论文标题
亚里曼尼亚人地标匹配及其解释为残留神经网络
Sub-Riemannian Landmark Matching and its interpretation as residual neural networks
论文作者
论文摘要
在形状分析中，找到一个与初始的点旋转到目标集的时间相关的向量场的问题。这是在差异形状匹配方案中问题的一个示例，可以被认为是差异图像匹配的空间离散化。在本文中，我们考虑通过限制可用矢量字段的集合来修改的地标匹配，从某种意义上说，向量字段通过一组控件参数化。我们确定问题的几何设置，称为亚riemannian Landmark匹配，并得出控件的运动方程。我们提供两种计算算法，并在数值示例中演示它们。特别是，实验突出了正规化项的重要性。一个强烈的动机是，亚里曼尼亚人地标匹配与神经网络有联系，尤其是将残留神经网络解释为连续控制问题的时间离散化。它允许形状分析从业人员在差异地标匹配方面考虑神经网络，从而在两个字段之间提供桥梁。
The problem of finding a time-dependent vector field which warps an initial set of points to a target set is common in shape analysis. It is an example of a problem in the diffeomorphic shape matching regime, and can be thought of as a spatial discretization of diffeomorphic image matching. In this paper, we consider landmark matching modified by restricting the set of available vector fields in the sense that vector fields are parametrized by a set of controls. We determine the geometric setting of the problem, referred to as sub-Riemannian landmark matching, and derive the equations of motion for the controls. We provide two computational algorithms and demonstrate them in numerical examples. In particular, the experiments highlight the importance of the regularization term. A strong motivation is that sub-Riemannian landmark matching have connections with neural networks, in particular the interpretation of residual neural networks as time discretizations of continuous control problems. It allows shape analysis practitioners to think about neural networks in terms of diffeomorphic landmark matching, thereby providing a bridge between the two fields.
