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Analytic Solutions of Integral Moving Least Squares for Polygon Soups

机译：多边形汤的整数移动最小二乘解析解

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摘要

This paper presents analytic solutions to the integral moving least squares (MLS) equations originally proposed by Shen et al. by choosing another specific weighting function that renders the numerator in the MLS equation unitless. In addition, we analyze the original method to show that their approximation surfaces (i.e., enveloping surfaces with nonzero epsilon values in the weighting function) often form zero isosurfaces near concavities behind the triangle-soup models. This paper also presents error terms for the integral MLS formulations against signed distance fields. Based on our analytic solutions, we show that our method provides both interpolation and approximation surfaces faster and more efficiently. Because our method computes solutions for integral MLS equations directly, it does not rely on numerical steps that might have numerical-accuracy issues. In particular, unlike the original method that deals with incorrect approximation surfaces by iteratively adjusting parameters, this paper proposes faster and more efficient approximations to surfaces without needing iterative routines. We also present computational efficiency comparisons, in which our method is 15-fold faster in computing integrations, even with conservative assumptions. Finally, we show that the surface normal vectors on the implicit surfaces formed by our analytic solutions are identical to the angle-weighted pseudonormal vectors.
机译：本文提出了由Shen等人最初提出的积分移动最小二乘（MLS）方程的解析解。通过选择另一个特定的加权函数，该函数使MLS方程中的分子变为无单位。此外，我们分析了原始方法，以表明它们的近似曲面（即加权函数中具有非零epsilon值的包络面）通常在三角形汤模型后面的凹面附近形成零等值面。本文还提出了针对有符号距离场的积分MLS公式的误差项。根据我们的解析解，我们证明了我们的方法可以更快，更有效地提供插值曲面和逼近曲面。因为我们的方法直接计算积分MLS方程的解，所以它不依赖于可能存在数值精度问题的数值步骤。特别是，与原始方法不同，该方法通过迭代地调整参数来处理不正确的近似曲面，而本文提出了无需迭代例程即可更快，更有效地近似曲面的方法。我们还提出了计算效率的比较，即使保守的假设，我们的方法在计算集成中也要快15倍。最后，我们证明了由解析解形成的隐式曲面上的曲面法线向量与角度加权的伪法线向量相同。

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