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.

中文翻译：

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将形状流形简化为用于形状分析的线性流形

在本文中，提出了一种将形状流形投影为线性流形的双射，以简化形状分析的非线性问题。形状由离散曲线的方向函数表示。这些形状是有限维形状流形的元素。我们从三个角度讨论形状流形：使用参考坐标系的外在、内在和全局。然后，我们构造另一个流形，其中参考系是傅里叶基，相关坐标是傅里叶变换得到的傅里叶系数。这种变换确保了两个流形的局部空间之间的双射。在构造的流形中，非线性结构由参考系描述。因此，我们仅使用相关坐标获得线性流形。