Tweedie hidden Markov random field and the expectation-method of moments and maximisation algorithm for brain MR image segmentation

ABSTRACT In this paper, a segmentation algorithm of brain magnetic resonance imaging is elaborated. The proposed method rests on Tweedie hidden Markov random field processing and the Expectation Method of moments and Maximisation algorithm. This new type of stochastic process stands for a Markov Random Field whose state sequence cannot be observed directly and whose spatial information is encoded through the mutual influences of neighbouring sites. Tweedie models are a special case of exponential dispersion models with power mean variance. In image processing problems as medical image segmentation no work has been recorded in terms of demonstrating the applicability of the Tweedie models with hidden Markov random field and different Tweedie parameters values p. Hence, our research stands for the pioneering work aiming at proving the efficiency of the Tweedie models in brain image segmentation. Expectation- Method of moments and Maximisation algorithm results from the combination of both EM (Expectation-Maximisation) algorithm and the method of moments. The algorithm was validated on synthetic data and tested on real images. The basic merit of our proposed algorithm of segmentation lies in its ability to identify the brain tumour. Experiments using Tweedie hidden Markov random field showed that the proposed method produces more intuitive results with a recognition rate of 93.72. In fact, the experimental results highlighted that our approach outperforms other approaches that are reported in literature.


Introduction
Medical imaging is regarded as a wealthy and fertile field through which radiologists are still attempting derive accurately vital information from image data. It corresponds to acquiring and restoring images from physical phenomena by Magnetic Resonance (MR) and radioactivity (Thierry 2015;Anne-Sophie 2003;Atam 2010). The reconstructed image may be processed to provide information about the human body (organ anatomy or their function and tumour). The most used imaging technique of the head is the Magnetic Resonance Imaging (MRI). First, MRI produces detailed pictures of the brain. These pictures are clearer and more detailed than the pictures produced by other imaging methods. Second, MRI can be used in the visualisation and characterisation of tumour in brain image. The segmentation of brain tumour is an intrinsic area of research concerning therapeutic intervention. Moreover, segmentation is considered as the most difficult step since tumours have a wide range of appearances and structures. In literature, there exist multiple methods of brain segmentation MRI (Anil et al. 2019;Cocquerez and Philipp (1995); Sobel (1961); Bo et al. (2010); Barbudhe (2014); Thierry (2015); Priyadarshi (2019); Yongyue et al. ()). In this respect, brain tumour segmentation problem can be approached from two angles: The first one is to consider the tumour as the only goal of segmentation and thus to use a method completely dedicated to this task. The second angle is to consider the tumour, and possibly oedema, as the goal of brain segmentation, and therefore segmentation is used in order to isolate all the brain structures. In general, there are many approaches of segmentation which handle the problem of image processing such as the approaches regions (Kapouleas 1990;Cocquerez and Philipp 1995;Anne-Sophie 2003) which are designed to locate homogeneous areas in the images, and the approaches contours (Anne-Sophie 2003; Ugale and Patil 2015) which characterise the presence of borders between regions like the derivative methods (Roberts 1965;Prewitt 1970;Sobel 1961;Cocquerez and Philipp 1995) and the deformable models (Cocquerez and Philipp 1995;Anne-Sophie 2003). In addition, the segmentation of brain MRI is created through supervised or unsupervised methods (Atizez et al. (2012(Atizez et al. ( , 2012).
In this paper, we are basically interested in the unsupervised segmentation of brain MRI. We set forward a hybrid brain MRI segmentation algorithm based on Tweedie hidden Markov random field (THMRF) processing (Yongyue et al. ;Edward and Zinoviy 2010) and the Expectation-Method of moments and Maximisation algorithm (EMM algorithm). The Expectation-Method of moments-Maximisation algorithm (EMM) is a combination between the EM algorithm (Dempster et al. 1977;Zitouni et al. 2018) and the method of moments. The proposed brain segmentation algorithm is called THMRF-EMM algorithm. It is an iterative technique for the segmentation of structure as brain tumour and for resolving estimation problems. It consists of three steps: the initialisation step which has been overcome by the K-Means algorithm. The Expectation step (E) which computes the expectation of the log-likelihood evaluated using the current estimate for the parameters. Eventually, the Method of moments-Maximisation step which computes the mean parameter μ maximising the expected loglikelihood found at the E step and estimates the dispersion parameter λ by using the Method of moments.
