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Randomized progressive iterative approximation for B-spline curve and surface fittings

Nian-Ci Wu, Chengzhi Liu

TL;DR

This work introduces Randomized Progressive Iterative Approximation (RPIA) for B-spline curve and surface fittings, extending the traditional LSPIA by updating only randomly selected blocks of control points at each iteration. RPIA retains the least-squares limit behavior in expectation, and its updates correspond to a randomized block coordinate descent framework, enabling significant memory and computational savings, especially for large-scale data. Theoretical convergence analyses show that, under full-column-rank conditions, the method converges in expectation to the LS solution for both curves and surfaces; tensor-formulation further supports parallelizable updates. Numerical experiments demonstrate that RPIA matches the final accuracy of LSPIA-family methods while achieving substantially lower CPU times due to locality and parallelizability, with additional potential extensions to weighted variants and subdivision surfaces. Overall, RPIA offers a scalable, efficient alternative for high-volume B-spline data fitting in CAD, graphics, and visualization contexts.

Abstract

For large-scale data fitting, the least-squares progressive iterative approximation is a widely used method in many applied domains because of its intuitive geometric meaning and efficiency. In this work, we present a randomized progressive iterative approximation (RPIA) for the B-spline curve and surface fittings. In each iteration, RPIA locally adjusts the control points according to a random criterion of index selections. The difference for each control point is computed concerning the randomized block coordinate descent method. From geometric and algebraic aspects, the illustrations of RPIA are provided. We prove that RPIA constructs a series of fitting curves (resp., surfaces), whose limit curve (resp., surface) can converge in expectation to the least-squares fitting result of the given data points. Numerical experiments are given to confirm our results and show the benefits of RPIA.

Randomized progressive iterative approximation for B-spline curve and surface fittings

TL;DR

This work introduces Randomized Progressive Iterative Approximation (RPIA) for B-spline curve and surface fittings, extending the traditional LSPIA by updating only randomly selected blocks of control points at each iteration. RPIA retains the least-squares limit behavior in expectation, and its updates correspond to a randomized block coordinate descent framework, enabling significant memory and computational savings, especially for large-scale data. Theoretical convergence analyses show that, under full-column-rank conditions, the method converges in expectation to the LS solution for both curves and surfaces; tensor-formulation further supports parallelizable updates. Numerical experiments demonstrate that RPIA matches the final accuracy of LSPIA-family methods while achieving substantially lower CPU times due to locality and parallelizability, with additional potential extensions to weighted variants and subdivision surfaces. Overall, RPIA offers a scalable, efficient alternative for high-volume B-spline data fitting in CAD, graphics, and visualization contexts.

Abstract

For large-scale data fitting, the least-squares progressive iterative approximation is a widely used method in many applied domains because of its intuitive geometric meaning and efficiency. In this work, we present a randomized progressive iterative approximation (RPIA) for the B-spline curve and surface fittings. In each iteration, RPIA locally adjusts the control points according to a random criterion of index selections. The difference for each control point is computed concerning the randomized block coordinate descent method. From geometric and algebraic aspects, the illustrations of RPIA are provided. We prove that RPIA constructs a series of fitting curves (resp., surfaces), whose limit curve (resp., surface) can converge in expectation to the least-squares fitting result of the given data points. Numerical experiments are given to confirm our results and show the benefits of RPIA.
Paper Structure (14 sections, 2 theorems, 82 equations, 13 figures, 8 tables, 2 algorithms)

This paper contains 14 sections, 2 theorems, 82 equations, 13 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The LSPIA method
  3. The case of curves
  4. The case of surfaces
  5. The RPIA method
  6. The case of curves
  7. The case of surfaces
  8. Convergence analyses of RPIA
  9. The case of curves
  10. The case of surfaces
  11. Numerical experiments
  12. Curve fitting
  13. Surface fitting
  14. Concluding remarks

Key Result

Theorem 4.1

Let $\left\{\mu_i(x):i \in[n] \right\}$ be a blending basis sequence and ${\bm A}$ be the corresponding collocation matrix on the real increasing sequence $\left\{x_h \right\}_{h=0}^m$. Suppose that ${\bm A}$ has a full-column rank, when the number of data points is larger than that of control point

Figures (13)

  • Figure 1: Graphical illustration of RPIA for least-squares curve fitting.
  • Figure 2: The data point sets to be fitted in Examples \ref{['ex:RPIA1']} (a), \ref{['ex:RPIA2']} (b), \ref{['ex:RPIA2']} (c), and \ref{['ex:RPIA4']} (d) with $m=20000$.
  • Figure 3: The cubic B-spline fitting curves given by RPIA with $m=20000$ and $n= 500$ for Example \ref{['ex:RPIA1']}.
  • Figure 4: The cubic B-spline fitting curves given by RPIA with $m=20000$ and $n= 500$ for Example \ref{['ex:RPIA2']}.
  • Figure 5: The cubic B-spline fitting curves given by RPIA with $m=20000$ and $n= 500$ for Example \ref{['ex:RPIA3']}.
  • ...and 8 more figures

Theorems & Definitions (15)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 4.1
  • Theorem 4.2
  • Example 5.1
  • Example 5.2
  • Example 5.3
  • ...and 5 more