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Lipschitz Extension Initialization for Moving Least Squares Reconstruction from Sparse Irregular Samples

Li Chen

arXiv:2609.02918v1eess.SPcs.GR

TL;DR

Sparse and irregular samples can leave MLS local neighborhoods without enough geometric information for stable reconstruction. The paper initializes MLS with a Lipschitz extension, and experiments report improved reconstruction quality, lower error, and greater surface stability, while broader validation remains future work.

  • Problem

    MLS can become unstable and produce larger reconstruction errors when sparse or highly irregular samples leave local neighborhoods without sufficient geometric information.

  • Method

    The paper constructs a stable Lipschitz extension as a preprocessing stage and uses its initialized field as additional geometric information for otherwise unchanged MLS reconstruction.

  • Results

    Numerical experiments report lower reconstruction error and a more stable reconstructed surface than direct MLS under sparse irregular sampling.

  • Takeaways & Limitations

    Lipschitz extensions can serve as effective geometric priors for mesh-free reconstruction methods within the studied sparse irregular sampling setting.

  • Takeaways & Limitations

    The study is preliminary; larger experiments across additional benchmark functions, real-world datasets, and higher-dimensional data remain for future work.

Abstract

from arXiv · show

The idea of using Lipschitz extensions [1,2], or Gradually Varied Functions (GVFs)[3], for mesh-free scattered data reconstruction was proposed by the author in 2012 [4]. However, its practical application to modern mesh-free reconstruction methods has not been fully explored. Motivated by recent advances in computational tools, including AI-assisted mathematical programming and software development, we revisit this idea and investigate the use of a Lipschitz extension as an initialization step for Moving Least Squares (MLS) reconstruction [5,6]. Our computational experiments indicate that this initialization significantly improves the stability and reconstruction accuracy of MLS under sparse and irregular sampling. This is a preliminary study intended to establish feasibility; a fuller evaluation with additional benchmarks and comparisons is left to future work.

1 Introduction

The paper revisits Lipschitz extensions as a preprocessing initialization for MLS reconstruction from sparse, irregular samples. Experiments indicate that this initialization improves reconstruction quality and stability while preserving MLS as the subsequent approximation method.

  • Motivation: Sparse or highly irregular sampling can degrade MLS performance because local neighborhoods lack sufficient geometric information.The paper frames this as the motivation for adding a globally defined initialization before local approximation.
  • Approach: The method constructs a stable Lipschitz interpolation using McShane-Whitney extensions, with GVFs providing a discrete implementation on graphs and digital domains.The continuous and discrete formulations are presented as corresponding ways to initialize scattered-data reconstruction.
  • Approach: The Lipschitz extension propagates information into unsampled regions while preserving original samples, supplying geometric information where MLS neighborhoods are insufficient.MLS is applied to the initialized field rather than directly to the sparse irregular samples.
  • Results: Numerical experiments show that Lipschitz initialization significantly improves reconstruction quality under sparse irregular sampling.With one hundred irregular samples, it substantially reduces error relative to direct MLS and produces a smoother, more stable surface.
  • Contribution: The framework separates stable information propagation from smooth MLS approximation and may serve as a broader geometric-prior preprocessing strategy.It is intended to complement rather than replace existing reconstruction algorithms.

2 Background

The background introduces Lipschitz extensions, GVFs, and MLS as the mathematical components underlying the proposed reconstruction framework. It emphasizes that MLS is smooth but vulnerable to sparse or irregular local sampling.

  • Overview: The framework combines McShane-Whitney extensions, GVFs, and MLS as its mathematical foundation.These components respectively support continuous extension, discrete propagation, and local smooth approximation.
  • McShane-Whitney Extension: The McShane-Whitney theorem extends a Lipschitz function from sample points to the entire space without increasing its Lipschitz constant.The extension depends only on distances to sample points, making it suitable for scattered data.
  • Gradually Varied Functions: GVFs provide a discrete graph-based realization of Lipschitz extensions by propagating values through neighboring vertices while maintaining local consistency.Adjacent values differ by at most one level in the basic GVF formulation, while more general range values satisfy a Lipschitz condition.
  • Moving Least Squares: MLS computes local polynomial approximations using distance-dependent nonnegative weights and generally produces smooth reconstructed surfaces.Its applications include graphics, mesh-free numerical methods, point-cloud processing, and geometric modeling.
  • Moving Least Squares: MLS quality depends strongly on sample distribution, with sparse or highly irregular neighborhoods causing unstable approximations and larger reconstruction errors.Dense, uniform samples provide adequate information for local polynomial fitting, whereas large unsampled regions do not.

3 Lipschitz Extension Initialization for Moving Least Squares

The proposed LI-MLS framework adds a globally defined Lipschitz initialization to unchanged MLS reconstruction. It uses propagated geometric information to improve local fitting under sparse irregular sampling while remaining modular and compatible with existing methods.

