Multiplicative Noise Removal via a Novel Variational Model

Abstract

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.

1. Introduction

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 [3], principal component analysis-based approaches [4, 5], and variational approaches [6]. We refer the reader to the literature [7, 8] and references herein for an overview of the subject.

In this paper, we focus on the issue of multiplicative noise removal. Specifically, we are interested in the denoising of SAR images. According to [9] and other references, the noise in the observed SAR image is a type of multiplicative noise which is called speckle. And the image formation model is

(1)

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) [16], 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. [17], 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 [18]. Recently, Shi and Osher [19] 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. [20] modified the SO-model by adding a quadratic term to get a simpler alternating minimization algorithm. Similarly with SO-model, Bioucas and Figueiredo [21] converted the multiplicative model into an additive one by taking logarithms and proposed Bayesian type variational model. Steidl and Teuber [22] 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 [23].

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 [24] 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.

2. The Proposed Model and Mathematical Analysis

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

Let denote -pixels instances of some random variables and . Adopting a conditionally independent multiplicative noise model, we have

(2)

where is an image of independent and identically distributed (i.i.d) noise random variables with mean one, following Gamma density:

(3)

After standard computation, we get

(4)

Under the MAP frameworks, the original image is inferred by solving a minimization problem with the form

(5)

We assume that follows a Gibbs prior: , where is a normalizing constant, and a nonnegative given function. Moreover, since is i.i.d, therefore we have= Then, the previous computation leads us to propose the following model for restoring images corrupted with Gamma noise:

(6)

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

As stated above, the choice of is important. To the best knowledge of our known, total variational functional has been brought into wide use ever since its introduction by Rudin et al. [6]. is defined by

(7)

which reads for functions with weak first derivatives in as

(8)

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.

As an image model, does not take into account that our visual sensitivity to the regularity or local fluctuation depends on the ambient intensity level [24]. Since all images are eventually perceived and interpreted by the Human Visual System (HVS), as a result, many researchers have found that human vision psychology and psychophysics play an important role in image processing. The classical example is the using of the Just Noticeable Difference Model (JND) in image coding and watermarking techniques [26, 27]. In these fields, the JND model is used to control the visual perceptual distortion during the coding procedure and watermark embedding. Weber's law was first described in 1834 by German physiologist Weber [28]. The law reveals the universal influence of the background stimulus on human's sensitivity to the intensity increment , or so called JND, in the perception of both sound and light:

(9)

According to Weber's law, when the mean intensity of the background is increasing with a higher value, then the intensity increment also has higher value. In literature [24], the authors proposed a nonconvex variational model for additive Gaussian noise removal:

(10)

where

(11)

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. [6].

Considering that our visual sensitivity to the local fluctuation depends on the ambient intensity level , we take the regularization term as follows:

(12)

According to the different purposes of image processing, we can design different . As stated previously, we adopt and propose the following multiplicative denoising variational model:

(13)

where the first two terms are the regularization terms, while the third one is the nonconvex data fidelity term following the MAP estimator for multiplicative Gamma noise. are regularization parameters, and in is the given data. The first regularization term is the TV functional, which preserves important structures such as edges, an important visual cue in human and computer vision. The second term is the well-known Weberized TV regularization term. To briefly explain the role of this term, we assume that has a gradient , then and the Weberized local variation is

(14)

which encodes the influence of the background intensity according to Weber's law (9).

The formulation (13) seems to include previous models.

• (i) When , this reduces to the SO-model [19] by letting .

• (ii) When , this reduces to the AA-model [18].

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 .

First, we give the admissible space for the restoration model (13). The regularization term

(15)

can be understood in the sense of the following coarea formula.

Lemma 1.

(Coarea formula). Let be a function and ; then

(16)

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 .

Proof.

Applying [26, Theorem ], we get the conclusion.

From Lemma 1, we give the following nature admissible space for the restoration model (13):

(17)

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).

Assume that is in , and is a minimizer of the restoration energy . Then is unique if

(18)

For the proof of the existence and uniqueness see the appendix for details.

3. Numerical Results

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.

3.1. Algorithm

To numerically compute a solution to the problem (13), as in [24, 29, 30], we apply the linearization technique to iteratively solve the associated Euler-Lagrange equation, which we call "lagged diffusivity fixed point iteration". Since total variation functional is nonsmooth, which caused the main numerical difficulty, we replace the total variation functional by a smooth approximation like

(19)

in (13). Here is the regularized parameter chosen near .

We first give a computational lemma.

Lemma 2.

Let be a function and

(20)

Then the formal Euler-Lagrange differential of is

(21)

Proof.

Applying Green's identity, we directly compute the first Gateaux derivative of and get the conclusion.

Applying the above lemma to our restoration functional, the formal Euler-Lagrange equation for any solution of problem (13) is as follows

(22)

Since , then the Euler-Lagrange equation (22) of minimizing can be rewritten equivalently as

(23)

Define . Then (23) can be rewritten as

(24)

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 .

