A new four-directional total variation (4-TV) model, applicable to isotropic and anisotropic TV functions, is proposed for image denoising. A dual based fast gradient projection algorithm for the constrained 4-TV image denoising problem is also reported which combines the well-known gradient projection and the fast gradient projection methods. Experimental results show that this model provides in most cases a better signal to noise ratio when compared to previous models like the reference TV, the total generalized variation, and the nonlocal total variation.
Variational models have found a wide variety of applications in image processing and computer vision, in particular in restoration tasks such as denoising, deblurring, and blind deconvolution. One of the major concerns in machine vision remains to preserve important image features (edges, lines, and also textures) while removing noise. Total variation (TV) based image restoration models were first introduced by Rudin et al. (ROF) in their pioneering work [
Many methods were proposed to solve (
The methods of much interest to this paper are the dual approach proposed by Chambolle [
In this paper, we provide the complete mathematical proof for both anisotropic and isotropic cases together with a proper definition of the 4-TV model. Moreover, we propose a fast constrained 4-TV algorithm for image denoising problem. To our knowledge, this is the first time that the double information in both time and space domains is jointly used.
The paper is organized as follows. In Section
We first consider the discrete ROF model which is a convex but nonsmooth minimization problem
Here, we only consider images defined on a rectangular domain, so we have
The isotropic TV is defined as
The discrete 4-TV model is an extended version of the conventional TV proposed by Sakurai et al. [
Total variation components [
Two-directional TV
Four-directional TV
The anisotropic 4-TV is defined by Sakurai et al. [
In order to get a more accurate 4-TV based model, we construct a new image
Let The corrupted image The new image
The images in (a) and (b) share the following relations:
Therefore, the anisotropic 4-TV can be defined as
Consequently, the discrete 4-directional TV model is defined by
By adopting the above 4-directional TV model, each iteration step uses double information in the space domain. For this reason, the new model can be expected to have more general and effective properties than the standard one in image denoising problem. In addition, both anisotropic and isotropic TV cases can be dealt with.
We consider the constrained 4-directional TV based denoising problem which corresponds to
The TV function is characterized by the nonsmoothness. The characteristic of the nonsmoothness is the key difficulty in problem equation (
First, we define
Here, we do not assume reflexive boundary conditions owing to constructing the new image space
Now, let
Then, we also have the relation:
We introduce the linear operation
So, we can write
So, (
Because the above function is concave in
Let
The optimal solution of the constrained 4-directional TV based denoising model in (
By neglecting the constant term
Let
The operator
We consider the function
Equation (
And the gradient of
Therefore,
From the above derivation, we see that the dual problem expressed by (
The only difference between the isotopic TV and anisotropic TV cases is contained in the relation:
The gradient of (
The Euclidian norm of the matrix-pairs
For every two groups of matrices
Now,
Thus, we have
The overall procedure to implement this constrained 4-directional gradient projection (4-GP) algorithm can be summarized as shown in Algorithm
In the constrained case, a group
Note that the difference between the isotopic TV and anisotropic TV is as follows:
It has been shown from the above derivations that the dual problem equation (
The original fast gradient projection algorithm can be traced back to the gradient mapping approach proposed by Nesterov [
Here, we used the constrained 4-directional fast gradient projection (4-FGP) algorithm for the denoising problem. The 4-FGP algorithm has a convergence rate in
These experiments were conducted on images widely used in the computer vision literature. We selected two samples among this trial set, the “Cameraman” and the “Moon” pictures, to illustrate the effectiveness of the proposed method. These two images, by their different contents, are representative of the large spectrum of data sets that can be considered. A comparison of our methods (4-GP and 4-FGP) was performed with the GP, FGP [
The
Comparison of the different methods on the “Cameraman” image. (a) Original image; (b) noisy image, PSNR = 19.60 dB; (c) denoised image with GP, PSNR = 26.19 dB; (d) denoised image with FGP, PSNR = 26.19 dB; (e) denoised image with TGV, PSNR = 26.38 dB; (f) denoised image with NLTV, PSNR = 27.13 dB; (g) denoised image with 4-GP, PSNR = 27.41 dB; and (h) denoised image with 4-FGP, PSNR = 27.41 dB.
