- Research Article
- Open Access
Multiplicative Noise Removal via a Novel Variational Model
© Li-Li Huang et al. 2010
- Received: 30 March 2010
- Accepted: 2 June 2010
- Published: 27 June 2010
Multiplicative noise appears in various image processing applications, such as synthetic aperture radar, ultrasound imaging, single particle emission-computed tomography, and positron emission tomography. Hence multiplicative noise removal is of momentous significance in coherent imaging systems and various image processing applications. This paper proposes a nonconvex Bayesian type variational model for multiplicative noise removal which includes the total variation (TV) and the Weberized TV as regularizer. We study the issues of existence and uniqueness of a minimizer for this variational model. Moreover, we develop a linearized gradient method to solve the associated Euler-Lagrange equation via a fixed-point iteration. Our experimental results show that the proposed model has good performance.
- Regularization Term
- Multiplicative Noise
- Just Noticeable Difference
- Coarea Formula
- Restoration Model
Image denoising is an inverse problem widely studied in signal and image processing fields. The problem includes additive noise removal and multiplicative noise removal. In many image formation model, the noise is often modeling as an additive Gaussian noise: given an original image , it is assumed that it has been corrupted by some Gaussian additive noise . The denoising problem is then to recover from the data . There are many effective methods to tackle this problem. Among the most famous ones are wavelets approaches [1, 2], stochastic approaches , principal component analysis-based approaches [4, 5], and variational approaches . We refer the reader to the literature [7, 8] and references herein for an overview of the subject.
where is the observed image, is the original SAR image, and is the noise which follows a Gamma Law with mean one. Speckle is one of the most complex image noise models. It is signal independent, non-Gaussian, and spatially dependent. Hence speckle denoising is a very challenging problem compared with additive Gaussian noise.
Multiplicative noise removal methods have been discussed in many reports. Popular methods include local linear minimum mean square error approaches [10, 11], anisotropic diffusion methods [12–15], and nonlocal means (NL-means) , which will not be addressed in this paper. We will focus on the variational approach-based multiplicative noise removal, especially that our researches will emphasis on TV-based methods.
To the best of our knowledge, there exist several variational approaches devoted to multiplicative noise removal problem. The first total variation-based multiplicative noise removal model (RLO-model) was presented by Rudin et al. , which used a constrained optimization approach with two Lagrange multipliers. Multiplicative model (AA-model) with a fitting term derived from a maximum a posteriori (MAP) was introduced by Aubert and Aujol . Recently, Shi and Osher  adopted the data term of the AA-model but to replace the regularizer by . Moreover, setting , then they derived the strictly convex TV minimization model (SO-model). Afterwards, Huang et al.  modified the SO-model by adding a quadratic term to get a simpler alternating minimization algorithm. Similarly with SO-model, Bioucas and Figueiredo  converted the multiplicative model into an additive one by taking logarithms and proposed Bayesian type variational model. Steidl and Teuber  introduced a variational restoration model consisting of the I-divergence as data fitting term and the total variation seminorm as regularizer. A variational model involving curvelet coefficients for cleaning multiplicative Gamma noise was considered in .
As information carriers, all images are eventually perceived and interpreted by the human visual system. As a result, many researchers have found that human vision psychology and psychophysics play an important role in the image processing. Among them, Shen  has proposed Weberized TV model to remove Gaussian additive noise which incorporated the well-known psychological results—Weber's Law.
However, the previous multiplicative removal models pay a little attention to this point. Inspired by the Weberized TV regularization method [24, 25], we propose a nonconvex variational model for multiplicative noise removal. Then we prove the existence and uniqueness of a minimizer for the new model. Moreover, we develop an iterative algorithm based on the linearization technique for the associated nonlinear Euler-Lagrange equation. Our experimental results show that the proposed model has good performance.
The outline of this paper is as follows. In Section 2, we derive a new nonconvex variational model to remove multiplicative Gamma noise under the MAP framework. Moreover, we carry out the mathematic analysis of the variational model in the continuous setting. In Section 3, we develop a linearized gradient method to solve the associated Euler-Lagrange equation via a fixed-point iteration and illustrate our algorithm by displaying some numerical examples. We also compare it with other ones. Finally, concluding remarks are given in Section 4.
In this section, we propose the multiplicative noise removal model from the statistical perspective using Bayesian formulation, for which we prove the existence and uniqueness of a solution.
2.1. MAP-Based Multiplicative Noise Modeling
Here, the first term is the image fidelity term which measures the violation of the relation between and the observation . The second term is the regularization term which imposes some prior constraints on the original image and to a great degree determines the quality of the recovery image. And is the regularization parameter which controls the tradeoff between the fidelity term and regularization term.
2.2. Our Variational Model
This definition for the TV functional does not require differentiability or even continuity of . In fact one of the remarkable advantages of using TV functional for image restoration is to preserve edges due to its jump discontinuities.
The essential idea of the above model (10) is that it replaces the TV functional by the functional , the well known perceptual law-Weber's law, in the classical TV image restoration model of Rudin et al. .
which encodes the influence of the background intensity according to Weber's law (9).
The current paper is devoted to the study of the mathematical properties of this new model, including issues related to the existence and uniqueness of the minimizer, and its computational approach.
2.3. Mathematical Properties of the Variational Model (13)
In this subsection, we first give the admissible space for the restoration model (13) and then investigate the existence and uniqueness of the minimizer to the model. Throughout the paper, we assume that is a Lipschitz open domain with a finite Lebesgue measure .
Since denotes the intensity value, thus . When , it is the singularity of both Weber's fraction (9) and the Weberized local variation (14). Hence, technically we should stay away from this point and assume that .
can be understood in the sense of the following coarea formula.
