New Local Region Based Model for the Segmentation of Medical Images

Segmentation of images having inhomogeneous intensities is always challenging. In this paper, we propose a model based on new local data statistics using local means and variances for detection of region of interest in medical images suffered from intensity inhomogeneity. This is done by introducing a new probability density function based on coefficient of variation, which is a best measure for inhomogeneous data. The new energy functional in the proposed model is then expressed in terms of level set function and is minimized for optimal energy. Minimization of the energy will lead to a partial differential equation, which is solved by using well known explicit method. Results of the proposed model are compared with other state of the art models and found that the proposed model outperform other existing models. Comparison is given in both qualitative and quantitative way. Furthermore, the proposed model is tested on different type of medical images like MRI, CT, Mammogram and skin lesion etc.


I. INTRODUCTION
Image segmentation is the process of semi-automatic or automatic detection of edges/region of interest based on intensity homogeneity, within an image for further analysis. Segmentation of images plays an important role in medical sciences. To deal with the medical images having tumors, an automatic detection of the tumor is one of the major and challenging task. A tumor, an abnormal growth of mass/tissues, needs to be detected critically and efficiently in order to increase the survival rate. The main difficulty with segmenting medical images having tumor is due to high variability in an image. Furthermore, different analysis itself indicates major modes of variation in intensities. Inhomogeneous intensity usually occurs in medical images such as Magnetic Resonance Imaging (MRI), Computed Tomography (CT), X-ray, Positron Emission Tomography (PET), Optical Coherence Tomography (OCT), Single Photon Emission Computed Tomography (SPECT), as a result of artifacts, the effects of illumination or technical limitations introduced by display objects.

