Novel Method for Brain Tumor Classification Based on Use of Image Entropy and Seven Hu’s Invariant Moments

Ruan et al. [11] proposes a method based on the use of partial volume modeling for brain tissue classification in MR images. The researchers consider two classes in a brain dataset. The first is called the pure class, and is composed of the three main types of brain tissue: gray matter, white matter, and cerebrospinal fluid. The second is called mixed class, and is composed mainly of mixtures. The proposed method proceeds in two steps, both of which are based on the use of Markov Random Field (MRF) models. The first step consists of segmenting the brain into pure classes and mixed class, while the second consists of reclassifying mixed class into pure classes using some knowledge of the obtained pure classes. This method has been evaluated on simulated and real-world brain MR images.

The method proposed by Bauer et al. [12] has been tested on the BRATS2012 database [13] composed of simulated and clinical images. The main objective of this method is not only to separate healthy tissues from the pathologic tissue but also, on one hand, to classify healthy tissues into three components: gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF), and on the other hand, to classify pathologic tissue into the following components: necrotic, active and edema. According to the authors, convincing results have been obtained (the average Dice coefficient equal to 0.73 for tumor and 0.59 for edema) within an acceptable computation time (approximately 4 to 12 minutes).

Cobzas et al. [14] propose a variational segmentation algorithm for brain tumors by using a high-dimensional feature set calculated from MRI data and registered atlases. The learning of the statistical model for tumor and normal tissue is implemented by using manually segmented data. According to the authors, the use of the conditional model to discriminate between normal and abnormal regions significantly improves the segmentation results compared to those of the traditional generative models.

The objective of the study proposed by Iftekharuddin et al. [15] is to develop a new technique for brain tumor segmentation. The authors use information extracted from the multiresolution texture data that combines fractal Brownian motion (fBm) and wavelet multiresolution analysis, and the proposed technique has been evaluated on pediatric brain MR images from St. Jude Children’s Research Hospital. Both the pixel intensity and the multiresolution texture features are exploited to obtain a segmented tumor. In the next step, the presence of tumor in an MR brain image is statistically validated by using the feedforward (FF) neural network as a classifier.

The main objective of the study proposed by Zhang et al. [16] is the development of an accurate system for detection of brain pathologies from MR images of the brain. The proposed method is based on the use of entropy of the wavelet and the Hu’s moment invariants for feature extraction, while the classification is performed first by the generalized eigenvalue proximal support vector machine (GEPSVM); afterwards, to improve the results of classification, the authors use the popular radial basis function (RBF) kernel. According to the authors of this study, the results obtained show that the proposed method is effective and can be used in real-world applications.

In the context of brain tumor classification, Chaplot et al. [17] have proposed a new method of classifying brain MR images for either normal or abnormal cases. The study is based on the comparison of results obtained by using the classifiers in two steps: first, neural network self-organizing maps and second, the support vector machine. Note that the authors have used wavelets for the input of the two classifiers. The proposed method has been tested on fifty-two brain MR images; according to the authors, the obtained results are very encouraging, and the classification rate of the support vector machine classifier is observed to be higher than that obtained by using the self-organizing maps classifier.

The proposed system is composed of three essential steps (see Figure 2), namely, preprocessing, feature extraction and brain tumor classification. The role of and various methods and techniques used for each step are described as follows:

4.1 Preprocessing

The only preprocessing operation we consider necessary in this study is image resizing; this operation is performed due to the nature of the feature extraction methods in the next step that are only applied to normalized images. As brain images have variable sizes, the preprocessing step consists of resizing all images of the database to all images of the database to 128x128 pixels in size. Note that the choice of the size 128x128 pixels as a normalized size for the images of the used database is related to the nature of the feature extraction methods used in this study. Resizing MR images to a size of 128x128 pixels allows us to obtain an appropriate number of zones selected by the sliding window used for each method.

1.jpg

Figure 1. Several samples of each class of the used database: a) Samples of class 01, b) Samples of class 02, c) Samples of class 03, d) Samples of class 04, e) Samples of class 05, f) Samples of class 06

2.jpg

Figure 2. General schema of the proposed system

4.2 Feature extraction

In this paper, the feature vector is composed of several parameters extracted from the brain MR image by two methods, the first based on the calculation of image entropy, and the second consisting of the calculation of the seven Hu’s invariant moments; the both methods are applied on each zone selected by sliding a window over the brain MR image. For reasons involving computation time and required memory, the sizes of the sliding windows used for the two methods are not the same; the size of the sliding window used for the entropy method is 16x16 pixels, while the size used for Hu’s invariant moments method is 64x64 pixels. Note that we have used sample 01 of class 02 and sample 03 of class 03 as examples to explain the principle of these two methods and to show the obtained features or parameters.

