A DENOISING OF BIOMEDICAL IMAGES

Today imaging science has an important development and has many applications in different fields of life. The researched object of imaging science is digital image that can be created by many digital devices. Biomedical image is one of types of digital images. One of the limits of using digital devices to create digital images is noise. Noise reduces the image quality. It appears in almost types of images, including biomedical images too. The type of noise in this case can be considered as combination of Gaussian and Poisson noises. In this paper we propose method to remove noise by using total variation. Our method is developed with the goal to combine two famous models: ROF for removing Gaussian noise and modified ROF for removing Poisson noise. As a result, our proposed method can be also applied to remove Gaussian or Poisson noise separately. The proposed method can be applied in two cases: with given parameters (generated noise for artificial images) or automatically evaluated parameters (unknown noise for real images).


INTRODUCTION
One of the important types of digital technique that has many applications in many fields of life is digital image.It is a type of a signal that is obtained from a real analogous signal by discretization and quantization.There are many devices can create digital images, such as digital camera, X-ray scanner, and so on.Ordinarily these devices can give unexpected effects.One of them is noise.Noise reduces image quality and efficiency of image processing.
The problem of noise removal from digital images is very actual today.In order to remove noise more effectively, we need to classify it.There are many types of noise, for example, Gaussian noise (almost for digital image by using digital camera), Poisson noise (for X-ray image), speckle noise (for ultrasonogram), and so on.
However, ROF model is usually used to remove only Gaussian noise.Of course it can also remove other types of noise, but not very effectively.Another popular noise in medical images is Poisson noise.For example, this noise appears in medical X-ray images.ROF model cannot treat this noise effectively.Therefore, Le T. (2007) developed so called modified ROF model.
Both of Gaussian and Poisson noises is popular, but their combination is also important (Luisier, 2011).This combination of noises usually appears in biomedical images, for example, in electronic microscopy images (Jezierska, 2011;Jezierska 2012).
As we talk above, ROF and modified ROF models ineffectively treat this combination.ROF model gives priority to Gaussian noise, but modified ROF model gives it to Poisson noise.
In order to treat this combination of noise, we will combine ROF model (for Gaussian noise) and modified ROF model (for Poisson noise).Our model will treat this combination by considering proportion of noise between them.
In experiments, we used a real image and add noise into them.We performed denoising of digital images by proposed method and other methods, such as ROF model, median filter (Wang, 2012) and Wiener filter (Abe, 2012).In order to evaluate an image quality after denoising, we used well-known criteria MSE (Mean Square Error), PSNR (Peak Signal-to-Noise Ratio) and SSIM (Structure SIMilarity) (Wang, 2004;Wang, 2006).We give priority to PSNR, because it is most popular and used to evaluate the quality of restored signal in signal processing in general, and in image processing, especially.

Denoising Model
According to results (Chang, 2005;Burger, 2008;Rudin, 1992;Chen, 2013;Scherzer, 2009), image smoothness is characterised by the total variation.The total variation of noisy image is always greater than the total variation of smoothed image. When We have to note that intensity levels of image colours are integer (for example, the intensity interval for an 8-bit grayscale image is from 0 to 255), so we regard u as an integer value, but this will ultimately be unnecessary (Le 2007).
In order to treat combination of Gaussian and Poisson noises, we assume the following linear combination According to (1), we obtain the denoising problem with constrained condition as following: where  is a constant value.We can transform this constrained optimization problem to the unconstrained optimization problem by using Lagrange functional where 0   is a Lagrange multiplier.This is our proposed model to remove mixed Poisson-Gaussian noise from digital image.We have to notice that, if

Model Discretization
In order to solve the problem (2), we can use the Lagrange multipliers method (Zeidler, 1985;Rubinov, 2003;Gill, 1974).However, in this paper, we will solve it by using the Euler-Lagrange equation (Zeidler, 1985).

Let function ( , )
f x y be defined in limited domain  ( , , , , ) where We use the result above to solve the problem (2).The solution of the problem (2) is given by the following Euler-Lagrange equation: where In order to discretize the equation ( 4), we add an artificial time parameter and consider the function    .We can write the discretized form of the equation ( 5) as following: where ; Here K is enough great number.In this paper, we use 500 K  .

Finding Optimal Parameters
We can use the procedure (6) to perform image denoising.In this procedure, values of parameters 12 , , ,     need to be given.In some cases, we have to define these parameters to perform image denoising automatically.Then parameters 12 ,,    in process (6) need to be changed into 12 ,, So we obtain new procedure that allows us to calculate values of these parameters automatically in iteration steps.

Optimal Parameters 1 and 2
Let ( , ) u  be a solution of the problem (2).Then we get the condition ( , ) This condition gives us the optimal parameters 12 , : .

