Image colorization by using graph bi-Laplacian

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.