# The Fusion of MRI and CT Medical Images Using Variational Mode Decomposition
## 2 Proposed Methodology  
Our proposed work aims to integrate the details of the soft tissue and dense bone structure provided by MRI and CT medical imaging technologies into a unique image. For this, we have proposed a multimodal medical image fusion based on a blend of VMD and LEM, as shown in Figure 2.  
The main steps involved in our fusion methodology are:  
1. VMD-based image decomposition;
2. A fusion strategy depending on the LEM;
3. Synthesizing the fused image. 1. VMD-Based Image Decomposition  
The traditional decomposition approaches, such as wavelets [37; 38], BEMD [39], and EWT [40], suffer from various problems such as boundary distortions and mode-mixing. With these issues, we may fail to achieve an appropriate fusion result. To address these problems, we employed VMD [41], a robust adaptive decomposition approach, highlighting meaningful details in the form of sub-images. The VMD finds applications in image denoising [42] and texture decomposition [43]. VMD is a non-stationary and adaptive signal processing technique. Unlike EMD and its variants, VMD is not a recursive analysis approach, and it decomposes the signal/image into bandlimited sub-bands based on its frequency content. This work uses VMD to obtain distinct and significant IMFs from the source images (MRI and CT). The derived IMFs reduce mode-mixing and boundary distortions, which are the major concerns in the above mentioned transform domain methods. With this VMD decomposition, we can extract  
Figure 2: Proposed MRI-CT medical image fusion scheme. prominent edge information. Initially, we decomposed the input images into six IMFs, which are illustrated in Figure 3.  
From Figure 3, it can be observed that the first IMF ((b) and (i)) captures prominent information from the source images, whereas the remaining IMFs encompass the line and edge information. We can note from Figure 3 that as the mode number increases, the visual details are not significant. Mathematical Details of VMD:  
The main goal of VMD is to subdivide an input signal () into a specific number of sub-bands (IMFs or Modes) (\(b_{l}\)), and each sub-band is bandlimited to specific frequencies in the spectral domain (Fourier domain) by maintaining sparsity. Each of the sub-bands is bandlimited to its center frequencies. VMD involves the following steps to get the bandlimited sub-bands [41]:  
1. For each sub-band, its analytical counterpart needs to be computed using Hilbert transform to get the one-sided frequency spectrum;  
2. An exponential is used to mix with each mode to shift its frequency spectrum to the baseband;  
3. Finally, the bandwidth of the mode estimates using the squared \(L^{2}\)-norm of the gradient. The constrained variational problem can be represented as below. \[ \begin{split}\underset{\{b_{l}\},\{\omega_{l}\}}{\min}\left\{ \sum\limits_{l}\|\partial_{t}\Big{[}\Big{(}\delta(t)+\frac{j}{\pi l}\Big{)}*b_{ l}(t)\Big{]}e^{-j\omega_{l}t}\|_{2}^{2}\right\}\\ \{b_{l}\}\text{ and }\{\mathbf{w}_{l}\}\end{split}  \tag{1}\]  
where \(l^{th}\) indicates the \(l^{th}\) sub-band and its center frequency, respectively. \(\delta(t)\) represents the Dirac distribution, \(*\) is the symbol of the convolution. The constrained problem in Equation (1) is solved using the quadratic penalty term and Lagrangian multipliers \(\lambda\) to make it an unconstrained problem given in Equation (2). \[L(\{b_{l}\},\{\omega_{l}\},\lambda)=\alpha\underset{l}{\sum}\|\partial_{t} \Big{[}\Big{(}\delta(t)+\frac{j}{\pi t}\Big{)}*b_{l}(t)\Big{]}e^{-i\omega_{l}t }\|_{2}^{2}+\|x(t)-\underset{l}{\sum}b_{l}(t)\|_{2}^{2}+\left\langle\lambda,x (t)-\underset{l}{\sum}b_{l}(t)\right\rangle \tag{2}\]  
where \(L\) represents augmented Lagrange matrix function, \(\alpha\) is the penalty factor parameter, \(\lambda\) indicates the Lagrange multiplier, and \(x(t)\) is the input signal. Now the solution of Equation (1) can be computed as the saddle point of Equation (2) using the method called an alternating direction method of multipliers (ADMM).