A Fast Algorithm for Edge-Preserving Variational Multichannel Image Restoration

Abstract

Variational models with $\ell_1$-norm based regularization, in particular total variation (TV) and its variants, have long been known to offer superior image restoration quality, but processing speed remained a bottleneck, preventing their widespread use in the practice of color image processing. In this paper, by extending the grayscale image deblurring algorithm proposed in [Y. Wang, J. Yang, W. Yin, and Y. Zhang, SIAM J. Imaging Sci., 1 (2008), pp. 248–272], we construct a simple and efficient algorithm for multichannel image deblurring and denoising, applicable to both within-channel and cross-channel blurs in the presence of additive Gaussian noise. The algorithm restores an image by minimizing an energy function consisting of an $\ell_2$-norm fidelity term and a regularization term that can be either TV, weighted TV, or regularization functions based on higher-order derivatives. Specifically, we use a multichannel extension of the classic TV regularizer (MTV) and derive our algorithm from an extended half-quadratic transform of Geman and Yang [IEEE Trans. Image Process., 4 (1995), pp. 932–946]. For three-channel color images, the per-iteration computation of this algorithm is dominated by six fast Fourier transforms. The convergence results in [Y. Wang, J. Yang, W. Yin, and Y. Zhang, SIAM J. Imaging Sci., 1 (2008), pp. 248–272] for single-channel images, including global convergence with a strong q-linear rate and finite convergence for some quantities, are extended to this algorithm. We present numerical results including images recovered from various types of blurs, comparisons between our results and those obtained from the deblurring functions in MATLAB's Image Processing Toolbox, as well as images recovered by our algorithm using weighted MTV and higher-order regularization. Our numerical results indicate that the processing speed, as attained by the proposed algorithm, of variational models with TV-like regularization can be made comparable to that of less sophisticated but widely used methods for color image restoration.