Markov random fields (MRFs) have been widely used for computer vision problems, such as image segmentation (Yongyue et al.), surface reconstruction (Vaidya and Boyer 2015) and depth inference (Ashutosh et al. 2008). The HMRF-EM framework was first elaborated for brain segmentation MR images (Yongyue et al.). This classical hidden Markov random field (HMRF) model admits the two following hidden Markov random fields X which is an unobservable Markov field, and Y which is an observable Markov random field. The significance of the MRF models (Thierry 2015;Yongyue et al. ;Quan 2012) refers to the spatial information given by the neighbouring sites.
It was departing from this fact that a new idea was born based on MRF models and particularly on THMRF model. The Tweedie distribution is an exponential dispersion model which represents a generalisation of natural exponential families (Jorgensen 1987(Jorgensen , 1997Letac 1970;Eunho et al. 2015). Besides, the class of Tweedie models exists for all values of Tweedie parameter p outside the interval ð0; 1Þ: It includes a high number of models such as Gaussian, gamma, compound poisson, inverse Gaussian models, etc. Furthermore, Tweedie models present limits for exponential dispersion models exhibiting certain classical stable convergence results. In fact, such data set can be described by Gaussian model but can be well described by a Tweedie distribution (Jorgensen 1987;Aymen et al. 2017).
Further applications of Tweedie distributions have been developed over the past few decades in many fields especially in biology. In fact, Tweedie models are related to Taylor's power law given by the British ecologist Lionel Roy Taylor (Taylor 1961) who studied the relation between the mean and the variance of population density for at least 22 biological species in 24 data sets (Xu 2001). In addition, he proved that the meanvariance relationship is established well by a linear equation logðvarianceÞ ¼ logðaÞ þ p logðmeanÞ; a > 0: (1) (2) is called Taylor's law or Taylor's power law. In physical applications, the relation between Taylor's law and Tweedie models is established in 2011s by (Kendal and Jorgensen 2011). Other applications of Tweedie models in actuarial (Smyth and Jorgensen 1994) and in ecology (Foster and Bravington 2013) are recorded. Tweedie models with Tweedie parameter p which makes you move from model to another one, offer a high approved results in the real-life applications. The novelty of the paper relies on generating the Tweedie models in brain image segmentation. Otherwise, the THMRF model is an undirected graphical model (Roberts 1965). It is a statistical model where the system being modelled is supposed to be Markov process with unknown parameters. Therefore, the challenge is to define the hidden parameters from the observable parameters (Yongyue et al. ;Quan 2012;Eunho et al. 2015). The segmentation algorithm has been validated on synthetic data and it has been tested on real MR brain images. On the other side, a comparison between the proposed algorithm and the classical HMRF-EM algorithm was mentioned. The results proved the performance and the best quality of the proposed segmentation method with THMRF model. In addition, we evaluated the finite sample performance by calculating such evaluation criteria as Misclassification Error (MCR), Mean Squared Error (MSE), Peak Signal to Noise Ratio (PSNR) and Image Fidelity (IF) (Ashutosh et al. 2008;El-Hachemi et al. 2013). This paper rests upon seven sections. In section 2, we recalled some basic concepts of exponential dispersion models. Section 3 is dedicated to specifying a new class of mixtures based on hidden Markov models namely the THMRF model. In section 4, we introduced a new approach by incorporating the THMRF model into EMM algorithm called the THMRF-EMM algorithm which is adopted to estimate the parameters of the proposed stochastic process. Section 5 is devoted to the proposed segmentation algorithm of brain MRI and to recall the evaluation criteria. In section 6, we applied the proposed algorithm with synthetic and real brain MR images. To further confirm the validity of the proposed approach, a comparative study with respect to HMRF-EM algorithm is performed. The different experiment results demonstrate the performance of the proposed unsupervised algorithm. The closing section crowns the whole work, wraps up the conclusion, furnishes the discussion and provides new perspectives for future works.

