The Fusion of MRI and CT Medical Images Using Variational Mode Decomposition
Abstract
:1. Introduction
2. Proposed Methodology
 VMDbased image decomposition;
 A fusion strategy depending on the LEM;
 Synthesizing the fused image.
Algorithm 1 
$\mathrm{Let}\mathrm{us}\mathrm{consider}\mathrm{the}\mathrm{IMFs}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{image}\mathrm{as}{\mathrm{IMFs}}_{A}^{i}$$,\mathrm{and}\mathrm{the}\mathrm{sec}\mathrm{ond}\mathrm{image}\mathrm{as}{\mathrm{IMFs}}_{B}^{i}.\mathrm{The}$ $\mathrm{local}\mathrm{information}L{E}_{\alpha}\left(x,y\right)$$\mathrm{of}IMF{s}_{\alpha}^{i}(\alpha =A,B)$ is evaluated using the following steps. $\mathrm{Input}:\mathrm{Decomposed}\mathrm{modes}\mathrm{of}\mathrm{images}{\mathrm{IMFs}}_{A}^{i}$$,{\mathrm{IMFs}}_{B}^{i}$. $\mathrm{Output}:\mathrm{Enhanced}\mathrm{decomposition}\mathrm{modes}{\mathrm{F}}^{i}{}_{IMF{s}_{A,B}}(x,y)$. $\mathrm{Step}1:\mathrm{Calculate}\mathrm{the}\mathrm{local}\mathrm{information}LE{M}_{\alpha}\left(x,y\right)$$\mathrm{of}\mathrm{individual}\mathrm{modes}{\mathrm{IMFs}}_{\alpha}^{i}(\alpha =A,B)$
$${\mathrm{LEM}}_{\alpha}\left(x,y\right)={\displaystyle \sum _{i=1}^{w}{\displaystyle \sum _{j=1}^{w}{\left[IMF{s}_{\alpha}^{i}\left(x+i,y+j\right)\right]}^{2}\times {W}_{k}(i,j)}}$$
$${\mathrm{W}}_{k}=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$$
$${\mathrm{L}}_{\alpha}\left(x,y\right)=\mathrm{max}\left\{LE{M}_{\alpha}\left(x+i,y+j\right)1\le i,j\le 3\right\}$$
$${X}_{1}\left(x,y\right)=\{\begin{array}{c}1,\mathrm{if}{L}_{A}(\mathrm{x},\mathrm{y}){L}_{B}(\mathrm{x},\mathrm{y})\\ 0,\mathrm{otherwise}\hfill \end{array}$$
$${X}_{2}\left(x,y\right)=\{\begin{array}{c}1,\mathrm{if}{L}_{B}(\mathrm{x},\mathrm{y}){L}_{A}(\mathrm{x},\mathrm{y})\\ 0,\mathrm{otherwise}\hfill \end{array}$$
$${\mathrm{F}}^{i}{}_{IMF{s}_{A,B}}={X}_{1}(x,y)\times IMF{s}_{A}^{i}(x,y)+{X}_{2}(x,y)\times IMF{s}_{B}^{i}(x,y)$$

