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.

We present a novel remeshing algorithm that avoids triangles with small (acute) angles and those with large (obtuse) angles. Our solution is based on an extension of Centroidal Voronoi Tesselation (CVT).We augment the original CVT formulation with a penalty term that penalizes short Voronoi edges, while the CVT term helps to avoid small angles. Our results show significant improvements in remeshing quality over the state of the art.

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.

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In this paper, we tackle the problem of tiling a domain with a set of deformable templates. A valid solution to this problem completely covers the domain with templates such that the templates do not overlap. We generalize existing specialized solutions and formulate a general layout problem by modeling important constraints and admissible template deformations. Our main idea is to break the layout algorithm into two steps: a discrete step to lay out the approximate template positions and a continuous step to refine the template shapes. Our approach is suitable for a large class of applications, including floorplans, urban layouts, and arts and design.

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Non-rigid registration of 3D shapes is an essential task of increasing importance as commodity depth sensors become more widely available for scanning dynamic scenes. Non-rigid registration is much more challenging than rigid registration as it estimates a set of local transformations instead of a single global transformation, and hence is prone to the overfitting issue due to underdetermination. The common wisdom in previous methods is to impose an l2-norm regularization on the local transformation differences. However, the l2-norm regularization tends to bias the solution towards outliers and noise with heavy-tailed distribution, which is verified by the poor goodness- of-fit of the Gaussian distribution over transformation differences. On the contrary, Laplacian distribution fits well with the transformation differences, suggesting the use of a sparsity prior. We propose a sparse non-rigid registration (SNR) method with an l1-norm regularized model for transformation estimation, which is effectively solved by an alternate direction method (ADM) under the augmented Lagrangian framework. We also devise a multi-resolution scheme for robust and progressive registration. Results on both public datasets and our scanned datasets show the superiority of our method, particularly in handling large-scale deformations as well as outliers and noise.

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