# The Fusion of MRI and CT Medical Images Using Variational Mode Decomposition
## 2 Proposed Methodology
Equation (3) can be further solved using an alternating direction method of multipliers (ADMM) [41]. Finally, the estimate of the \(l^{th}\) sub-band is computed as [44]:  
Figure 3: IMFs obtained after VMD decomposition: (**a**) MRI image, (**b**) and (**c**-**g**) are approximation and detail images of (**a**), respectively. (**h**) CT image, (**i**) and (**j**-**n**) are approximation and detail images of (**h**), respectively. \[\hat{b}_{l}{}^{n+1}(\omega)=(\hat{x}(\omega)-\sum\limits_{j\
eq l}\hat{b}_{j}( \omega)+\frac{\hat{\lambda}(\omega)}{2})\frac{1}{1+2\alpha(\omega-\omega_{l})^{2}} \tag{3}\]  
Similarly, the center frequency is updated as:  
\[\omega_{l}^{n+1}=\frac{\int\limits_{0}^{\infty}\omega\Big{|}\hat{b}_{l}(\omega) \Big{|}^{2}d\omega}{\int\limits_{0}^{\infty}\Big{|}\hat{b}_{l}(\omega)\Big{|}^ {2}d\omega} \tag{4}\]  
In this work, we used the two-dimensional (2D)-VMD [45] method to decompose the MRI and CT images. As stated above, 2D-VMD is a helpful method in extracting useful information such as edges and curves from the source images. Furthermore, VMD is a reliable method to deal with noisy images. Therefore, it can improve the quality of the fusion process even without employing additional preprocessing techniques. B. Fusion Strategy Depending on LEM  
As discussed before, the VMD adaptively decomposes the input images into bandlimited sub-bands called IMFs. Indeed, these IMFs characterize the image features of source images. To highlight and extract relevant features in the fused image, we require appropriate fusion rules. As discussed in Section 1, many fusion rules [46], such as minima, maxima, averaging, and PCA, have been widely explored for this purpose over the past few years. Among them, minima and maxima cause brightness distortions, averaging rule blurs the fused image, and PCA degrades the spectral information [15]. Furthermore, the fusion rules mentioned above may produce low spatial resolution issues [47]. The LEM-based [47] fusion rule is adopted to tackle the issues discussed above in this work. We have demonstrated the influence of these fusion rules visually in Figure 4 and quantitatively in Table 2. As shown in Figure 4, the VMD with LEM fusion rule achieves visually satisfying results compared to VMD with other fusion rules. Similarly, as shown in Table 2, the fusion metric values calculated over 10 data sets proved the efficacy of the chosen LEM fusion rule. Figure 4: Visual quality analysis of various fusion rules on MRI-CT image pair. (**a**) MRI image, (**b**) CT image, (**c**) VMD-AVG, (**d**) VMD-MAX, (**e**) VMD-MIN (**f**) VMD-LEM. The technical details of the LEM fusion rule are discussed as follows. The principal idea behind using LEM is to extract and preserve vital information with the help of local information constraints from both the images pixel by pixel [47]. The entire process of LEM is described in Algorithm 1. ```
Let us consider the IMFs of the first image as IMFs\({}_{A}^{i}\), and the sec end image as IMFs\({}_{\text{B}}^{i}\). The local information \(LE_{n}(x,y)\) of \(IMFs_{n}^{i}(\alpha=A,B)\) is evaluated using the following steps. Input : Decomposed modes of images IMFs\({}_{A}^{i}\), IMFs\({}_{\text{B}}^{i}\). Output : Enhanced decomposition modes \(\text{F}^{i}_{IMFs_{\alpha,k}}(x,y)\). Step 1 : Calculate the local information \(LEM_{n}(x,y)\) of individual modes IMFs\({}_{n}^{i}(\alpha=A,B)\) \[\text{LEM}_{n}(x,y)=\sum_{i=1}^{w}\sum_{j=1}^{w}\left[IMFs_{n}^{i}(x+i,y+j) \right]^{2}\times W_{k}(i,j)\] (5) where, \(W_{k}\) is given by: \[W_{k}=\left[ \begin{array}{ccc}1&1&1\\ 1&1&1\\ 1&1&1\end{array} \right]\] Step 2 : Choose the maximum value in the local information \(LEM_{n}(x,y)\) \[\text{L}_{n}(x,y)=\max\{LEM_{n}(x+i,y+j)|1\leq i,j\leq 3\}\] (6) Step 3: Calculate the binary decision weight maps \[X_{1}(x,y)=\left\{ \begin{array}{ll}1,&\text{if }L_{A}(x,y)>L_{B}(x,y)\\ 0,&\text{otherwise}\end{array} \right.\] (7) \[X_{2}(x,y)=\left\{ \begin{array}{ll}1,&\text{if }L_{B}(x,y)>L_{A}(x,y)\\ 0,&\text{otherwise}\end{array}