Optimal Parameter 
In order to find an optimal parameter  , we multiply (3) by () uv  and integrate by parts over  .Finally, we obtain the formula to find the optimal parameter  : ; The

Optimal Parameter 
In order to evaluate this parameter  , we use the result of Immerker (1996): , where is the mask of an image.
Operator * is a convolution operator, where We have to notice, that the parameter  is just evaluated at first time of the iteration process.

Image Quality Evaluation
In order to evaluate image quality after denoising, we use criteria MSE (Mean Square Error), PSNR (Peak Signal-to-Noise Ratio) and SSIM (Structure SIMilarity) (Wang 2004(Wang , 2006)): , where For example, is between 20 and 25, then an image quality is acceptable, for example, for the wireless transmission (Thomos, 2006).

Initial solution
Because the iteration process uses the initial solution to perform finding solution, so the restoration result of automatically evaluated parameters case also depends on this initial solution.This dependency affects to restoration result, but it is not too much.
The initial solution can be given by one from two methods: directly given or given through a set of initial parameters If the initial values of parameters 12 ,,    are given, the obtained solution is not very good.Because when we set up the initial parameters to find initial solution, the priority of processing of Gaussian and Poisson noise on initial solution is fixed and the result will depend on these initial parameters.
If the initial solution is constant value, the total variation and the differences by x-direction and y-direction will be 0.This is very bad for our iteration process.
If we make an artificial image by randomizing, the restoration result is bad.Because the random function will affect to the noise property.
The best case is the initial solution need to be enough different with noisy image but not much.In experiment, we make this solution by using average neighbour pixels (closed similar with noise assessment method of Immerker).

Experiments
We use an example to test our model in the case of processing a real image.In this case, we use an image of human skull with the size 300x300 pixel (Figure 1a).We zoom, crop and show the part of the original image under processing (Figure 1b -1f).First, we create the noisy image by adding Gaussian noise (Figure 1c) and second, create noisy image by adding Poisson noise (Figure 1d).
In v .Obviously, intensity value of (2)   v ought to be between 0 and 255.If the intensity value of some pixels are out of this interval, they need to be reset to intensity value of respective pixel of the original image u , that means (2) ij ij vu  .In this case, number of them is 5 (0.0056%).The variance of Poisson noise can be calculated as average value 2 10.0603   .Now, we consider Gaussian noise.Its variance need to be 40.2412(because we explained above, variance of Gaussian noise is four times over variance of Poisson noise).We denote this Gaussian noisy image as (1)  v .As above case, intensity value of (1)   v also need to be between 0 and 255.In this case, there are 5780 pixels out of this interval, respectively 6.42% of all image pixels.
We create resulting noisy image (Figure 1e) by combining first noisy and second noisy images with proportion 0.5 for Gaussian noisy image (1)   v and 0.5 for Poisson noisy image  The Table 1 shows the result of denoising for real image in cases: given parameters and automatically evaluated parameters.
We have to notice, that in the case of the real image, the value of QPSNR of denoising for given ideal parameters is better, than the value of QPSNR of denoising for automatically evaluated parameters, but the value of QSSIM is inversed.
Based on experimental results, we can see that the restoration result of automatically evaluated parameters case depends on initial solution.
We use convolution operator to make a new image.The Table 2 shows the dependency of restoration result on initial solution, where: paper, we only consider function u that always has limited total variation [] T Vu.The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XL-5/W6, 2015 Photogrammetric techniques for video surveillance, biometrics and biomedicine, 25-27 May 2015, Moscow, Russia obtain ROF model for removing Gaussian noise.In the case of 12 0, 0   we get the model for removing mixed Poisson- Gaussian noise.
for an 8-bit greyscale images.The greater value PSNR Q , the better image quality.If PSNR Q evaluate image quality by comparing similarity of both images.Its value is between -1 and 1.The greater value SSIM Q , the better image quality.

Figure 1 .
Figure 1.Denoising of real image: a) original image, b) cropped image, c) with Gaussian noise, d) with Poisson noise, e) with mixed noise, f) after denoising.
the best denoising result for the case (d) with respect two most important criteria (PSNR and MSE) .
Rudin solved the problem [ ] min The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XL-5/W6, 2015Photogrammetric techniques for video surveillance, biometrics and biomedicine, 25-27 May 2015, Moscow, Russia This contribution has been peer-reviewed.doi:10.5194/isprsarchives-XL-5-W6-73-201574 International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XL-5/W6, 2015 Photogrammetric techniques for video surveillance, biometrics and biomedicine, 25-27 May 2015, Moscow, Russia order to calculate proportion between intensities of Gaussian and Poisson noises, we calculate the variance of Poisson noise.The value of variance of Gaussian noise is calculated via Poisson noise variance.Let the variance of Gaussian noise be four times greater than the variance of

Table 1 .
Quality comparison of noise removal methods for real image of human skull.

Table 2 .
Dependency of restoration result on initial solution.