In this paper, we introduce a surface reconstruction method that can perform gracefully with non-uniformly-distributed, noisy, and even sparse data. We reconstruct the surface by estimating an implicit function and then obtain a triangle mesh by extracting an iso-surface from it. Our implicit function takes advantage of both the indicator function and the signed distance function. It is dominated by the indicator function at the regions away from the surface and approximates (up to scaling) the signed distance function near the surface. On one hand, it is well defined over the entire space so that the extracted iso-surface must lie near the underlying true surface and is free of spurious sheets. On the other hand, thanks to the nice properties of the signed distance function, a smooth iso-surface can be extracted using the approach of marching cubes with simple linear interpolations. More importantly, our implicit function can be estimated directly from an explicit integral formula without solving any linear system. This direct approach leads to a simple, accurate and robust reconstruction method, which can be paralleled with little overhead. We call our reconstruction method Gauss surface reconstruction. We apply our method to both synthetic and real-world scanned data and demonstrate the accuracy, robustness and efficiency of our method. The performance of Gauss surface reconstruction is also compared with that of several state-of-the-art methods.

The alternating direction method of multipliers (ADMM) is a popular approach for solving optimization problems that are potentially non-smooth and with hard constraints. It has been applied to various computer graphics applications, including physical simulation, geometry processing, and image processing. However, ADMM can take a long time to converge to a solution of high accuracy. Moreover, many computer graphics tasks involve non-convex optimization, and there is often no convergence guarantee for ADMM on such problems since it was originally designed for convex optimization. In this paper, we propose a method to speed up ADMM using Anderson acceleration, an established technique for accelerating fixed-point iterations. We show that in the general case, ADMM is a fixed-point iteration of the second primal variable and the dual variable, and Anderson acceleration can be directly applied. Additionally, when the problem has a separable target function and satisfies certain conditions, ADMM becomes a fixed-point iteration of only one variable, which further reduces the computational overhead of Anderson acceleration. Moreover, we analyze a particular non-convex problem structure that is common in computer graphics, and prove the convergence of ADMM on such problems under mild assumptions. We apply our acceleration technique on a variety of optimization problems in computer graphics, with notable improvement on their convergence speed.

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