Efficient Pixel-Wise SVD Required for Image Processing Using the Color Line Feature

In color image processing based on mathematical optimization, a color-line image feature has been considered and many methods that give good results have been proposed. A color-line is a linear color distribution (correlation line) observed in a local image region, and is numerically represented by the sparsity of a data matrix generated from the neighboring pixel values. However, the calculation requires a lot of processing time because each data matrix is processed by inverse calculation or singular value decomposition (SVD) with some operations on the decomposed singular values. In this paper, in order to address this problem, we propose a method that can effectively compute SVD for each data matrix. Using the experimental knowledge that matrices obtained from neighboring regions (each centered at an adjacent pixel) are similar to each other, we intentionally design the SVD by using an iterative method (Arnoldi iteration), and propagate the converged singular vectors at a pixel to the next pixel as the initial vectors of the iteration. This propagation can drastically reduce the number of iterations required for convergence. Additionally, the singular values and vectors are obtained in descending order, which is advantageous when a matrix is reconstructed after reducing small singular values, so we can truncate the calculation when a singular value becomes lower than a threshold value. To show the effectiveness, we apply the proposed method to a denoising method, arranged structure-tensor total variation (ASTV), and show that the processing time is shortened by 95% compared to the naive method without losing numerical accuracy.