Simplifying a shape manifold as linear manifold for shape analysis
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ORIGINAL PAPER
Simplifying a shape manifold as linear manifold for shape analysis Peng Chen1 · Xutao Li2 · Jianxing Liu1 · Ligang Wu1 Received: 12 October 2019 / Revised: 11 October 2020 / Accepted: 16 November 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
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. Keywords Shape analysis · Shape manifold · Shape retrieval · Shape interpolation
1 Introduction Shape analysis studies the objects in a scene that is a pivotal way to understand objects. The shapes are the boundaries of objects that express the external form or appearance of the objects that are invariant by the transformation of translation, rotation and scaling [15]. The goal of shape analysis is to find a metric distance, which is used to measure the dissimilarity between shapes. The related methods are widely used in computer vision and biomedical engineering [9,12,14,16,24]. A common framework for shape analysis, firstly proposed in [17], is to construct a shape manifold, where each element corresponds to one shape. Usually, shape manifold is a kind of Riemannian manifold, where a metric distance between shapes is computed by a particular Riemannian metric along
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Ligang Wu [email protected] Peng Chen [email protected] Xutao Li [email protected] Jianxing Liu [email protected]
1
Department of Control Science and Engineering, Harbin Institute of Technology, Harbin 150001, China
2
Department of Electronics Engineering, Shantou University, Shantou 515063, China
geodesic. The metric distance is a dissimilarity score in shape comparison. Furthermore, it is a minimization process to find a geodesic on shape manifold. This technique is used in shape deformation and shape synthesis restricted by the deformation energy minimization. In [17], the shapes are considered as planar and closed curves and represented as their direction functions or curvature functions, which leads an infinite-dimensional Hilbert manifold for the shape space. A set of Fourier basis represent the tangent of shape manifold, and a shooting method finds geodesic on an infinit
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