Results on the “Moon” image. (a) Original image; (b) noisy image, PSNR = 24.82 dB; (c) denoised image with GP, PSNR = 29.35 dB; (d) denoised image with FGP, PSNR = 29.35 dB; (e) denoised image with TGV, PSNR = 30.08 dB; (f) denoised image with NLTV, PSNR = 30.23 dB; (g) denoised image with 4-GP, PSNR = 30.35 dB; (h) denoised image with 4-FGP, PSNR = 30.36 dB.
The parameters were set to
For the TGV method [
The MATLAB codes of the TGV method [
The PSNR values obtained in the above cases for the different methods are indicated in Figures
From Figures
As it was expected, the convergence is much faster when using 4-FGP instead of 4-GP. A detailed analysis of the convergence process makes it clear that the number of iterations is image dependent and much higher for GP than for FGP; for the “Cameraman” image, this number is equal to 124 for 4-GP and to 41 for 4-FGP; for the “Moon” image, they are, respectively, equal to 103 and 51. The time computation varies accordingly; it goes for the “Cameraman” image from about 55 seconds for 4-GP to approximately 18 seconds for 4-FGP and for the “Moon” image from about 53 seconds for 4-GP to approximately 27 seconds for 4-FGP.
From Figure
Comparison of the convergence behavior for the different methods with respect to PSNR. (a) The “Cameraman” image (standard deviation of zero-mean Gaussian white noise: 0.11) and (b) the “Moon” image (standard deviation of zero-mean Gaussian white noise: 0.07).
In the examples described above, the values of the noise and
The results are provided in Figure
Evolution of PSNR with
By taking into account the previous results, the noise effect was analyzed. The selected parameters were
Evolution of PSNR when varying the noise level for the different methods. (a) The “Cameraman” image (
The performances of the FGP, TGV, and NLTV methods are inferior to the performance obtained with 4-FGP method when the value of noise is large and superior to the performance of the 4-FGP method when the value of noise is rather low.
These methods were compared on three other images and additional comments are provided in this section. The objective was to see if they were stable enough to be generalized to any type of images. Two images, “Lena” and “Woman,” were used by Chambolle in [
PSNR (dB) for GP, FGP, TGV, NLTV, 4-GP, and 4-FGP on different images (“Lena,” “Woman,” and “Louvre” images, standard deviation of zero-mean Gaussian white noise 0.1 and
Image | Noisy image | GP | FGP | TGV | NLTV | 4-GP | 4-FGP |
---|---|---|---|---|---|---|---|
Lena | 20.43 | 26.22 | 26.23 | 26.55 | 26.60 | 26.67 | 26.67 |
Woman | 20.50 | 27.48 | 27.48 | 27.70 | 27.78 | 28.39 | 28.38 |
Louvre | 20.35 | 27.02 | 27.02 | 26.91 | 27.45 | 27.50 | 27.49 |
In this paper, 4-GP and 4-FGP methods were proposed for image denoising. We added the diagonal components to the conventional TV model and we provided the complete mathematical proof of the relevance of the new model. Moreover, the 4-FGP algorithm makes for the first time use of the double information in both time and space domains. Experimental results show that the 4-GP and 4-FGP methods lead to better denoising results in most cases. In the future work, we will address the weighted 4-GP and 4-FGP frames.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Basic Research Program of China under Grant 2011CB707904, by the National Natural Science Foundation of China under Grants 61201344, 61271312, and 61073138, by the project sponsored by SRF for ROCS, SEM, by the SRFDP under Grants 20110092110023 and 20120092120036, and by Natural Science Foundation of Jiangsu Province under Grant BK2012329 and by Qing Lan Project. This work was also supported by INSERM postdoctoral fellowship.