design interpolation methods

This work addresses the longstanding challenge that existing spline interpolation methods struggle to simultaneously achieve exact interpolation and strict arc-length parameterization. The paper proposes an iterative optimization algorithm capable of constructing spline curves in arbitrary-dimensional spaces that exactly interpolate prescribed data points while being strictly parameterized by arc length. This approach represents the first method to unify exact interpolation with rigorous arc-length parameterization, thereby overcoming the traditional trade-off between parametrization quality and interpolation accuracy inherent in conventional splines. Numerical experiments in two-dimensional space demonstrate the algorithm’s effectiveness and high precision, highlighting its promising applications in geometric modeling, trajectory generation, and related fields.

This paper addresses spline interpolation on compact Riemannian manifolds—such as spheres and cylinders—i.e., non-Euclidean geometric domains. We propose the first general, smooth spline modeling framework applicable to arbitrary compact Riemannian manifolds. Our method formulates interpolation within a Gaussian Markov random field (GMRF) kriging framework, with numerical solution achieved via discretization of the Laplace–Beltrami operator and finite-element approximation. Key contributions include: (1) the first extension of spline interpolation systems to general compact Riemannian manifolds; (2) support for locally anisotropic covariance modeling; and (3) incorporation of domain deformation to enhance geometric adaptability. Experiments on spherical and cylindrical domains demonstrate substantial improvements over classical spherical harmonic methods, with superior robustness and broad applicability across diverse manifold geometries.

For interpolation of large-scale scattered data using Matérn-type radial basis functions (RBFs), this paper proposes an efficient algorithm based on multilevel residual correction and samplet coordinate representation. The method constructs a hierarchical system by incorporating samplets—local discrete signed measures with vanishing moments—into the Matérn multiresolution framework, ensuring uniform boundedness of the condition number and reducing overall complexity to (O(N log^2 N)). Key components include multilevel residual decomposition, samplet coordinate transformation, diagonal scaling preconditioning, and sparse approximation of the generalized Vandermonde matrix. Theoretical analysis establishes both the well-conditionedness of the resulting system and controllable interpolation error. Numerical experiments in two and three dimensions confirm near-linear-logarithmic scaling in both matrix assembly and solver time, significantly outperforming conventional RBF approaches.

This work addresses the problem of local parameterization of implicit surfaces—such as neural implicit fields and point clouds—at arbitrary query points. We propose a geodesic spline-based parameterization method that relies solely on the signed distance function (SDF) and its projection operator, without requiring meshes, surface normals, or other auxiliary geometric data. Our method employs a two-stage radial sampling and B-spline interpolation framework: first, neighborhood points are sampled along geodesic directions; second, a conformal, low-distortion explicit spline surface mapping is constructed. To our knowledge, this is the first unified local parameterization scheme supporting diverse geometric inputs—including neural implicit representations and unstructured point clouds. Experiments demonstrate significant improvements in robustness and generality for local texture mapping and interactive curve drawing on implicit surfaces. The approach establishes a new paradigm for real-time editing and visualization of implicit geometry.

This study addresses the challenge of overfitting in exact interpolation under noisy observations, which compromises the generalization capability of multivariate surface modeling. The authors establish a unified slice-based training/testing protocol, implementing Clough–Tocher cubic and Multiquadric radial basis function (RBF) interpolations using SciPy/NumPy. To ensure rigor and reproducibility, they incorporate fixed random seeds, repeated data splits, and Bootstrap-based uncertainty quantification into a standardized evaluation framework. Experimental results demonstrate that both methods achieve high accuracy in noise-free settings, yet exhibit overfitting when noise is present. Notably, cubic interpolation consistently outperforms RBF across RMSE, MAE, and R² metrics, showing greater stability. The findings highlight that structured interpolation can effectively recover physically meaningful process behavior even when anomalous measurements are retained, offering a promising approach for modeling noisy data.

Multiharmonic splines suffer from cubic computational complexity O(N³) and theoretical inconsistency in high-dimensional regression due to violated manifold assumptions. Method: This paper proposes a multilevel clustered harmonic spline cascade architecture—the first to introduce a “bundle-style” cascaded structure—integrating clustered harmonic spline kernel construction, block-sparse matrix fast algorithms, and forward-backward joint optimization to achieve intrinsic-dimensionality-agnostic modeling. The method rigorously preserves the self-consistency of stochastic function theory while enabling differentiable, scalable end-to-end training without pre-specifying manifold geometry. Contribution/Results: The proposed approach reduces computational complexity to near-linear scale, simultaneously ensuring theoretical optimality and high-fidelity gradient propagation. Empirical evaluations demonstrate its effectiveness on large-scale, high-dimensional regression tasks, bridging the gap between statistical rigor and practical trainability.

To address the low accuracy and poor efficiency of envelope computation during dynamic deformation of planar regions, this paper proposes an arc-spline approximation method based on medial axis transform and Minkowski-space embedding. We innovatively map the medial axis into Minkowski space, establishing a geometric correspondence between the deforming domain’s boundary and a closed curve in the cyclic image space; consequently, envelope computation is reformulated as an optimal arc-spline approximation problem on this closed curve. A scan-line algorithm is integrated to enable efficient sampling and redundant-branch pruning. The method achieves high geometric fidelity while reducing time complexity to the theoretical optimum, significantly enhancing both computational stability and efficiency. It is broadly applicable to CAD/CAM, motion planning, and deformation analysis.

This work addresses the problem of continuous interpolation and approximation of multivariate scattered data under optional monotonicity constraints. The authors propose a training-free, Lipschitz-continuous approximation method that constructs tight upper and lower bounds, integrating local interpolation with smoothing strategies to ensure strict adherence to prescribed monotonicity while avoiding discontinuities. The key contribution lies in achieving shape-preserving, continuous, and computationally efficient instance-wise approximation. By leveraging GPU-based parallelization, the method achieves substantial performance gains. The proposed algorithms have been implemented in the open-source Python package LipFit, making them readily applicable to large-scale constrained scattered data fitting tasks.

Sparse spatiotemporal observations—e.g., groundwater storage in Bangladesh—hinder accurate continuous field reconstruction. Method: We propose a hybrid spatiotemporal modeling framework integrating deep learning with geostatistical interpolation. Departing from conventional “aggregate-then-model” or “model-then-interpolate” paradigms, our approach innovatively employs time-series dynamic clustering to guide spatial interpolation, explicitly encoding geological uncertainty’s influence on temporal evolution. It combines grid-to-grid/point deep learning prediction with kriging interpolation and identifies dynamically similar yet non-neighboring observation regions via adaptive similarity metrics. Contribution/Results: Experiments demonstrate significantly higher temporal forecasting accuracy than pure spatial interpolation; proximity does not imply dynamic similarity; and the method exhibits superior interpolation robustness and physical interpretability—particularly for latent driving variables such as groundwater storage.

This work addresses the challenge of modeling circular density data characterized by periodicity and relative structure by proposing a novel periodic spline approach within the Bayes space framework. By applying the centered log-ratio transformation, densities are mapped into an L² subspace subject to a zero-integral constraint, enabling the construction of spline bases that simultaneously respect periodicity and Hilbert space structure. The method unifies smoothing and penalized spline estimation in a matrix formulation for computational efficiency. It represents the first integration of periodic splines with Bayes space theory, preserving the relative nature and interpretability of densities while facilitating subsequent functional data analysis. Experiments on wind direction data demonstrate that the proposed approach yields smooth, plausible, and interpretable density estimates, offering a new paradigm for modeling complex circular density data.