We present a new method for solving total variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the nondi erentiability of the quantity jruj in the de nition of the TV-norm before we apply a linearization technique such as Newton's method. This is accomplished by introducing an additional variable for the ux quantity appearing in the gradient of the objective function, which can be interpreted as the normal vector to the level sets of the image u. Our method can be viewed as a primal-dual method as proposed by Conn and Overton 9] and Andersen 3] for the minimization of a sum of Euclidean norms. In addition to possessing local quadratic convergence, experimental results show that the new method seems to be globally convergent.
The total variation (TV) denoising method is a PDE-based technique that preserves edges well but has the sometimes undesirable staircase effect, namely, the transformation of smooth regions (ramps) into piecewise constant regions (stairs). In this paper we present an improved model, constructed by adding a nonlinear fourth order diffusive term to the Euler-Lagrange equations of the variational TV model. Our technique substantially reduces the staircase effect, while preserving sharp jump discontinuities (edges). We show numerical evidence of the power of resolution of this novel model with respect to the TV model in some 1D and 2D numerical examples. Introduction.The degradation of an image is usually unavoidable during its acquisition and at the early stages of its processing, and it renders the later phases difficult and inaccurate. The classical algorithms for image denoising have been mainly based on least squares, and, consequently, their outputs may be contaminated by Gibbs phenomena and do not approximate images containing edges well. To overcome this difficulty a technique based on the minimization of the total variation (TV) norm is proposed in [12]. This technique preserves edges well, but the images resulting from the application of this technique in the presence of noise are often piecewise constant; thus the finer details in the original image may not be recovered satisfactorily, and ramps (affine regions) will give stairs (piecewise constant regions); see Figure 8.2. In this paper we present an improved model, constructed by adding a nonlinear fourth order diffusive term to the Euler-Lagrange equations of the variational TV model. This technique substantially reduces the staircase effect, while preserving sharp jump discontinuities (edges).The paper is organized as follows. In section 2, we describe the original TV model and the nonlinear equations associated with it. In section 3, we describe the staircase effect caused by the TV model and briefly review several techniques proposed in the literature to deal with it. In section 4 we aim to reduce the staircase effect by using a variational formulation for the denoising problem with a functional obtained by adding a nonlinear second order term to the TV functional. In the next section we analyze the fourth order nonlinear Euler-Lagrange equation associated with the latter variational problem in the 1D case and notice that their discretizations can be ill-posed. We then propose to drop the term that might cause the ill-posedness, thus deviating from the variational formulation to obtain the nonlinear fourth order diffusion equation that we propose in this paper. We next present a fixed point scheme for the solution of
In this paper we show that the lagged di usivity xed point algorithm introduced by Vogel and Oman in 10] to solve the problem of Total Variation denoising, proposed by Rudin, Osher and Fatemi in 9], is a particular instance of a class of algorithms introduced by Eckhardt and Voss in 11], whose origins can be traced back to Weiszfeld's original work for minimizing a sum of Euclidean lengths 12]. There have recently appeared several proofs for the convergence of this algorithm 2], 3], 6]. Here we present a proof of the global and linear convergence using the framework introduced in 11] and give a bound for the convergence rate of the xed point iteration that agrees with our experimental results. These results are also valid for suitable generalizations of the xed point algorithm.
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