The exponential dispersion models
Exponential dispersion models (EDMs) generated by a probability measure ν constitute some statistical models in which the probability distributions are characterised by a special form (Jorgensen 1987(Jorgensen , 1997Letac 1970). It is a twoparameter family of distributions consisting of a linear exponential family with an additional dispersion parameter λ: It has been established as a rich model with wide potential applications, and it generalizes the exponential family and includes many standard models such as Gaussian, gamma, inverse Gaussian, compound poisson, stable . . .
A detailed study of their properties was reported by Jørgensen (Jorgensen 1987(Jorgensen , 1997. He has provided a new version of EDMs called the reproductive EDMs. It is an extension of the Gaussian distribution with parameter θ and the dispersion parameter λ. The reproductive EDM is defined by its probability density function Pðx; θ; λÞ ¼ cðx; λÞe λ½θxÀ kðθÞ� ; x 2 R where c and k are suitable functions and ðθ; λÞ 2 ΘðνÞ � ð0; þ1Þ: The function k is called the cumulant function of EDMs. It is strictly convex, infinitely differentiable and its differential is given by k 0 ðθÞ ¼ μ called the mean of Pð:; θ; λÞ. The mapping from the parameter θ to the mean μ is invertible, so we may write ψðμÞ ¼ θ where ψ is the inverse function of k 0 : In addition, μ ! V ðμÞ ¼ ½ψ 0 ðμÞ� À 1 ¼ k 00 ðψðμÞÞ is called the variance function of natural exponential family F. A new parametrisation is called the mean parametrisation of reproductive EDMs and the probability density function can be expressed as Pðx; μ; λÞ ¼ cðx; λÞe λ½ψðμÞxÀ kðψðμÞÞ� : In this study, we consider a class of EDMs which is infinitely divisible. Additionally, the densities ð1Þ and ð2Þ are presented with respect to Lebesgue measure or Counting measure. The expectation and the variance of a random variable X following Pð:; μ; λÞ are, respectively, indicated in terms of EDMs include several well known family models as the Tweedie with variance function given by We clearly observe that for ðp ¼ 0Þ Gaussian, ðp ¼ 1Þ poisson, ðp ¼ 2Þ gamma and ðp ¼ 3Þ inverse Gaussian. Otherwise, in the Table 1 given below, we present some examples of reproductive absolutely continuous EDMs.

Tweedie hidden Markov random field models: THMRF models
Let S be the set of N pixels. Let X ¼ ðX s Þ s2S and Y ¼ ðY s Þ s2S be two random fields taking their values in Ω ¼ fw 1 ; . . . ; w K g and R respectively. We denote by V the neighbouring system of S which is defined as where V s is the set of neighbouring sites s having the properties s‚V s and s 2 V t , t 2 V s : The most common neighbourhood systems are: the 4 connected neighbourhoods about a pixel consisting of the pixel and its north, south, east and west neighbourhoods and the 8 connected neighbourhoods about a pixel consisting of the pixel in a 3 � 3 window whose centre is the given pixel. In what follows, Y ¼ y is the observed image and X ¼ x is the label class image. The problem of the statistical segmentation is to recover the unobservable X ¼ ðX s Þ s2S from the observed Y ¼ ðY s Þ s2S : Hence, let us specify HMRFs. We say that the pairwise ðX; YÞ is a HMRF if X is a MRF and Y is a random field. The distribution of ðX; YÞ is characterised by the density probability distribution f ðx; yÞ and the conditional distribution f ðyjxÞ which verifies the two following hypotheses: and ðH 2 Þf ðy s jxÞ ¼ ðy s jx s Þforeachs 2 S: The random field X is a MRF with respect to the neighbouring system V; if, f ðxÞ > 0; "x 2 Ω and its conditional distribution satisfies the following property f ðx s jx SÀ s Þ ¼ f ðx s jV s Þ: According to the Hammersley-Clifford Theorem (Besag 1974;Yongyue et al.), the joint probability distribution of the MRF X can equivalently be characterised by a Gibbs distribution expðÀ UðxÞÞ is a normalising constant, C is the set of cliques c (a clique being either a singleton or a set of mutually neighbouring pixels) and U is the energy of MRF defined by which is a sum of clique potentials V c ðxÞ over all possible cliques c 2 C. Next, the Potts model is used to represent the clique potential. As a matter of fact, the energy U is denoted as where, s and t are neighbouring sites forming a pair-site clique, C H is the set of couples of pixels horizontally neighbours, C v is the set of couples of pixels vertically neighbours, and δðx s ; x t Þ is the Kronecker delta defined by The joint probability distribution of ðX; YÞ is provided by Now, we shall consider that f ðy s jx s Þ is a Tweedie dispersion distribution (Jorgensen 1987(Jorgensen , 1997Edward and Zinoviy 2010) with parameters ðμ xs ; λ xs Þ. Then, the likelihood function of the observed pixel intensity Y can be formulated as Therefore, we adopt the posterior distribution f ðxjyÞ which is factorised as f ðx; yÞαf ðyjxÞf ðxÞ and further

