To avoid the staircase artifacts, an adaptive image denoising model is proposed by the weighted combination of Tikhonov regularization and total variation regularization. In our model, Tikhonov regularization and total variation regularization can be adaptively selected based on the gradient information of the image. When the pixels belong to the smooth regions, Tikhonov regularization is adopted, which can eliminate the staircase artifacts. When the pixels locate at the edges, total variation regularization is selected, which can preserve the edges. We employ the split Bregman method to solve our model. Experimental results demonstrate that our model can obtain better performance than those of other models.

1. Introduction

Images are inevitably contaminated by noise during acquisition and transmission and so image denoising is one of the most fundamental tasks in image processing and computer vision [1]. Image denoising aims at preserving edges and fine-scale structural information while removing the noise in the image. During the past two decades, a great deal of research [24] has gone into excogitating models for removing noise while preserving edges and other fine-scale image details. Recently, partial differential equation (PDE) and variational-based methods have gained popularity for image denoising [5, 6]. Among them, the best known one is TV model proposed by Rudin et al. [7]. Although this second-order PDE can succeed in preserving edges, it suffers from the staircase effect in the smooth regions. To prevent the staircasing, a modified model proposed by Blomgren et al. [8] was to automatically adapt the gradient-based exponent to fit the data term. It has the following form: Gradient-based exponent has value 1 near the edges and is close to 2 in the smooth region, so that near the edges it behaves exactly like the ROF model and behaves like Tikhonov regularization method in the smooth regions. However, this model is nonconvex and difficult to solve. Another approach to overcome the limitation of TV is to introduce higher-order derivatives into the energy function [9, 10]. One of the most classical fourth-order PDEs is introduced by You and Kaveh (Y-K), which measured the oscillations in the noisy image according to the two different functions [11]. But, a major challenge higher-order PDEs face is numerical computation. At the same time, another challenge is to blur the edges in the denoising. In addition, they can also produce speckle noise in the recovered image. Recently, some hybrid models were proposed which combined the second-order partial differential equation models and the fourth-order PDEs [12, 13]. Gholami and Hosseini [14] proposed a novel method to combine total variation regularization and Tikhonov regularization to reconstruct piecewise smooth signals.

Inspired by their work, we propose an adaptive image denoising model by the weighted combination of Tikhonov regularization and total variation regularization. Unlike the model [14] which firstly divided the image into piecewise constant component and smooth component, our model directly uses the gradient information to define the weight functions which judge whether the pixels locate at the edges or the smooth regions. When the pixels belong to the smooth regions, Tikhonov regularization is adopted, which can eliminate the staircase artifacts. When the pixels locate at the edges, total variation regularization is selected, which can preserve the edges. Our model does not solve the high-order PDE and is also a convex combination. All numerical experiments verify the efficiency of our proposed model.

The remainder of this paper is organized as follows. In Section 2, we briefly review Tikhonov regularization and TV regularization and then discuss their advantages and limitations. In Section 3, we give the proposed model and numerical implementation in detail. The experimental results are given in Section 4. Finally, this paper is concluded in Section 5.

In this paper, the observed noisy image can be defined as ; here, is the original image, and is an additive Gaussian white noise with the mean 0 and variance . Mathematically speaking, the denoising problem is an ill-posed inverse problem. One of the most known techniques to solve this problem is regularization method and energy minimization as follows: Here, the first term is a regularization term, while the second term is fidelity term which ensures that the denoised image is close to the observed image, and is the regularization parameter which balances the regularization term and the fidelity one. Much research has been put forward to remove the noises from the noisy image by either exploiting a particular regularization technique [15, 16] or taking the advantages of a set of regularizations using a combination of them [14, 17, 18]. In the image denoising, Tikhonov regularization and total variation regularization are two kinds of regularization techniques and they are widely used in the past three decades.

2.1. Tikhonov Regularization

Tikhonov regularization method is one of the earliest used regularizations, in which the cost function contains an -norm regularization term of the magnitude of the image gradient [19]. The model based on the Tikhonov regularization method is defined as follows: where is norm of the bounded variation.

