Simplifying a shape manifold as linear manifold for shape analysis

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.