MR-BRAIN IMAGE SEGMENTATION USING GAUSSIAN MULTIRESOLUTION ANALYSIS AND THE EM ALGORITHM

Mohammed F. Tolba, Mostafa G. Mostafa Faculty of Computer and Information Sciences, Ain Shams University, Abbassia, Cairo 11566, Egypt.

Email: mtolba@asunet.shams.edu.eg, mostafa@asunet.shams.edu.eg

Tarek F. Gharib, Mohammed A-Megeed Faculty of Computer and Information Sciences, Ain Shams University, Abbassia, Cairo 11566, Egypt

Email: gharib@asunet.shams.edu.eg, mamegeed@hotmail.com

Keywords: Medical image, MRI, Image segmentation, EM algorithm, Multiresolution analysis, Gaussian mixture model, Maximum likelihood estimator.

Abstract: We present a MR image segmentation algorithm based on the conventional Expectation Maximization (EM) algorithm and the multiresolution analysis of images. Although the EM algorithm was used in MRI brain segmentation, as well as, image segmentation in general, it fails to utilize the strong spatial correlation between neighboring pixels. The multiresolution-based image segmentation techniques, which have emerged as a powerful method for producing high-quality segmentation of images, are combined here with the EM algorithm to overcome its drawbacks and in the same time take its advantage of simplicity. Two data sets are used to test the performance of the EM and the proposed Gaussian Multiresolution EM, GMEM, algorithm. The results, which proved more accurate segmentation by the GMEM algorithm compared to that of the EM algorithm, are represented statistically and graphically to give deep understanding.

1 INTRODUCTION

Magnetic resonance image (MRI) of the human brain, among a variety of imaging modalities such as positron emission tomography (PET), computer tomography (CT), ultrasound, radiography, and mammography, was the most common type of medical images used in the medical diagnosis. For that, they are the subject of the research in the area of medical image analysis. This is due to its importance in the advanced imaging applications such as medical image rendering and image guided procedures. We used this type of images in the performance analysis of the algorithms developed here; since they are complex enough to show the advantages and disadvantages of any image segmentation algorithm.

To obtain the MRI, the patient is subject to a radio frequency pulse. As a result of this the magnetic nuclei pass into a high energy state, and then immediately relieve themselves of this stress by emitting radio waves through a process called relaxation (Pal, 1993). The MRI of the normal brain

can be divided into three regions other than the background, white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF) or vasculature (Rajapakse, 1997). Because most brain structures are anatomically defined by boundaries of these tissues classes, a method to segment tissues into these categories is an important step in quantitative morphology of the brain.

Image segmentation is to divide the image into disjoint homogenous regions or classes, where all the pixels in the same class must have some common characteristics. According to the nature of the image the approach of segmentation may be either region-based approaches (Motensen, 1995)(Yen Wan, 2000), or pixel-based approaches, where the segmentation is done according to the pixels features, such as pixel intensity (Saeed, 1998)(Kaufhold, 1997)(Nevin, 2002)(Mostafa, 2001)(A-Megeed, 2002). A will known statistical approach of image segmentation based on pixel intensity is the Expectation Maximization (EM) algorithm (Ambroise, 1995)(Moon, 1996), which is used to estimate the parameters of different classes in the image. To overcome the drawbacks of the EM

algorithm, we propose a multiresolution algorithm, the Gaussian Multiresolution EM Algorithm, GMEM, which proved high reliability and performance under different noise levels, and in the same time it keeps the advantages of the conventional EM algorithm.

The EM algorithm for image segmentation based

on modeling the image as a Gaussian Mixture Model, GMM, where, the parameters of the model are not knowing a prior (missing data), and it utilize the estimation theory to use the pixel intensity (incomplete data) to estimate the missing data (Moon, 1996)(Dempster, 1997).

