An Algorithm for image removals and decompositions without inverse matrices

Partial Differential Equation (PDE) based methods in image processing have been actively studied in the past few years. One of the effective methods is the method based on a total variation introduced by Rudin, Oshera and Fatemi (ROF) [L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992) 259-268]. This method is a well known edge preserving model and an useful tool for image removals and decompositions. Unfortunately, this method has a nonlinear term in the equation which may yield an inaccurate numerical solution. To overcome the nonlinearity, a fixed point iteration method has been widely used. The nonlinear system based on the total variation is induced from the ROF model and the fixed point iteration method to solve the ROF model is introduced by Dobson and Vogel [D.C. Dobson, C.R. Vogel, Convergence of an iterative method for total variation denoising, SIAM J. Numer. Anal. 34 (5) (1997) 1779-1791]. However, some methods had to compute inverse matrices which led to roundoff error. To address this problem, we developed an efficient method for solving the ROF model. We make a sequence like Richardson's method by using a fixed point iteration to evade the nonlinear equation. This approach does not require the computation of inverse matrices. The main idea is to make a direction vector for reducing the error at each iteration step. In other words, we make the next iteration to reduce the error from the computed error and the direction vector. We describe that our method works well in theory. In numerical experiments, we show the results of the proposed method and compare them with the results by D. Dobson and C. Vogel and then we confirm the superiority of our method.