Printer Friendly

A mixed noise removal method based on total variation.

Due to the technology limits, digital images always include some defects, such as noise. Noise reduces image quality and affects the result of image processing. While in most cases, noise has Gaussian distribution, in biomedical images, noise is usually a combination of Poisson and Gaussian noises. This combination is naturally considered as a superposition of Gaussian noise over Poisson noise. In this paper, we propose a method to remove such a type of mixed noise based on a novel approach: we consider the superposition of noises like a linear combination. We use the idea of the total variation of an image intensity (brightness) function to remove this combination of noises.

Keywords: total variation, ROF model, gaussian noise, poisson noise, image processing, biomedical image, eulerlagrange equation

Povzetek: Clanek predlaga izvirno kombinacijo Gaussovega in Poissonovega filtra za filtriranje suma v slikah.

1 Introduction

Image denoising has attracted a lot of attention in recent years. In order to suppress noise effectively, we need to know its type. There are many types of noises, for example, Gaussian (digital images), Poisson (X-Ray images), Speckle (ultra sonograms) noises and so on.

One of the most famous effective methods is the total variation model [2-4, 10, 12, 17, 18, 22, 26]. The first person who suggested it to solve the denoising problem is Rudin [17]. He used the total variation as a universal tool in image processing. His denoising model is well-known as the ROF model [3, 17]. The ROF model is targeted to efficiently remove Gaussian noise only.

This model is often used to remove not only Gaussian noise, but also other types of noise. For example, the ROF model suppresses Poisson noise not so effectively. Le T. [9] proposed another model, well-known as the modified ROF model to remove Poisson noise only.

Gaussian and Poisson noises both are widespread in real situations, but their combination is important too, for example, in electronic microscopy images [7, 8]. In these images, both types of noises are combined as a superposition. In physical process, Poisson noise usually is added first, before Gaussian noise. Luisier F. with co-authors proposed the theoretically strong PURE-LET method [11] (Poisson-Gaussian unbiased risk estimate--linear expansion of thresholds) to remove this type of combination of noises.

However, such kind of noises usually can be considered as dependent on the image acquisition systems. At the same time, in many papers devoted to the image denoising problem the idea of Poisson-Gaussian noise combination is considered, even though such is not the case.

From other side, many noise reduction approaches have been developed, particularly, wavelet-based transforms, etc. It needs to draw attention, noise reduction approaches that have been developed based on wavelet transform are only for Gaussian or Poisson noise.

In order to remove mixed noise, let us assume that the superposition of noises can be equivalent to some unknown linear combination of them.

We can combine ROF and modified ROF models to suppress the linear combination of noises. The obtained model is supposed to remove the mixed noise better than ROF or modified ROF models separately. Additionally, it is simpler than PURE-LET, because we try to find only the proportion between Poisson and Gaussian noises in the mixed noise.

In experiments, we use images and add noise into them. The image quality is compared with some other denoising methods such as ROF, modified ROF models, and PURE-LET method to remove the superposition of noises. In our paper [19], we proposed to remove the linear combination of Poisson and Gaussian noises and compared results with Wiener [1] and median [23] filters, and with Beltrami method [29]. Our method gives better results for Gaussian and Poisson noises separately, and for the combination of noises too. Hence, in this paper, we do not compare our approach with these methods.

In order to compare image quality after restoration, we use criteria PSNR (Peak Signal-to-Noise Ratio), MSE (Mean Square Error), SSIM (Structure SIMilarity) [24, 25]. The PSNR criterion is the most important, because it is always used to evaluate images and signals quality.

In this paper, we try to represent and discuss only the case limited by the greyscale artificial and real images with artificial noise. According to it, we can use only criteria above based on the full-reference image quality evaluation approach.

In the case of greyscale real images with unknown noises, we need to use the no-reference approach to evaluate the quality of denoising. In general, it is complicated theoretical problem to develop a criterion for it.

Our investigation based on BRISQUE criterion [13] (Blind/Referenceless Image Spatial QUality Evaluator) in this case was discussed in paper [20].