2 DATA MODELING

Traditionally, statistical models are used to represent the image data, they consider the image as a random process, in particular; a mixture of random processes. The processes are assumed to be represented by independent identically distributed functions (iid), in particular, a Gaussian distribution functions. The model is then represented by (Ambroise, 1995)(Bezedek, 1993):

∑=Φ=

K

kkiki xGpxf

1)|()|( θ

Where K is the number of processes or classes that need to be extracted from the image, θk ∀ k=1,2,…,K is a parameter vector of class k and its of the form [µk, σk] such that µk, σk are the mean and standard deviation of the distribution k, respectively, pk is the mixing proportion of class k (0< pk <1, ∀ k=1, …, K and Σk pk=1), xi is the intensity of pixel i, and Φ= { p1, …, pK, µ1, …, µK, σ1, …, σK} is called the parameters vector of the mixture (Ambroise, 1995).

The parameter vector Φ, is the missing data that defines the mixture. The EM algorithm utilize the maximum log-likelihood estimator to estimate the value of Φ. Let (Hsu, 1997)

)|()|,...,()( 121 ∏ = Φ=Φ=Φ ni in xfxxxfL

The joint probability distribution, L(Φ), represents the likelihood that the values x1, x2, …, xn will be observed when Φ is the true value of the parameter. Then we define ΦML to be the maximizing value of L(Φ) that is

L(ΦML) = arg max (L(Φ)) Hence ΦML is called the maximum log-likelihood

estimator of Φ.

3 THE EXPECTATION MAXIMIZATION ALGORITHM

Once the true values of the parameters are found, then the class of each pixel in the image can be easily determined by computing the probability of this pixel to be belong to each class in the image. The EM algorithm is often used for finding the unknown parameters of a mixture model. that maximizes the log-likelihood of the sample.

3.1 The General Statement of the EM Algorithm

The EM algorithm as developed and discussed by Moon (Moon, 1996) and Ambroise and Goaert (Ambroise, 1995) is achieved by executing two steps iteratively.

1- The Expectation (E) Step: Compute the expected value of zik use the current

estimate of the parameter vector Φ .

)|()θ|(

)(

(t)k

)()(

ti

it

ktik

xfxGpzΦ

=

Where zik is the posteriori probability that given xi, xi comes from class k. the NxK matrix Z=[zik]of posteriori probability satisfies the constraints, (0≤zik≤1, ∑k zik =1, ∑i zik >0, 1≤i≤N , 1≤k≤K). xi is

the value of the pixel i. )|( k

(t)θxG i is the

probability of pixel i given it is a member of class k.

2- The Maximization (M) Step: Use the data from the expectation step as if it were

actually measured data (Moon, 1996).

∑

∑==

=+Ni

tik

Ni i

tikt

zxz

k1

)(1

)()1(µ

∑

∑=+

=

=+

Ni

tik

Ni

ti

tik

z(xzt

k1

)(1

2)1()( )-)1(2 µσ

Nzp

Ni

tikt

k∑= =+ 1

)()1(

Each iteration of the EM algorithm optimizes

alternatively the criterion according to the parameters of the mixture with a fixed pixel class probabilities and then according to the pixel class probabilities Z=[zik] with the parameters Φ fixed (Ambroise, 1995). The EM starts with initial guess

of the parameters of the distributions and the proportions of the distributions in the image Φ(0).

The EM algorithm is always followed by a classification step. Since the EM is producing just the missing parameters, that is Φ, those parameters are then used by a classifier which we define it as

),(G(maxarg )= kik

i xk θ

It assigns a class membership to a pixel i depending on its intensity, xi, to the class which its parameter vector maximize the Gaussian density function.

3.2 The EM Algorithm Steps

We can summarize the EM algorithm for image segmentation in the following steps 1. The number of classes K and the Image I are

provided to the system.

2. The initial estimation of parameter Φ(0) is estimated based on the histogram of the image and the number of classes. For example, in case of MRI segmentation, this requires a general guess for the means and variances for the three classes (WM, GM, and CSF). White matter will have highest mean and CSF will have lowest mean. Because of the variation between different MRI acquisition systems, the histogram of the image should be used. The priory class probabilities pk are assumed to be equally likely.

