Elsevier

Signal Processing

Volume 90, Issue 10, October 2010, Pages 2891-2897
Signal Processing

Fast communication
Impulse noise removal by a nonmonotone adaptive gradient method

https://doi.org/10.1016/j.sigpro.2010.04.017Get rights and content

Abstract

Image denoising is a fundamental problem in image processing. This paper proposes a nonmonotone adaptive gradient method (NAGM) for impulse noise removal. The NAGM is a low-complexity method and its global convergence can be established. Numerical results illustrate the efficiency of the NAGM and indicate that such a nonmonotone method is more suitable to solve some large-scale signal processing problems.

Introduction

Images are often corrupted by impulse noise in which the noisy pixels are assumed to be randomly distributed in the image. An important characteristic of impulse noise is that only part of the pixels are contaminated by the noise and the rest are free. There are two common types of impulse noise: one is the salt-and-pepper noise and the other is the random-valued impulse noise. For images corrupted by salt-and-pepper noise (respectively, random-valued noise), the noisy pixels can take only the maximal and minimal pixel values (respectively, any random value) in the dynamic. The goal of noise removal is to suppress the noise while preserving image details. The median filter was once the main method for removing impulse noise [1]. Over the years, several improved methods for impulse noise removal with different noise detectors were proposed, for example, the adaptive median filter (AMF) [2] and adaptive center-weighted median filter (ACWMF) [3], etc. These nonlinear filters can detect the noisy pixels even at a high noise level. However, they cannot restore such pixels satisfactorily because they do not take into account local image features such as the possible presence of edges. Hence details and edges are not recovered well, especially when the noise level is high.

Recently, a two-phase method was proposed in [4], [5]. The first phase is the detection of the noise pixels by using the adaptive median filter (AMF) [2] for salt-and-pepper noise while for random-valued noise, it is accomplished by using the adaptive center-weighted median filter (ACWMF) [3]. Let the true image denoted by X, and A={1,2,3,,M}×{1,2,3,,N} be the index set of X. Let NA denote the set of indices of the noise pixels detected in the first phase. Let Vi,j denote the set of the four closest neighbors of the pixel at position (i,j)A and yi,j be the observed pixel value of the image at position (i, j), and u=[ui,j](i,j)N denote a column vector of length c ordered lexicographically. Here c is the number of elements of N. Let φα be an edge-preserving functional and set Si,j1=(m,n)Vi,jNφα(ui,jym,n), Si,j2=(m,n)Vi,jNφα(ui,jum,n). Then, the second phase is the recovering of the noise pixels by minimizing the following functional:Gα(u)=(i,j)N|ui,jyi,j|+τ2(i,j)N(2·Si,j1+Si,j2),where first summation is a data-fitting term and the second summation is a regularization term, and τ>0 is a parameter. An example of edge-preserving potential function is φα(t)=α+t2,α>0, which corresponds to the popular smoothly approximated total variation (TV) regularization term. The explanation of the extra factor “2” in the second summation in (1) can be seen in [7].

The two-phase method can restore large patches of noisy pixels because it introduces pertinent prior information via the regularization term. However, the functional to be minimized in the second phase is nonsmooth, and it is costly to get the minimizer. The relaxation method in [4], [5] is convergent but slow. To improve the computational efficiency, it was proposed in [6] to drop the nonsmooth data-fitting term, as it is not needed in the 2-phase method, where only noisy pixels are restored in the minimization. Therefore, there are a lot of optimization methods can be extended to minimize smooth edge-preserving regularization (EPR) functional. A Newton method was proposed in [6], a quasi-Newton method was presented in [7] and a class of conjugate gradient methods were considered in [7], [8] to minimize the following smooth functional:Gα(u)=(i,j)N(2·Si,j1+Si,j2).

This paper proposes an effective nonmonotone method to solve the above minimization problem. Section 2 describes our globally convergent nonmonotone adaptive gradient method (NAGM) in detail. Section 3 gives numerical results to illustrate the convergence and efficiency of the proposed method. Finally we have a conclusion section.

Section snippets

Adaptive gradient method

Given a starting point u0 and using the notation gk=f(uk), the gradient methods for minuRnf(u) are defined by the iteration uk+1=uktkgk, k=0,1,…, where the stepsize tk>0 is determined through an appropriate selection rule. In the classical steepest descent (SD) method, the stepsize tk>0 is obtained by minimizing the function f(u) along the ray {uktgk:t>0}. In 1988, Barzilai and Borwein (BB) [9] developed an ingenious gradient method in which stepsize tk(k>0) is determined by tkBB1=sk1Tsk1y

Numerical experiments

In this section, the numerical results are presented to demonstrate the performance of our proposed nonmonotone adaptive gradient method (NAGM) for salt-and-pepper impulse noise removal. The simulations are preformed in Matlab 7.4 (R2007a) on a PC with an Intel Core 2 Duo CPU at 3.0 GHz and 2 GB of memory. The test images are all 512 ×512 gray level images except that the “man” is a 1024 ×1024 image. In the Algorithm 1, we fix t0=1/f(u0), M=5, β=0.9, θ=104, ρmax=1/ρmin=1030, σ=0.5. According

Conclusion

In this paper, we present a nonmonotone adaptive gradient method (NAGM) to minimize the smooth regularization functional for impulse noise removal. The NAGM is a low-complexity method and its global convergence can be established. Numerical results illustrate the efficiency of the NAGM and indicate that such a nonmonotone method is more suitable to solve some large-scale signal processing problems.

Acknowledgments

This work was partly supported by the Research Grant Council of Hong Kong, a postdoctoral fellowship from the Department of Applied Mathematics at the Hong Kong Polytechnic University, a Grant from the Ph.D. Programs Foundation of Ministry of Education of China (no. 200805581022), the National Natural Science Foundation of China (no. 10926029 and 60804008), the Natural Science Foundation of Jiangxi Province, China (2009GQS0007), the Science Foundation of Jiangxi Educational Committee

References (14)

  • A. Bovik

    Handbook of Image and Video Processing

    (2000)
  • H. Hwang et al.

    Adaptive median filters: new algorithms and results

    IEEE Trans. Image Process.

    (1995)
  • T. Chen et al.

    Adaptive impulse detection using center-weighted median filters

    IEEE Signal Process. Lett.

    (2001)
  • R.H. Chan et al.

    Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization

    IEEE Trans. Image Process.

    (2005)
  • R.H. Chan et al.

    An iterative procedure for removing random-valued impulse noise

    IEEE Signal Process. Lett.

    (2004)
  • R.H. Chan, C.-W. Ho, C.Y. Leung, M. Nikolova, Minimization of detail-preserving regularization functional by Newton's...
  • J.F. Cai et al.

    Minimization of a detail-preserving regularization functional for impulse noise removal

    J. Math. Imag. Vis.

    (2007)
There are more references available in the full text version of this article.

Cited by (0)

These authors contributed equally to the paper.

View full text