We know that minimizing formula (3) yields the Euler-Lagrange equation Seen from (4), Tikhonov regularization technique is actually isotropic diffusion. Hence, while removing the noise in the observed image , prominent structures like edges and jumps in the denoised image are blurred. However, the first- and second-order derivative operators are only used in Tikhonov regularization, so it has computational advantages [20].

2.2. Total Variation Regularization

In addition, there are other regularization techniques which preserve the edge information in the recovered image [21]. Among these edge-preserving regularization methods [7], the best known one is the total variation regularization. Rudin et al. proposed using the norm of the gradient of the image, instead of the norm, so the formulation for the variational model (ROF) is as follows: subject to Here is a bounded domain of the image with Lipschitz boundary. is the space of functions with bounded variation. represents the area of the image. By the gradient descent method, we get the Euler-Lagrange equation of TV denoising model from (5) as follows: When , formula (5) is the isotropic TV of . and are the horizontal and the vertical gradient operators, respectively. The isotropic TV regularization performs well in preserving edges while eliminating noise. However, the fine-scale details such as textures are often smoothed out with noise in the image. Esedoǵlu and Osher [21] proposed anisotropic total variation (ATV) regularization as . The uniqueness and existence of solutions of ATV are shown in [22]. However, the ATV is still TV-type regularization which is not good at preserving fine-scale details and may cause staircase artifacts in the smooth regions.

3. The Proposed Method

3.1. The New Model

Generally, the natural images contain the smooth regions with sharp edges. According to what is mentioned above, the TV-type regularization can effectively remove noise and preserve sharp edges, but it easily causes staircase effects in the flat regions. On the other hand, Tikhonov regularization can eliminate the staircase effect although it blurs the edges. Therefore, in order to eliminate staircase effect and preserve fine-scale details, we combine these two regularizations and take the energy function of the image for image denoising as follows: Here the functions are defined as where is constant and is defined as the image gradient modulus value. The function decreases monotonically from 1 to 0. When the pixels belong to edges, the value of is large, so . When the pixels belong to flat regions, is small, so . From the proposed model, we can see that when , which means the pixel locates in the edge, TV regularization term is adopted; when , which means the pixel belongs to the smooth region, Tikhonov regularization is used. So we can see from our model (8) the following.(1)In the region containing edges, when is close to 0, the new model adopts the total variation (TV) model, so the model is good at preserving the edges of the image.(2)In the smooth areas of the image, is close to 1, and the new model will highlight Tikhonov regularization model, so the model can eliminate the staircase effect.(3)In the flat areas of image containing less image features and much noise, Tikhonov regularization model and total variation model work together.

3.2. Numerical Implementation

To solve (8), we use split Bregman method proposed by Goldstein and Osher [23]. Split Bregman is a flexible and effective tool for solving various inverse problems. Split Bregman idea is to use splitting operator and Bregman iteration to solve the constrained minimization problems. First, we convert (8) into the constrained minimization problem by introducing an auxiliary variable , so where . is used to replace . We can see that the first and second terms are not directly crossed. Second, by introducing the variable , the constrained minimization problem (10) can be converted to the following unconstrained problem: where the penalty parameter is a position constant. Using split Bregman method, the unconstrained problem (11) can be solved by the following iterative equations: To solve the first problem in (12), we let The partial derivatives of (13) are showed as follows: Therefore the Euler-Lagrange equation of (14) is According to the gradient descent method, we can derive We also derive the Euler-Lagrange equation of the second problem in (12) as follows: This is a linear equation of , so additional operator split (AOS) iteration is used to solve (17).

In sum, the new model can be implemented as follows.

Algorithm 1. The proposed algorithm for image denoising is as follows.

Step  1. Input the observed image .

Step  2. Initialization: , , and .

Step  3. Compute according to (16).

Step  4. Compute according to (17).

Step  5. Compute according to the third term of (12).

Until the stop condition is satisfied.

