Image colorization by using graph bi-Laplacian

Fang Li*, Kwok Po NG

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

9 Citations (Scopus)


Image colorization aims to recover the whole color image based on a known grayscale image (luminance or brightness) and some known color pixel values. In this paper, we generalize the graph Laplacian to its second-order variant called graph bi-Laplacian, and then propose an image colorization method by using graph bi-Laplacian. The eigenvalue analysis of graph bi-Laplacian matrix and its corresponding normalized bi-Laplacian matrix is given to show their properties. We apply graph bi-Laplacian approach to image colorization by formulating it as an optimization problem and solving the resulting linear system efficiently. Numerical results show that the proposed method can perform quite well for image colorization problem, and its performance in terms of efficiency and colorization quality for test images can be better than that by the state-of-the-art colorization methods when the randomly given color pixels ratio attains some level.

Original languageEnglish
Pages (from-to)1521–1549
Number of pages29
JournalAdvances in Computational Mathematics
Issue number3
Early online date12 Mar 2019
Publication statusPublished - 1 Jun 2019

Scopus Subject Areas

  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Graph bi-Laplacian
  • Graph Laplacian
  • Image colorization
  • Linear system
  • Optimization


Dive into the research topics of 'Image colorization by using graph bi-Laplacian'. Together they form a unique fingerprint.

Cite this