3. Performing the E-step and M-step iteratively until convergence, at each iteration the E-step compute the class probability of each pixel based on the current estimation of Φ(t). M-step computes the new expectation of Φ(t+1) based on values computed in the pervious E-step. After convergence the maximum estimator of Φ is produced.

4. Use ΦML in a classifier to generate the classification matrix C.

5. Assign color or label to each class based on the classification matrix C and generates the segmented image.

3.3 The Drawbacks of the EM Algorithm

Although the EM algorithm is used in MRI of human brain segmentation, as well as image segmentation in general, it fails to utilize the strong

spatial correlation between neighboring pixels. For example, if a pixel, i, all its surrounding pixels, neighbor pixels, are being classified to belong to the same class say Ka, but if it has an intensity closer to the mean of another class, say Kb, the classifier would incorrectly classify this pixel to belong to class Kb.

This drawback is due to that the EM is based on

the GMM which assumes that all the pixels distributions are identical and independent; however it has an advantage that it reduces the computational complexity of the segmentation task by allowing the use of the well-characterized Gaussian density function (Saeed, 1998).

4 THE GAUSSIAN MULTIRESOLUTION EM ALGORITHM

In this paper we propose a new image segmentation algorithm, namely; Gaussian Multiresolution EM algorithm, GMEM, which is based on the EM algorithm and the multiresolution analysis of images. It keeps the advantages of the simplicity of the EM algorithm and in the same time overcome its drawbacks by taking into consideration the spatial correlation between pixels in the classification step.

We mean by the term “spatial correlation”, that

the neighboring pixels are spatially correlated because they have a high probability of belonging to the same class. We think that utilizing the spatial correlation between pixels is the solution key to overcome the drawbacks of the EM algorithm. Therefore, we propose to modify the EM algorithm so that it takes in its consideration the effect of the neighbor pixels when classifying the current pixel, by utilizing the multiresolution technique.

4.1 The Multiresolution Analysis

The multiresolution-based image segmentation techniques, which have emerged as a powerful method for producing high-quality segmentations of images (Saeed, 1998)(Mostafa, 2001), are combined here with the EM algorithm to overcome the EM drawbacks and in the same time take its advantages. The Multiresolution analysis is based on the aspect that “all the spaces are scaled versions of one space” (Daubechies, 1992), where successive coarser and coarser approximations to the original image are

obtained. This is interpreted as representing the image by different levels of resolution. Each level contains information about different features of the image. Finer resolution, i.e., higher level, shows more details, while coarser resolution, i.e., lower level, shows the approximation of the image and only strong features can be detected. Working with the image in multiresolution enables us to work with the pixel as well as its neighbors, which makes the spatial correlation between pixels easy to implement.

In this work we have generated two successive scales of the image, namely, parent and grandparent images. We used an approximation filter, in particular, a Gaussian filter, to generate such low-resolution images. The Gaussian filter is a low pass filter used to utilize the low frequency components of neighboring pixels (Walpole, 1998). We used the Gaussian filter in a manner similar to a moving window. Where a standard Gaussian filter of size n x n is created and in the same time the original image is divided into parts each of which has the same size as the filter size. The filter is then applied to each part of the image separately. This can be interpreted as a windowed convolution where the window size is the same as the filter size. and also this agree with the concept of the distinct block operation (Gonzalez , 2000), where the input image is processed a block at a time. That is, the image is divided into rectangular blocks, and some operation is performed on each block individually to determine the values of the pixels in the corresponding block of the output image, the operation in our case is the Gaussian filter. Each time we apply the filter on a part of the image the result is placed as a pixel value in a new image in a similar location to that where it was obtained. Later we use this new image as the parent of the original image. In the following we illustrate this in more details.

In Figure. 1, the original image I0 at scale J=0,

say of size 9x9 is divided into parts each part of size 3x3, then a Gaussian filter of size 3x3 is applied to the first part of the image I0(1:3,1:3) the result of the windowed convolution, say a11, is placed in location (1,1) in the new image, I1. This step is repeated to each part of the image which generate a sequence of coefficients, a11..a13, a21 ..a23, and a31 ..a33, these coefficients are placed in the new image by the same order as they obtained. The new created image is of size 3x3 represents a lower-resolution approximation of the original image and acts as a parent image, at scale J=1, of the original image I0, at scale J=0, where each nine neighboring pixels in I0 are used to generate one pixel in I1. By the same way, we used a Gaussian filter of size 5x5 to create the grandparent image I2 from I0 at scale J=2.