Inhomogeneous intensity is a smooth change in intensity
The associate editor coordinating the review of this manuscript and approving it for publication was Essam A. Rashed . within the original uniform region. The presence of inhomogeneous intensity in medical images can lead to misjudgment by doctors and researchers. The segmentation results can then be used to get additional diagnostic data. In this paper, our main focus is to develop a new active contour model for segmentation of medical images having intensity inhomogeneity.
Variety of mathematical models have been proposed for segmentation of images, which are mainly classified into two groups (i) Edge based active contour models (EBMs) (ii) and Region based active contour models (RBMs). The first group of models are using edge information (gradient) for detection of edges in the images [2]. These models are very sensitive to initialization and outliers (noise) in the image. While region based models use region statistics (like least square data fitting) for detection of region of interest in images [3].
One of the most known EBM is the geodesic active contour model [4], which has been efficiently applied for images with high dissimilarity in gradient at the object's edges. But the model may not produce satisfactory results in images having noise or blur [2]. Also this model is sensitive to initialization and may fall into local minima [17]. The above discussed shortcomings in the efficient use of EBM have limited their applicability in the field of image segmentation and lead VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ to the use of RBMs. Owing to the less sensitivity towards noise in the images [5] and no data required about image gradient, such models successfully allow the segmentation of objects with weak or no edges [15]. The significant features of RBMs can also automatically detect the interior contours and shows less dependency on the location of initial contour [15]. The first model of RBMs was introduced by Mumford and Shah (MS) [12], which illustrates that piecewise smooth intensity is appropriate for images having intensity inhomogeneity. However, MS model has some limitations due to which it is hard to minimize the energy functional. Since firstly, it is non-convex and may have several local minima. Secondly, the energy functional of MS model contains two unknowns both are of different nature. That is image (function of N dimensions) and curve ((N-1) dimensions) [16]. The Chan-Vese (CV) model is a special case of MS model, which divides given image into two parts i.e background and foreground [5]. CV model uses the global information of homogeneous regions outside and inside the evolving curve to identify objects whose boundaries may not be defined by gradient [6]. As a result, CV model has obtained outstanding segmentation result in images with weak or blur boundaries. Due to taking constant mean value in each region in a global framework, this model may not work efficiently in images with inhomogeneous intensity and images having clutter or complex background [6]. Li et al. [8] suggested local binary fitting (LBF) model for segmentation of images with inhomogeneous intensity.
LBF model utilizes two fitting functions that locally approximate the image intensities on the two sides of the contours. This model may produce better segmentation results of objects boundary completely due to local intensity information [1]. Though, it is delicate to the initial value on contours and is easy to capture into a local minimum [8], [10] and also not able to extract out the boundaries of objects with low contrast [11] which limit its realistic applications.
Wang et al. [1] proposed an active contour model driven by Local Gaussian Distribution Fitting (LGDF) energy with a level set function and local means and variances as variables. The local intensity means and variances are strictly derived from a variational principle, instead of being defined empirically. So it has shown certain ability of handling images having intensity inhomogeneity, noise and also identifying regions with different intensity variations. This model may not work very well in images having intensity inhomogeneity with complex/cluter background, which can be seen in experimental section IV. This model may not give good segmentation results in images of large sizes or may be very slow in convergence.
Badshah et al. [13] presented a selective segmentation model (CVES) based on coefficient of variation. The fitting term is based on coefficient of variation which is used to segment the image where regions are overlapping and have inhomogeneous intensities. CVES model works better for images with overlapping regions and intensity inhomogeneity [13]. But images having severe intensity inhomogeneity and is not able to segment the edges of the desired object.
Zhang et al. [14] proposed a region (local and global) based signed pressure force (SPF) model for image segmentation with a new level set method called selective binary and Gaussian filtering regularized level set (SBGFRLS) method. SBGFRLS model used the signed pressure force function as a region detector function. The main features of this model is that firstly the contour efficiently stop on fuzzy edges. Secondly, this model does not depend on initial contour and can detect the objects with exterior and interior boundaries. Thirdly, this model has both properties of selective local and global segmentation. Due to this property this model not only detect the required object but also detect the other objects as well.
Fang et al. [18] proposed a new model i.e. HRSPF model, which is driven by weighted hybrid region based SPF to segment images having inhomogeneous intensities and noise. In which they defined two weighted functions i.e. global region based SPF (GRSPF) and local region based SPF (LRSPF) functions which are based on normalized global intensity and normalized absolute local intensity differences respectively. By combining these functions and also introduced a force propagation function, this model is robust and is able to segment real images having inhomogeneous intensities and noise.
Liu et al. [19] proposed GLSPF model i.e. global and local signed energy based pressure force model. First a global signed energy based pressure force (GSPF) is introduced, which can enhance the robustness to initial contour. Secondly also introduced a local signed energy based pressure force (LSPF), which can handle images having inhomogeneous intensities and noise. The global and local image information is used for the global and local force propagation functions, respectively. Then automatically used the global and local variances to balance the weights of the GSPF and the LSPF, thus solve the problem of setting of parameters. Also applying a regularization term which is used to avoid the process of re-initialization and a penalty term which is used to smooth the level set function. This model is able to segment images having intensity inhomogeneity and noise.
Segmentation of images having inhomogeneous intensities is always challenging. In this paper, we propose a model based on new local data statistics using local means and variances for detection of region of interest in medical images suffered from intensity inhomogeneity. This is done by introducing a new probability density function based on coefficient of variation, which is a best measure for inhomogeneous data. The new energy functional in the proposed model is then expressed in terms of level set function and is minimized for optimal energy. Minimization of the energy will lead to a partial differential equation, which is solved by using well known explicit method.
In this paper, our main focus is to develop an active contour model for segmentation of medical images, which have intensity inhomogeneity with cluttered /complex background.
We have also improved the computational cost for segmentation of medical images having large sizes. The proposed model is based on new local data statistics by introducing new probability density function in terms of coefficient of variation. It is a well known fact that coefficient of variation fits data in a better way in images having intensity inhomogeneity as compare to variances. Furthermore, to extend the domain to the whole domain, the model is expressed in terms of the level set function. For optimal value of local intensity means, variances and level set function the energy functional of proposed model is minimized through Euler Lagrange equation. The results of the proposed model are compared quantitatively and qualitatively with other existing state of the art models i.e SBGFRLS, LGDF and CVES models. The proposed model has produced more desirable results in terms of Computational (CPU) time, Similarity coefficients and Evaluation metrics. The proposed model is tested on two types of medical data sets, where it produced better segmentation results. Firstly, the proposed model works very well with images having intensity inhomogeneity with complex background i.e. minimum, maximum or averaged intensity background. Secondly, the proposed model improves the computational cost and converges fast and also works well with images of large sizes.
The rest of this paper is designed in the following way: Discussion on existing models is given in Section II. In Section III the proposed model is described in details with implementation of an algorithm. Final experimental results and comparisons of medical images are given in Section IV along with quantitative results. Conclusion is presented in Section VI.