Parameters estimation: The THMRF-EMM algorithm
In this section, we are going to present the hybrid algorithm EMM (Expectation-Method of moments-Maximisation) for estimating THMRF parameters. So, firstly it is important to calculate the conditional expectation of the complete data log-likelihood Q (Dempster et al. 1977;Zitouni et al. 2018). According to (7) where Ω is the set of all possible configurations of labels, Since ψ is a bijective function, we make the transformation t xs ¼ ψðμ xs Þ: So, a new parametrisation by ðt xs ; λ xs Þ of the conditional expectation of the complete-data log-likelihood Q is obtained. Therefore, in order to determine the maximum likelihood estimator of μ x s ; we calculate the maximum likelihood of t xs . Hence, So, in the ðl þ 1Þ iteration, the estimator of the mean parameter μ x s is given by μ ðlþ1Þ where s ¼ 1; . . . ; N: For such models, the maximum likelihood estimator of dispersion parameter λ x s can not be found. Further explanation is maintained in (Jorgensen 1997). We also need to consider another method for estimating λ: We adopt the method of moments in order to approximate the unknown parameter λ xs . Hence, by applying the latter, we realise that the estimator of λ x s in the ðl þ 1Þ iteration takes the form λ ðlþ1Þ

Segmentation of brain MR image
Medical image processing is one of the most crucial subfields in not only statistics, but also engineering and radiology. It permeates several domains such as image segmentation, and image registration. Its significance refers to its ability to extract information about the human body, and more specifically about tissue characterisation,organs, anatomic structures, lesions and tumours. This information helps doctors delineate and track the progress of diseases. Basically, there are three major types of imaging commonly used for the diagnosis of brain tumours (Anne-Sophie 2003; Atam 2010). The Computed Tomography is a diagnostic imaging test used to create detailed images of internal organs, bones, soft tissue and blood vessels. The scintigraphic imaging is one of the methods used in order to study the transmission, the metabolism and duration life of a substance in the body as well as the functioning of the body. It is an exploration technique that exploits the radioactive properties of the matter. It rests on injecting to the patient a certain dose of radioactive product. The injected products differ according to the pathology and the organ concerned. The scintigraphic image is then a representation of the Radiation emitted by the injected radioactive elements. Finally, the Magnetic Resonance Imaging (MRI) is a tomographic imaging technique that produces images of internal physical and chemical characteristics of an object from externally measured Nuclear Magnetic Resonance (NMR) signals. It is the modality of choice for evaluating patients who have symptoms and signs suggesting a brain tumour (complex structure). World health organisation (WHO) classifies brain tumours into 4 grades. Grade I and Grade II are benign brain tumours. Grade III and Grade IV are malignant tumours. The major tumours types are meningiomas, glioblastoma and medulloblastoma.
In this paper, we are basically interested in brain MR image unsupervised segmentation. Our objective is to segment brain tumours by using Tweedie HMRF. As a matter of fact, in order to classify pixels from brain MR image we shall use the undirected graphical models like the Tweedie HMRF (Edward and Zinoviy 2010;Eunho et al. 2015) and a hybrid iterative algorithm called Expectation-Method of Moments-Maximisation (EMM).

