Analytic Solutions of Integral Moving Least Squares for Polygon Soups

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.