As it was requested in paper [18], the sources of these two samples are as follows:

• Sample 01 of class 02: Case courtesy of A.Prof Frank Gaillard, Radiopaedia.org, rID: 22205
• Sample 03 of class 03: Case courtesy of Dr. Ian Bickle, Radiopaedia.org, rID: 50807

4.2.1 Image entropy

The notion of entropy has been widely used for brain pathology classification [16, 19-23] and can be defined as the concept of the spread of states a system can be in. A system that occupies a small number of such states is characterized by low entropy, while a system that occupies a large number of states is characterized by high entropy.

In the case of an image, interpreting the preceding entropy definition leads us to consider the states as the grayscale levels an individual pixel can have. In particular, there are 256 states for a grayscale image when the pixels are represented using 8 bits.

 The entropy of the image reaches the maximum value if the spread of the states is maximized, this holds if all such states are equally occupied, as they are in an image that has been perfectly histogram-equalized. On the other hand, if the image has been processed using a threshold, then only two states are occupied, and consequently the entropy is low. If all the pixels have the same value, the entropy of the image is zero. In conclusion, the entropy of an image varies proportionally to its information content. The entropy changes from high entropy of a full grayscale image to low entropy of a threshold binary image and to zero entropy of a single-valued image.

The entropy E of an image is defined as

$E=-\sum\limits_{k=0}^{M-1}{{{P}_{k}}{{\log }_{2}}({{P}_{k}})}$                              (1)

where, M is the number of grayscale levels, and Pk is the probability associated with grayscale level k.

4.2.2 Seven Hu’s invariant moments

Invariant moments have been widely applied to image processing [16, 22] and have proven useful in a variety of applications due to their invariance with respect to image translation, scaling and rotation. They were first introduced by Hu [24], who derived six absolute orthogonal invariants and one skew orthogonal invariant based upon algebraic invariants, which are independent not only of position, size and orientation but also of parallel projection. Invariant moments have been proven to be the adequate measures for tracing image patterns affected by image translation, scaling and rotation under the assumption of noise-free images with continuous functions. The quantities used to construct the invariant moments are defined in the continuous form, but for practical implementation they are computed in the discrete form. For a two-dimensional continuous function f(x,y), the moment (sometimes called the "raw moment") of order (p +q) is defined as

${{M}_{pq}}=\int_{-\infty }^{\infty }{\int_{-\infty }^{\infty }{{{x}^{p}}{{y}^{d}}f(x,y)dxdy}}$                     (2)

Mpq is a two-dimensional moment of function f(x,y), where the indices p and q are both natural numbers. Adapting this to a grayscale image with pixel intensities I(x,y), the raw moments Mpq become

${{M}_{pq}}=\sum\limits_{x}{\sum\limits_{y}{{{x}^{p}}{{y}^{q}}I(x,y)}}$                       (3)

One should note that the Mpq moments are not invariant with respect to f(x,y) changing by translation, rotation and scaling. The moment that is invariant to translation can be obtained by calculating the moment of order (p+q) called central moment and given by the following equation:

${{\mu }_{pq}}=\int_{-\infty }^{\infty }{\int_{-\infty }^{\infty }{{{\left( x-\overline{x} \right)}^{p}}{{\left( y-\overline{y} \right)}^{q}}f(x,y)dxdy}}$              (4)

where, $\bar{x}$ and $\cdot \bar{y}$ are the coordinates of the gravity center of the image, calculated using Eq. (3) and given by

$\overline{x}=\frac{{{M}_{10}}}{{{M}_{00}}}\begin{matrix}   {} & {}  \\\end{matrix}\overline{y}=\frac{{{M}_{01}}}{{{M}_{00}}}$                             (5)

In the case of a grayscale image with intensity I(x,y), Eq. (4) becomes

${{\mu }_{pq}}=\sum\limits_{x}{\sum\limits_{y}{{{\left( x-\overline{x} \right)}^{p}}{{\left( y-\overline{y} \right)}^{q}}I(x,y)}}$                (6)

The central moments can be extended to be both translation- and scale-invariant by using the following equation:

${{\eta }_{pq}}=\frac{{{\mu }_{pq}}}{\mu _{00}^{\left( 1+\frac{p+q}{2} \right)}}$                                   (7)

The obtained moments are called the normalized central moments. To enable rotation invariance, based on the above moments (normalized central moments), Hu introduced the seven invariant moments given by the following equations:

${{\Phi }_{1}}={{\eta }_{20}}+{{\eta }_{02}}$                                 (8)

${{\Phi }_{2}}={{\left( {{\eta }_{20}}-{{\eta }_{02}} \right)}^{2}}+4\eta _{11}^{2}$                           (9)

${{\Phi }_{3}}={{\left( {{\eta }_{30}}-3{{\eta }_{12}} \right)}^{2}}+{{\left( 3{{\eta }_{21}}-{{\eta }_{03}} \right)}^{2}}$                (10)