2 Combined denoising model

Let in [R.sup.2] space a bounded domain [OMEGA][subset][R.sup.2] be given. Let functions u(x,y) [member of] R and v(x,y) [member of] R, respectively, be ideal (without noise) and observed (noisy) images, (x,y) [member of] Q . For smooth function u, its total variation can be defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [nabla]u = ([u.sub.x],[u.sub.y]) is a gradient, [u.sub.x] = [partial derivative]u/[partial derivative]x, [u.sub.y] = [partial derivative]u/[partial derivative]y, [absolute value of [nabla]u] = [square root of [u.sup.2.sub.x] + [u.sup.2.sub.y]. In this paper, we consider that the function u always has limited total variation [V.sub.T][u] < [infinity].

According to [2, 3, 17, 18], an image smoothness is characterized by the total variation of an image intensity function. The total variation of the noisy image is always greater than the total variation of the corresponding smooth image. In order to solve the problem [V.sub.T][u] [right arrow] min, we need to use the following condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, we obtain the ROF model to remove Gaussian noise in the image [17, 18]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [lambda] > 0 is Lagrange multiplier. This is a solution of the unconstrained optimization problem.

In order to remove Poisson noise, Le T. built another model based on ROF model [9] as the optimization problem [V.sub.T][u] [right arrow] min with the following constraint

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This model resulted in the following unconstrained optimization problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [beta] > 0 is a coefficient of regularization. This is the known modified ROF model to remove Poisson noise.

In order to build a model for removing the mixed Poisson-Gaussian noise, we also solve the same optimization problem [V.sub.T][u] [right arrow] min, but with a different constraint as follows.

This constraint is very similar to constraints above. We consider, the noise variance is unchangeable (Poisson noise is not changed and Gaussian noise only depends on noise variance):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where p(v | u) is a conditional probability of the real image v with the ideal image u given.

The probability density function of Gaussian noise is

[p.sub.1](v|u) = exp (-[(v - u).sup.2]/2[[sigma].sup.2])/([sigma][square root of 2[pi]]),

and the probability distribution of Poisson noise is

[p.sub.2](v|u) = exp(-u)[u.sup.v] / v!.

We have to notice that intensity functions of images u and v are integer (for example, for 8-bits greyscale image the range of intensity is from 0 to 255).

In order to combine Gaussian and Poisson noises, we consider the following linear combination

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

According to (1), we obtain the denoising problem as a constrained optimization problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [kappa] is a constant value. We transform this problem into unconstrained optimization problem by using Lagrange functional

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

to find the solution as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [tau] > 0 is Lagrange multiplier.

If [[lambda].sub.1] = 0 and [beta] = [[lambda].sub.2][tau], we obtain the modified ROF model to remove Poisson noise. If [[lambda].sub.2] = 0 and [lambda] = [[lambda].sub.1][tau]/[[sigma].sup.2], we obtain the ROF model to remove Gaussian noise. If [[lambda].sub.1] > 0, [[lambda].sub.2] > 0 , we obtain our model to remove mixed Poisson-Gaussian noise.

3 Discrete denoising model

The problem (2) can be solved by using Lagrange multipliers method [5, 16, 28].

We use Euler-Lagrange equation [28]. Let a function f(x,y) be defined in a limited domain [OMEGA] [subset] [R.sup.2] and be second-order continuously differentiated by x and y, where (x,y)[member of][OMEGA]. Let F(x,y,f,[f.sub.x],[f.sub.y]) be a convex functional, where [f.sub.x] = [partial derivative]f / [partial derivative]x, [f.sub.y] = [partial derivative]f / [partial derivative]y. Then the solution of the following optimization problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

satisfies the following Euler-Lagrange equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We use the above result to solve the obtained model. Then the solution of the problem (2) satisfies the following Euler-Lagrange equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where [mu] = 1/[tau]. We rewrite (3) in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

In order to obtain the discrete form of the model (4), we add an artificial time parameter and consider the function u = u(x,y,t) in the following diffusion equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Then the discrete form of the equation (5) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where K is enough great number, K = 500.

4 Optimal model parameters

In practice, parameters [[lambda].sub.1], [[lambda].sub.2], [mu], [sigma] in procedure (6) are usually unknown. We have to change [[lambda].sub.1], [[lambda].sub.2], [mu] into [[lambda].sup.k.sub.1], [[lambda].sup.k.sub.2], [[mu].sup.k] to evaluate them on every iteration step k.