II. EXISTING LITERATURE
Here we give a brief discussion of existing state of the art models for segmentation of images, both for global and selective segmentation. Some of these models are taken as motivation for proposing a novel model and some are discussed for comparison with proposed model.

A. CHAN VESE (CV) MODEL
Chan and Vese (CV) [5] is a region based model which is using piecewise constant Mumford and Shah model [12] by restricted it to two regions. For given gray image I 0 : → R, the following minimization problem is proposed min where is the level set and χ 1 and χ 2 are characteristic functions which are used as region descriptor. The optimal values of b i can be obtained by The following Euler Lagrange equation is solved to get optimal as: for i = 1, 2, with the homogeneous Neumann boundary condition and χ 1 is the derivative of characteristic functions. The CV model works well in images with noise. This model may not produce very good results in images having intensity inhomogeneity and is also computationally expensive [6].

B. LOCAL BINARY FITTING (LBF) ENERGY MODEL
CV model [5] uses constant average intensities in different regions so works well in images with homogeneity and may not produce good results in inhomogeneous images. For segmentation of images having inhomogeneous intensity, Li et al. proposed a model based on local binary fitting energy (LBF) [8]. The energy functional in terms of level set is given by: where G σ is a Gaussian kernel with standard deviation σ which control the size of local region, g i for i = 1, 2 are local intensity fitting functions. The optimal values of g i can be obtained by: The following partial differential equation is solved for optimal as: The localization property of LBF energy allows it to give better segmentation results in images having inhomogeneous intensities. And because of localization: the model may easily stuck at local minima and is very sensitive to the initialization, which may lead towards wrong segmentation for different initialization of [7]. Also this model does not give good segmentation result in noisy images and having severe intensity inhomogeneity. VOLUME 8, 2020

C. THE LOCAL GAUSSIAN DISTRIBUTION FITTING (LGDF) ENERGY MODEL
LBF model [8] uses information of the local intensity to segment images with inhomogeneous intensities, but may easily get stuck within local minima. So Wang et al. [11] proposed an active contour model driven by local Gaussian distribution fitting (LGDF) energy with local means and variances as variables. The energy functional in terms of level set formulation is given by: for i = 1, 2, where first term is the data fitting term, G σ (x −y) is a Gaussian kernel which is positive function with fixed parameter σ and logarithm function is used to convert the maximization problem into minimization. Second and third terms are the length and re-initialization terms which can be defined as L( ) = |∇χ( (x))|dx and P( ) = 1 2 (|∇ (x)| − 1) 2 dx respectively. Also f i for i = 1, 2 are the probability density functions and are given by: Minimizing the energy functional (5) w.r.t σ i and u i for i = 1, 2, we get: By keeping fixed σ i (x) 2 and u i (x), we minimize (5) w.r.t and can be obtained ∂ ∂t LGDF model works efficiently in the segmentation of images with different regions having inhomogeneous intensities. Firstly, this model may not work very effectively with images having severe inhomogeneous intensities, complex background (i.e. with maximum or minimum intensity background) and severe noise. Secondly, LGDF model may stuck within local minima [9]. Thirdly the computational cost of LGDF model increases with an increase in the size of images.

D. THE COEFFICIENT OF VARIATION EQUIPPED SELECTIVE (CVES) MODEL
Badshah et al. [13] proposed a selective segmentation model based on geometrical constraints, while the fitting term is based on coefficient of variation to segment images having overlapping regions and images with inhomogeneous intensity. The following minimization problem min is as follows: where i = 1, 2 and b i can be found by: By keeping b i fixed, the minimization of energy functional (7) w.r.t is given as: By adding balloon term αd(x, y)g(|∇I 0 |)|∇ | in above (8) is used to speed up the convergence of the evolution.
CVES model has a better performance in images having overlapping regions and images having inhomogeneous intensity [13]. This model may produce leakages through edges in images with severe inhomogeneous intensity.

