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Efficient manifold evolution algorithm using adaptive B-Spline interpolation

Muhammad Ammad, Leevan Ling

TL;DR

The paper addresses evolving point clouds on smooth manifolds using a Lagrangian approach driven by curvature-influenced velocity fields, e.g., $\vec{V}=-\kappa\hat{\mathbf{n}}$, to replace re-interpolation with adaptive B-Spline interpolation. It develops a cover-based local B-Spline framework where control points carry geometric meaning, and updates occur alongside data points via a partitioned stencil strategy with open-uniform basis, chord-length parameterization, and knot-insertion refinement. Geometric quantities such as normals and curvature are computed from the B-Spline interpolant, while a cost analysis shows substantial efficiency gains over re-interpolation. Numerical experiments on curvature-driven flows and reaction-diffusion-coupled boundaries demonstrate accurate geometry updates with dynamic point redistribution and highlight the method's potential to scale to higher-dimensional manifolds and multi-physics settings.

Abstract

This paper explores an efficient Lagrangian approach for evolving point cloud data on smooth manifolds. In this preliminary study, we focus on analyzing plane curves, and our ultimate goal is to provide an alternative to the conventional radial basis function (RBF) approach for manifolds in higher dimensions. In particular, we use the B-Spline as the basis function for all local interpolations. Just like RBF and other smooth basis functions, B-Splines enable the approximation of geometric features such as normal vectors and curvature. Once properly set up, the advantage of using B-Splines is that their coefficients carry geometric meanings. This allows the coefficients to be manipulated like points, facilitates rapid updates of the interpolant, and eliminates the need for frequent re-interpolation. Consequently, the removal and insertion of point cloud data become seamless processes, particularly advantageous in regions experiencing significant fluctuations in point density. The numerical results demonstrate the convergence of geometric quantities and the effectiveness of our approach. Finally, we show simulations of curvature flows whose speeds depend on the solutions of coupled reaction--diffusion systems for pattern formation.

Efficient manifold evolution algorithm using adaptive B-Spline interpolation

TL;DR

The paper addresses evolving point clouds on smooth manifolds using a Lagrangian approach driven by curvature-influenced velocity fields, e.g., , to replace re-interpolation with adaptive B-Spline interpolation. It develops a cover-based local B-Spline framework where control points carry geometric meaning, and updates occur alongside data points via a partitioned stencil strategy with open-uniform basis, chord-length parameterization, and knot-insertion refinement. Geometric quantities such as normals and curvature are computed from the B-Spline interpolant, while a cost analysis shows substantial efficiency gains over re-interpolation. Numerical experiments on curvature-driven flows and reaction-diffusion-coupled boundaries demonstrate accurate geometry updates with dynamic point redistribution and highlight the method's potential to scale to higher-dimensional manifolds and multi-physics settings.

Abstract

This paper explores an efficient Lagrangian approach for evolving point cloud data on smooth manifolds. In this preliminary study, we focus on analyzing plane curves, and our ultimate goal is to provide an alternative to the conventional radial basis function (RBF) approach for manifolds in higher dimensions. In particular, we use the B-Spline as the basis function for all local interpolations. Just like RBF and other smooth basis functions, B-Splines enable the approximation of geometric features such as normal vectors and curvature. Once properly set up, the advantage of using B-Splines is that their coefficients carry geometric meanings. This allows the coefficients to be manipulated like points, facilitates rapid updates of the interpolant, and eliminates the need for frequent re-interpolation. Consequently, the removal and insertion of point cloud data become seamless processes, particularly advantageous in regions experiencing significant fluctuations in point density. The numerical results demonstrate the convergence of geometric quantities and the effectiveness of our approach. Finally, we show simulations of curvature flows whose speeds depend on the solutions of coupled reaction--diffusion systems for pattern formation.

Paper Structure

This paper contains 12 sections, 1 theorem, 37 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Lagrangian Framework for Moving Curves
  3. Adaptive method for point cloud evolution using B-Splines
  4. Designing a cover for point clouds
  5. Interpolation of parametric curves
  6. Distance-based Control Point Refinement
  7. Curvature and normal computation
  8. Geometry-driven point evolution and control point optimization
  9. Point removal, insertion, and redistribution
  10. Computational cost analysis of evolving point cloud data
  11. Numerical Results and Simulations
  12. Conclusion

Key Result

Lemma 3.1

Let $p$ be a positive integer and let $\Psi$ be a knot vector containing at least $p + 2$ knots. If $\tilde{\Psi}$ is a knot vector such that $\Psi \subseteq \tilde{\Psi}$, then the spline space associated with $\Psi$ is a subspace of the spline space associated with $\tilde{\Psi}$, i.e., $\mathbb{S

Figures (13)

  • Figure 1: This flowchart provides a detailed workflow for the evolution of a point cloud. It includes steps for initialization, partitioning, interpolation using B-Splines, checking tolerances, updating point positions, and refining control points, culminating in obtaining an approximation to the final curve.
  • Figure 2: Partitioning of point cloud data into non-overlapping cores (red segments), which consist of subsets of consecutive points, and their associated stencils (black lines), which extend beyond the cores by including additional boundary points to ensure a smooth transition across the curves.
  • Figure 3: Illustration of parameterization and interpolation in the given data. (Top) The input data points, denoted as $q_i$ and labeled as $\bullet$, with $d_i$ representing the distances between consecutive points. (Bottom) The parameterization values, $u_1, u_2, \dots, u_6$, are computed based on chord length. The red curve represents the smooth interpolation of the x-coordinates, with discrete points labeled as $\circ$, while the blue curve represents the smooth interpolation of the y-coordinates, with discrete points labeled as $\square$.
  • Figure 4: Comparison of B-Spline basis functions and their relation to interpolation (a) Periodic B-Spline basis functions (top) provide smooth cyclic transitions across the knot vector domain. The associated curve (bottom) passes through the extended cover points, but the first and last control points lie outside the data domain, complicating the computation of geometric quantities at the boundaries. (b) Open uniform B-Spline basis functions (top) result in a curve (bottom) that passes through all extended cover points, with control points lying entirely within the data domain, including boundary points, facilitating geometric computations.
  • Figure 5: A demonstration of control point refinement using knot insertion. The original control points $\vec{P}_j$ are labeled as $\circ$, and the refined points $\vec{P}_j^{\prime}$, obtained through knot insertion, are labeled as $*$. This refinement results in a control polygon more closely aligned with the B-Spline curve.
  • ...and 8 more figures

Theorems & Definitions (4)

  • Definition 3.1: marsh2005applied
  • Definition 3.2: marsh2005applied
  • Definition 3.3: scharf2003computing
  • Lemma 3.1: lyche2008spline