4. Experiments

To evaluate the performance of the proposed model, in this section, we have compared our model with other models in terms of the visual quality of denoised images, peak signal to noise ratio (PSNR), and structural similarity index measure (SSIM). PSNR is defined by with where denotes the image size, is the recovered image, and is the noise-free original image. MSE represents the noise variance in the recovered image. PSNR is used to measure the noise removal effect and the larger value indicates the smoother recovered image. However, PSNR sometimes disaccords with human visual judgments. So, SSIM (structural similarity) is used to assess the noise removal quality because the SSIM criterion is closer to the human vision system [24]. SSIM is defined by where and are the mean and variance of , respectively, is the covariance of and , and and are two constants to avoid instability. SSIM measures the structure similarity between the original image and the recovered image. A stopping condition for all experiments is defined as follows: Here is a given positive number. We set in the paper. The other denoising parameters in the experiments are set as follows: the time step and grid step size , , and to 25. We find that has a deep influence on the denoising results. All experiments were implemented by MATLAB R2009a and performed on 64-bit Windows 7 on the desk with an Intel CPU of 1.7 GHz and 4 GB memory.

4.1. Experiments with Synthetic Images

Figure 1 shows the comparison of some denoising methods on a 256 256 synthetic image. Figure 1(a) is the original image and Figure 1(b) is the noisy image with Gaussian noise whose standard deviation is 30 and mean value is zero. Figures 1(c)1(h) show the denoising results of nonlocal means model [2], Y-K model [11], Perona-Malik model [4], total variation (TV) method [7], modified TV (MTV) [25], and our proposed model. Figure 2 shows the locally enlarged images of Figure 1. From Figures 1 and 2, we can observe that our model yields better results in the visual sense than other models in the visual sense, since it eliminates the staircase effect of TV model and preserves edges as well as TV model. In other words, our model can take advantage of both the filter based on Tikhonov regularization and the one based on total variation regularization.

Table 1 shows the PSNR, SSIM, and computational times of Figure 1. From Table 1, we also see that our model also outperforms other models in PSNR and SSIM. The experiments of our model are solved iteratively like TV, PM, and MTV. Seen from Table 1, the proposed method uses less computational time than Y-K and NLM, but the computational time of our proposed method is higher than those of TV, PM, and MTV.

4.2. Experiments with Natural Images

To verify the performance of our model comprehensively, we have also implemented the quantitative and qualitative evaluation on the natural images from http://decsai.ugr.es/cvg/dbimagenes/. Figures 3, 4, and 5 exhibit the denoising results on the noisy images “lena,” “peppers,” and “house” with . The figures show that the denoising results of our model look better than other models. The results of TV model and modified TV cause the staircase effect. Blurring edges come out in Perona-Malik model. The result of Y-K has speckles. Figure 6 shows the partial enlarged images of Figure 3. Tables 2 and 3 show the PSNR and SSIM of our experiments, from which we can see that the proposed algorithm in all noise power has the highest PSNR and SSIM among the six methods. In addition, we show that the experimental results on the “lena” image with , , , , and in Figure 7 and Tables 4 and 5 give the quantitative experimental results of PSNR and SSIM, respectively. These experimental results demonstrate that our model has better performance.

4.3. Comparison with BM3D

Block-matching and 3D filtering (BM3D) [3] is the state-of-the-art denoising method, which enhances the sparsity by grouping similar 2D image blocks into 3D groups based on the fact that an image has a locally sparse representation in transform-domain. We compare the proposed model with BM3D whose code is downloaded from http://www.cs.tut.fi/~foi/GCF-BM3D/. As shown in Figure 8 and Table 6, we find that the proposed model is not necessarily better than BM3D in removing the noises from the images. But, our model is still novel considering that it is a significant improvement on the other models based on PDE.

5. Conclusion

To eliminate the so-called staircase effect in many known image denoising models and speckles in the high-order PDE, we propose an improved model for image denoising with a weighted combination of Tikhonov regularization and total variation regularization. Numerical experiments show that our proposed model can recover the images from the noisy images, has the highest PSNR and SSIM among the six methods and can preserve important structures, such as edges and corners.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


This work is supported by the National Science Foundation of China (no. 61070227) and NASF-Guangdong Joint Foundation (Key Project) (no. U1135003).