The proposed segmentation algorithm
The proposed segmentation algorithm is an iterative method created in order to classify sites into two tissue classes. According to scientific experts, brain MR image can be divided into 2 tissue classes either tumour tissue C 1 or healthy tissue C 2 : The latter is a clustering algorithm that produces robust and good results. The proposed algorithm is characterised as follows: (1)-Choice the number of image regions ðK ¼ 2Þ: (2)-Initialisation of the parameters Θ ¼ ðμ; λÞ by using the K-Means method.
(3)-Alternating the following 2 steps until the following constraint is obtained -E step: compute the posterior probabilities f ðxjyÞ.
-MM step: calculate the estimator of λ x s by using the method of moments and maximise the conditional expectation with respect to the mean μ x s .
(4)-Calculate the posterior probability of a pixel at location y s for each class C 1 and C 2 defined as: Pðy s ; μ xs ; λ xs Þ ¼ cðy s ; λ xs Þexp λ xs ψðμ xs Þy s À kðψðμ xs ÞÞ À � À Uðx s Þ � � ; (24) "x s 2 f1; 2g: Further, Pðy s ; μ xs ; λ xs Þαf ðx s jy s Þ: The decision of whether a pixel is tumour tissue or healthy tissue is taken according to the maximum likelihood criteria. Then, the pixel at location y s is said to be tumour if otherwise, it is considered as normal tissue. Then, iteratively from one pixel to the other, a decision is taken according to equation (26).

Evaluation criteria
In order to better prove the efficiency of the proposed algorithm, and examine the quality of our segmentation results, some metrics are defined below ( where, Imax is the maximum intensity of pixel value of the image. In order to measure the segmentation accuracy, we also define the misclassification ratio (MCR)

MCR ¼ the number of misclassified pixels the number of all pixels from the original image
: (28) The image fidelity (IF) measures the similarity between the original image I and the segmented image b I which is defined by It is helpful to evaluate the ability of the proposed segmentation algorithm and to discriminate tissues and especially tumours. This section highlights the experimental results of the proposed segmentation algorithm with synthetic and real brain MR images and a comparison with classical HMRF-EM algorithm.

The proposed segmentation algorithm with synthetic image (K¼ 2 regions)
Simulation is used to generate synthetic images. Therefore, in order to validate the proposed algorithm, the latter was applied with synthetic images. Synthetic images consist of two strongly noisy regions ðK ¼ 2Þ where the noise is a Tweedie noise with mean 3 and variance 0.8 (see Figure 1). The original image is of size ð143 � 143Þ pixels with intensity means of 80 and 200 and proportions of 0.6045 and 0.3955 (see Figure 1). We tested the developed algorithm with synthetic images. Hence, for the selected Tweedie parameter p; the pixels in each region have been assigned to a value obtained according to one of these statistical models as the Gaussian model, the compound Poisson model (Abdelaziz and Afif 2013) with Tweedie parameter p 2�1; 2½; the inverse Gaussian model and the gamma model. The performance evaluation is performed by using the proposed segmentation approach on the synthetic data set and compared to the Gaussian model and traditional HMRF-EM algorithm. The segmented synthetic images using the proposed classification method are outlined in Figure 2 According to the segmented images ( Figure 2) with various models, the two classes were better identified and basically few pixels were labelled falsely by the proposed segmentation algorithm.
The proposed segmentation algorithm with compound Poisson model ðp ¼ 1:5Þ yielded the segmentation of synthetic image with better performance than the result of the proposed classification method with Gaussian model using classical  HMRF-EM algorithm. In other words, the proposed classification method with gamma and inverse Gaussian identifies the 2 classes with few pixels classified falsely in a way that is not better than compound poisson but it can be considered as better than the traditional HMRF-EM algorithm. It can be inferred from the segmentation results that the proposed segmentation method based on compound Poisson model with parameter p 2�1; 2½ values has achieved the best results. Thus, when we compare the proposed technique to classical HMRF-EM algorithm, we notice that the experimental results (Table 2) demonstrate that the proposed segmentation algorithm provides better results than classical HMRF-EM algorithm. For more quantitative results and in order to assess the ability of the segmentation method, and improve the classification as well as prove the high performance of the proposed method as opposed to the classical HMRF-EM algorithm, we calculated the MCR and invested other evaluation criteria mentioned in Section 5.2. In Table 2 below, we summarised the numerical results of the evaluation parameters value by varying the Tweedie parameter p.