Algorithm 2 
Input: Image A (MRI), Image B (CT). Output: The fused image F. Step 1: Image decomposition using VMD: $\mathrm{Employ}\mathrm{VMD}\mathrm{on}\mathrm{the}\mathrm{source}\mathrm{images}(\mathrm{A}\mathrm{and}\mathrm{B})\mathrm{to}\mathrm{obtain}{\mathrm{IMF}}_{\mathrm{S}}$ which are represented as
$$VMD(A)=\left\{IMF{s}_{A}^{1},IMF{s}_{A}^{2}\dots IMF{s}_{A}^{i}\right\}.,i=(1,2,\dots N);\phantom{\rule{0ex}{0ex}}VMD(B)=\left\{IMF{s}_{B}^{1},IMF{s}_{B}^{2}\dots IMF{s}_{B}^{i}\right\}.,i=(1,2,\dots N)$$
$\left(\mathrm{a}\right)\mathrm{Estimate}\mathrm{the}\mathrm{local}\mathrm{information}LE{M}_{\alpha}\left(x,y\right)$$\mathrm{from}\mathrm{each}\mathrm{sub}\mathrm{band}IMF{s}_{\alpha}^{i}(\alpha =A,B)$ using Equation (5). $\left(\mathrm{b}\right)\mathrm{Consider}\mathrm{the}\mathrm{maximum}\mathrm{value}{\mathrm{L}}_{\alpha}\left(x,y\right)$$\mathrm{of}LE{M}_{\alpha}\left(x,y\right)$ by Equation (6). $\left(\mathrm{c}\right)\mathrm{Evaluate}\mathrm{the}\mathrm{binary}\mathrm{decision}\mathrm{weight}\mathrm{maps}{X}_{1}\left(x,y\right)$$,{X}_{2}\left(x,y\right)$ with Equations (7) and (8). $\left(\mathrm{d}\right)\mathrm{Fuse}\mathrm{the}\mathrm{decomposed}\mathrm{modes}{\mathrm{F}}^{i}{}_{IMF{s}_{A,B}}(x,y)$ using Equation (9). Step 3: Reconstruct the fused image by summing all the fused subbands obtained from Step 2.
$$F={\displaystyle \sum _{i=1}^{N}{F}^{i}{}_{\alpha}(x,y)},i=1,\dots N$$

3. Results and Discussion
3.1. Subjective Assessment
3.2. Objective Assessment
4. Conclusions and Future Scope
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Image Fusion Types  Fusion Methods  Advantages  Drawbacks  

Spatial domain  Average, minimum, maximum, morphological operators [11], Principal Component Analysis (PCA) [14], Independent Component Analysis (ICA) [29]  Easy to implement. Computationally efficient  Reduces the contrast, produces brightness or color distortions. May give desirable results for a few fusion datasets.  
Transform domain  Pyramidal methods  Contrast Pyramid [30], Ratio of the lowpass pyramid [31], Laplacian [19]  Provides spectral information  May produce artifacts around edges. Suffer from blocking artifacts 
Wavelet transform  Discrete wavelet transform (DWT) [15], Shift invariant discrete wavelet transform (SIDWT) [32], Dualtree complex wavelet transform (DcxDWT) [20]  Provides directional information  May produce artifacts around edges because of shift variant nature. Computationally expensive and demands large memory.  
Multiscale geometric analysis (MGA)  Curvelet [24], Contourlet [33], Shearlet [34], Nonsubsampled Shearlet transform (NSST) [28]  Provides the edges and texture region  Loss in texture parts, high memory requirement, demands high run time. 
Metrics  Methods  

VMDAVG  VMDMAX  VMDMIN  VMDLEM  
EI  48.439  58.322  36.487  71.751 
MI  4.384  4.376  3.486  4.391 
VIFF  0.335  0.397  0.063  0.428 
${Q}_{P}^{AB/F}$  0.307  0.356  0.198  0.443 
SSIM  0.599  0.232  0.563  0.621 
AG  4.845  5.714  3.735  6.973 
RMSE  0.0296  0.005  0.036  0.020 
PSNR  15.926  14.553  15.869  18.580 
Metrics  Methods  

ASR  CVT  DTCWT  MSVD  CSMCA  NSST  Proposed Method  
EI  85.184  91.417 (1)  88.853  77.183  87.219  81.907  90.390 (2) 
MI  3.948 (2)  3.548  3.656  3.490  3.811  3.703  4.079 (1) 
VIFF  0.321  0.290  0.280  0.344 (2)  0.319  0.267  0.406 (1) 
${Q}_{P}^{AB/F}$  0.535  0.478  0.500  0.427  0.536 (2)  0.373  0.538 (1) 
SSIM  0.563  0.376  0.499  0.548  0.629 (2)  0.520  0.697 (1) 
AG  8.561  9.140 (1)  8.933  8.332  8.674  8.368  9.008 (2) 
RMSE  0.034  0.034  0.034  0.034  0.035  0.027 (2)  0.020 
PSNR  16.328  16.749  17.166  13.28  17.393 (2)  13.976  21.342 (1) 
Metrics  Methods  