Equations: (24) can be expressed in operator notation

(25)

where is the linear diffusion operator whose action on a function is given by

(26)

The fixed point iteration is then

(27)

Finite difference method is used commonly for discretization of partial differential equation (PDE). Equations (25) can be approximately computed by the first-order accurate finite difference schemes described as follows [6]:

(28)

where . Here, we denote the space step size by . These schemes yield approximate form of (26):

(29)

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.

Input:  .

Initialization:  .

(1) Compute a descent direction for at .

(2) .

(3) Check stopping criteria (see [29]): or or .

In step 1, we set and yield

(30)

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

We remark that there are two regularization parameters and in the proposed algorithm, which controls the tradeoff between the image fidelity term and the regularization term. When , we note that our model (13) is the AA-model [18] as follows:

(31)

Borrowing the idea of [6], we dynamically compute the value of according to the variance of the recovered noise which matches that of our prior knowledge. The Gamma-distributed noise has the mean and variance as follows:

(32)

The solution procedure uses a parabolic equation with time as an evolution parameter. This means that we solve

(33)

for . We merely multiply the first equation of (33) by and integrate by parts over . If steady state has been reached, the left side of the first equation of (33) vanishes, and then we have

(34)

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.

RLO Method. The solution of RLO-model [17] is obtained by using the following gradient projection iterative scheme [17] (the subscripts are omitted):

(35)

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 [17]).

AA Method. The solution of AA-model [18] is obtained by using the gradient projection method:

(36)

In our experiments, take the same value in the RLO method. The regularization parameter is dynamically updated according to (34).

HMW Method. The solution of HMW-model [20] (we note that HMW-model is equivalent to SO-model as .)

(37)

is obtained by using the following alternating minimization algorithm:

(38)

The corresponding nonlinear Euler-Lagrange equation of -subproblem of (3.12)

(39)

was solved by using the Newton method. The Chambolle projection algorithm was employed in the denoising -subproblem of (3.12) [20]. 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 [20].

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. [5] 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.

3.5. Results

The six test images (size: ) used in the experiments, including five grey level images and one color image, are shown in Figures 1(a)–6(a) and Figures 9(a)–10(a), respectively. In our tests, each pixel of an original image is degraded by a noise which follows a Gamma distribution with density function in (3) and is specified to have mean 1 and standard deviation . The noise level is controlled by the value of in the experiments. The noisy images with different levels () are shown in Figures 1(b)–6(b) and Figures 9(b)–10(b), respectively. We display the denoising results obtained by our approach as well as by the RLO, AA, HMW, and LPG-PCA methods.

Figure 1

The original, noisy, and restored "Lena" images: (a) noiseless image; (b) noisy image with ; denoised images by (c) RLO method; (d) AA method; (e) HMW method ; (f) the proposed method .

Figure 2

The original, noisy, and restored "Lena" images: (a) noiseless image; (b) noisy image with ; denoised images by (c) RLO method; (d) AA method; (e) HMW method ; (f) the proposed method .

Figure 3

The original, noisy, and restored "Cameraman" images: (a) noiseless image; (b) noisy image with ; denoised images by (c) RLO method; (d) AA method; (e) HMW method ; (f) the proposed method .

Figure 4

The original, noisy, and restored SAR images: (a) noiseless image; (b) noisy image with ; denoised images by (c) RLO method; (d) AA method; (e) HMW method ; (f) the proposed method .

Figure 5

The original, noisy, and restored "SynImag1" images: (a) noiseless image; (b) noisy image with ; denoised images by (c) RLO method; (d) AA method; (e) HMW method ; (f) the proposed method .

Figure 6

The original, noisy, and restored " SynImag2" images: (a) noiseless image; (b) noisy image with ; denoised images by (c) RLO method; (d) AA method; (e) HMW method ; (f) the proposed method .

We measure the quality of restoration by the peak signal-to-noise ratio (PSNR), the improved signal-to-noise ratio (ISNR), and the relative error (ReErr) of the restored image, defined by

(40)

where are the original, the restored, the observed image, and the size of the image, respectively.

Figures 1–6 show the denoising results of the six noisy gray level images by different methods. The subfigures (c–f) are the denoised images by the different methods. In these experiments, it is clear that the restoration results obtained by the proposed method are visually better than those by the HMW, AA, and RLO methods, especially when the noise variance is large, that is, when is small. Although the most denoised images by the HMW method have better visual quality, we note that some details, such as camera and buildings in Figure 3(e), become hazy. Such hazy details make the reconstructed image visually unpleased in some areas. Next we check the homogeneity of regions of interest in the image and analyze the loss (or the preservation) of contrast. In Figures 7 and 8, we show several lines of the original, noisy, and restored images. It is clear from the figures that the lines restored by the proposed method are better than those restored by the other three methods.

Figure 7

The 135th line of the original, noisy, and restored images of the "Cameraman" image. (a) The noisy slice; the slice restored by (b) the RLO method; (c) the AA method; (d) the HMW method; (e) the proposed method. Here the blue line is the original image, and the red line is the restored image.