Grounded on Table 2, it is clearly detected that the proposed segmentation algorithm works better with the synthetic image. The lowest MSE, MCR and the highest PSNR value illustrates the performance of the proposed clustering method, which implies less errors and higher quality of the segmented images (Kapouleas 1990). The value obtained of the MCR with compound Poisson model ðp 2�1; 2½Þ and Gaussian model is smaller than that obtained with gamma model and inverse Gaussian model. However, between inverse Gaussian and gamma model, there exists a tiny variation. The MCR and the MSE reveal that the performance of the strategy to classify tissues with compound poisson is much better than the classical HMRF-EM algorithm. Therefore, it's obvious that the lower PSNR and MSE value is furnished by classical HMRF-EM algorithm with Gaussian model. From the IF value between 0:6 and 1 the traditional HMRF-EM algorithm provides the least IF value 0:636. Generally, the correct and plausible segmentation result is procured by compound poisson model using the proposed technique.

The proposed segmentation algorithm with synthetic THMRF image (K¼ 3 regions)
To further scrutinize the performance of the proposed segmentation algorithm, the second experiment considers a 140 � 140 synthetic image created by THMRF with 3 components (Figure 3).
The intensities for the three classes are 30,140 and 255 respectively. The gain of performance of the method is then significant when the observations are noisy. The Tweedie noisy image with mean 2 and variance 0:76 is portrayed in Figure 4 It is an image with added Tweedie noise with mean 2 and dispersion parameter 4. To compare the influence and the performance of the proposed method in the   synthetic THMRF image, we applied this method in synthetic THMRF image and compared it to the classical HMRF-EM algorithm (see Figure 5). Table 3 helps us derive the average values of MSE, PSNR, MCR and IF. The HMRF-EM algorithm produced an average MSE value of 7:1921; which is quite higher than the proposed segmentation algorithm with compound Poisson and the other models, as it produces an average MSE value of 1:3347.
For different values of p, the MCR value is low in the case of THMRF-EMM algorithm based on compound Poisson model ðp 2�1; 2½Þ. Moreover, the THMRF-EMM algorithm with compound Poisson model produces the highest PSNR value 13:2371 in comparison with other candidate models. Besides, the IF value is close to 1. These criteria values reflect the relevance of the proposed method in segmenting synthetic THMRF image into 3 classes.
Note that, varying the intensity of noise (variance) on the synthetic image has influence on the PSNR values but always we have good results with our proposed method. Thus, we infer that the proposed technique out performs the HMRF-EM algorithm.

The proposed segmentation algorithm with real brain Magnetic Resonance Image (MRI)
In this section, we shall display the results of parameters estimation and unsupervised segmentation obtained from THMRF model and the proposed segmentation algorithm adapted to real brain MR images (see Figure 5).
Hence, we choose three real brain MR images: (a) is a coronal T2 weighted turbo spin echo (TSE) magnetic resonance imaging of the brain of size ð259 � 377Þ pixels with a fast growing, high grade tumour located in the cerebellum called Medulloblastoma, (b) is a sagittal T1 weighted magnetic resonance imaging of size ð703 � 571Þ pixels with aggressive tumour arising from a layer of tissue that covers the brain and spin called meningioma, (c) is a T1 weighted spin echo (se) magnetic resonance imaging of size ð417 � 463Þ pixels with a high malignant tumour called glioblastoma (see Figure 5). Moreover, the efficiency of our approach is elaborated by comparing the proposed method to a traditional HMRF-EM algorithm.  In the diagnosis of brain tumours, radiologists inject gadolinium because with this contrast the MR image can reflect the presence of brain tumour more clearly (see Figure 6).
