Abstract
The computation of quaternion singular triplets has become one of the core targets of color image processing. However, the existing algorithms are far from meeting people’s expectations on the computation speed. A novel harmonic multi-symplectic Lanczos algorithm is presented for approximating extreme quaternion singular triplets, which performs real operations entirely and stores only four parts of quaternion matrices or vectors. The underlying theory is to preserve an algebraic structure during the partial bidiagonalization, the argumentation, and the restarted bidiagonalization. Both the smallest and largest quaternion singular triples are computed with high precision and in short calculation time. The proposed algorithm is applied to color video semantic segmentation. Numerical examples on synthetic and color image data sets illustrate that the proposed algorithm is superior to the state-of-the-art algorithms in terms of residual calculation and computational time.
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Data Availability
The datasets generated during and/or analyses during the current study are available in the website https://math.nist.gov/MatrixMarket and the paper [31].
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Acknowledgements
Thanks to Professor Michael K. Ng for the wonderful suggestions on the presentation of the paper. We would like to say thanks to Professor Ken Hayami and the anonymous referees for their useful commons and suggestions that are very helpful to improve the original presentation. The authors would like to dedicate this article to Prof. Musheng Wei on his 75th birthday.
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This paper is supported in part by the National Natural Science Foundation of China under grants 12171210, 12090011 and 11771188.
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Z. Jia and M. Zhao wrote the first version of this paper; X. Liu and J. Zhu implemented the numerical experiments; X. Liu and Z. Jia contributed equally to this work; all authors reviewed the last version.
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Jia, Z., Liu, X., Zhu, J. et al. Harmonic multi-symplectic Lanczos algorithm for quaternion singular triplets. Numer Algor 93, 1309–1335 (2023). https://doi.org/10.1007/s11075-022-01469-7
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DOI: https://doi.org/10.1007/s11075-022-01469-7