Figure 8

The 128th line of the original, noisy, and restored images of the "SynImag1" image. (a) The noisy slice; the slice restored by (b) the RLO method; (c) the AA method; (d) the HMW method; (e) the proposed method. Here the blue line is the original image, and the red line is the restored image.

Figure 9

The original, noisy, and restored color "Lena" images: (a) noiseless image; (b) noisy image with ; denoised images by (c) LPG-PCA method; (d) the proposed method .

Figure 10

The original, noisy, and restored color "Lena" images: (a) noiseless image; (b) noisy image with ; denoised images by (b) LPG-PCA method; (c) the proposed method .

Table 1 lists the PSNR, ISNR, and ReErr results by different methods on the six noisy gray level images. From Table 1, we can see that the PSNRs and ISNRs of the images restored by using our method are more than those restored by using the other three methods, and ReErrs are less than the other four methods, except for the denoising results of Figure 4(b) obtained by the HMW method. In addition to the quality of the restored images, we also find that the proposed algorithm is efficient. Table 2 shows the number of iterations and the computational times required by the different algorithms. From Table 2, we see that the RLO algorithm takes more time than the other three methods. The spending time of our algorithm is in a middle position in comparison with the other three methods.

Table 1 The PSNR (dB), ISNR (dB), and ReErr of the restored images using four methods.

Table 2 The number of iterations (It no.), and computational times of four methods.

We now compare the LPG-PCA method with the proposed method on denoising the noisy color images. Figures 9–10 show the denoising results by the two methods. Table 3 lists the PSNR, ISNR, and ReErr results, the number of iterations, and the computational times of the two algorithm. We see that although LPG-PCA method has the lower PSNR and ISNR measures and higher ReErrs than our method, their denoised images have better visual quality. The LPG-PCA method well preserves the image edges without introducing staircase effect. However, staircase effect is an innate defect of the TV regularization method. From Table 3, we also see that the LPG-PCA method consumes much time than our method to obtain comparable good images.

Table 3 The PSNR (dB), ISNR (dB), ReErr, number of iterations (It no.) and computational times of the restored images using two methods.

4. Conclusion

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.

Appendix

Proof of Theorems 1 and 2

To prove the existence Theorem 1, we first give a Maximum Principle type result for the energy form (13).

Lemma A.3.

Suppose that with ; the solution of the problem (13) has the following property:

(A.1)

Proof.

The similar assertion appears in [18]; we detail the argument for completeness. Let , and . We remark that is strictly increasing for . Hence, we have that

(A.2)

Moreover, we have that (see [34, Lemma in Section ], [21, Lemma in Section ])

(A.3)

Therefore we deduce that

(A.4)

and the equality holds if and only if , a.e. Since is a minimizer in , the equality must hold and thus , a.e. We get in the same way that

(A.5)

and thus .

Based on aforementioned lemma, now we are ready to give the proof of Theorem 1.

Proof of Theorem 1.

Without loss of generality, we assume that . Applying ideas in [24], let us denote that . We note that the admissible space is nonempty since . Let be a minimizing sequence for problem (13). Thanks to Lemma 2, we can assume that . This implies that is bounded ( is bounded).

For a sequence , we have , where is a constant. Since and reaches its minimum value when , we get that is bounded in .

Thus, up to a subsequence, there exists u in such that in -strong. Furthermore, after a refinement of the subsequence if necessary, we can assume that

(A.6)

Using the Lebesgue Dominated Convergence Theorem, then we have

(A.7)

Next we prove that is lower semicontinuity. Firstly, from the properties of the [7], we have

(A.8)

Secondly, the lower semicontinuity of Weberized TV can also be proved. Let us define and ; then

(A.9)

Let be a vector-valued function such that . We have

(A.10)

and also

(A.11)

Since is compactly supported, the right side of the above inequality belongs to . Therefore, again by the Lebesgue Dominated Convergence Theorem,

(A.12)

Now take sup over to get

(A.13)

Combining (A.1), (A.2), and (A.3), we have

(A.14)

It is easy to see that . Since is a minimizing sequence, we therefore have shown that is in fact a minimizer.

Based on the theory of optimization, an objective function possesses a unique minimizer when it is strictly convex and coercive [35]. Since the negative log-likelihood and Weberized TV prior are not convex, as a result, the restoration energy in (13) is not convex and uniqueness is no longer a direct product of convexity. We address the problem of the uniqueness of the solution of problem (13), which relies on the formal Euler-Lagrange equation of (13):

Proof of Theorem 2.

Let us denote

(A.15)

Define a new reference energy for the restoration energy as follows:

(A.16)

It is easy to derive that (23) is exactly the Euler-Lagrange equilibrium equation for . We have

(A.17)

We deduce that if (18) holds, then and is strictly convex. Now that the TV Radon measure is semiconvex, so the objective function is globally strictly convex and possesses a unique minimizer.