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An Adaptive Lagrangian B-Spline Framework for Point Cloud Manifold Evolution

Muhammad Ammad, Leevan Ling

TL;DR

The paper presents a meshless, Lagrangian framework for evolving point-cloud representations of smooth surfaces in $\mathbb{R}^3$ using overlapping tensor-product B-spline patches. It introduces conditioning-aware local interpolation, Gauss-Seidel control-point refinement, and adaptive knot insertion with density control to maintain interpolation accuracy while driving mean curvature- and field-influenced motions. The approach achieves substantial computational savings by patch-wise evolution and reuse of the initial surface, demonstrated across mean curvature flow, strongly coupled surface-field dynamics on a torus, and a tumor-growth model on an evolving surface. These results establish a versatile, efficient tool for dynamic manifold approximation and multiphysics coupling on evolving interfaces. The framework is extensible to more complex topologies and noisy data, with future work targeting topology updates and meshless ALE integration.

Abstract

We extend our recent curve-evolution framework based on localized B-spline interpolation to present an adaptive Lagrangian framework for the geometric evolution of point-cloud data representing smooth, codimension-one surfaces in $\mathbb{R}^3$. The method constructs overlapping, localized tensor-product B-spline patches, enabling direct, meshless surface evolution from discrete samples. Within each patch, the differentiable B-spline representation yields analytic, high-order estimates of intrinsic geometric invariants, supporting curvature-driven and geometry-coupled flows. The organization of control points facilitates coherent updates of both surface samples and spline coefficients under intrinsic velocity fields. A conditioning-aware formulation of the local interpolation system, combined with a Gauss-Seidel refinement of control points, maintains interpolation quality throughout the evolution. Adaptive knot insertion and point redistribution, guided by geometric error indicators and local sampling density, preserve surface resolution and regularity during deformation. Numerical experiments demonstrate efficient and accurate reproduction of surface evolution phenomena, including mean-curvature flow, anisotropic deformations, and coupled surface-field dynamics, establishing localized B-spline methods as precise and versatile tools for dynamic manifold approximation.

An Adaptive Lagrangian B-Spline Framework for Point Cloud Manifold Evolution

TL;DR

The paper presents a meshless, Lagrangian framework for evolving point-cloud representations of smooth surfaces in using overlapping tensor-product B-spline patches. It introduces conditioning-aware local interpolation, Gauss-Seidel control-point refinement, and adaptive knot insertion with density control to maintain interpolation accuracy while driving mean curvature- and field-influenced motions. The approach achieves substantial computational savings by patch-wise evolution and reuse of the initial surface, demonstrated across mean curvature flow, strongly coupled surface-field dynamics on a torus, and a tumor-growth model on an evolving surface. These results establish a versatile, efficient tool for dynamic manifold approximation and multiphysics coupling on evolving interfaces. The framework is extensible to more complex topologies and noisy data, with future work targeting topology updates and meshless ALE integration.

Abstract

We extend our recent curve-evolution framework based on localized B-spline interpolation to present an adaptive Lagrangian framework for the geometric evolution of point-cloud data representing smooth, codimension-one surfaces in . The method constructs overlapping, localized tensor-product B-spline patches, enabling direct, meshless surface evolution from discrete samples. Within each patch, the differentiable B-spline representation yields analytic, high-order estimates of intrinsic geometric invariants, supporting curvature-driven and geometry-coupled flows. The organization of control points facilitates coherent updates of both surface samples and spline coefficients under intrinsic velocity fields. A conditioning-aware formulation of the local interpolation system, combined with a Gauss-Seidel refinement of control points, maintains interpolation quality throughout the evolution. Adaptive knot insertion and point redistribution, guided by geometric error indicators and local sampling density, preserve surface resolution and regularity during deformation. Numerical experiments demonstrate efficient and accurate reproduction of surface evolution phenomena, including mean-curvature flow, anisotropic deformations, and coupled surface-field dynamics, establishing localized B-spline methods as precise and versatile tools for dynamic manifold approximation.
Paper Structure (18 sections, 2 theorems, 68 equations, 8 figures, 1 algorithm)

This paper contains 18 sections, 2 theorems, 68 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Fundamentals of Lagrangian and Spline Geometry
  3. Lagrangian Dynamics of Evolving Manifolds
  4. Tensor-Product B-Spline Surfaces and Classical Interpolation
  5. Differential Geometry of B-spline Surfaces
  6. Localized B-spline Surface Evolution for Dynamic point-clouds
  7. Constructing Overlapping Patches for Local Surface Approximation
  8. B-spline Interpolation Technique for Dynamic Surface Modeling
  9. Conditioning-Aware Reformulation for B-Spline Surface Interpolation
  10. Control Point Refinement Through Distance Minimization
  11. Optimizing Control Points for Geometry-Driven Surface Evolution
  12. Optimizing point-cloud Density for Surface Evolution
  13. Computational Cost Analysis
  14. Numerical Results and Simulations
  15. Mean Curvature Flow
  16. ...and 3 more sections

Key Result

Proposition 3.1

Let $M(\omega)\in\mathbb{R}^{N\times N}$ be the square interpolation matrix constructed from the normalized parameter pairs $\{(u_r(\omega),v_r(\omega))\}$ obtained by rotation $T(\omega)\in\mathrm{SO}(2)$ followed by axis-wise min--max scaling, with $\omega\in[0,2\pi]$. Define and assume that $M(\omega^*)$ is nonsingular. Then the interpolation system admits the unique solution $P=M(\omega^*)^{

Figures (8)

  • Figure 1: Visualization of the patch construction on a point cloud sampled from a spherical surface. Each distinct color denotes a disjoint core patch $\mathcal{P}_k^{\mathrm{core}}$, with no point shared across patches. The highlighted circular region illustrates an extended patch $\mathcal{P}_k$, which includes its core (marked by uniform color) together with additional neighboring points from adjacent patches.
  • Figure 2: B-spline surface interpolation pipeline. (a) Input data points in patch $\mathcal{P}_k \subset \mathbb{R}^3$ (b) rotation to local frame; (c) projection to a parametric plane (d) uniform knot vector construction (e) resulting B-spline interpolant.
  • Figure 3: $\ell^2$-norm error in estimating (a) unit normal vectors and (b) mean curvature at $t = 0$, using local B-spline fitting on scattered surface data. Each curve corresponds to a different point-cloud resolution with total points $N \in \{948, 1806, 2964, 3816, 4890\}$. The x-axis represents the number of core points per patch $m_c$, while the y-axis shows the corresponding $\ell^2$-norm error.
  • Figure 4: Comparison between the numerically computed and analytic radius for the evolution of a sphere under mean curvature flow. The dashed blue curve shows the radius estimated at each time step by our point-cloud B-spline method, while the red line represents the exact solution given by Equation \ref{['eq:sphere_exact']}.
  • Figure 5: Evolution of a unit sphere under mean curvature flow, shown at representative times $t = 0$, $0.05$, $0.10$, $0.15$, and $0.20$. The point cloud shrinks isotropically as the surface contracts, illustrating the expected behavior for this classical test case.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Proposition 3.1: Square system solved at the best-conditioned rotation
  • proof
  • Definition 3.1: Control Point Deviation
  • Definition 3.2: Greville-Based Deviation Approximation
  • Lemma 3.1: Knot Insertion for Tensor-Product Surfaces lyche2008spline