# Sharpening of Worldview-3 Satellite Images by Generating Optimal High-Spatial-Resolution Images
## 2 Materials and Methods
### Spectral Distortion Reduction Method of Sharpened Images  
Generally, high-spatial-resolution satellite images provide panchromatic images and multispectral images that have a fourfold difference in spatial resolution. Because the existing pansharpening techniques were developed for high-spatial-resolution satellite images whose spatial resolution differs by fourfold, they may introduce additional spectral information distortion when applied to Worldview-3 satellite images, which have greater differences in spatial resolution. Thus, this study aimed to develop and apply a technique that could reduce the spectral distortion based on the spectral difference between the final sharpened image and the original image. Figure 3 shows a flowchart describing the spectral distortion reduction method.  
Figure 3: Workflow of the spectral distortion technique.  
The spectral distortion reduction method has two phases: The first phase generates images with reduced distortion, and the second phase produces the final sharpened images by injecting high-frequency information. The first phase of the spectral distortion reduction method is performed based on the difference between the sharpened image and the original image. The sharpened image is transformed into an image with the same number of pixels as the original image, and an image that has the distortion information is produced through the difference between the two images. The image containing the distortion information is defined by Equation (17):  
\[\mathbf{D}_{L}=\mathbf{SWIR}-\widehat{\mathbf{SWIR}}_{L} \tag{17}\]  
In Equation (24), \(\mathbf{SWIR}\) refers to the original SWIR image, \(\widehat{\mathbf{SWIR}}_{L}\) refers to the sharpened SWIR image converted into the size of the original SWIR image, and \(\mathbf{D}_{L}\) refers to the image that has the distortion information with size of SWIR image.  
The generated image containing the distortion information is then transformed into an image with the same number of pixels as the sharpened image, thereby creating the image \(\mathbf{DR}\), which is used to remove distortion from the sharpened image through differencing with the original sharpened image. The distortion-removed image \(\mathbf{DR}\) is generated via Equation (18):  
\[\mathbf{DR}=\widehat{\mathbf{SWIR}}-\mathbf{D} \tag{18}\]  
where \(\mathbf{D}\) is the resized version of \(\mathbf{D}_{L}\), which is the resampled image converted into the size of \(\widehat{\mathbf{SWIR}}\). Because the spectral distortion-removed image generated in the first phase is produced based on image differencing, spatial information such as boundaries between objects in the image is likely to be distorted. In the second phase, high-frequency information is extracted from the original sharpened image and injected into the spectral distortion-removed image, thereby producing the final spectral distortion-removed-sharpened image. The method for extracting the high-frequency information from the original sharpened image is similar to the MRA-based technique, in which the result of applying the MTF filter to the original sharpened image is subtracted from the original sharpened image to extract the high-frequency information. This extraction is defined in Equation (19):  
\[\mathbf{HF}=\widehat{\mathbf{SWIR}}-\widehat{(\mathbf{SWIR}}_{L})_{H} \tag{19}\]  
In Equation (19), \(\widehat{(\mathbf{SWIR}}_{L})_{H}\) refers to the SWIR image converted into the size of the sharpened image after the MTF filter is applied to the original sharpened image, and \(\mathbf{HF}\) refers to the high-frequency information extracted through differencing between the original sharpened image and \(\widehat{(\mathbf{SWIR}}_{L})_{H}\).  
Finally, the extracted high-frequency information is injected into the spectral distortion-removed image, thereby producing the final sharpened image \(\mathbf{SWIR}\). The final sharpened image can be calculated using Equation (20):  
\[\widehat{\mathbf{SWIR}}=\mathbf{DR}+\mathbf{HF} \tag{20}\]