# An Integrated Dictionary-Learning Entropy-Based Medical Image Fusion Framework
## 3 Proposed Framework
### Gaussian-Filter Based Decomposition  
Gaussian filter (also known as Gaussian smoothing) is an effective method that is widely used in image processing to reduce image noise and reduce detail. Gaussian smoothing is often used as a pre-processing stage in computer vision algorithms in order to enhance image structures in  
Figure 1: The Proposed Image Fusion Framework.  
different scale-space representation [57; 58; 59]. Gaussian filter is a low pass filter that has good effect of reducing the image's high-frequency components. So Gaussian filter is widely used in image fusion for decomposing source images into low-frequency and high-frequency components.  
The proposed fusion method processes high-frequency and low-frequency components of source images separately. Then the fused results of high-frequency and low-frequency components are merged. The low-frequency components contain structural information, and contour information to constitute the base of an image. The high-frequency components give the details of an image. The high-frequency components are added upon the low-frequency components to refine the image. It is important to decompose high-frequency and low-frequency components of source images accurately and completely. The accuracy and completeness of decomposition directly affects the fusion result. Only accurate and complete image decomposition can ensure the fused image contains more detailed information. Gaussian filter can decompose the detailed information and noises of source image into high-frequency components [60]. The sparse-representation based solution not only denoises, but also enhances the detailed information of fused image.  
In this paper, a two-dimension Gaussian function is used for smooth. The Gaussian function is shown in Equation (1).  
\[G(x,y)=\frac{1}{2\pi\sigma^{2}}e^{-(x^{2}+y^{2})/2\sigma^{2}}\ \, \tag{1}\]  
where \(x\) is the distance from the origin in the horizontal axis, \(y\) is the distance from the origin in the vertical axis, and \(\sigma\) is the standard deviation of the Gaussian distribution. \(\sigma\) is variable, that can effect how much does it smooth the picture. The standard deviation \(\sigma\) of Gaussian filter ranges from 2 to 4 depending on the size and type of source images (e.g., \(\sigma=3\) is usually for over 256 \(\times\)256 size image). When applied in a smooth image, it produces a surface whose contours are concentric circles with a Gaussian distribution from the center point. Values from this distribution are used to build a convolution matrix which is applied to the original image. Each pixel's new value is set to a weighted average of that pixel's neighborhood [61]. When the Gaussian filter algorithm is implemented as a low-pass filter to smooth the input source image, low-frequency component \(\mathbf{I}_{l}\) of the input source image can be got. Then high-frequency component can be got by Equation (2).  
\[\mathbf{I}_{h}=\mathbf{I}-\mathbf{I}_{l}\ \, \tag{2}\]  
where \(\mathbf{I}\) is the input source image and \(\mathbf{I}_{h}\) is the high-frequency component. After high-frequency and low-frequency components of the source image are received, different fusion scheme is implemented to them, respectively.  
If it does not do the decomposition first, it may misselect the sparse coefficient, and even cause the accuracy of fused result [27]. Source images shown in Figure 2 are used in the comparison of non-decomposition and decomposition. Figure 3a,b show the results of non-decomposition and decomposition respectively. The same pixels marked in blue frame are selected from each result to do the detailed comparison. The result of non-decomposition is blur, but decomposition obtains the clear result.