Here the level set is is the perimeter of the set and the space is of functions of bounded variation consisting of all functions with .
Applying [26, Theorem ], we get the conclusion.
When , we note that (16) is precisely the classical co-area formula. It means that , when . We shall work with from now on.
Secondly, we give a theorem on the existence and uniqueness of the solution of the problem (13), respectively.
Theorem 1 (Existence).
Suppose that with ; then problem (13) has at least one minimizer in the admissible space .
Theorem 2 (Uniqueness).
For the proof of the existence and uniqueness see the appendix for details.
In this section, we present some numerical examples to demonstrate the performance of our method. We also compare it with some existing other ones. All experiments were performed under Windows XP and MATLAB v7.1 running on a desktop with an Intel (R) Pentium (R) Dual E2180 Processor 2.00 GHZ and 0.99 GB of memory.
in (13). Here is the regularized parameter chosen near .
We first give a computational lemma.
Applying Green's identity, we directly compute the first Gateaux derivative of and get the conclusion.
with the Neumann adiabatic condition along the boundary of the image domain. It is formally identical to the classical TV denoising equation [6, 29], except that the fitting constant now depends on . Notice that since .
and matrix operators (cf. (25)) which are symmetric and positive definite and sparse. In our computational experiments, is set to be .
What follows is a generic algorithm for the minimization of in (13). The superscript denotes iteration count. are user-defined tolerance, is an iteration limit, and denotes the norm.
(1) Compute a descent direction for at .
(3) Check stopping criteria (see ): or or .
Equation (30) follows from (27) and (24), respectively. The conjugate gradient method applied to solve the above linear diffusion equations to get the and the stopping criterion of the inner conjugate gradient iteration is that the residual should be less than . In our computational experiments, we set , and .
3.2. Parameters Choice
Then, we determine the best value of from their tested values such that the peak signal-to-noise ratio (PSNR, see definition here in after) of the restored image is the maximal.
3.3. Other Methods
We have compared our results with some other variational multiplicative denoising methods.
In our experiments, and time step size are set to be and 0.2, respectively. The two Lagrange multipliers and are dynamically updated to satisfy the constraints (as explained in ).
In our experiments, take the same value in the RLO method. The regularization parameter is dynamically updated according to (34).
was solved by using the Newton method. The Chambolle projection algorithm was employed in the denoising -subproblem of (3.12) . Then the restored image is computed by . Here, the rule to determine the two regularization parameters and the stopping criterion of the HMW method are chosen as suggested in .
In our computational experiments, we use the initial guess in RLO and AA method and in HMW method. RLO and AA algorithms are terminated once they reached maximal PSNR.
3.4. Denoising of Color Images
In this subsection, we extend our approach to solve the multichannel version of (13). The general framework of the variational approach for color images processing based on the linear RGB color models can be classified into two categories—the channel-by-channel approach and the vectorial approach. Compared with the first approach, the second approach can exploit the spatial correlation and the spectral correlation in processing color images. So the vectorial approach has already been used in most of the literature for RGB images, such as the work of [31–33] solved multichannel total variation (MTV) regularization reconstruction problem. Considering that our multiplicative denoising variational model includes the Weberized TV regularizer, we choose the channel-by-channel approach in this paper for color image multiplicative noise removal due to its simplicity and robustness.
Recently, Zhang et al.  proposed an additive denoising scheme by using principal component analysis (PCA) with local pixel grouping (LPG). We refer to this method as LPG-PCA method. For a better preservation of image local structures, a pixel and its nearest neighbors are modeled as a vector variable, whose training samples are selected from the local window by using block matching-based LPG. The LPG-PCA denoising procedure is iterated one more time to further improve the denoising performance, and the noise level is adaptively adjusted in the second stage.
In our experiments, we only compare the denoising results of the noisy color images obtained by our approach with those obtained by the LPG-PCA method. We do it for the following two reasons: first, the LPG-PCA method using the channel-by-channel approach has been extended to solve the color image denoising problem; second, the multiplicative noise can be converted into additive noise by logarithmic transformation. In the LPG-PCA method, we make the size of the variable block and training block 2 and 20, respectively. We use as the initial guess. Then, the restored image is computed by exponential transform.
where are the original, the restored, the observed image, and the size of the image, respectively.
The PSNR (dB), ISNR (dB), and ReErr of the restored images using four methods.
The number of iterations (It no.), and computational times of four methods.
CPU time (s)
CPU time (s)
CPU time (s)
CPU time (s)
The PSNR (dB), ISNR (dB), ReErr, number of iterations (It no.) and computational times of the restored images using two methods.
In this paper, we have studied a new nonconvex variational model for multiplicative noise removal under MAP framework. Then we prove the existence and uniqueness of a minimizer for the new model. Moreover, we develop an iterative algorithm based on the linearization technique for the associated nonlinear Euler-Lagrange equation and we demonstrate the good performance of the model on some numerical results. In the future, we will focus on resolving the two remaining problems. First, we will prove the uniqueness issue of the proposed model using the instead of in . Second, we will give the convergence proof of our algorithm.
The authors thank the authors of [5, 20] for sharing their programs. Moreover, they would like to express their gratitude to the anonymous referees and editor for making helpful and constructive suggestions. This work was supported in part by the Natural Science Foundation of China under Grant nos. 60802039 and 60672074, by the National 863 High Technology Development Project under Grant no. 2007AA12Z142, by the Doctoral Foundation of Ministry of Education of China under Grant no. 200802880018, and by the Scientific Innovation project of Nanjing University of Science and Technology no. 2010ZDJH07.
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