A Bendlet Transform Framework for Magnetic Resonance Imaging Brain Tumor Classification with Optimized Feature Selection

The Human Brain is the most essential and prime organ in our body. In the central nervous system, the intermediate organ is the human brain, along with the spinal cord. The main functions of the human brain and spinal cord in our body are to coordinate and control all our activities, such as thinking and intelligence, sensory processing, voluntary movements, and autonomic control, and this processed signal is sent to the remaining parts of the body. The human brain comprises three important regions: the cerebrum, cerebellum, and medulla oblongata. The functions of the cerebrum are thinking, memory, intelligence, and voluntary actions. Secondly, the function of the cerebellum is balance and coordination of muscles. At last, the functions of the Medulla oblongata control breathing and heartbeat. The detailed explanation of the three regions labelled is shown in Figure 1.

The Cerebrum is the largest part of the human brain and consists of two hemispheres. The names of the hemisphere is inner most and outermost. The innermost hemisphere is made up of white matter. Similarly, the outermost is commonly called the cerebral cortex, which is made up of grey matter. The white matter is for thinking and memory, and the grey matter is for signal transmission. Moreover, each hemisphere in the Cerebrum includes four lobes, namely the frontal, parietal, temporal, and occipital lobes. The frontal lobe is responsible for planning and decision making; likewise, the remaining three lobes are sensory information, emotional response, and visual processing. The connection between the spinal cord and the brainstem with help of cerebrum. The Cerebellum is the second largest region of the human brain, which is located below the cerebrum and at the back of the brainstem. The main function of the cerebellum is to coordinate voluntary movements. Next to the Medulla Oblongata, which is located at the lowest part of the brainstem. It is used to connect the brain to the spinal cord and is responsible for involuntary activities.

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Figure 1. Anatomy of the human brain

In the human brain, the abnormal growth of cells is called a brain tumor, which affects all brain activities because the human brain is responsible for all human activities. The brain tumors are classified into three types depending on the stages which are affected the brain. The primary brain tumors are benign and malignant, which originate in the brain. The examples of these tumors are Glioma, Meningioma, and pituitary. Similarly, the secondary tumors are malignant and spread over the brain to other organs. The examples of the tumors are lung, breast, kidney, and melanoma. The benign tumors are non-cancerous as well as slow growing and don’t spread over the brain, whereas malignant tumors are cancerous and fast growing over the brain. The diagnosis of a brain tumor is made by using Magnetic Resonance Imaging (MRI) and Computed Tomography (CT) imaging. In addition, detection and classification of brain tumors is done by deep learning-based algorithms, as shown in Figure 2.

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Figure 2. Types of segmented tumors

The classification of brain tumors is a difficult task caused by irregular shape, heterogeneous texture, varying size, and uneven boundaries of tumor areas in MRI images. Along with, various tumor categories frequently give the same intensity distributions and overlapping structural characteristics, which affect accurate discrimination and classification. Compared with existing techniques such as Wavelet, Curvelet, Shearlet, and Contourlet transforms, which primarily focus on multiscale and directional feature representation, these techniques reveal several limitations in representing complex tumor regions. The Wavelet transform is limited in directional sensitivity for curved tumor boundaries, while the Curvelet transform is inefficient in capturing local bending and geometric variations. Moreover, the Shearlet transform is also affected by computational complexity and curvature adaptability; meanwhile, the contourlet transform is also unable to find accurate curvature details and uneven tumor regions. Overall, the drawback of existing transforms yields to minimize the capability of edge variations and curvature characteristics in brain tumors, computational complexity, and classification accuracy. To address these drawbacks, the proposed Bendlet transform integrates bending parameters that effectively capture the curvature-based multidirectional features and enhance the performance of tumor representation and classification.

Initially, the MRI brain image as input data is taken and applied for the preprocessing steps to subtract the noise, then improve the contrast in order to increase the image quality. In addition, the preprocessed input image is given to a segmentation process, which is a U-Net based model that gives the exact tumor area from the preprocessed brain image. Moreover, distinguished features are effectively extracted by using the Bendlet transform from the segmented images, which yields curvature information of the tumor. In addition, the extracted features set may contain some redundant information. To avoid these issues, the Whale Optimization algorithm is used for optimization, which is to select more relevant features from the redundant information and also enhance the efficiency. At last, classification of tumors from the selected feature by using fully connected networks [24].

Figure 3 shows the proposed block diagram of tumor detection and classification system. This proposed framework ensures accurate tumor localization, optimized feature representation, and valid tumor classification.

Figure 3. Block diagram of the proposed tumor detection and classification system

3.1 Bendlet transform

The Bendlet transform is a powerful tool that is used to represent images with curved contours, such as Brain images [3]. The transform starts by taking the input image or two-dimensional signal $f(x)$, where $x \in \mathrm{R} 2$, as the function to be analyzed. A mother bendlet function $\psi(x)$ is first selected, which serves as the generating function for constructing all bendlet atoms. This transform is built on four primary operators that define the 'bendlet' shape used to analyse the image, namely, Anisotropic scaling, Translation, Shearing $s \in$ R, and bending parameter $b \in \mathrm{R}$, which captures curvature information; and the translation parameter $t \in \mathrm{R} 2$, which localizes the analysis in space [24].