4.1 Optimal parameters [[lambda].sub.1] and [[lambda].sub.2].

Let (u, [tau]) be a solution of problem (2). Then we obtain the following condition [partial derivative]L(u,[tau])/ [partial derivative]u = 0. This condition give us optimal [[lambda].sub.1] and [[lambda].sub.2]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The discrete form for k = 0,1, ..., K is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4.2 Optimal parameter [mu]

In order to find the optimal [mu], we multiply (3) by (v - u) and integrate by parts over domain [OMEGA]. Finally, we obtain the formula to find the optimal

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The discrete form is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

4.3 Optimal parameter [sigma]

The parameter [sigma] is calculated at the first step of the iteration process. We use the method of Immerker [6]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with the mask [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for convolution operator *,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4.4 Initial solution

In the iteration procedure (6), the result depends on initial parameters [[lambda].sup.0.sub.1], [[lambda].sup.0.sub.2], [[mu].sup.0]. If [[lambda].sup.0.sub.1], [[lambda].sup.0.sub.2], [[mu].sup.0] are given first, then its unsuitable values define not so good solution [u.sub.ij] and later, not so good evaluation of a probability distribution parameters. If [[lambda].sup.0.sub.1], [[lambda].sup.0.sub.2], [[mu].sup.0] are randomized, the result is unacceptable too, because of the additional noise added in the image.

Of course, initial values of [[lambda].sup.0.sub.1], [[lambda].sup.0.sub.2], [[mu].sup.0] need to be closed to required values. We evaluate [[lambda].sup.0.sub.1], [[lambda].sup.0.sub.2], [[mu].sup.0] as average values of neighbour pixels of the image, for example, by the method of Immerker.

5 Image quality evaluation

In order to evaluate the image quality after denoising, we use criteria PSNR, MSE and SSIM [24, 25]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For example, [K.sub.1] = [K.sub.2] = [10.sup.-6], L is an image intensity with L = 255 for 8-bits greyscale image.

The greater the value of [Q.sub.PSNR], the better the image quality. If the value of [Q.sub.PSNR] belongs to the interval from 20 to 25, then the image quality is acceptable, for example, for wireless transmission [21].

The [Q.sub.MSE] is a mean squared error and is used to evaluate the difference between two images. The lower the value of [Q.sub.MSE], the better the result of restoration. The value of [Q.sub.MSE] directly related to the value of [Q.sub.PSNR].

The value of [Q.sub.SSIM] is used to evaluate an image quality by comparing the similarity of two images. This value is between -1 and 1. The greater the value of [Q.sub.SSIM], the better the image quality.

6 Experiments and discussion

In this paper, we consider cases as in [19] and additionally the superposition of noises. The image size is changed from 300x300 pixels to 256x256 pixels specified in PURE-LET method [11]. We process the artificial image with artificial noise and the real image with artificial noise. The artificial image is noise-free and we need to add noise with high intensity (the image to be very noisy) to reduce its quality. Therefore, we specify 0.6 for proportion of Gaussian noisy image and 0.4 for proportion of Poisson noisy image. The real image (captured by a digital device) already includes some noise. We specify 0.5 for proportion of Gaussian noisy image and 0.5 for proportion of Poisson noisy image too.

We need to point the attention in the case of Gaussian noise our method sometimes can be better than ROF, because the method to evaluate the variance of Gaussian noise can be better than one included in the original ROF model in many cases. In the case of superposition of noises, our method sometimes can be better than PURELET, because parameters of our method are usually more optimal than in original model too.

6.1 Artificial image with artificial noise

We use artificial image with artificial mixed noise for the first test. The image includes eight bars (Fig. 1a). Other images (Fig. 1b-j) show noisy and denoised images and zoomed out part of them.

Artificial noise is generated by linear combination, and by superposition of Poisson and Gaussian noises.

For both cases, we consider Poisson noise first. Its probability density is [p.sub.2](v | u), and variance is [[sigma].sub.2] = [square root of [u.sub.ij]] at every pixel (i,j), i = 1, ..., [N.sub.1], j = 1, ..., [N.sub.2]. Poisson noise variance is an average value [[bar.[sigma]].sub.2] =11.7939. If the grey value of a pixel after adding of Poisson noise is out of the interval from 0 to 255, it needs to be reset to [v.sup.(2).sub.ij] = [u.sub.ij]. For this image, there are no pixels out of this interval. Next, we consider the variance of Gaussian noise is four times greater than the variance of Poisson noise [[sigma].sub.1] = 4[[bar.[sigma]].sub.2] =47.1757.

