用于从稀疏不规则样本进行移动最小二乘重建的Lipschitz扩展初始化
Lipschitz Extension Initialization for Moving Least Squares Reconstruction from Sparse Irregular Samples
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中文总结 AI 辅助
本研究提出将Lipschitz扩展用作移动最小二乘重建的初始化步骤，实验表明该方法可提升稀疏不规则采样下MLS重建的稳定性与精度，为后续更全面评估奠定基础。
中文摘要 AI 辅助
使用Lipschitz扩展（或渐变异函数GVF）进行无网格散乱数据重建的想法，由作者于2012年提出，但该方法在现代无网格重建方法中的实际应用尚未得到充分探索。受AI辅助数学编程与软件开发等计算工具最新进展的推动，本文重新探讨该想法，研究将Lipschitz扩展用作移动最小二乘（MLS）重建的初始化步骤。计算实验表明，该初始化能显著提升MLS在稀疏不规则采样下的稳定性与重建精度。本研究为建立可行性的初步研究，更全面的评估（含额外基准测试与对比）留待未来工作开展。
英文摘要
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.
