We focus on image restoration where the reconstructed image is found by minimizing an objective functional that consists of a quadratic data-fidelity term regularized with the standard sparsity-enforcing norm:
where is the observation, models the measurement process and is some transformation, e.g. wavelet, such that . We propose a novel algorithmic approach to solve this optimization problem iteratively. Our idea amounts to approximating the result of the restoration as a linear combination of basic thresholds (e.g. soft-thresholds) weighted by unknown coefficients at each iteration:
The few coefficients (typically less than 10) of this expansion are obtained by minimizing the equivalent low-dimensional -regularized objective functional, which can be solved efficiently with standard convex optimization tools, e.g. iterative reweighted least squares (IRLS). We prove that, under simple conditions that the elementary thresholding functions should satisfy, global convergence of the iterated LET algorithm is guaranteed.
Figure 1: Iterative LET Scheme
Experiments on several test images over a wide range of noise levels and different types of convolution kernels clearly indicate that the proposed framework usually outperform state-of-the-art algorithms in terms of both CPU time and number of iterations.
[1] Pan, H. & Blu, T.,"An Iterative Linear Expansion of Thresholds for $ell_1$-based Image Restoration", IEEE Transactions on Image Processing, Vol. 22 (9), pp. 3715-3728, September 2013. |
[2] Pan, H., & Blu, T.,“Sparse Image Restoration Using Iterated Linear Expansion of Thresholds”, Proceedings of the 2011 IEEE International Conference on Image Processing (ICIP'11), Brussels, Belgium, pp. 1905-1908, September 11--14, 2011. |