Using these parameters, the corresponding transformation operators are constructed. The anisotropic scaling matrix Aa applies parabolic dilation, the shearing matrix Ss adjusts orientation, and the bending operator Bb introduces a nonlinear curvature modification. These operators are combined with translation to generate the bendlet function.

$\Psi_{a, b, s, t}(X)=a^{-\frac{3}{4}} \Psi\left(A_a^{-1} S_s^{-1} B_b^{-1}(x-t)\right)$                  (1)

where, the normalization factor a^−3/4 ensures energy preservation in L²(R²).

The bendlet coefficients are then obtained by computing the inner product between the input signal and each bendlet function,

$\left(f, \Psi_{a, b, s, t}\right)$              (2)

which measures the presence of features at a specific scale, orientation, curvature, and spatial location. Eqs. (1) and (2) represent the process that is repeated for all combinations of the parameters a, s, b, and t. The resulting set of coefficients forms a multiscale, multi-orientation, and curvature-sensitive representation of the signal, providing detailed geometric information about edges and curved structures within the image, where ‘a’ represents anisotropic scaling parameter, captures features at multiple resolutions, s gives directional information, b gives curved structures, and t represents localizing the transform to specific regions of the image.

Figure 4 represents the flowchart of the Bendlet transform Algorithm. The bendlet transform is a second-order shearlet-based framework used to represent images with curved, discontinuous contours. While classical shearlets efficiently represent [25] anisotropic features such as edges with straight boundaries, bendlets incorporate curvature modeling to improve the sparse representation of curved singularities.

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Figure 4. Flowchart of the Bendlet transform algorithm

The Shearlet transform is defined as:

$\varphi_{a, s, t}(x)=a^{-\frac{3}{4}} \varphi\left(A_a^{-1} S_s^{-1}(x-t)\right)$                 (3)

Step 1:

The basic wavelet theory is defined by 1D:

$\varphi_{a, t}(x)=a^{-\frac{1}{2}} \varphi\left(\frac{x-t}{a}\right)$                   (4)

Step 2:

The same idea applies in 2D:

$A_a=\left(\begin{array}{cc} a & 0 \\ 0 & a^{\frac{1}{2}} \end{array}\right)$                (5)

$x_1$ direction scales by a

$x_2$ direction scales by $a^{-\frac{1}{2}}$

The determinant is given by:

$\operatorname{det}\left(A_a\right)=a . a^{\frac{1}{2}}=a^{\frac{3}{2}}$

Step 3:

General rule for Matrix Scaling:

$\emptyset_A(x)=\emptyset\left(A^{-1} x\right)$                 (6)

Then:

$\left\|\emptyset_A\right\|^2 L^2=\int_{R^2}\left|\emptyset\left(A^{-1} x\right)\right|^2$                   (7)

Now perform a change of variable:

$y=A^{-1} x \rightarrow x=A y$                (8)

The Jacobian gives:

$d x=|\operatorname{det}(A)| d y$                (9)

Therefore,

$\left\|\emptyset_A\right\|^2 L^2=|\operatorname{det}(A)| \int|\emptyset(y)|^2 d y$                 (10)

$\left\|\emptyset_A\right\|^2 L^2=|\operatorname{det}(A)|\|\emptyset\|^2 L^2$                   (11)

To preserve energy, multiply by:

$|\operatorname{det}(A)|^{-1 / 2}$

Step 4:

Apply this to $A_a$:

$\operatorname{det}\left(A_a\right)=a^{\frac{3}{2}}$

The normalization factor must be:

$\left|\operatorname{det}\left(A_a\right)\right|^{-\frac{1}{2}}=\left(a^{\frac{3}{2}}\right)^{-\frac{1}{2}}=a^{-\frac{3}{4}}$               (12)

Step 5: Add shearing

The shearing matrix is given by:

$S_s=\left(\begin{array}{ll}1 & s \\0 & 1\end{array}\right)$                  (13)

Now,

$\operatorname{det}\left(S_s\right)=1$

Shearing does not change area. Therefore, it does not affect normalization.

Step 6: Add translation

$x \rightarrow x-t$

Final derived equation is:

$\emptyset_{a, s, t}(x)=\left|\operatorname{det}\left(A_a\right)\right|^{-1 / 2} \emptyset\left(A_a{ }^{-1} S_s{ }^{-1}(x-t)\right)$                 (14)

$\emptyset_{a, s, t}(x)=a^{-3 / 4} \emptyset\left(A_a{ }^{-1} S_s{ }^{-1}(x-t)\right)$                  (15)

3.2 Whale Optimization algorithm

The Whale Optimization algorithm (Figure 5) is a technique to solve single- objective optimization problems. It will utilize three operators for the simulation: search for prey, encircling prey, and Bubble-net foraging behavior. The algorithm starts with the initialization of a random population of agents (whales), including the control parameters, such as the maximum number of iterations, the linearly decreasing coefficient a, and the spiral constant b. In each iteration, search agents update their positions relative to the effective solution among them and are identified as X, which is denoted as Prey.

