  Solving an optimization problem whose objective function is the sum of two
convex functions has received considerable interests in the context of image
processing recently. In particular, we are interested in the scenario when a
non-differentiable convex function such as the total variation (TV) norm is
included in the objective function due to many variational models established
in image processing have this nature. In this paper, we propose a fast fixed
point algorithm based on the quasi-Newton method for solving this class of
problem, and apply it in the field of TV-based image deblurring. The novel
method is derived from the idea of the quasi-Newton method, and the fixed-point
algorithms based on the proximity operator, which were widely investigated very
recently. Utilizing the non-expansion property of the proximity operator we
further investigate the global convergence of the proposed algorithm. Numerical
experiments on image deblurring problem with additive or multiplicative noise
are presented to demonstrate that the proposed algorithm is superior to the
recently developed fixed-point algorithm in the computational efficiency.