E. ACTIVE CONTOURS WITH SELECTIVE LOCAL OR GLOBAL SEGMENTATION: A NEW FORMULATION AND LEVEL SET (SBGFRLS) METHOD
Zhang et al. [14] proposed a region (local and global) based model for image segmentation with a new level set method called selective binary and Gaussian filtering regularized level set (SBGFRLS) method. Firstly selective step was used to penalize the level set function to binary and then for the regularization Gaussian filter was used. The main purpose of Gaussian filter is that the evolution become more stable and also makes level set function smooth. Furthermore SBGFRLS model combines both the properties of Geodesic active contour (local segmentation) and Chan-Vese (global segmentation) models. The final evolution equation for the SBGFRLS model is defined as: where and α controls the propagation of the contour. This SBGFRLS model is more efficient and accurate than geodesic active contour and CV models [14]. This model may not yield very good results in images having inhomogeneous intensity and complex background.
where b 1 and b 2 are the average intensities as defined in [5], w l and w g are two weighted variables used to balance the effects of the functions of LRSPF and GRSPF, spf GR and spf LR represents the GRSPF and LRSPF functions respectively and max spf = max(|spf GR (I 0 (x)), spf LR (I 0 (x))|) is the maximum absolute value of the GRSPF and the LRSPF functions.
This HRSPF model has the capability to control the pressure sign and implicitly control the evolution of the curve. And also contour expands and shrinks if it is inside and outside the object of interest respectively [18]. This model is not able to segment the region of interest having severe inhomogeneous intensities and severe noise.

G. A NOVEL ACTIVE CONTOUR MODEL GUIDED BY GLOBAL AND LOCAL SIGNED ENERGY BASED PRESSURE FORCE (GLSPF) MODEL
Liu et al. [19] proposed a novel GLSEPF model based on both global and local signed energy based pressure force model. Both global and local signed energy based pressure forces (GSPF and LSPF) are introduced, which can enhance the robustness to initial contour and can handle images having inhomogeneous intensities and noise. To avoid the problem of parameters setting, automatically used the global and local variances, which are used to balance the weights of the GSPF and the LSPF. Then combine both regularization and penalty terms. The final evolution equation for the GLSEPF model is defined as: where w l and w g are two weighted variables, b 1 and b 2 are the average intensities, ∇E g (I 0 (x)) is the difference of the global energy and E l 1 (x), E l 2 (x) are two local energy function. This model is able to segment images having intensity inhomogeneity as well as noise [19]. But images having severe intensity inhomogeneity and severe noise, this model is not able to segment the region of interest.

III. THE PROPOSED MODEL
The LGDF model [11] can use the fitting energy of the local Gaussian distribution to separate regions with same average intensity but different variances. But LGDF model [11] is not able to acquire precise results when the images have severe intensity inhomogeneity and severe noise. In medical images it is hard to get precise segmentation results of images with severe intensity inhomogeneity and time consuming because these images with inhomogeneous intensities are always complex and large in size. In order to reduce the impact of severe inhomogeneous intensity, we propose a novel active contour model in which the local image intensities are based on means and variances which can rapidly segment the images having severe inhomogeneous intensities and also obtain accurate results. The coefficient of variation is the ratio of standard deviation to the mean and is widely used to get unit free measure of dispersion in the data. It can be very useful for comparing the variability between groups of observations. The coefficient of variation can be used both as a fidelity term and as a good region detector. We introduce a new local statistic (squared coefficient of variation) in local framework i.e. a new local intensity data fitting with the following probability density function: where σ i and u i for i = 1, 2 are spatially varying local variances and means respectively. To convert the model from maximization to minimization we take the logarithm of the probability density function, so we have the following energy functional for minimization as: where G σ (x − y) is a kernel, which takes a non-zero value in the neighborhood of x and zero otherwise. Thus we propose the following minimization problem in terms of the level set function and using length term and re-initialization of level set: ×exp −(u 2 (x) − I 0 (y)) 2 2σ 2 (x) 2 u 2 (x) 2 χ 2 ( (y))dy dx +ν |∇χ 1 ( (x))|dx + µ where u i and σ i for i = 1, 2 may be computed by using following: and Now keeping u i and σ i fixed and minimizing the final energy functional E(u 1 , u 2 , σ 2 1 , σ 2 2 , ) w.r.t , we get the following Euler-Lagrange equation for : −χ 1 G σ (y − x) log(σ 1 (y)) + log(u 1 (y)) + (u 1 (y) − I 0 (x)) 2 2σ 2 1 (y)u 2 1 (y) dy +χ 1 G σ (y − x) log(σ 2 (y)) + log(u 2 (y)) + (u 2 (y) − I 0 (x)) 2 2σ 2 2 (y)u 2 2 (y) dy with initial and boundary conditions. Eq. (18) may be considered as the steady state solution of the following partial differential equation with homogeneous Neumann boundary conditions. The partial differential equation (19) is solved by simple and conditionally stable explicit method. The main steps of the proposed method are given in the following algorithm: A. ALGORITHM

Algorithm 1 An Algorithm for the Evolution of Level Set
Through Proposed Model 1) Input: Parameters α, µ, ν etc.
3) Use updated values of , find σ i , u i and solve (19) to update . 4) If required results are obtained stop the iterations otherwise go to step 1.

