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Article

Isometry Invariant Shape Recognition of Projectively Perturbed Point Clouds by the Mergegram Extending 0D Persistence

Materials Innovation Factory, Computer Science Department, University of Liverpool, Liverpool L69 3BX, UK
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This paper is an extended version of the shorter paper published in the 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020), Prague, Czech Republic, 24–28 August 2020.
Academic Editors: Rocio Gonzalez Diaz and Matthias Zeppelzauer
Mathematics 2021, 9(17), 2121; https://doi.org/10.3390/math9172121
Received: 6 July 2021 / Revised: 24 August 2021 / Accepted: 25 August 2021 / Published: 1 September 2021
Rigid shapes should be naturally compared up to rigid motion or isometry, which preserves all inter-point distances. The same rigid shape can be often represented by noisy point clouds of different sizes. Hence, the isometry shape recognition problem requires methods that are independent of a cloud size. This paper studies stable-under-noise isometry invariants for the recognition problem stated in the harder form when given clouds can be related by affine or projective transformations. The first contribution is the stability proof for the invariant mergegram, which completely determines a single-linkage dendrogram in general position. The second contribution is the experimental demonstration that the mergegram outperforms other invariants in recognizing isometry classes of point clouds extracted from perturbed shapes in images. View Full-Text
Keywords: shape recognition; Topological Data Analysis; machine learning; computer vision shape recognition; Topological Data Analysis; machine learning; computer vision
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MDPI and ACS Style

Elkin, Y.; Kurlin, V. Isometry Invariant Shape Recognition of Projectively Perturbed Point Clouds by the Mergegram Extending 0D Persistence. Mathematics 2021, 9, 2121. https://doi.org/10.3390/math9172121

AMA Style

Elkin Y, Kurlin V. Isometry Invariant Shape Recognition of Projectively Perturbed Point Clouds by the Mergegram Extending 0D Persistence. Mathematics. 2021; 9(17):2121. https://doi.org/10.3390/math9172121

Chicago/Turabian Style

Elkin, Yury, and Vitaliy Kurlin. 2021. "Isometry Invariant Shape Recognition of Projectively Perturbed Point Clouds by the Mergegram Extending 0D Persistence" Mathematics 9, no. 17: 2121. https://doi.org/10.3390/math9172121

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