A novel Gateaux derivatives with efficient DCNN-Resunet method for segmenting multi-class brain tumor

Abstract

In hospitals and pathology, observing the features and locations of brain tumors in Magnetic Resonance Images (MRI) is a crucial task for assisting medical professionals in both treatment and diagnosis. The multi-class information about the brain tumor is often obtained from the patient’s MRI dataset. However, this information may vary in different shapes and sizes for various brain tumors, making it difficult to detect their locations in the brain. To resolve these issues, a novel customized Deep Convolution Neural Network (DCNN) based Residual-Unet (ResUnet) model with Transfer Learning (TL) is proposed for predicting the locations of the brain tumor in an MRI dataset. The DCNN model has been used to extract the features from input images and select the Region Of Interest (ROI) by using the TL technique for training it faster. Furthermore, the min-max normalizing approach is used to enhance the color intensity value for particular ROI boundary edges in the brain tumor images. Specifically, the boundary edges of the brain tumors have been detected by utilizing Gateaux Derivatives (GD) method to identify the multi-class brain tumors precisely. The proposed scheme has been validated on two datasets namely the brain tumor, and Figshare MRI datasets for detecting multi-class Brain Tumor Segmentation (BTS).The experimental results have been analyzed by evaluation metrics namely, accuracy (99.78, and 99.03), Jaccard Coefficient (93.04, and 94.95), Dice Factor Coefficient (DFC) (92.37, and 91.94), Mean Absolute Error (MAE) (0.0019, and 0.0013), and Mean Squared Error (MSE) (0.0085, and 0.0012) for proper validation. The proposed system outperforms the state-of-the-art segmentation models on the MRI brain tumor dataset.

Graphical abstract

Appendices

Appendix A: Snakes

The geodesic active counter (GAC) model is used to retrieve objects from images (also called Snakes). GAC models are given in Eq. (30) by author [63]. Let, $W_{k}=W$ is function of counter interacts of the image, where k=1,2,...,k.

$$\begin{aligned} \varepsilon (\Gamma ) = \int _{0}^{1} g(\mid \bigtriangledown _{q}W (\Gamma _{q}) \mid ) dq \end{aligned}$$

(29)

Equation (29) has been describe in below Eq. (30)

$$\begin{aligned} \varepsilon (\Gamma )= & {} \int _{0}^{1} g\mid \bigtriangledown _{q}W \Gamma _{q}\mid ^{2} dq + w_{1} \int _{0}^{1} g \mid \Gamma _{q}\mid ^{2} dq \nonumber \\ {}{} & {} + w_{2} \int _{0}^{1} g \mid \Gamma _{q}\mid ^{2} dq \end{aligned}$$

(30)

where, $\Gamma ^{t} = \left\{ x \epsilon D:\Phi (q,t)=0 \right\}$. The idea is that g is small near object boundaries. The first term describes the external energy which pushes the contours toward the object’s border, whilst the second and third terms are associated with the internal energy that regulates the smoothing of the contours. So, getting Eq. (2).

$$\begin{aligned} \varepsilon (\Gamma )= & {} \int _{0}^{1}\frac{1}{2}w_{1}(q) \mid \Gamma _{qq} \mid ^{2} + \frac{1}{2}w_{2}(q)\mid \Gamma _{q} \mid ^{2}\nonumber \\ {}{} & {} + W(\Gamma (q))dq \end{aligned}$$

where $g(\bigtriangledown _{q}) = \frac{1}{1 + \gamma \mid \bigtriangledown _{q} \mid }$

As a result, the image and the prospective function $W:D \rightarrow R$ interact since it is a lower limit for some threshold and I(x) is the $\varepsilon (\Gamma )$.

Therefore, getting Eq. (3), $W(x)=\frac{1}{1+ \left\| \bigtriangledown Z \sigma *I(x))\right\| ^{2}}$

Now, the result reduces $\varepsilon$ value which attracts $\Gamma$ to the margins. The descent flow is a fourth order nonlinear parabolic equation (4) we get.

$$\begin{aligned} \frac{\delta \Gamma }{\delta t}= -(w_{2}\Gamma _{qq})_{qq} + (w_{1}\Gamma _{q})_{q} + \bigtriangledown W (\Gamma (q,t)) \end{aligned}$$

Appendix B: Brain tumor boundary detection

The simplification of Eq (8). Assume that $C (z,t): D \times [0, \alpha ) \rightarrow \mathbb {R}^{n}$ is a collection of planar arcs in the given image, and that $\sigma _{1}, \sigma _{2}$ are the mean intensities of I both outside and inside C(z,t) respectively. Then

$$\begin{aligned}{} & {} \sigma _{1} = \sigma _{1}(x,\omega ) = \frac{1}{\mid \omega \mid } \int _{\omega } I(z) dz \nonumber \\{} & {} \sigma _{2} = \sigma _{2}(x,\omega ) = \frac{1}{\mid \omega \mid } \int _{\omega ^{c}} I(z) dz \end{aligned}$$

(31)

$$\begin{aligned} \mid \omega \mid = \int _{\omega } dz \end{aligned}$$

(32)

where, $\omega ^{c}$ is exterior, and $\omega$ is interior of C(z,t). Based on the distance, C is able to distinguish between the background and foreground areas.

$$\begin{aligned} Sup E = Sup(\sigma _{1}- \sigma _{2}) \end{aligned}$$

(33)

Sup is Sup is C on $E^{1,2}(D)= \left\{ C \epsilon L^{2}(D);\bigtriangledown C \epsilon L^{2}(D)\right\}$ We rephrase Eq. (33) as a minimization issue in order to increase the applicability of its skills to a wide range of images. We also add an edge function f to enhance its capability for detection of boundary.

$$\begin{aligned} E^{'}(z, \omega ) = -\frac{1}{2} f(z)(\sigma _{1}-\sigma _{2})^{2} \end{aligned}$$

(34)

We include a regularization to enhance Eq. (34) even further $L = \oint _{c} dt$ to get Eq. (8) $E(z, \omega ) = -\frac{1}{2} f(z) (\sigma _{1} - \sigma _{2})^{2} + \beta \oint _\partial \omega dt$