5) Output: Segmented result.
Various steps of the proposed model are illustrated in block diagram as shown in Fig. 1. In the next section we give experimental results of the proposed model on different type of medical images and compared the qualitative and quantitative results of the proposed model with a different state of the art models.

IV. EXPERIMENTAL RESULTS
In this section, experimental results of the proposed model on different types of medical images like mammogram images, MRI, Cardiac Magnetic Resonance (CMR) images is mention. All experimental tests are carried out by using Matlab 2010b. The proposed model on medical images taken from the website and two different datasets i.e. Montreal Neurological Institute's Brain Images of Tumors for Evaluation (MNI BITE) [20] and Radau et al. [21]. All these images have different level of intensity inhomogeneity with complex backgrounds. The experimental results of the proposed model are compared with three different existing state of the art models SBGFRLS, LGDF, CVES, HRSPF and GLSPF qualitatively and quantitatively.
The proposed model is using the following parameters σ , α and ν, which effects the final segmentation results by changing their values. In all experiments we used the parameters t = 0.1, λ 1 = λ 2 = 1, µ = 1, ν = 0.001 × 256 × 256 and the rest of parameters are adjusted according to the image. The Gaussian kernel G σ is a scale parameter with standard deviation σ and α is also a scaling constant which are used to control the region scalability from small neighborhoods to the whole domain of the image and size of local region respectively. A small value of σ may cause undesirable results, detect unwanted regions and convergence is slow, whereas for large value of σ may cause high computational complexity. So the value of σ should be appropriately selected according to the image.