Generally, the distinct block operation may require image padding, since the image is divided into blocks. These blocks will not always fit exactly over the image. In this work we used the symmetric padding where the boundaries of the image at which the image is padded are replicated (Misiti , 2000).

4.2 Multiresolution Image Segmentation

Once the parent and grandparent images have been created, we move to the next step by solving the segmentation problem using the different scales of the image.

(Saeed, 1998) made three assumptions while

they were trying to solve the segmentation problem using multiresolution analysis of image. Those assumptions are, first, the pdf of pixel (x,y) at resolution J is dependent upon its neighbors, second, it is dependent upon the parent pixel (x′,y′) at resolution J+1 and its neighbors, third, it is dependent upon the grandparent pixel (x″,y″) at resolution J+2 and its neighbors. Thus, their model attempted to utilize the dependence of pdf across both scale and space towards the goal of a more robust segmentation algorithm. Their aim is to modify the Gaussian Mixture density function such that they penalize the likelihood of pixel membership to a certain class when its neighbors, parent, and parent’s neighbors have a low probability of belonging to this same class (Saeed, 1998). We tried different approaches to utilize the multiresolution image analysis with the conventional EM algorithm. Thus, we did not use the assumptions in (Saeed, 1998), instead we made another assumption that “the classification of a pixel (x,y) at resolution J is dependent upon both the classification of its parent pixel (x’,y’) at resolution J+1 and the classification of its grandparent pixel (x″,y″) at resolution J+2”. We made this assumption because we think that the parent and grandparent pixels

Figure 1: Illustration of the use of theGaussian-window

represent the averaging of the interested pixel and its neighbors. So, the classification of the parent or grandparent represent the approximated class of these pixels together.

The implementation of the GMEM algorithm, therefore, is done as follows: we apply the EM algorithm on both the parent and grandparent images to produce three segmented images in three successive scales of the original image. In other words, we used the EM algorithm to segment the image of each scale independent on the others. The EM algorithm is followed by a classifier, therefore, the output of this step is three classification matrices C0, C1, and C2 representing the segmentation of the original image, its parent, and its grandparent images, respectively. Those matrices, obtained from the segmentation of the different resolutions of the image are then used to find the final classification of the image.

The final classification step is done by assigning

weights to each classification matrix obtained from the previous step. The assigning of the weights reflects our confidence in the segmentation decision of the corresponding level. The final classification step computes the confidence of each class and returns the class of the highest confidence, i.e., the winner class. For example consider that for a pixel I(x,y) the values of the classification matrices C0, C1, and C2 were k2, k1, and k1, respectively. And the weights assigned to them were (0.4), (0.35), and (0.25), respectively. Then the output of the reclassification step will be k1.

These weights need not to be fixed, but in our

study the weights have been assigned such that they ensure the following points: 1. If the pixels at all the levels of the analysis

I(x,y), I1(x',y'), and I2(x",y") belong to the same class then the final classification is as any of them, i.e. if C0(x,y)=C1(x',y')=C2(x",y") then C(x,y)=C0(x,y).

2. If the classification of a pixel at any two different levels of analysis then the final classification is as any one of both of them, i.e. if C0(x,y) = C1(x',y') or C1(x',y') = C2(x",y") or C0(x,y) = C2(x",y") then C(x,y) = C0(x,y) or C(x,y) = C1(x,y) or C(x,y) = C0(x,y), respectively.

3. If the classifications of a pixel at all levels do not match then the final classification is as one of them defined as a leading level.

Where (x',y') is the parent of (x,y) and (x",y")is the grand parent of (x,y).

Point two ensures that no pixels in the image I will be mistakably assigned to some class K1 while its parent and grand parent belong to another class say K2.

