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Simplifying a shape manifold as linear manifold for shape analysis

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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.

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Acknowledgements

This work was supported in part by the National Key R&D Program of China (2019YFB1312001), the National Natural Science Foundation of China (61471229, 62022030, 62033005 and 61673130), the state grid Heilongjiang electric power company limited funded project (No. 522417190057) and the Self-Planned Task of State Key Laboratory of Advanced Welding and Joining (HIT), the Natural Science Foundation of Guangdong Province of China (2019A1515011950) and Heilongjiang Provincial Natural Science Foundation of China (F2018012 and QC2017072).

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Authors and Affiliations

  1. Department of Control Science and Engineering, Harbin Institute of Technology, Harbin, 150001, China

    Peng Chen, Jianxing Liu & Ligang Wu

  2. Department of Electronics Engineering, Shantou University, Shantou, 515063, China

    Xutao Li

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  1. Peng Chen
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  2. Xutao Li
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Correspondence to Ligang Wu.

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Chen, P., Li, X., Liu, J. et al. Simplifying a shape manifold as linear manifold for shape analysis. SIViP 15, 1003–1010 (2021). https://doi.org/10.1007/s11760-020-01825-x

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