Localized B-Spline Interpolation

• Localized B-spline interpolation is a method that utilizes compact support basis functions to achieve exact fitting or smooth approximation of sparse and irregular data.
• It employs adaptive techniques such as local knot insertion, mesh refinement, and domain-informed shift-variant basis functions to enhance computational efficiency and reconstruction fidelity.
• The approach supports dynamic updates and multi-dimensional applications, including medical imaging and geometric modeling, while ensuring robust error control and locally adaptive refinement.

Localized B-spline interpolation refers to spline interpolation techniques designed to provide exact fitting or smooth approximation of data using basis functions with compact support, specifically constructed and adapted to local or irregular sampling, complex domains, or regions exhibiting inhomogeneity or dynamic evolution. Such localized frameworks overcome limitations of global, uniform-grid-based B-spline methods, enabling high-fidelity surface or manifold reconstruction even for sparse, non-uniform, or hierarchically sampled datasets.

1. Mathematical Foundation of Localized B-spline Interpolation

Localized B-spline interpolation builds on the Cox–de Boor recursion to define piecewise-polynomial basis functions over adaptive knot vectors:

$$N_{i,0}(u) = \begin{cases} 1, & u_i \le u < u_{i+1} \ 0, & \text{otherwise} \end{cases}$$

$$N_{i,k}(u) = \frac{u-u_i}{u_{i+k}-u_i} N_{i,k-1}(u) + \frac{u_{i+k+1}-u}{u_{i+k+1}-u_{i+1}} N_{i+1,k-1}(u)$$

For surfaces,

$$S(u,v) = \sum_{i=0}^n \sum_{j=0}^m P_{i,j} N_{i,p}(u) N_{j,q}(v)$$

where the support of each basis function is compact: each point $(u,v)$ depends only on at most $(p+1)(q+1)$ control points (Wang et al., 2021).

Localized variants introduce knot vectors and mesh refinement strategies adapted to sparsity, inhomogeneity, or computational convenience. Open-uniform local knot vectors, domain-informed shift-variant basis, or mesh structures such as criss-cross triangulations and locally refined (LR) meshes serve as the backbone for constructing basis functions with strictly local impact.

2. Adaptive and Localized Algorithms

KPI: Key-Point Interpolation

The KPI procedure (Wang et al., 2021) employs a two-stage algorithm for highly irregular, sparse data:

1. Kriging-based Data Augmentation: Sparse, scattered samples $Q=\{x_i, Z(x_i)\}$ are used to construct a dense grid $\tilde{Q}$ via ordinary Kriging, preserving both mean and spatial covariance structure.
2. Key Point Selection and B-spline Fitting: Key points (original samples and control points inferred from local support neighborhoods) drive a localized constrained least-squares fit, with knot insertion selectively refined wherever two key points interfere. The fitting is localized: only $O(pq)$ control points per sample are affected, and explicit constraints enforce exact interpolation at key locations.

Domain-Informed Shift-Variant B-splines

"Domain-Informed Spline Interpolation" (Behjat et al., 2018) constructs shift-variant B-spline generators $\psi_{n,k}(x)$ utilizing known, possibly overlapping subdomain indicator functions $d_j(x)$. Each generator is adapted to the local domain mix, yielding an interpolant consistent with anatomical or physical boundaries:

$$\psi_{n,k}(x) = \dot{\beta}_{k}^{(n)}(x) + \ddot{\beta}_{k}^{(n)}(x)$$

where dominant ($\dot{\beta}$) and residual ($\ddot{\beta}$) kernels are crafted to increase local coherence and partition of unity.

LR B-splines and Hierarchical Refinement

Locally Refined (LR) B-splines are constructed by adaptive mesh splitting which preserves local linear independence (N2S property) and minimal support (Patrizi et al., 2020). The refinement algorithm operates only on marked splines whose local error exceeds a tolerance, splits local knot-vectors, updates the mesh, and corrects nesting through additional tensor expansion. Resulting bases admit true locality, polynomial reproduction of degree $p$, and quasi-interpolants with optimal error rates.

Hierarchical B-spline spaces (Buffa et al., 2015) proceed via nested sequences of open knot vectors, maintaining parent–child relations for refinement. The multiscale quasi-interpolant is recursively constructed by successively projecting the residual onto finer mesh spaces, ensuring that each level's correction localizes to the region of interest.