Point three means that a leading weight must be assigned to one of the analysis levels, so that it makes the final classification as the classification of that level if all levels are different.

Points two and three utilize the multiresolution analysis of the image, while the multiresolution analysis has no effect on point one.

4.3 The GMEM Algorithm Steps

The GMEM algorithm can be summarized in the following steps and as depicted in the flowchart shown in Figure. 2.

1- Start with an image Io as input and generates its parent I1 and grandparent I2 using the Gaussian moving windows of sizes 3x3 and 5x5, respectively.

2- Apply the conventional EM algorithm for image segmentation on the images Io, the parent I1, and the grandparent I2. The outputs of this step are the classification matrices C0, C1, and C2, respectively.

3- Reclassify the original image I. using the weights specified previously to generate the final classification matrix C. That represents the classification of the image I0 after taking into account the spatial correlation between pixels.

4- Assign colors or labels to each class and generates the segmented image S.

GWindow 3X3 GWindow 5X5

EM EM EM

C(x,y) = 0.40* C0(x,y) + 0.35* C1(x,y) +0.25*C2(x,y)

Assigning Colors to Classes

Input:I0

I1 I2

C1

C

C2 C0

Output:S

Figure 2: The GMEM flowchart, the input is the image to be segmented, I0 and the output is the segmented image S

Figure 3: The synthetic images (a) and (b) show original images with 4 classes with means (50, 100, 150, 200) and variances 100, and 400, respectively.

(a) (b)

5 EXPERIMENTAL RESULTS AND DISCUSSIONS

In order to test the performance, and introduce good comparisons between the EM algorithm and the GMEM introduced in this paper, we used two data sets. The first set of these data is the simulated data, as shown in Fig. 3, which are synthetic images with certain specifications which have been chosen so that they enable us to test the different characteristics and the performance of the algorithms. Since the structure of the MRI of the human brain is very difficult to be simulated by synthetic image, we used manually segmented MR images as a second data set and are shown in Fig. 4. The both data sets have made to test the EM and the GMEM algorithms against the noise level and against the complex structure of the MRI of the human brain, moreover, to test the effect of assigning the leading weight to the different level of analysis for the GMEM algorithm.

5.1 The Results of the Synthetic Data

We created a synthetic image consists of four different classes, The gray level intensity of the pixels of the four classes belong to a different four Gaussian distributions with means, 50, 100, 150, and 200, respectively and all with the same variance. The classes placed such that we get two types of edges or boundaries, horizontal and vertical. The variance of the classes can be interpreted as a noise added to the image and the variance value is as the noise level. Therefore, we used the same variance value for all the classes. Two values of variance are used, 100, and 400, to create two test images. Where figure. 3 (a), shows the image with noise of variance 100, and figure. 3 (b) shows image with noise of variance 400. The variance 100 represent low noise level and 400 a high noise level.

We applied the EM algorithm on the two images. The accuracy of the algorithm can be easily computed using the confusion matrix (Kohavi, 1998), where the overall accuracy (OA) of the segmentation computed by dividing the total number of correct classified pixels over the total number of pixels, i.e.

∑ ∑∑= x yx yxxxAC ),(),( The accuracy of a class, x, is computed by

dividing the number of correct classified pixels of this class over the total number of pixels that have been assigned to this class, i.e.

∑= y yxxxACx ),(),(

Figure. 5(a), and figure. 5(b) display the segmented images of the images in the figure. 3(a), and figure 3(b), respectively, obtained by the EM algorithm. The overall accuracies are computed and are reported in TABLE I.

We have two observations here. First, the accuracy of the segmentation is dropped from 99% to 81.9% when the noise variance increased from 100 to 400. Which assure that this algorithm is sensitive to the noise and makes this algorithm not suitable for noisy images. Second, many pixels are misclassified, although, all its surrounded pixels are correctly classified. So the EM fails to utilize the strong spatial correlation between neighboring pixels. This is due to it is based on the Gaussian Mixture Model topology which assumes that all the pixels are independently and identically distributed.