The proposed segmentation algorithm has been applied on real brain MR images of the head with external noise. Only on selected images given by Figure 6 the method is applied. From this perspective, additional testing with real data is needed in order to check the performance of the proposed technique with THMRF models. In order to visualise the difference between the proposed segmentation algorithm and the traditional HMRF-EM algorithm, the proposed technique has been applied on various brain images volume. Relying on the opinion of a doctor, the original images involve a lot of irrelevant regions. For this reason, we attempt to define a square region of interest just including the brain tumour in the MR images (see Figure 8).
This selection can also make the segmentation of brain tumour more efficient. Therefore, for each real brain MR images, we classified the image pixels into two THMRF classes corresponding to the two tissue classes: class of brain tumour and class of normal tissue. For the proposed segmentation algorithm, the initial model parameters ðμ; λÞ are estimated by the K-Means algorithm based on the true classification. By varying the parameter p of Tweedie distributions (compound Poisson p 2�1; 2½ and Gaussian) with proposed algorithm and the Gaussian model with traditional HMRF-EM algorithm, the model parameters ðμ; λÞ are estimated and the convergence of the proposed algorithm is achieved after 20 iterations. According to Figure 9, we can see clearly that the proposed segmentation algorithm extracts and defines the area of brain tumour. Hence, when we use the estimators of parameters ðμ; λÞ instead of the real brain MR image the segmentation is accepted.
According to Figure 9, the effectiveness of the proposed algorithm THMRF-EMM is evaluated. Although the noise is very strong, the satisfactory segmentation results can also be obtained by using the proposed method. We can notice that the proposed algorithm with compound Poisson model gets a better segmentation result than that of classical HMRF-EM algorithm with Gaussian model. For each image, using proposed algorithm with compound Poisson model, we observe a location of tumour (black part) even though they are misclassified pixels. The proposed algorithm defines and selects the complex structure (brain tumour) on such real brain MR images. For each images ða; b; c; d; e; f ; g; h; i; j; k; lÞ in Figure 9 the black region represent brain tumour and the grey region with some badly classified pixels is the non tumour region. Consequently, from the segmented result denoted (l) in Figure 9, the brain tumour is not well located with HMRF-EM algorithm. We can not determine the abnormalities. In addition, using proposed technique with compound Poisson marks the presence of a tumour with a little difference in most images with respect to the segmented result using HMRF-EM algorithm with Gaussian model. Yet, from the segmented images, the accuracy of the developed algorithm is superior to the classical HMRF-EM algorithm in most images. In the next part of this section, we shall investigate the power of the proposed algorithm in terms of approximating complex structure (the abnormalities part) by creating the histogram of segmented images. Figure 10 outlines the different results obtained from traditional HMRF-EM algorithm and the proposed algorithm with brain MR image (c) presented in Figure 7.
The histogram of segmented image using THMRF-EMM algorithm corresponds to accurate mixture of two component densities. The intensities of tumour is 0. Therefore, in the following histogram, using the proposed method THMRF-EMM, some of the samples clustered on the left side of the histogram exhibit approximately 120 to 180 intensities. They stand for the normal tissue and the majority of the rest sample values are clustered on the right side of the histogram around 0. Roughly speaking, there exist 2 pics which implies two classes: class of tumour and class of normal tissue. On the other side, it can be inferred from the histogram of segmented image using traditional HMRF-EM algorithm that pixels are clustered into 3 classes. Hence, there exist some of the sample values which are clustered on the left side of the histogram having approximately 250 intensities and others are clustered nearly into two groups. Some have around 0 intensities which represent the tumour, and the others in the centre have approximately 110 to 185 intensities. Departing from the following histogram, we can assert that there is a significant difference between both methods in terms of effectiveness to separate the pixels into two regions. This is confirmed further by the quantitative segmentation results in terms of the evaluation criteria between segmented brain MR images depicted in Figure 9 and original brain MR images provided in Figure 8, which are portrayed in Table 4 below.  Table 4, it is clearly noticed that statistical metrics values of proposed segmentation approach are much closer to the ideal segmented image. The smallest MSE and MCR values as well as the highest PSNR and IF values specify the high quality of segmented results with the proposed method THMRF-EMM compared to the traditional HMRF-EM algorithm. The results mentioned in Table 4 exhibit the performance of compound Poisson model with the proposed method to segment brain MR image in terms of MCR. Indeed, the lowest MCR values for the 3 brain MR images. Therefore departing from the MCR, it is observed that the proposed algorithm outperforms HMRF-EM algorithm. Even so, the unsupervised algorithm is very sensitive to initial parameters values.

