近期，实验室博士生陈鹏作为第一作者，导师吴立刚教授作为通讯作者的论文“Simplifying a shape manifold as linear manifold for shape analysis”已在国际期刊Signal, Image and Video Processing发表。

在该文章中，针对经典的无穷维形状流形空间，根据纤维丛理论设计了一种简化形状流形结构的方法，将无穷维流形简化为有限维流形，并且通过空间转换的方式将流形上的非线性度量转化线性度量。本文利用纤维丛理论，将流形局部空间的坐标系与坐标分离，使得非线性结构群从作用于局部坐标转换到局部坐标系，由此形状流形的局部坐标之间实现了线性化。此外，根据曲线数据离散化的过程，我们提出形状流形局部的无穷维函数空间也将相应地被离散化为有限维的空间。参考前期工作，我们将形状流形局部空间的基底定义为具有傅里叶形式的基底，数据坐标由傅里叶变换计算得出。由此我们构建了一个从无穷维形状流形到有限维线性空间的双射变化。最后，通过形状插值、形变迁移和形状检索等应用，验证了该方法的有效性。

Abstract

In this paper, a bijection, which projects the shape manifold as a linear manifold, is proposed to simplify the nonlinear problems of shape analysis. Shapes are represented by the direction function of discrete curves. These shapes are elements of a finite-dimensional shape manifold. We discuss the shape manifold from three perspectives: extrinsic, intrinsic and global using the reference coordinate system. Then, we construct another manifold, in which the reference frame is the Fourier basis and the associated related coordinate is the Fourier coefficients obtained by Fourier transformation. This transformation ensures a bijection between the local spaces of two manifolds. In the constructed manifold, the nonlinear structure is described by the reference frames. Consequently, we obtain a linear manifold only using the related coordinate. The performance of our method is illustrated by the applications of shape interpolation, transportation of shape deformation and shape retrieval.