Figures 5(c), 5(d), 5(e), 5(f), 5(g) and 5(h) show

the segmented images by the GMEM algorithm and their overall accuracies are also displayed in TABLE I. Figures 5(c) and 5(d) show the results when the leading weight assigned to the level 0 of the analysis, the original image. Figures 5(e) and 5(f), show the results when the leading weight assigned to the level 1 of the analysis, the parent image. Figures 5(g) and 5(h), show the results when the leading weight assigned to the level 2 of the analysis, the grandparent image. The overall accuracies of the GMEM algorithm are displayed in TABLE I, where we mean by GMEM_0, the accuracy of the GMEM when the leading weight assigned to level 0 of the analysis, and GMEM_1, and GMEM_2 when the leading weight assigned to level 1 and level 2, respectively. These accuracies show that the segmentation by the GMEM is more accurate than that obtained by the conventional EM algorithm for all the possible cases of the leading level. Also, the segmentation under the very high noise level is still accurate and so we can deduce that the GMEM algorithm is less sensitive to the noise level and can be used for noisy images. Other observation is that

the most misclassified pixels are very near to the edges. These errors are due to that the parent and grandparent images contain only low frequency components of the image and hence the edges are rarely appears in those images. In other words, if a pixel (x,y) in the resolution J is laying on a boundary between two classes or more, and belong to one of them say k1, then its parent pixel (x',y') at resolution J+1 is formed by another eight pixels around this pixel and its grandparent pixel (x'',y'') at resolution J+2 is formed by another twenty-four pixel also around this interested pixel. Most probably many of those pixels will be from different classes other than k1, and hence the classification of the parent or the grandparent pixels must be effected by those pixels and the output of their classification could be another class say k2 which is different than the true class of the original pixel (x,y). Thus, the classification of the parent and grandparent pixels of a pixel laying on an edge or boundary will be most probably wrong and not trusted. Therefore, much of the errors occurred by the GMEM algorithm are due to it used the classification of the parent and grandparent images to reclassify the pixels near or on the edges of the original image.

Figure 7, shows a chart represent the relation between the accuracies obtained by the EM algorithm and those of the GMEM algorithm when applied to the images in figure 3.

5.2 The Results of the Manually Segmented MR Images

The synthetic images used previously have certain specifications that make the test and comparisons between the different algorithms easy to be done. However, the shape of the real MRI of the human brain seems different than those images, and the application of the EM and GMEM on a real MRI can not give us a quantitive measure about how they are successful for such complicated image. We used the output, segmented image, by the EM when applied to real MRI as a manual segmentation of a MRI, and we used its classification matrix to measure and test the accuracy of the algorithm. By the same way as the synthetic images, we used the Gaussian distribution to generate this image with the help of the classification matrix. Where three classes are generated their pixels values belong to three Gaussian distributions with means 50, 100, and 150, respectively, and all with the same variance. To simulate different levels of noise, two images created with variances 100, and 400 to represent low, and high noise level, and are shown in figures 4(a) and 4(b), respectively.

Figures 6(a), and 6(b), show the segmentation results obtained by the EM algorithm. The results obtained by the GMEM algorithm are shown in figures 6(c), 6(d), 6(e), 6(f), 6(g), and 6(h). Figures 6(c) and 6(d), show the results when the leading weight assigned to level 0 of the analysis. Figures 6(e) and 6(f), show the results when the leading weight assigned to level 1 of the analysis. Figures 6(g) and 6(h), show the results when the leading weight assigned to level 2 of the analysis. The accuracies of the EM and the GMEM algorithm are displayed in TABLE II. We found that the accuracy of segmentation obtained by GMEM algorithm much exceeds that obtained by EM algorithm in all cases tested. Although, the accuracy of segmentation of the EM dropped from 97% to 84% when the variance of the image increased from 100 to 400 in the case of synthetic images, we find the accuracy of GMEM just decreased from 97%, the best overall accuracy, when the leading level was the level 0, to 93.3%, the worst accuracy amongst the GMEM results, when the leading level was level 1, for the same images. Moreover; in the results of EM, there are many misclassified pixels in the background. These misclassified pixels are hardly found in the corresponding results of GMEM, where we find an increase in the segmentation accuracy in all noise level cases for class 1, which represent the background and the CSF of the brain tissue, these results are summarized in TABLE III. We refer this advance in the segmentation performance to the use of the multiresolution analysis of the image, which utilize the spatial correlation between neighbored pixels of the class membership. TABLE IV summarizes the accuracies of the algorithms for class 2. In these results we find that the accuracy of EM algorithm has advanced that of the GMEM algorithm when the noise level was low (variance = 100), but with the increase of the noise level the accuracy of the EM decreased faster than that of the GMEM algorithm. We refer this behavior of the EM algorithm because this class has many edges which make the algorithm that utilizes the pixels

