Total Variation Based Noise Removal Research Articles

Image denoising using the Gaussian curvature of the image surface

A number of high-order variational models for image denoising have been proposed within the last few years. The main motivation behind these models is to fix problems such as the staircase effect and the loss of image contrast that the classical Rudin–Osher–Fatemi model [Leonid I. Rudin, Stanley Osher and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), pp. 259–268] and others also based on the gradient of the image do have. In this work, we propose a new variational model for image denoising based on the Gaussian curvature of the image surface of a given image. We analytically study the proposed model to show why it preserves image contrast, recovers sharp edges, does not transform piecewise smooth functions into piecewise constant functions and is also able to preserve corners. In addition, we also provide two fast solvers for its numerical realization. Numerical experiments are shown to illustrate the good performance of the algorithms and test results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1066–1089, 2016

Improving source camera identification using a simplified total variation based noise removal algorithm

In this paper a new method for photo-response non-uniformity (PRNU) noise extraction is proposed. Photo-response non-uniformity noise patterns are a reliably method for digital camera identification. Especially with a large number of images the process of camera identification can be time consuming. The proposed method aims to increase the speed of PRNU extraction without losing accuracy when compared to the state-of-the-art method. Currently wavelet based denoising is used as the standard for PRNU extraction. Our proposed method is based on a simplified version of the Total Variation based noise removal algorithm. Results show that extraction is about 3.5 times faster with our method than with the wavelet based denoising algorithm. While initially only an increase in speed was the goal, results indicate that the Total Variation based noise removal algorithm is not only faster, but also more accurate than the state-of-the-art method.

An effective method for solving nonlinear equations and its application

The linearized partial differential equation from the nonlinear partial differential equation which was proposed by Rudin, Osher and Fatemi [L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms] for solving image decomposition was introduced by Chambolle [A. Chambolle, An algorithm for total variation minimization and applications] and R. Acar and C.R. Vogel [R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems]. In this paper, we propose a method for solving the linearized partial differential equation and we show numerical results for denoising which demonstrate a significant improvement over other previous works.

A new variational model for removal of combined additive and multiplicative noise and a fast algorithm for its numerical approximation

Variational image restoration models for both additive and multiplicative noise (MN) removal are rarely encountered in the literature. This paper proposes a new variational model and a fast algorithm for its numerical approximation to remove independent additive and MN from digital images. Two previous works by L. Rudin, S. Osher, and E. Fatemi [Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992), pp. 259–268] and Z. Jin and X. Yang [Analysis of a new variational model for multiplicative noise removal, J. Math. Anal. Appl. 362 (2010), pp. 415–426] are used to develop the new model. As a result, developing a fast numerical algorithm is difficult because the associated Euler–Lagrange equation is highly nonlinear and standard unilevel iterative methods are not appropriate. To this end, we develop an efficient nonlinear multigrid approach via a robust fixed-point smoother. Numerical tests using both synthetic and realistic images not only confirm that our new model delivers quality results but also that the proposed numerical algorithm allows a very fast numerical realization of the model.

Convergence of a fixed point iteration method for the OSV model

In image processing, image denoising and texture extraction are important problems in which many new methods recently have been developed. One of the most important models is the OSV model [S. Osher, A. Solé, L. Vese, Image decomposition and restoration using total variation minimization and the H - 1 norm, Multiscale Model. Simul. A SIAM Interdisciplinary J. 1(3) (2003) 349–370] which is constructed by the total variation and H - 1 norm. This paper proves the existence of the minimizer of the functional from the OSV model and analyzes the convergence of an iterative method for solving the problems. Our iteration method is constructed by a fixed point iteration on the fourth order partial differential equation from the computation of the associated Euler–Lagrange equation, and the limit of our iterations satisfies the minimizer of the functional from the OSV model. In numerical experiments, we compare the numerical results of our works with those of the ROF model [L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992) 259–268].

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.

Image decompositions using bounded variation and generalized homogeneous Besov spaces

This paper is devoted to the decomposition of an image f into u + v , with u a piecewise-smooth or “cartoon” component, and v an oscillatory component (texture or noise), in a variational approach. Y. Meyer [Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series, vol. 22, Amer. Math. Soc., Providence, RI, 2001] proposed refinements of the total variation model [L. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992) 259–268] that better represent the oscillatory part v: the weaker spaces of generalized functions G = div ( L ∞ ) , F = div ( BMO ) , and E = B ˙ ∞ , ∞ −1 have been proposed to model v, instead of the standard L 2 space, while keeping u ∈ BV , a function of bounded variation. Such new models separate better geometric structures from oscillatory structures, but it is difficult to realize them in practice. D. Mumford and B. Gidas [D. Mumford, B. Gidas, Stochastic models for generic images, Quart. Appl. Math. 59 (1) (2001) 85–111] also show that natural images can be seen as samples of scale invariant probability distributions that are supported on distributions only, and not on sets of functions. In this paper, we consider and generalize Meyer's ( BV , E ) model, using the homogeneous Besov spaces B ˙ p , q α , − 2 < α < 0 , 1 ⩽ p , q ⩽ ∞ , to represent the oscillatory part v. Theoretical, experimental results and comparisons to validate the proposed methods are presented.

An improvement of Rudin–Osher–Fatemi model

In this article, we investigate some mathematical properties of a new algorithm proposed by Meyer [Y. Meyer, Oscillating patterns in image processing and in some nonlinear evolution equations, The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, University Lectures Series, vol. 22, Amer. Math. Soc., Providence, RI, 2001] to improve the Rudin–Osher–Fatemi model (ROF) [L. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992) 259–268] in order to separate objects and textures contained in an image. He pointed out the crucial role played by a certain norm called the G-norm or “dual norm,” denoted ‖ ‖ ∗ , and the main drawback for the ROF model: any image is considered to have a textured component. We are then interested in minimizing the functional ‖ u ‖ BV + λ ‖ v ‖ ∗ . The main Theorem 6.1 is about invariance and stability properties of the new algorithm. It was first implemented by Osher and Vese [L. Vese, S.J. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, UCLA C.A.M. Report 02-19, 2002]. In particular, we point out the role played by particular functions called extremal functions and characterize them.

Nonlinear total variation based noise removal algorithms

A constrained optimization type of numerical algorithm for removing noise from images is presented. The total variation of the image is minimized subject to constraints involving the statistics of the noise. The constraints are imposed using Lanrange multipliers. The solution is obtained using the gradient-projection method. This amounts to solving a time dependent partial differential equation on a manifold determined by the constraints. As t → ∞ the solution converges to a steady state which is the denoised image. The numerical algorithm is simple and relatively fast. The results appear to be state-of-the-art for very noisy images. The method is noninvasive, yielding sharp edges in the image. The technique could be interpreted as a first step of moving each level set of the image normal to itself with velocity equal to the curvature of the level set divided by the magnitude of the gradient of the image, and a second step which projects the image back onto the constraint set.