3. Computational Structures and Implementation

Localized B-spline interpolation relies on sparse, banded linear algebra structures. Typical data structures include:

• kd-trees, bounding volume hierarchies: Fast nearest-neighbor queries for Kriging or mesh assignment.
• Sparse banded matrices: Arising from clearly defined local support for B-splines and LR variants.
• Indexed arrays: Tracking each basis function’s neighborhood.
• Local patch organization: Overlapping patches in adaptive manifold evolution allow segmentation, localized parametrization, and direct mapping of control points and data samples (Ammad et al., 16 Jan 2026).

Table: Comparison of Algorithmic Complexity

Method	Setup Cost	Per-step Update	Localization Strategy
KPI (Wang et al., 2021)	$O(N \log N)$	$O(L p^2 q^2)$	Key-point & banded Cholesky
Local B-spline (Ammad et al., 2 Oct 2025)	$O(N m^2)$	$O(N m^2 T/\Delta t)$	Local stencils, Gauss–Seidel
LR B-spline (Patrizi et al., 2020)	$O(\sum_\ell |B_\ell|)$	$O(\ell)$	N2S-structured refinement

4. Geometric Properties and Error Control

Localized interpolation preserves geometric fidelity, enabling analytic or high-order estimation of curvature, tangent, and normal vectors:

$$\kappa = \frac{|f_1'(u) f_2''(u) - f_2'(u) f_1''(u)|}{(f_1'(u)^2 + f_2'(u)^2)^{3/2}}$$

For surfaces, mean curvature can be computed from the B-spline representation derivatives by standard fundamental form coefficients (Ammad et al., 16 Jan 2026). Local error control is achieved through:

• Interpolation Error: Measured at data points; triggers Gauss–Seidel refinement.
• Control-point Deviation: Greville abscissae used to monitor the deviation of control points from the interpolated surface.
• Adaptive Refinement: Knot insertion at regions of maximal error, local point redistribution, or selective activation of parent–child refinement in hierarchical bases.

Uniform error estimates such as $$\|f-Q[f]\|_{L^\infty(Ω)} \leq C h^{p+1} \|D^{p+1}f\|_{L^\infty(Ω_B)}$$ (for LR or hierarchical splines) guarantee convergence and local polynomial reproduction under mild grading assumptions (Patrizi et al., 2020, Buffa et al., 2015).

5. Extensions to Dynamic and Multi-dimensional Contexts

Localized B-spline interpolation generalizes seamlessly to higher dimensions and dynamic settings:

• Manifold Evolution: Tensor-product B-spline patches for codimension-one surfaces support adaptive knot insertion, local Lagrangian evolution, and analytic geometric invariant computation (Ammad et al., 16 Jan 2026, Ammad et al., 2 Oct 2025).
• Time-Dependent Fields: Quadcubic interpolation (Walker, 2019) enables localized $C^1$ continuous interpolation in four dimensions, suitable for time-varying field reconstruction.
• Adaptive Refinement for Evolving Domains: Dynamic surface boundaries and moving sensor arrays accommodated by real-time reassignment of patches and points.
• Domain-informed Interpolation: Sample coherence with anatomical or geophysical structure improved via localized adaptation of basis functions (Behjat et al., 2018).

6. Representative Applications and Empirical Results

Localized B-spline techniques find application in geostatistics, medical imaging, dynamic surface reconstruction, real-time graphics, and physical simulation:

• KPI for Temperature Data: Interpolates sparse weather station data, achieves numerical zero error at interpolation sites ($\leq 10^{-14}$), and preserves local dynamic features with reduced control points (Wang et al., 2021).
• Curve and Surface Modeling: Compactly-supported, locally refinable interpolators permit intuitive, multi-resolution editing in interactive graphics (Schmitter et al., 2017).
• Point Cloud Manifold Evolution: Lagrangian patchwise B-spline evolution faithfully approximates curvature-driven flows, anisotropic deformations, and surface-field dynamics (Ammad et al., 16 Jan 2026).
• Signal and Segmentation in Inhomogeneous Domains: Domain-informed B-spline interpolation enforces anatomical consistency in neuroimaging and improves local fitting in complex domains (Behjat et al., 2018).

Localized B-spline interpolation, through its rigorous mathematical framework and adaptive, data-driven implementation strategies, enables scalable, precise, and robust reconstruction of surfaces, manifolds, and fields from sparse, non-uniform, or dynamically evolving data, outperforming global approaches in both computation and geometric fidelity.