Figure 4: The manually segmented MRI. (a) and (b) show original images with 3 classes representing the background and the CSF, the gray mater, and the white mater, and noise levels 100 and 400, respectively.

(a) (b)

dependency is more accurate than that which utilizes the spatial correlation between pixels. However the increase of the noise level makes the EM algorithm less accurate than the GMEM algorithm due to its sensitivity to noise. The accuracies summarized in TABLES II, III, and IV, are represented graphically in figures 8, 9, 10, respectively.

6 CONCLUSIONS

A new multiresolution algorithm for image segmentation has been proposed in this paper, namely, the Gaussian multiresolution EM algorithm (GMEM). The proposed algorithm is based on the conventional Expectation Maximization (EM) algorithm, and the multiresolution analysis of images. The EM has prevailed many other segmentation techniques because of its simplicity and performance. However, it is found to be very sensitive to noise level. Where, a drop of about 18% of its segmentation accuracy when noise increased from low to high levels, from variance = 100 to variance = 400. To overcome this drawback the GMEM algorithm uses the multiresolution analysis, where, the multiresolution analysis enables the algorithms to utilize the spatial correlation between neighboring pixels. The GMEM algorithm uses the Gaussian filter and the distinct block operation to generate low resolution images from the original image, where two images generated at two successive scales, the parent and the grandparent images.

The proposed algorithm has been tested using both synthetic data and manually segmented magnetic resonance images (MRI). Moreover, a comparison between the performance of this new algorithm and the conventional EM algorithm has been presented. We found that the accuracy of the segmentation done by the proposed algorithm increased significantly over that of the conventional EM algorithm. In case of the synthetic data, about 15% increase in the segmentation (OA) is obtained for high noised images (variance = 400). In case of the real MR images with ground truth and added high Gaussian noise, an increase in the segmentation OA of about 9% is obtained. These results show that our new multiresolution algorithm provide superior segmentations over the one-scale image segmentation algorithms.

The drawback found in the GMEM algorithm is that, when it is applied to pixels laying on the boundaries between classes or on edges, it generates many misclassified pixels, and this is because the parent and grandparent images contain only low

frequencies and hence the edges are rarely appear in these images. Much of the error occurred because we used the classification of the parent and grand parent images to reclassify the pixels near the edges.

In this paper we have tested the different combination of choosing the leading level of the analysis, and we reported the results. The number of levels of the multiresolution analysis have been chose by experience, however we think that using a scientific technique such as the data fusion will give us more reliable segmentation algorithm. Moreover, the GMEM algorithm has been proposed for MR images of the human brain, but we think that it can be used for many other medical imaging types or images in general and we think it will continue producing accurate segmentation.

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TABLE I: THE ACCURACIES OF EM AND GMEM

WHEN APPLIED TO SYNTHETIC IMAGES

TABLE II: THE ACCURACIES OF EM AND GMEM

WHEN APPLIED TO MANUALLY SEGMENTED MRI

100 400 100 400 EM 99% 81.9% EM 97% 84% GMEM_0 99% 96.4% GMEM_0 97% 93% GMEM_1 100% 97% GMEM_1 96.9% 93.3% GMEM_2 100% 98% GMEM_2 96.8% 93%

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 5: Segmentation results of the synthetic images. (a) and (b) show the segmented images by the EM algorithm. (c) and (d) the segmented images by the GMEM algorithm when the leading weight assigned to the 1st level of the analysis, (e) and (f) the segmented images by the GMEM when the leading weight assigned to the 2nd level, (g) and (h) the segmented images when the leading weight assigned to the 3rd level.

