterized models interpolate even noisy labels over large neighborhoods rather than narrow spikes (Gamba et al., 2022). The same study also shows that smooth interpolation is not identical to good generalization: with fully noisy labels, smooth interpolation can still occur while generalization is absent (Gamba et al., 2022).

In anisotropic MRI, smooth interpolation is implemented in latent space. Autoencoding for Semantic Interpolation trains a convolutional autoencoder so that the latent codes of adjacent slices can be combined by convex combinations,

$P_i$8

and decoded into intermediate slices, while an overview loss forces an actual middle slice to be predictable from its neighbors (Sander et al., 2022). The reported effect is semantically smooth through-plane interpolation that outperforms cubic B-spline interpolation in SSIM and PSNR on cardiac cine, neonatal brain, and adult brain MRI (Sander et al., 2022).

In thermal field theory, smooth interpolation can mean bridging asymptotic regimes rather than fitting data points. A single integrand-level construction is proposed to interpolate between Born $P_i$9 rates and LPM-resummed $$\iint_{U_i}\rho(x,y)\,dx\,dy=P_i,\qquad i=1,\ldots,n,$$0 rates by replacing the Born mass-shell structure with an LPM-style transverse Hamiltonian while keeping compatible kinematics and subtraction terms (Ghiglieri et al., 2021). This makes the resulting rates applicable across a broader range of masses and momenta than either regime alone.

In robotic trajectory generation, smooth interpolation is explicitly multi-layered. For a 3T1R parallel kinematic milling robot, tool-tip positions are fit with quintic B-splines, orientations are represented by unit quaternions and interpolated in quaternion space, synchronization is performed through modifier polynomials and piece-wise Bézier curves, and temporal trajectories are optimized by minimum-jerk, time-optimal piece-wise Bézier curves in task space and then in joint space (Akhbari et al., 18 Feb 2026). This is a concrete example in which geometric smoothness, temporal smoothness, and actuator feasibility are treated as separate interpolation subproblems.

Taken together, these developments show that smooth interpolation is not a single algorithmic doctrine but a family of exact constraints, variational principles, and geometric constructions adapted to the topology of the data domain and the physics of the application. A persistent theme is that exact interpolation and smoothness are not mutually exclusive, but the mechanism that reconciles them varies sharply across scalar fields, graphs, manifolds, measures, reciprocal space, and learned representations (Bates et al., 2008, Huber et al., 2017, Chewi et al., 2020).