  • 3 Lipschitz Extension Initialization for Moving Least Squares: LI-MLS combines Lipschitz initialization with MLS without modifying the MLS approximation algorithm itself.The preprocessing step supplies additional geometric information before local approximation.
  • 3 Lipschitz Extension Initialization for Moving Least Squares: The two-stage workflow separates global information propagation from local smooth approximation, allowing integration into existing MLS implementations with minimal modification.The same preprocessing strategy is expected to apply to other mesh-free reconstruction techniques.
  • 3 Lipschitz Extension Initialization for Moving Least Squares: The procedure first constructs a globally defined Lipschitz extension that propagates information into unsampled regions while preserving original sample values.McShane-Whitney extensions or discrete GVF implementations can provide this initialized field.
  • 3 Lipschitz Extension Initialization for Moving Least Squares: MLS then uses the original samples together with the initialized field as additional geometric information for smooth local approximation.The implementation may sample points from the initialized field and weight their influence through λ.
  • 3 Lipschitz Extension Initialization for Moving Least Squares: LI-MLS reduces to classical MLS, so its purpose is improved stability under sparse irregular sampling rather than changing the MLS fitting procedure.The added prior term preserves the flexibility of the original formulation.

4 Initial Numerical Experiments

The experiments compare direct MLS, McShane-Whitney extension, and LI-MLS on a smooth benchmark with 100 irregular samples. LI-MLS improves reconstruction quality and smoothness over direct MLS while combining Lipschitz information propagation with MLS approximation.

  • Sparse Irregular Sampling: Direct MLS produces a poor, visually nonsmooth approximation when only 100 irregular sample points are available.The irregular distribution leaves local neighborhoods with insufficient information for reliable fitting.
  • Sparse Irregular Sampling: Applying MLS after Lipschitz initialization produces a smoother result than the extension alone and a better visual reconstruction than direct MLS.The extension supplies a stable field and additional geometric information before local MLS fitting.
  • Error Analysis: LI-MLS significantly reduces reconstruction error compared with direct MLS under sparse, irregular sampling.The experiments use RMSE against the original analytical function; the supplied passages do not include the numerical values.
  • Error Analysis: The McShane-Whitney extension interpolates the samples well but remains nonsmooth, with kinks or nondifferentiable ridges.Its min-of-cones construction yields the lowest RMSE, but not the smooth reconstruction targeted here.
  • Discussion: Classical MLS is already accurate for dense, regular samples, whereas Lipschitz initialization is most useful when samples are sparse and irregular.The study presents this conclusion as evidence that stable Lipschitz extensions can improve the robustness of mesh-free reconstruction.
  • Discussion: The framework separates stable information propagation from smooth local approximation rather than requiring one method to perform both tasks.Interpolated extension points may be subsampled using random, uniform, adaptive, or feature-based strategies to reduce computational cost.

5 Experiments for a Physically Motivated Wave-Field Benchmark

The wave-field benchmark tests Lipschitz-initialized reconstruction on a localized oscillatory field with multiple peaks and valleys under sparse irregular sampling. With 300 samples, direct MLS fails to recover the central wave structure, motivating comparison with McShane initialization.

  • Benchmark function: The benchmark uses a Gaussian-modulated plane wave whose envelope localizes the field and whose cosine term creates multiple peaks and valleys.This combines localized structure with oscillatory behavior as a physically motivated reconstruction test.
  • Experimental setup: 300 samples were used because 100 samples were insufficient to represent the wave field’s multiple central oscillations.The experiment preserves relatively sparse sampling while testing whether initialization helps smooth mesh-free fitting.
  • Reconstruction comparison: Direct MLS applied to the 300 samples does not recover the multiple waves in the image center and instead produces a smooth interpretation with apparent surrounding noise.The corresponding MLS reconstruction is shown in Figure 8.
  • Reconstruction comparison: The experiment compares direct MLS with McShane extension and a subsequent MLS reconstruction, with the latter shown as the proposed combined outcome.Figure 9 shows McShane extension from the same samples, while Figure 10 presents the MLS-after-extension result.

6 Conclusions and Future Work

The paper concludes that Lipschitz initialization can improve MLS reconstruction from sparse irregular samples by propagating geometric information before smooth approximation. The framework is presented as modular and promising, but broader validation and adaptive design choices remain future work.

  • Conclusions: The LI-MLS framework combines McShane-Whitney Lipschitz extension with MLS to propagate information into unsampled regions before smooth local approximation.It separates stable information propagation from smooth approximation rather than modifying or replacing MLS.
  • Conclusions: On a benchmark smooth function, Lipschitz initialization lowers reconstruction error and produces a more stable reconstructed surface than direct MLS under sparse irregular sampling.The reported experiments support Lipschitz extensions as geometric priors for mesh-free reconstruction.
  • Conclusions: The preprocessing framework is modular and can be integrated with existing mesh-free reconstruction algorithms.Its purpose is to improve information available before approximation, not to replace MLS or alter its approximation procedure.
  • Future work: The study remains preliminary because larger benchmarks, real-world and higher-dimensional data, scalability tests, and adaptive choices of Lipschitz constants and neighborhoods are still needed.Future work also includes relating Lipschitz extensions and GVFs to higher-order smooth reconstruction frameworks.
  • Future work: The broader proposed perspective is that stable Lipschitz extensions can bridge interpolation and approximation for incomplete and irregular data.This implication is stated as a direction suggested by the work rather than as a completed validation across broader settings.
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