Nonisometric Surface Registration via Conformal Laplace–Beltrami Basis Pursuit
Nonisometric Surface Registration via Conformal Laplace–Beltrami Basis Pursuit
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DOI：
10.1007/s10915-020-01390-y
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发表时间：
2018-09
影响因子：
2.5
通讯作者：
Stefan C. Schonsheck;M. Bronstein;Rongjie Lai
中科院分区：
文献类型：
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作者：
Stefan C. Schonsheck;M. Bronstein;Rongjie Lai
曲面配准是几何加工中最基本的问题之一。已经开发了许多方法来解决表面几乎是等距的情况下的这个问题。然而，计算本质上不太相似的表面之间的对应关系更具挑战性。本文提出了一种变分模型，通过共形变形将两个非等距零形的拉普拉斯-贝尔特拉米（LB）本征系统对齐。这种方法使我们能够在非等距形状之间计算几何上有意义的点对点映射。我们的模型基于一种新的基追踪方案，我们同时计算“目标形状”的保形变形及其变形LB特征系统。我们使用一种混合增广拉格朗日方法的近端交替最小化算法来求解该模型，该算法仅在几个地标点的情况下就能产生精确的对应关系。为了克服变分问题的非凸性所带来的一些困难，我们还提出了一种重新初始化方案。大量的数值实验证明了该方法在处理具有大变形的非等距表面时的有效性和鲁棒性，同时考虑了底层流形上的噪声和给定地标或特征函数内的误差。
Surface registration is one of the most fundamental problems in geometry processing. Many approaches have been developed to tackle this problem in cases where the surfaces are nearly isometric. However, it is much more challenging to compute correspondence between surfaces which are intrinsically less similar. In this paper, we propose a variational model to align the Laplace-Beltrami (LB) eigensytems of two non-isometric genus zero shapes via conformal deformations. This method enables us to compute geometrically meaningful point-to-point maps between non-isometric shapes. Our model is based on a novel basis pursuit scheme whereby we simultaneously compute a conformal deformation of a ’target shape’ and its deformed LB eigensystem. We solve the model using a proximal alternating minimization algorithm hybridized with the augmented Lagrangian method which produces accurate correspondences given only a few landmark points. We also propose a re-initialization scheme to overcome some of the difficulties caused by the non-convexity of the variational problem. Intensive numerical experiments illustrate the effectiveness and robustness of the proposed method to handle non-isometric surfaces with large deformation with respect to both noises on the underlying manifolds and errors within the given landmarks or feature functions.