A. QUALITATIVE RESULTS
In this section we compared our proposed model qualitatively with other existing state of the art models. In all experiments from Fig. 2 to 8, Fig. (a) shows the original image with initial contour, (b), (c), (d), (e), (f) and (g) shows segmented results of SBGFRLS, LGDF, CVES, HRSPF, GLSPF and proposed models, respectively. We have used optimal (best) values of the parameters involved in each model. Fig. 2 shows the mammogram image in which region of interest is a tumor with complex background. In Fig. 2b, the tumor is not detected correctly after 2000 iterations, in Fig. 2c the tumor is detected successfully, but some extra/unwanted region is also detected after 50 iterations. In Fig. 2d the segmented result is not satisfactory and have multiple local minima after 1000 iterations. Fig. 2e and 2f shows the results of WRSPF and GRSPF models respectively, which lead to unsuccessful results, because these models are unable to segment the region of interest in the given image. In Fig. 2g, experimental result of the proposed model is given, where the tumor is detected efficiently in only 15 iterations. Result showed the better performance of the proposed model in Fig. 2 over the existing models as SBGFRLS, LGDF, CVES, HRSPF and GLSPF models take a large number of iterations to converge. Figure 3 demonstrates the segmentation results of real brain MRI image taken from free available data set the Montreal Neurological Institute's Brain Images of Tumors for Evaluation (MNI BITE) database [20]. In Fig. 3b and Fig. 3c the tumor is not segmented accurately and also some unwanted regions is segmented as well by SBGFRLS model and LGDF models respectively. In Fig. 3d, it can be seen that CVES model is able to detect the edges of tumor but it takes large number of iterations and CPU time. Fig. 3e and 3f shows the segmentation results of HRSPF and GLSPF respectively, where these models are not able to complete the task, give inaccurate results and also detect other undesired regions. Whereas our proposed model efficiently and accurately extracts the edges of tumor and it takes less number of iterations to converge (Fig. 3g) as compared to other existing models. Figure 4 and Fig. 5 shows the experimental results of the segmentation of brain tumor having intensity inhomogeneity by SBGFRLS, LGDF, CVES and the proposed model. The original image with initial contour in Fig. 4a and Fig. 5a are brain MRI images of two different patients taken from the local hospital of Pakistan. In both images the intensities in background is inhomogeneous and the intensities of tumors are very close to background. The final segmentation results of SBGFRLS model are shown in Fig. 4b and Fig. 5b, where it can be shown that in both MRI images the results were unsatisfactory after 600 and 2000 number of iterations respectively. Figure 4c and Fig. 5c shows the final segmentation result of LGDF model after 960 and 900 iterations respectively. Both images show that the segmented results are not satisfactory because tumors as well as some extra region other than tumors are also detected. Figure 4d and Fig. 5d shows the final segmentation result of CVES model which is obtained after 1000 and 500 iterations respectively, the results are not satisfactory and also detect unwanted regions. Fig. 4e, 5e and 4f, 5f shows the segmentation results of HRSPF and GLSPF models respectively, where both models fail to detect the region of interest and detect the unwanted regions/boundaries. The final segmentation results of the proposed model are shown in Fig. 4g and Fig. 5g,  where both tumors of interest has been segmented accurately after 30 and 22 iterations respectively. From these results, it can be seen that the proposed model works very well with brain MRI images having intensity homogeneity and also detect the region of interest in less number of iterations and CPU time.   accurately. The proposed model has given the required results in less number of iterations. Figure 7 is a CMR image taken from the free available data set of Radau et al. [21] having intensity inhomogeneity and has minimum intensity in background. The better performance of the proposed model over other models can be seen in Fig. 7g.
In Fig. 8, experimental results of all the six models are given on a MRI images where the region of interest is the tumor. All models have performed better segmentation results, but the proposed model has produced the required results in less number of iterations and computational (CPU) time in seconds as shown in Fig. 8g. The efficiency of the images is extremely affected by the parameters σ and α.  We have also tested the proposed model on different real dataset of MRI slices of different patients having tumor of different size taken from web (MNI BITE) as well as from hospital in Peshawar, Pakistan. In Fig. 9 the images in first and second columns are in the axial plane of different patient taken from MNI BITE dataset [20] whereas the images in third and forth columns are in sagittal and axial plane respectively and are of same patient taken from local hospital in Peshawar, Pakistan. The final segmented results of MRI images are given in Fig. 9 showed that the proposed model has successfully segmented tumor in all MRI images. The   (Table 1). is revealed from the comparison that proposed model is fast in convergence as compared to SBGFRLS, LGDF, CVES, HRSPF and GLSPF models. All the experiments are done with best values of parameters, which are considered by hit and trial method. Furthermore, the proposed model is compared with SBGFRLS, LGDF, CVES, HRSPF and GLSPF models with images having intensity inhomogeneity and complex background of large sizes. In Table 2, number of iterations and CPU time in seconds is given for images of different sizes. It can be observed from the table that the proposed model has segmented images of large sizes in less iterations and CPU time as compare to the other models discussed.
Next, quantitatively better performance of the proposed model in comparison with SBGFRLS, LGDF, CVES, HRSPF and GLSPF models is discuss. The segmentation efficiency of each model is checked by Jaccard similarity index (JSI), which is given by: JSI (A 1 , A 2 ) = (|A 1 ∩A 2 |) (|A 1 ∪A 2 |) , where A 1 is the final segmented image and A 2 is the ground truth which is obtained by manual segmentation. A model whose JSI value is close to 1 may be considered as a better segmentation model. Table 3 shows the JSI values of all four models for images given in Fig. 2 to 8, where the proposed model has given better JSI value as compared to other three models. For finding JSI value in all experiments the optimal value of parameters has used. Thus it is concluded that proposed model has produced better segmentation results in images having inhomogeneous intensity with complex background, which usually occur in medical images.
Remark 1: The proposed model is tested on a set of images from two different types of databases, where the JSI has a confidence interval 0.9 ± 0.07. Table 4 shows the quantitative analysis of the experimental results in terms of evaluation metrics. The evaluation metrics are used to evaluate the performance of SBGFRLS, LGDF, CVES, HRSPF, GLSPF and proposed models by comparing tumors with the ground truths created by human experts, including Sensitivity (SE), Specificity (SP), Accuracy (AC), Dice Similarity Coefficient (DSC). These metrics are defined as: where W TP , W TN , W FP and W FN indicate the number of true positives, true negatives, false positives and false negatives respectively.

