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Two-dimensional shape retrieval using the distribution of extrema of Laplacian eigenfunctions

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Abstract

We propose a new method using the distribution of extrema of Laplacian eigenfunctions for two-dimensional (2D) shape description and matching. We construct a weighted directed graph, which we call signed natural neighbor graph, to represent a Laplacian eigenfunction of a shape. The nodes of this sparse graph are the extrema of the corresponding eigenfunction, and the edge weights are defined by signed natural neighbor coordinates derived from the local spatial arrangement of extrema. We construct the signed natural neighbor graphs defined by a small number of low-frequency Laplacian eigenfunctions of a shape to describe it. This shape descriptor is invariant under rigid transformations and uniform scaling, and is also insensitive to minor boundary deformations. When using our shape descriptor for matching two shapes, we determine their similarity by comparing the graphs induced by corresponding Laplacian eigenfunctions of the two shapes. Our experimental shape-matching results demonstrate that our method is effective for 2D shape retrieval.

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Notes

  1. We do not consider the first Laplacian eigenfunction because this eigenfunction is constant.

  2. For the results generated by ShapeDNA that are shown in this paper, we tested using different numbers of Laplacian eigenvalues to do shape retrieval and chose the best results.

  3. The six shape classes from the Kimia-25 database consist of different numbers of shapes, and the largest number is five. For convenience, we compute the retrieval rate for the whole database by counting the correct matches among the first four retrieved shapes for all the 25 shapes.

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Acknowledgments

We thank Martin Reuter for making available his “ShapeDNA-tria” software, Jonathan Shewchuk for his “Triangle” program and Gabriel Peyré for his toolboxes—“Toolbox Fast Marching” and “Toolbox Graph”.

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Authors and Affiliations

  1. School of Computer Science and Technology, Shandong University, 1500 Shunhua Road, Jinan, 250101, China

    Dongmei Niu, Yuanfeng Zhou & Caiming Zhang

  2. Lawrence Livermore National Laboratory, 7000 East Avenue, L-422, Livermore, CA, 94551, USA

    Peer-Timo Bremer & Peter Lindstrom

  3. Department of Computer Science, Institute for Data Analysis and Visualization, University of California, Davis, One Shields Avenue, Davis, CA, 95616, USA

    Bernd Hamann

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  1. Dongmei Niu
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  2. Peer-Timo Bremer
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  3. Peter Lindstrom
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  5. Yuanfeng Zhou
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Correspondence to Dongmei Niu.

Additional information

Dongmei Niu acknowledges fellowship support from the China Scholarship Council (CSC). Caiming Zhang appreciates the supports from the National Nature Science Foundation of China (61373078) and NSFC Joint Fund with Guangdong (U1201258).

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Niu, D., Bremer, PT., Lindstrom, P. et al. Two-dimensional shape retrieval using the distribution of extrema of Laplacian eigenfunctions. Vis Comput 33, 607–624 (2017). https://doi.org/10.1007/s00371-016-1211-6

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