[FIGURE 1 OMITTED]

For the linear combination of noises, we denote the intensity function of Gaussian noisy image as [v.sup.(1)]. As above, values of [v.sup.(1)] need to be between 0 and 255. If the grey value of a pixel after adding of Gaussian noise is out of the interval from 0 to 255, it needs to be reset to [v.sup.(1).sub.ij] = [u.sub.ij]. In this case, there are 1075 pixels out of this interval (1.64%).

[FIGURE 2 OMITTED]

The final noisy image (linear combination of noises in Fig. 1c) is created with proportion 0.6 for Gaussian noisy image [v.sup.(1]) and proportion 0.4 for Poisson noisy image [v.sup.(2]): v = 0.6[v.sup.(1)] + 0.4[v.sup.(2)].

Then we define proportion for linear combination as [[lambda].sub.1] / [[lambda].sub.2] = (0.6 x 47.1757)/(0.4 x 11.7939) = 6/1. Coefficients of linear combination are defined as [[lambda].sub.1] = 6/7 = 0.8571, [[lambda].sub.2] = 1/7 = 0.1429.

Values of [Q.sub.PSNR], [Q.sub.MSE], and [Q.sub.SSIM] of the noisy image (linear combination of noises) are, respectively, 19.4291, 741.5963, and 0.1073.

In the case of the image with superposition of noises, we add Gaussian noise over Poisson noisy image. The intensity function of this Gaussian noisy image is [v.sup.(1)] too. As above, the grey values of [v.sup.(1)] need to be between 0 and 255. If the grey value of a pixel after adding of Gaussian noise is out of the interval from 0 to 255, it needs to be reset to [v.sup.(1).sub.ij] = [v.sup.(2).sub.ij].

There are 1220 pixels out of this interval (1.86%). The noisy image (superposition of noises, Fig. 1g) is also Gaussian noisy image v = [v.sup.(1)]. In this case, we don't know [[lambda].sub.1] and [[lambda].sub.2], therefore we use the algorithm with automatically defined parameters.

Values of [Q.sub.PSNR], [Q.sub.MSE], and [Q.sub.SSIM] of the noisy image are, respectively, 14.9211, 2093.9827, and 0.0439.

Tables 1-4 show results for linear combination of noises, Gaussian noise, Poisson noise, and superposition of noises for the artificial image.

6.2 Real image with artificial noise

The artificial noise is generated by linear combination and superposition of Poisson and Gaussian noises.

For both cases, we consider Poisson noise first. Poisson noise variance is an average value [[bar.[sigma]].sub.2] = 9.0882. If the grey value of a pixel after adding of Poisson noise is out of the interval from 0 to 255, it needs to be reset to [v.sup.(2).sub.ij] = [u.sub.ij]. Here there are no pixels out of this interval.

For Gaussian noise, we consider the variance of Gaussian noise is four times greater than the variance of Poisson noise [[sigma].sub.1] = 4[[bar.[sigma]].sub.2] = 36.3529. The real image is a human skull [14] (Fig. 2a). Others (Fig. 2b-j) show noisy and denoised images and zoomed out part of them.

For the case of linear combination of noises, we denote the intensity function of Gaussian noisy image as [v.sup.(1)]. As above, the grey values of intensity function [v.sup.(1)] also need to be between 0 and 255. If the grey value of a pixel after adding of Gaussian noise is out of the interval from 0 to 255, it needs to be reset to [v.sup.(1).sub.ij] =[u.sub.ij]. In this case, there are 5355 pixels out of this interval (8.17%). The final image (linear combination of noises, Fig. 2c) is created with proportion 0.5 for Gaussian noisy image [v.sup.(1)] and proportion 0.5 for Poisson noisy image [v.sup.(2)]: v = 0.5[v.sup.(1)] + 0.5[v.sup.(2)]. The proportion for linear combination is: [[lambda].sub.1] / [[lambda].sub.2] = (0.5 x 36.3529) / (0.5 x 9.0882) = 4 /1.

