Medical Image denoising with Generalized Gaussian Mixture modeling

Denoising is a classical challenging problem in signal and image processing. In this paper, propose a novel generalized gaussian mixture model(GGMM) to denoise medical images. Firstly, extend GMM to GGMM for modeling the corrupted medical images. Secondly, use Bayesian method and the minimum mean square error (MMSE) to derive a non-linear mapping function for processing the noisy. Lastly, use the Expectation-Maximization (EM) to estimate the parameters of the GGMM. MRI images from the Simulated Brain Database (SBD) are used to conduct experiments with several state-of-the-art methods. Experimental results show that our algorithm is effectiveness and superiority


Introduction
Because the signal-to-noise ratios (SNR) of MRI, CT and MR images are low, whenever possible, we try to perform denoising while maintaining image quality (1) .There are many approaches to image denoising, such as the spatial filtering methods and transform domain filtering methods.Firstly, a traditional way to denoise from image data is to employ spatial filters.Spatial filters can be further classified into non-linear and linear filters.Secondly, in the spectral domain, the observed image has a higher SNR at lower frequencies and a lower SNR at higher frequencies.This makes it possible for conventional spectral methods to denoise through using low-pass filters.Luo (1,2) proposed an averaging reconstructed part images of the spectrum of the image approach to denoise.Fang (3) proposed a multiscale sparsity based tomographic denoising (MSBTD) method to denoise Spectral-Domain Optical Coherence Tomography (SDOCT) volumetric data.Thirdly, another popular denoising approach based on the transformed domain is wavelet denoising (4,5) .The latter consists of discarding least significant wavelet coefficients (hard thresholding) or shrinking less significant wavelet coefficients more than significant coefficients (soft thresholding) to achieve noise reduction.
Mixture models are normally used to model complex data sets in image denoising by assuming that each observation has arisen from one of the different groups or components.Gaussian mixture models (GMMs) (6) are a particularly widely used instance of finite mixture models methods, not least because they have the advantage of simplifying analytical treatment of the problem.While the assumption of Gaussian works well in many situations, it does not always, especially in medical image analysis (7) .Firstly, only if the SNR is sufficiently high it is can closely approximated by a Gaussian distribution because according to physics of MRI the noise distribution follows a Rician distribution (8) .Secondly, the assumptions are far less appropriate for MRI images of the body, not least the liver, breast, and the colorectum.
To remedy the situation, a more flexible density function that can model a large range of statistical behaviors should be considered , especially for the heavy-tailed image.The density family stemming from the generalized Gaussian Model (GGM) is known to model successfully a wide variety of signal sources since the GGM has a symmetric, unimodal density characteristic with varying tail length (9) .Cho (10) proposed a multivariate generalized Gaussian distribution model which fits the actual distribution of wavelet coefficients in clean natural images.But there are few woks about Generalized Gaussian Mixture Model (GGMM) to medical image denoising.
The motivation of this paper is to extend GGM (10) and GMMs (6) to the GGMM for medical image denoising.Our work is different from GMM method (6) in three points:the first one is that our method uses the GGMM instead the GMM to model; the second one is that our method DOI: 10.12792/iciae2013.025estimates the joint distribution of the whole image instead the whole patch's pixels; and the last one is that our method estimates the number of mixture model adaptively instead manually.

GGD model
A model suitable for describing unimodal non-Gaussian amplitude distributions may be achieved by varying the shape parameter ( v ), mean ( µ ) and standard deviation ( σ ) of the generalized Gaussian Distribution(GGD) model， which is defined as , is a generalized measure of the variance that defines the dispersion or scale of the distribution, defined as (.) G defines the complete Gamma function given by ( ) Laplacian distribution as v = 1 and an ordinary Gaussian distribution as v = 2 (see Fig. 1).With the limiting cases, the PDF converges to a uniform distribution as v → ∞, and the PDF approaches to an impulse function as 0 v → .

Denoising Using GGMM
In most cases, the degraded images can be modeled by an additive Gaussian white noise, thus the noisy image is given by: is the noise-free image, N is the noise which is random variable described by zero mean Gaussian distribution with variance matrix 2 N I σ .And we assume that the noise-free image X can be modeled by the GGMM: where the parameters are  Using the Bayesian prior distribution and minimum mean square error (MMSE), we derive a non-linear mapping function for processing the noisy.According to the eq(4) and eq(5), we have Therefore, the distribution of noisy image is also GGMM, and The parameters of the noisy image can be estimated by the EM algorithm.And we can get the parameters of the noisy-free image by the following equations: According to the MMSE criterion, the estimator of any data x can be computed as 2 ˆarg min ( | )( ) arg min ( ) The derivative of J is ( ) ( )(2 2 ) 0 Therefore, the estimator of x can be computed as ( ) ˆ( ) The parameters estimation of the GGMM is more complex than in the case of GMMs.Difficulty lies in the estimate of the shape parameters According to Ould[11], the shape parameters can be estimated by the kurtosis.Kurtosis is a descriptor of the peakedness of the probability distribution of a real-valued random variable.It is defined as the fourth central moment divided by the square of the variance.
where log-likelihood function is defined as: For the d -dimensional dataset, the penalty function is: The estimator of the number of components is * arg min ( ) The well-known approach to estimate the parameters of the mixture model is to maximize the likelihood through the EM algorithm.Algorithm1: Medical Image Denoising with the GGMM Input: Noise Image Y ,the maximum K Output: Denoised Medical Image X (1) Initialization of the mixture parameters Y Θ ; (2) While(k< K ) (3) {Expectation step (E-step), is represented by the computation of the (n)th iteration conditional expectation probability based on the (n-1)th iteration parameters ( 1) ( 1) (4) Maximization step (M-Step), allows numerical maximization of the log-likelihood function.Update the parameters：

Experiment
The images from Brain Web are used to evaluate our algorithm [13]  The denoising effect must be evaluated by means of proper figure of merits, or quality indexes.The first adopted quality index is the peak signal to noise ratio PSNR, defined by:

Conclusions
We have presented a GGMM approach for medical image denoising.Our method uses the GGD to fit each non-Gaussian medical image, which is more suitable for medical image processing than the classical Gaussian model.We employ the EM algorihtm and BIC information criterion to estimate the distribution parameters and the number of mixture model.We uses MMSE rule to calculate the noisy image.Experimental results illustrate that our method can suppresses noise efficiently.

Fig
Fig.1 Normalized GGD model model selection for the GGMM, we use the BIC[12] criterion.

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Fig.1 Some MRI images of SDB In this paper, we compare denoising performance among the GMM method[6], Gaussians Scale Mixtures in the Wavelet Domain method (GSM)[14], Principal Component Analysis with Local Pixel Grouping [15](PCALPG) method and our GGMM method.Those algorithms were simulated by matlab2009 and all experiments were performed on a PC with 1.73 GHz Intel, 1024MB of RAM.Corrupted image of the Fig.2(a) by different noise levels with standard deviation 10,15,20 and 25 are show in Fig.3 (a), (b), (c) and (d), respectively.Denoising images of the Fig.3(d) by the GMM，PCALPG， GSM and GGMM methods are shown in Fig.4(a), (b), (c) and (d) , respectively.
Fig.3 Corrupted Image of the Fig.2(a) by Different Noise Levels m are the number of rows and columns of the image, respectively.L is the number of the image gray levels.The PSNR indices of the 4 methods for the 20 image with different noise levels are shown in Table1.From the Fig.4and table1, we can know that our GGMM is effective.
Table 1 PSNR index for Different methods for the 20 image with different noise level