${{\Phi }_{4}}={{\left( {{\eta }_{30}}+{{\eta }_{12}} \right)}^{2}}+{{\left( {{\eta }_{21}}+{{\eta }_{03}} \right)}^{2}}$                  (11)

$\begin{align}  & {{\Phi }_{5}}=\left( {{\eta }_{30}}-3{{\eta }_{12}} \right)\left( {{\eta }_{30}}+{{\eta }_{12}} \right)\left[ {{\left( {{\eta }_{30}}+{{\eta }_{12}} \right)}^{2}}-3{{\left( {{\eta }_{21}}+{{\eta }_{03}} \right)}^{2}} \right] \\  & \begin{matrix}   {}  \\\end{matrix}+\left( 3{{\eta }_{21}}-{{\eta }_{03}} \right)\left( {{\eta }_{21}}+{{\eta }_{03}} \right)\left[ 3{{\left( {{\eta }_{30}}+{{\eta }_{12}} \right)}^{2}}-{{\left( {{\eta }_{21}}+{{\eta }_{03}} \right)}^{2}} \right] \\ \end{align}$                               (12)

$\begin{align}  & {{\Phi }_{6}}=\left( {{\eta }_{20}}-{{\eta }_{02}} \right)\left[ {{\left( {{\eta }_{30}}+{{\eta }_{12}} \right)}^{2}}-{{\left( {{\eta }_{21}}+{{\eta }_{03}} \right)}^{2}} \right] \\  & \begin{matrix}   {} & {}  \\\end{matrix}+4{{\eta }_{11}}\left( {{\eta }_{30}}-{{\eta }_{12}} \right)\left( {{\eta }_{21}}+{{\eta }_{03}} \right) \\ \end{align}$                                    (13)

$\begin{align}  & {{\Phi }_{7}}=\left( 3{{\eta }_{21}}-{{\eta }_{03}} \right)\left( {{\eta }_{30}}+{{\eta }_{12}} \right)\left[ {{\left( {{\eta }_{30}}+{{\eta }_{12}} \right)}^{2}}-3{{\left( {{\eta }_{21}}+{{\eta }_{03}} \right)}^{2}} \right] \\  & \begin{matrix}   {}  \\\end{matrix}+\left( {{\eta }_{30}}-3{{\eta }_{21}} \right)\left( {{\eta }_{21}}+{{\eta }_{03}} \right)\left[ 3{{\left( {{\eta }_{30}}+{{\eta }_{12}} \right)}^{2}}-{{\left( {{\eta }_{21}}+{{\eta }_{03}} \right)}^{2}} \right] \\ \end{align}$                           (14)

4.3 Brain tumor classification

In this study, the classification is achieved using a neural network composed of three layers; namely, the input layer whose number of neurons is equal to the size of the used feature vector (N_IL=92); the output layer whose number of neurons is equal to the number of brain tumor classes to be recognized (N_OL=6), and one hidden layers whose number of neurons is experimentally chosen equal to fifty-five (55) (N_HL=55). The initial connection weights are randomly chosen in the range of [-1, 1]. The transfer function used is the familiar sigmoid function. The large use of the back propagation training method for a great number of pattern recognition problems and its simplicity encouraged us to use it for training the neural network. The principle of this method is based on the gradient descent algorithm, the learning rate is experimentally chosen and it allows the algorithm to converge more easily if it is properly chosen by the experimenter. In our case, we have opted for a rate η= 0.8.

This study explores the subject of automating the process of detection and classification of brain tumors from brain MR images. The importance of this type of study is due to solutions they propose to overcome multiple difficulties that have thus far prevented the implementation of a universal system for detection and automatic classification of brain tumors, such difficulties being mainly due to variability in tumor size and location in the brain. This study proposes a novel method for the classification of brain tumors by interpreting brain MR images. The main objective was to improve the results obtained in the study that we presented in the 2nd International Conference on Advanced Systems and Electrical Technologies (IC_ASET'2018) [10]. The proposed method is completely novel and it based on using of a new feature vector composed of several parameters obtained by applying two methods: the calculation of image entropy and Hu’s seven invariant moments. The first method consists of the calculation of entropy of each zone selected by sliding window 16x16 pixels in size, while the second method consists of the calculation of Hu’s seven invariant moments of each zone selected by other sliding window 64x64 pixels in size. The obtained results are very encouraging and they are better than those obtained in the work [10]. The system correctly classifies 97.77% of images of the used database, achieving an attractive recognition rate that can encourage further research. The results can be further improved by analyzing the images that have given the system difficulties when assigning them to their proper classes. This analysis will allow the identification of the reasons for misclassification and thus the proposal of necessary solutions for each stage of the system.