Hence, coefficients of linear combination are defined as [[lambda].sub.1] = 4/5 =0.8, [[lambda].sub.2] = 1/5 = 0.2. Values of [Q.sub.PSNR], [Q.sub.MSE], and [Q.sub.SSIM] of final noisy image are, respectively, 23.6878, 278.1619, and 0.5390.

For superposition of noises, we add Gaussian noise over Poisson noisy image. We denote the intensity function of Gaussian noisy image as [v.sub.(1)]. As above, grey values of [v.sub.(1)] need to be between 0 and 255. If the grey value after adding of Gaussian noise is out of the interval from 0 to 255, it needs to be reset to [v.sup.(1).sub.ij] = [v.sup.(2).sub.ij]. In this case, there are 5621 pixels out of this interval (8.58%). The final noisy image (superposition of noises, Fig. 2g) is also the Gaussian noisy image v = [v.sup.(1)].

In this case, we don't know [[lambda].sub.1] and [[lambda].sub.2], therefore we use the algorithm to find them. Values of [Q.sub.PSNR], [Q.sub.MSE], and [Q.sub.SSIM] of the final noisy image (superposition) are, respectively, 17.8071, 1077.3831, and 0.3242.

Tables 5-8 show results for linear combination of noises, Gaussian noise, Poisson noise, and superposition of noises for the real image.

6.3 About of initial solution

In order to create the initial image, we use the convolution operator. The table 9 shows the dependency of restored result for the initial image, where:

(a) Initial parameters [[lambda].sup.0.sub.1] = 0, [[lambda].sup.0.sub.2] = 1, [mu] = 1;

(b) Initial parameters [[lambda].sub.0.sub.1] = [[lambda].sup.0.sub.2] = 0.5, [mu] = 1;

(c) Initial solution [u.sup.0] is given as a randomized matrix;

(d) Initial solution [u.sup.0] = [LAMBDA] * v is given as an average value of neighbour pixels by the convolution operator with the mask [LAMBDA] = (1/9) of the size 3x3.

Table 9 shows the best result of denoising is (d) by criteria PSNR and MSE.

The result (c) by SSIM looks different in contract to ones in Tables 1-8. It illustrates incorrectness of a randomized initial solution (accidental and not stable, if a probability distribution is unknown).

Next, we have to notice that the non-optimal result (a) has been used in experiments for Table 5. It appears to be enough for the good result with automatically defined model parameters.

At last, the variant (b) initially looks better than (a) for kind of better assumption of [[lambda].sup.0.sub/1] = [[lambda].sup.0.sub.2] = 0.5 to process the real image. Nevertheless, our assumption about [mu] = 1 is very far from the good one, and evidently the limit of the number of steps K = 500 is insufficient in this case.

As a result, the variant (d) is the best idea for initial solution.

7 Conclusion

In this paper, we proposed a novel method that can effectively remove the mixed Poisson-Gaussian noise. Furthermore, our proposed method can be also used to remove Gaussian or Poisson noise separately. This method is based on the variational approach.

The denoising result strongly depends on values of coefficients of linear combination [[lambdas].sub.1], and [[lambda].sub.2]. These values can be set manually or can be defined automatically. When processing real images, we can use the proposed method with automatically defined parameters.

Although our method concentrates on removing the linear combination of noise, but it also efficiently removes the superposition of noises. In this case, we consider the superposition of noises is equivalent to some linear combination of them with coefficients found in iteration process.

In this paper we show that our simple model "feels" well the wide range of proportion of two types of noises. As a result, it appears to be the good basis for removing superposition of such noises.

It is evident, the iteration process (6) used here is insufficiently effective in comparing with other possible computational schemes. In this paper, we try to compare our approach to image denoising with PURE-LET method only in possible reduction of our model complexity, not in others.

We would like to express our great thanks to developers of PURE-LET method for kindly granted us the original executable module of it.

8 Acknowledgements

This work is partially supported by Russian Foundation for Basic Research, under grants 13-07-00529, 14-0700964, 15-07-02228, 16-07-01039, and by Ministry of Education and Training, Vietnam, under grant number B2015-01-90.

References

[1] Abe C., Shimamura T. Iterative Edge-Preserving adaptive Wiener filter for image denoising. ICCEE, 2012, Vol. 4, No. 4, P. 503-506.