A random number, PPP, is generated between 0 and 1 to decide the whale’s movement strategy. If the random number p < 0.5, the algorithm checks the magnitude of A.

If |A| < 1, the whale updates its position toward the best solution, by calculating a random value within an interval 0 to 1is called the exploitation phase. This mechanism is called the shrinking encircling mechanism. If |A| ≥ 1, the whale moves towards a randomly selected whale, enabling Exploration of the search space.

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Figure 5. Flowchart of Whale Optimization algorithm

If the random number P ≥ 0.5, then it updates its position by calculating the distance between the whale and the prey, which is called the spiral equation, to simulate the bubble net attacking behaviour. After the fitness value is calculated, the best solution is calculated. As a result, the optimal solution is the one with the lowest fitness value of all search agents. This process repeats until the termination condition is met, at which point the algorithm outputs the best solution obtained.

3.3 Mathematical model for Whale Optimization algorithm

Key Steps and Equations

1. Encircling prey

Always, whales choose the best solution and move around it. The equation is given by:

$\vec{D}=\left|\vec{C} . \vec{X}^*(t)-\vec{X}(t)\right|$                 (16)

$\vec{X}(t+1)=\vec{X}^*(t)-\vec{A} . \vec{D}$                 (17)

where,

$\vec{X}^*(t)$ is the best prey

$\vec{X}(t)$ is the current whale position

$\vec{A} . \vec{D}$ is the coefficient vectors

t is the current iteration

The calculation of $\vec{A}$ and $\vec{C}$:

$\vec{A}=2 \vec{a} . \vec{r}-\vec{a}$

$\vec{C}=2 . \vec{r}$

2. Bubble-net attack

Steps used by Whales: shrinking encircling mechanism and spiral updating position.

Shrinking Encircling Mechanism

It is similar to the encircling mechanism, but the condition for moving closer to the prey is:

$\vec{A}<1$

Spiral updating position

The whales move around the prey, which looks like a helix-shaped path:

$\vec{X}(t+1)=\overrightarrow{D^{\prime}} . e^{b l} . \cos (2 \pi l)+\vec{X}^*(t)$                  (18)

where,

$\overrightarrow{D^{\prime}}=\left|\vec{X}^*(t)-\vec{X}(t)\right|$                 (19)

b is a constant defining a spiral shape

l is a random number

The total probability p is:

$\vec{X}(t+1)=\left\{\begin{array}{c}\vec{X}^*(t)-\vec{A} . \vec{D}, \quad P<0.5 \\\vec{D}^{\prime} . e^{b l} . \cos (2 \pi l)+\vec{X}^*(t), \quad P \geq 0.5\end{array}\right.$

3. Search for prey

When $|\vec{A}|>1$, the whales move away from the value:

$\vec{D}=\left|\vec{C} . \vec{X}_{{rand }}-\vec{X}(t)\right|$                (20)

$\vec{X}(t+1)=\vec{X}_{ {rand }}-\vec{A} . \vec{D}$                 (21)

$\vec{X}_{ {rand}}$ is a random whale position.

Table 1 represents the Hypertuning parameters of the Whale optimization algorithm with Population Size, Maximum Iterations, Control Parameter, Coefficient Vector, Spiral Constant, and Random Number.

In Table 1, the values of hypertuning parameters of the whale optimization algorithm are based on convergence stability, feature selection efficiency, and computational complexity by using the trial and error method.

Table 1. Hypertuning parameters

3.4 Customized Convolutional Neural Network

The Convolutional Neural Network (CNN) designed for brain tumors with MRI images is shown in Figure 6. It takes an input of MRI images via a Conv2D layer with 32 filters for extracting features like edges and textures. Maxpooling is applied to reduce spatial dimensions, Batch Normalization can stabilize and speed up the training process, and it drops out (0.25) in order to prevent overfitting.

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Figure 6. Architecture of a fully connected neural network

This architecture uses convolutional blocks for two times repeatedly. These blocks can have Conv2D layers at different paths, Maxpooling, Batch Normalization, and the outputs combined by using a concatenation layer. The multi path structure enables the model to capture features from different scales, like Inceptional style design. After concatenation, a deeper feature extraction can be done using 5 × 5 and 3 × 3 Conv2D layers and 128 filters, Maxpooling, Batch Normalisation, and Dropout. The results are then flattened to a one-dimensional vector and passed into a fully connected layer, where it can include 256 neuron layers and 0.5 Dropout for regularisation. At last, we can get images with 4 categories: Normal, Glioma, Pituitary or Meningioma, which are classified by the output layer using a SoftMax activation function. Overall, this model integrates convolutional feature extraction, Multiscale learning, and a regularization technique in order to ensure precise multiclass brain tumor classification.