We present a method to match three dimensional shapes under non-isometric
deformations, topology changes and partiality. We formulate the problem as
matching between a set of pair-wise and point-wise descriptors, imposing a
continuity prior on the mapping, and propose a projected descent optimization
procedure inspired by difference of convex functions (DC) programming.
Surprisingly, in spite of the highly non-convex nature of the resulting
quadratic assignment problem, our method converges to a semantically meaningful
and continuous mapping in most of our experiments, and scales well. We provide
preliminary theoretical analysis and several interpretations of the method.

本文提出了一种通过描述符匹配、连续性限制和 DC 编程来解决三维形状匹配问题的方法，该方法可以在处理非等距变形、拓扑变化和不完整数据的情况下，有效地收敛到一种有意义的连续匹配方案，具有较好的可扩展性。

通过核匹配实现高效的可变形形状对应

Efficient Deformable Shape Correspondence via Kernel Matching

Establishing correspondence between shapes is a fundamental problem in
geometry processing, arising in a wide variety of applications. The problem is
especially difficult in the setting of non-isometric deformations, as well as
in the presence of topological noise and missing parts, mainly due to the
limited capability to model such deformations axiomatically. Several recent
works showed that invariance to complex shape transformations can be learned
from examples. In this paper, we introduce an intrinsic convolutional neural
network architecture based on anisotropic diffusion kernels, which we term
Anisotropic Convolutional Neural Network (ACNN). In our construction, we
generalize convolutions to non-Euclidean domains by constructing a set of
oriented anisotropic diffusion kernels, creating in this way a local intrinsic
polar representation of the data (`patch'), which is then correlated with a
filter. Several cascades of such filters, linear, and non-linear operators are
stacked to form a deep neural network whose parameters are learned by
minimizing a task-specific cost. We use ACNNs to effectively learn intrinsic
dense correspondences between deformable shapes in very challenging settings,
achieving state-of-the-art results on some of the most difficult recent
correspondence benchmarks.

本研究介绍了一种基于各向异性扩散核的内在卷积神经网络 (Anisotropic Convolutional Neural Network, ACNN)，该网络通过学习复杂形态变化的不变性，能够在极具挑战性的设置中高效地学习可变形形状间的内在密集对应关系，并在最近的对应基准测试中取得了最新的最佳结果。