[2] Chan T.F., Shen J. Image processing and analysis: Variational, PDE, Wavelet, and stochastic methods. SIAM, 2005.

[3] Chen K. Introduction to variational image processing models and application. International journal of computer mathematics, 2013, Vol. 90, No. 1, P.1-8.

[4] Getreuer P. Rudin-Osher-Fatemi total variation denoising using split Bregman. IPOL 2012: http://www.ipol.im/pub/art/2012/g-tvd/.

[5] Gill P.E., Murray W. Numerical methods for constrained optimization. Academic Press Inc., 1974.

[6] Immerker J. Fast noise variance estimation. Computer vision and image understanding, 1996, Vol. 64, No.2, P. 300-302.

[7] Jezierska A. An EM approach for Poisson-Gaussian noise modelling. EUSIPCO 19th, 2011, Vol. 62, Is. 1, P. 13-30.

[8] Jezierska A. Poisson-Gaussian noise parameter estimation in fluorescence microscopy imaging. IEEE International Symposium on Biomedical Imaging 9th, 2012, P. 1663-1666.

[9] Le T., Chartrand R., Asaki T.J. A variational approach to reconstructing images corrupted by Poisson noise. Journal of mathematical imaging and vision, 2007, Vol. 27, Is. 3, P. 257-263.

[10] Li F., Shen C., Pi L. A new diffusion-based variational model for image denoising and segmentation. Journal mathematical imaging and vision, 2006, Vol. 26, Is. 1-2, P. 115-125.

[11] Luisier F., Blu T., Unser M. Image denoising in mixed Poisson-Gaussian noise. IEEE transaction on Image processing, 2011, Vol. 20, No. 3, P. 696-708.

[12] Lysaker M., Tai X. Iterative image restoration combining total variation minimization and a second-order functional. International journal of computer vision, 2006, Vol. 66, P. 5-18.

[13] Mittal A., Moorthy A.K., and Bovik A.C. No reference image quality assessment in the spatial domain, IEEE Trans. Image Processing 21 (12), 4695-4708 (2012).

[14] Nick V. Getty images: http://well.blogs.nytimes .com/2009/09/16/what-sort-of-exercise-can-make-you-smarter/.

[15] Rankovic N., Tuba M. Improved adaptive median filter for denoising ultrasound images. Advances in computer science, 2012, P.169-174.

[16] Rubinov A., Yang X. Applied Optimization: Lagrange-type functions in constrained non-convex optimization. Springer, 2003.

[17] Rudin L.I., Osher S., Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D., 1992, Vol. 60, P. 259-268.

[18] Scherzer O. Variational methods in Imaging. Springer, 2009.

[19] Thanh N. H. Dang, Dvoenko Sergey D., Dinh Viet Sang. A Denoising Method Based on Total Variation. Proc. of 6th Intern. Symposium on Information and Communication Technology (SoICT-2015). P. 223-230. ACM, NY, USA.

[20] Thanh D.N.H., Dvoenko S.D. A method of total variation to remove the mixed Poisson-Gaussian noise. Pattern Recognition and Image Analysis, 26 (2), 285-293 (2016). DOI: 10.1134/S 1054661816020231.

[21] Thomos N., Boulgouris N.V., Strintzis M.G. Optimized Transmission of JPEG2000 streams over Wireless channels. IEEE transactions on image processing, 2006, Vol. 15, No.l, P .54-67.

[22] Tran M.P., Peteri R., Bergounioux M. Denoising 3D medical images using a second order variational model and wavelet shrinkage. Image analysis and recognition, 2012, Vol. 7325, P. 138-145.

[23] Wang C., Li T. An improved adaptive median filter for Image denoising. ICCEE, 2012, Vol. 53, No. 2.64, P. 393-398.

[24] Wang Z. Image quality assessment: From error visibility to structural similarity. IEEE transaction on Image processing, Vol. 13, No. 4, P. 600-612. 2004.

[25] Wang Z., Bovik A.C. Modern image quality assessment. Morgan & Claypool Publisher, 2004.

[26] Xu J., Feng X., Hao Y. A coupled variational model for image denoising using a duality strategy and split Bregman. Multidimensional systems and signal processing, 2014, Vol. 25, P. 83-94.