V. APPLICATION OF PROPOSED MODEL ON DIFFERENT MEDICAL IMAGES A. TEST ON MRI AND MAMMOGRAM IMAGES
The proposed model is tested on MRI and Mammogram images taken from real data set collected and Website with severe inhomogeneous intensity. Our proposed model efficiently and successfully extract the region of interest (tumor) and give clear segmented results as shown in Fig. (10a-10d). Figure 11 shows the final segmentation results on set of brain MRI images taken from the data set of the local hospital in Pakistan. The first column shows the original image with initial contour and the second column shows the final segmented results. These images are corrupted with severe intensity inhomogeneity and have low contrast and weak edges. Due to these factors the segmentation of these medical images with tumor are very hard to segment. But clearly it can be seen that our proposed model is able to accurately detect and segment the edges of region of interest i.e. tumors and also outperformed very well in all images as shown in Fig. 11.
In Fig. 12 the brain MRI image of Fig. 4 can be used for five different positions of initial contours that can be set inside and across the edges of a region of interest. First row in Fig. 12, shows the original image with different initial contours. Row 12b shows the segmentation results of SBGFRLS model, row 12c shows the segmentation results of LGDF model, row 12d shows the segmentation results of CVES model, row 12e shows the segmentation results of HRSPF model, row 12f shows the segmentation results of GLSPF model and last row 12g shows the final segmentation results of proposed model. From the segmentation results it demonstrate that our proposed model efficiently segment the region of interest having intensity inhomogeneity for different positions of initial contours, whereas the existing models are even not able to segment the region of interest as well as sensitive to the initial contour.

B. TEST ON ULTRASOUND AND CMR IMAGES
Further, images of ultrasound acquired from the Website and CMR from dataset Radau et al. [21] are also considered to assess the performance of the proposed model. The proposed model is able to segment the region of interest properly and showing its capability of dealing with inhomogeneous medical images as shown in Fig. 13. Figure 14 illustrates the final segmentation results of real noisy images by proposed model. Fig. 14a shows the original image having tumor and Fig. 14b, 14c, 14d, 14e, 14f, 14g, 14h shows the image which are corrupted by zero mean Gaussian noise with different standard deviation values i.e. 0.05, 0.08, 0.005, 0.008, 0.1, 0.4 and 0.6 respectively. The final segmentation results shows that the proposed model efficiently detect tumor in all images in the presence of Gaussian noise. Even    in presence of severe noise the proposed model is also able and performed well to segment the desired tumor. Figure 8 shows the brain MRI image of Fig. 15 that can be used with salt and pepper noise and weak boundaries. Fig. 15a, 15b, 15c, 15d, 15e shows the salt and pepper noise with different noise densities, i.e. 0.05, 0.08, 0.09, 0.1, 0.2 and 0.3 respectively. The proposed model is capable of handling and accurately segment tumor having salt and pepper noise with different noise densities and yields adequate results of segmentation for both Gaussian noise as well as for salt and pepper noise.

D. PARAMETER SENSITIVITY
In this section, we give a brief discussion about the sensitivity of the proposed model on a parameter σ . σ is one of the important parameters, which can affect the segmentation results of medical images. In order to test the dependence of     is between 11.7 and 13.8, where optimal Jaccard value is obtained as shown in Fig. 16c.

VI. CONCLUSION
In this paper, a novel active contour model is proposed to segment structure of tumor in medical images intensity inhomogeneity. The proposed model with new statistic is developed by introducing coefficient of variation in probability density function. By using relative statistic, squared coefficient of variation in the data fitting term to attain local intensity information of the image which gives better segmentation results in medical images having inhomogeneous intensity. The proposed model has produced good results in challenging images having tumor with severe inhomogeneous intensities and also having complex background. The proposed model is tested on different type of other medical images to show that the object of interest is approximately similar to the ground truth and also the results are compared with the existing state of the art models in terms of both qualitatively and quantitatively. The quantitative results of the proposed model indicate that the segmented region of interest is approximately similar to the ground truth and prove the best values of JSI, sensitivity, specificity, accuracy and dice similarity coefficient. The proposed model is tested on different medical images taken from different databases. The proposed model is also able to handle the images having severe noise.

A. LIMITATIONS
The proposed model has certain limitations. Firstly, the proposed model should be improved for large size medical MATI ULLAH is currently pursuing the Ph.D. degree with the Department of Basic Sciences, University of Engineering and Technology Peshawar, Pakistan. His research interests include joint image restoration and segmentation model of color images. VOLUME 8, 2020