Lanczos method for large-scale quaternion singular value decomposition

Zhigang Jia, Kwok Po NG*, Guang Jing Song

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

48 Citations (Scopus)


In many color image processing and recognition applications, one of the most important targets is to compute the optimal low-rank approximations to color images, which can be reconstructed with a small number of dominant singular value decomposition (SVD) triplets of quaternion matrices. All existing methods are designed to compute all SVD triplets of quaternion matrices at first and then to select the necessary dominant ones for reconstruction. This way costs quite a lot of operational flops and CPU times to compute many superfluous SVD triplets. In this paper, we propose a Lanczos-based method of computing partial (several dominant) SVD triplets of the large-scale quaternion matrices. The partial bidiagonalization of large-scale quaternion matrices is derived by using the Lanczos iteration, and the reorthogonalization and thick-restart techniques are also utilized in the implementation. An algorithm is presented to compute the partial quaternion singular value decomposition. Numerical examples, including principal component analysis, color face recognition, video compression and color image completion, illustrate that the performance of the developed Lanczos-based method for low-rank quaternion approximation is better than that of the state-of-the-art methods.

Original languageEnglish
Pages (from-to)699-717
Number of pages19
JournalNumerical Algorithms
Issue number2
Publication statusPublished - 1 Oct 2019

Scopus Subject Areas

  • Applied Mathematics

User-Defined Keywords

  • Color images
  • Lanczos method
  • Partial bidiagonalization
  • Quaternion SVD


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