Resting upon
On the other side, we infer from the higher PSNR and the lower MSE values that the compound Poisson model with proposed classification method provides the best segmented results with respect to the Gaussian with traditional HMRF-EM algorithm and the other candidate models with the proposed approach. Basically, based upon the qualitative results, the histogram of segmented images and Figure 9, the proposed algorithm yields satisfactory results in brain MR images. It is appreciated by the experts in most cases.
In the following, in order to more prove the efficiency of the proposed method with compound Poisson a comparative study with respect to a recent method called unsupervised image segmentation based on Beta-Liouville mixture models and Markov Random Field (Muhammad Azam et al. 2021) is carried out. Hence, we have calculated the MCR criteria which summarised in the Table 5 below.
Tables 5 demonstrate the performance of the proposed method. The more small and interesting MCR value is this recorded using compound Poisson model. So, it is clearly the latter outperforms recent method based on Beta-Liouville mixture models.

Discussions and conclusion
At this stage of analysis, we would assert that we elaborated an iterative hybrid algorithm for segmenting real brain MR image. Besides, iteratively statistical classification of pixels was provided based on THMRF models and the EMM algorithm. In addition, each tissue derived from the brain MR image had probability  density distribution from THMRF models. The first 4 candidate models from THMRF models were applied on both synthetic images. We first experimented the proposed method on synthetic data. The segmentation was performed on synthetic images with compound Poisson HMRF model and these were evaluated by calculating different evaluation criteria as MSE, MCR, PSNR and IF. Then, the proposed segmentation algorithm was exercised on a variety of real brain MR images including diverse type of tumours and it was compared with traditional algorithm HMRF-EM algorithm. The proposed method solved the problem (select tumour) by obtaining better segmentation results than traditional HMRF-EM algorithm from PSNR, IF and MSE values. Yet, there is a little difference between proposed method and the Gaussian with classical HMRF-EM algorithm from MCR values. Nevertheless, we can state that the proposed algorithm outperformed the classical algorithm. Although, the parameters estimation is a difficult task and the algorithms are very sensitive to initial parameters' values. The model parameters ðμ; λÞ for candidate models are estimated by the proposed THMRF-EMM algorithm and the initialisation has occurred by using K-Means algorithm. Notice that the classical HMRF model has already come with interesting applications which have been mentioned in the introduction. Thus, in this study we have applied the classical HMRF-EM algorithm with Gaussian Figure 9. Segmented brain MR images: (a),(e) and (i) Segmented brain MR images using compound Poisson with proposed algorithm, (b),(f) and (j) Segmented brain MR image using gamma with proposed algorithm THMRF-EMM, (c),(g) and (k) Segmented brain MR images using inverse Gaussian with proposed algorithm, (d), (h) and (l) Segmented brain MR image using Gaussian with HMRF-EM algorithm. model in synthetic image and real brain MR image. The comparisons between the classical HMRF-EM algorithm and the proposed segmentation method shows that the proposed segmentation algorithm can give more accurate segmentation results in both segmentation and evaluation metrics with compound poisson model p 2�1; 2½ with synthetic images. In addition, the proposed algorithm has provided accepted segmented results in real brain MR images. This implies that, the use of undirected HMRF Model as Tweedie HMRF models and the unsupervised estimation method in real brain MR image is a new strategy which gives accepted but not ideal classification result into two tissue brain regions: tumour tissue and healthy tissue in most cases because of such external factors as time. To this extent, promising as it seems, our work is a step that may be refined, taken further and built upon as it paves the way and lays the ground for future research to overcome certain deficiencies, overcome some short comings and address. On the other side the initialisation of parameter vector which is still the major difficult topic in order to get the corresponding segmentation.