[27] Zhu Y. Noise reduction with low dose CT data based on a modified ROF model. Optics express, 2012, Vol. 20, No. 16, P. 17987-18004.

[28] Zeidler E. Nonlinear functional analysis and its applications: Variational methods and optimization. Springer, 1985.

[29] Zosso D., Bustin A. A Primal-Dual Projected Gradient Algorithm for Efficient Beltrami Regularization. Computer Vision and Image Understanding, 2014: http://www.math.ucla.edu/~zosso/.

Dang N.H. Thanh

Tula State University; 92 Lenin Ave., Tula, Russian Federation

Hue College of Industry, 70 Nguyen Hue st., Hue, Vietnam

E-mail: dnhthanh@hueic.edu.vn

Dvoenko Sergey D.

Tula State University; 92 Lenin Ave., Tula, Russian Federation

E-mail: dsd@tsu.tula.ru

Dinh Viet Sang

Hanoi University of Science and Technology; 1 Dai Co Viet st., Hanoi, Vietnam

E-mail: sangdv@soict.hust.edu.vn

Received: March 24, 2016
Table 1: Quality of noise removing for the artificial
image with linear combination of noises.

                              [Q.sub.PSNR]   [Q.sub.SSIM]   [Q.sub.MSE]

Noisy                         19.4291        0.1073         741.5963
ROF                           34.1236        0.8978         25.1606
Modified ROF                  32.4315        0.8703         37.8791
PURE-LET                      33.0309        0.9277         32.3587

Proposed method               41.1209        0.9841         4.9905
[[lambda].sub.1] = 0.8571,
[[lambda].sub.2] = 0.1429,
[mu] = 0.5003,
[sigma] = 47.1757

Proposed method               41.0998        0.9840         5.0478
with automatically
defined parameters
[[lambda].sub.1] = 0.8414,
[[lambda].sub.2] = 0.1586,
[mu] = 0.5112,
[sigma] = 41.0314

Table 2: Quality of noise removing for the artificial
image with Gaussian noise.

                              [Q.sub.PSNR]   [Q.sub.SSIM]   [Q.sub.MSE]

Noisy                         15.1406        0.0457         1990.8
ROF                           31.4797        0.8364         21.2502
Modified ROF                  28.4591        0.7871         27.5694
PURE-LET                      28.9451        0.7986         25.9883

Proposed method               35.8011        0.9598         16.8122
[[lambda].sub.1] = 1,
[[lambda].sub.2] = 0,
[mu] = 0.3033,
[sigma] = 47.1757

Proposed method               35.7589        0.9596         17.2658
with automatically
defined parameters
[[lambda].sub.1] = 0.9715,
[[lambda].sub.2] = 0.0285,
[mu] = 0.3021,
[sigma] = 46.0314

Table 3: Quality of noise removing for the artificial
image with Poisson noise.

                              [Q.sub.PSNR]   [Q.sub.SSIM]   [Q.sub.MSE]

Noisy                         26.6776        0.3640         139.7396
ROF                           36.4958        0.9381         14.5715
Modified ROF                  44.6347        0.9897         2.2001
PURE-LET                      37.4485        0.9404         10.5692

Proposed method               44.6343        0.9897         2.2014
[[lambda].sub.1] = 0,
[[lambda].sub.2] = 1,
[mu] = 0.8012,
[sigma] = 0.0001

Proposed method               44.6156        0.9896         2.2466
with automatically
defined parameters
[[lambda].sub.1] = 0.0524,
[[lambda].sub.2] = 0.9476,
[mu] = 0.7923,
[sigma] = 2.0544

Table 4: Quality of noise removing for the artificial
image with superposition of noises.

                              [Q.sub.PSNR]   [Q.sub.SSIM]   [Q.sub.MSE]

Noisy                         14.9211        0.0439         2093.983
ROF                           31.2913        0.8346         48.3008
Modified ROF                  30.5471        0.8232         56.5601
PURE-LET                      33.9889        0.9298         25.9534

Proposed method               37.3366        0.9677         12.0066
with automatically
defined parameters
[[lambda].sub.1] = 0.8014,
[[lambda].sub.2] = 0.1986,
[mu] = 0.4812,
[sigma] = 40.0314

Table 5: Quality of noise removing for the real
image with linear combination of noises.