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 6: Segmentation results of manually segmented MRI. (a) and (b) show the results of the EM algorithm, for noise level 100 and 400 respectively, (c) and (d) the segmentation results by the GMEM when the leading weight assigned to the 1st level of the analysis, (e) and (f) the segmented images by the GMEM when the leading weight assigned to the 2nd level, (g) and (h) the segmented images when the leading weight assigned to the 3rd level.

TABLE III:

THE ACCURACIES OF THE SEGMENTATION OF CLASS 1 OF MANUALLY SEGMENTED MRI

TABLE IV: THE ACCURACIES OF THE

SEGMENTATION OF CLASS 2 OF MANUALLY SEGMENTED MRI

100 400 100 400 EM 98% 90% EM 94% 80.5% GMEM_0 99% 97% GMEM_0 93% 88.6% GMEM_1 98.5% 96.6% GMEM_1 92.6% 89% GMEM_2 98.5% 95.9% GMEM_2 92.4% 88.5%

Mostafa G. Mostafa, Mohammed F. Tolba, Tarek F.

Gharib, Mohammed A-Megeed, 2001. Medical Image Segmentation Using a Wavelet-Based Multiresolution EM Algorithm. IEEE International Conference of Electronics Technology and Application, IETA'2001, Cairo, Egypt.

Mohammed A-Megeed, "Medical Image Segmentation and Visualization", Master thesis, Department of Scientific Computing, Faculty of Computers and Information Sciences, Ain Shams Univ., Cairo, Egypt, June, 2002.

C. Ambroise and G. Govaert, 1995. Spatial Clustring and the EM algorithm. Univ. of technolgic de Compiegane, Aussois, France.

A. P. Dempster, N. M. Laird, D. B. Rubin, 1997. Maximum Likelihood from Incomplete Data Via The EM Algorithm. J.Roy. Statist. Soc.,Vol 38-B, pp.1-38.

Ronald E. Walpole, Raymound H. Myers, Sharon L. Myers, 1998. Probability and Statistics for Engineers and Scientists 6th Edition. Prentice Hall, Inc.

Rafael C. Gonzalez & Richard E. Woods., 2000. Image Processing Toolbox User’s Guide. The Mathworks. Inc.

M. Misiti, Y. Misiti, G. Oppenheim, J. Poggi., 2000. Wavelet Toolbox User's Guide, 2nd Edition. The Mathworks. Inc.

Kohavi, Provost, 1998, hhtp://www.cs.uregina.ca/~dbd/ cs831/notes/confusion_matrix/cofusion_matrix.htm

Class 2 A ccuracy

70

75

8085

90

95

100 400Noise Level

Accu

racy

%

EM GMEM Level 0GMEM Level 1 GMEM Level 2

Figure 10: Chart represents a comparison between the accuracies summarized in TABLE IV of class 2 of the manually segmented MR images, when obtained by the EM algorithm and the GMEM algorithm

Overall Accuracy

7580

8590

95100

100 400Noise Level

Acc

urac

y %

EM GMEM Level 0GMEM Level 1 GMEM Level 2

Figure 8: Chart represents a comparison between the (OA) summarized in TABLE II obtained by the EM algorithm and the GMEM algorithm when applied on manually segmented MR images.

Class 1 A ccuracy

85

90

95

100

100 400Noise Level

Accu

racy

%

EM GMEM Level 0GMEM Level 1 GMEM Level 2

Figure 9: Chart represents a comparison between the accuracies summarized in TABLE III of class 1 of the manually segmented MR images, when obtained by the EM algorithm and the GMEM algorithm

Overall Accuracy

020406080

100120

100 400Noise Level

Acc

urac

y %

EM GMEM Level 0GMEM Level 1 GMEM Level 2

Figure 7: Chart represents a comparison between the (OA) summarized in TABLE I obtained by the EM algorithm and the GMEM algorithm when applied on a synthetic images.