ASR  CVT  DTCWT  MSVD  CSMCA  NSST  Proposed Method  
EI  67.026  79.944 (2)  75.086  64.169  70.435  75.318  80.087 (1) 
MI  4.279  3.904  4.030  4.227  4.346 (1)  4.116  4.339 (2) 
VIFF  0.272  0.254  0.249  0.286  0.297 (2)  0.241  0.356 (1) 
${Q}_{P}^{AB/F}$  0.472  0.421  0.435  0.392  0.481 (1)  0.421  0.480 (2) 
SSIM  0.593  0.276  0.413  0.301  0.537  0.600 (1)  0.599 (2) 
AG  6.662  7.887 (2)  7.421  6.812  6.877  7.471  7.980 (1) 
RMSE  0.029  0.029  0.029  0.028  0.029  0.024 (2)  0.021 (1) 
PSNR  16.857  17.171  17.720  15.804  17.892 (1)  13.981  17.794 (2) 
Metrics  Methods  

ASR  CVT  DTCWT  MSVD  CSMCA  NSST  Proposed Method  
EI  51.347  63.877  58.355  49.732  51.899  65.474 (2)  65.802 (1) 
MI  4.186  3.878  3.995  4.090  4.284 (2)  4.214  4.549 (1) 
VIFF  0.356  0.362  0.365  0.348  0.412 (2)  0.340  0.484 (1) 
${Q}_{P}^{AB/F}$  0.465 (2)  0.418  0.431  0.380  0.461  0.446  0.478 (1) 
SSIM  0.674 (2)  0.338  0.507  0.417  0.663  0.590  0.694 (1) 
AG  5.065  6.231  5.719  5.197  5.045  6.349 (1)  6.326 (2) 
RMSE  0.028  0.029  0.029  0.026  0.028  0.022 (2)  0.018 (1) 
PSNR  17.396  17.268  17.649  16.392  18.644 (1)  14.096  18.024 (2) 
Metrics  Methods  

ASR  CVT  DTCWT  MSVD  CSMCA  NSST  Proposed Method  
EI  57.800  64.531  61.820  50.850  58.592  62.404  64.582 
MI  3.666  3.360  3.446  3.694  3.657  3.740  3.830 
VIFF  0.376  0.362  0.358  0.365  0.401  0.364  0.498 
${Q}_{P}^{AB/F}$  0.541  0.483  0.500  0.399  0.531  0.439  0.542 
SSIM  0.651  0.350  0.503  0.614  0.634  0.586  0.657 
RMSE  0.029  0.029  0.029  0.029  0.029  0.022  0.020 
AG  5.772  6.390  6.148  5.427  5.771  6.217  6.412 
PSNR  16.803  16.972  17.242  16.000  17.757  16.021  20.291 
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Polinati, S.; Bavirisetti, D.P.; Rajesh, K.N.V.P.S.; Naik, G.R.; Dhuli, R. The Fusion of MRI and CT Medical Images Using Variational Mode Decomposition. Appl. Sci. 2021, 11, 10975. https://doi.org/10.3390/app112210975
Polinati S, Bavirisetti DP, Rajesh KNVPS, Naik GR, Dhuli R. The Fusion of MRI and CT Medical Images Using Variational Mode Decomposition. Applied Sciences. 2021; 11(22):10975. https://doi.org/10.3390/app112210975
Chicago/Turabian StylePolinati, Srinivasu, Durga Prasad Bavirisetti, Kandala N V P S Rajesh, Ganesh R Naik, and Ravindra Dhuli. 2021. "The Fusion of MRI and CT Medical Images Using Variational Mode Decomposition" Applied Sciences 11, no. 22: 10975. https://doi.org/10.3390/app112210975