                              [Q.sub.PSNR]   [Q.sub.SSIM]   [Q.sub.MSE]

Noisy                         23.6878        0.5390         278.1619
ROF                           27.3974        0.8295         118.3975
Modified ROF                  25.5644        0.7513         197.5403
PURE-LET                      25.7781        0.8105         191.0341

Proposed method               27.6641        0.8331         110.9451
[[lambda].sub.1] = 0.8,
[[lambda].sub.2] = 0.2,
[mu] = 0.0524,
[sigma] = 36.3529

Proposed method               27.6039        0.8325         112.8984
with automatically
defined parameters
[[lambda].sub.1] = 0.7804,
[[lambda].sub.2] = 0.2196,
[mu] = 0.0512,
[sigma] = 34.2311

Table 6: Quality of noise removing for the real
image with Gaussian noise.

                              [Q.sub.PSNR]   [Q.sub.SSIM]   [Q.sub.MSE]

Noisy                         18.0693        0.3337         1014.3
ROF                           24.0246        0.7299         257.4095
Modified ROF                  23.2511        0.7019         311.8742
PURE-LET                      23.8712        0.7989         265.6153

Proposed method               24.2011        0.8029         242.5101
[[lambda].sub.1] = 1,
[[lambda].sub.2] = 0,
[mu] = 0.0956,
[sigma] = 36.3529

Proposed method               24.1882        0.8028         247.8894
with automatically
defined parameters
[[lambda].sub.1] = 0.9538,
[[lambda].sub.2] = 0.0462,
[mu] = 0.0902,
[sigma] = 35.0633

Table 7: Quality of noise removing for the real image
with Poisson noise.

                              [Q.sub.PSNR]   [Q.sub.SSIM]   [Q.sub.MSE]

Noisy                         28.4991        0.7625         91.8683
ROF                           31.0567        0.9457         50.9818
Modified ROF                  31.1992        0.9022         48.9375
PURE-LET                      30.8955        0.8678         53.1066

Proposed method               31.1334        0.8986         49.7922
[[lambda].sub.1] = 0,
[[lambda].sub.2] = 1,
[mu] = 0.0541,
[sigma] = 0.0001

Proposed method               31.1316        0.8986         50.1094
with automatically
defined parameters
[[lambda].sub.1] = 0.0491,
[[lambda].sub.2] = 0.9509,
[mu] = 0.0567,
[sigma] = 4.2012

Table 8: Quality of noise removing for the real image
with superposition of noises.

                              [Q.sub.PSNR]   [Q.sub.SSIM]   [Q.sub.MSE]

Noisy                         17.8077        0.3242         1077.383
ROF                           23.1936        0.7062         311.6856
Modified ROF                  23.0413        0.7033         319.3831
PURE-LET                      23.6278        0.7072         282.0349

Proposed method               23.7292        0.7094         275.5229
with automatically
defined parameters
[[lambda].sub.1] = 0.7704,
[[lambda].sub.2]= 0.2296,
[mu] = 0.1102,
[sigma] = 36.3412

Table 9: Dependency of denoising on initial solution.

                     (a)        (b)        (c)        (d)

[[lambda].sub.1]   0.7804     0.8094     0.8733     0.8032
[[lambda].sub.2]   0.2196     0.1906     0.1267     0.1968
[mu]               0.0512     0.0573     0.0653     0.0565
[sigma]            34.2311
[Q.sub.PSNR]       27.6039    27.2214    26.5611    27.6523
[Q.sub.MSE]        112.8984   120.4355   132.0264   107.5431
[Q.sub.SSIM]       0.8325     0.8317     0.8395     0.8392
COPYRIGHT 2016 Slovenian Society Informatika
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2016 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Thanh, Dang N.H.; Sergey D., Dvoenko; Sang, Dinh Viet
Publication:Informatica
Article Type:Report
Date:Jun 1, 2016
Words:5564
Previous Article:Editors' introduction to the special issue on "The Sixth International Symposium on Information and Communication Technology--SoICT 2015".
Next Article:A multi-criteria document clustering method based on topic modeling and pseudoclosure function.
Topics:

Terms of use | Privacy policy | Copyright © 2019 Farlex, Inc. | Feedback | For webmasters