# Image Interpolation with Regional Gradient Estimation
## 2 Fundamental Issues and Ideas  
Image interpolation refers to estimating unknown high-resolution values based on known low-resolution pixels. We denote the low-resolution (LR) image as \(I_{\text{LR}}\) (\(i\), \(j\)), with a size of \(M\times N\), and the high-resolution (HR) image with a size of \(2M\times 2N\) as \(I_{\text{HR}}\) (\(2i\), \(2j\)). Figure 1 shows the first step of the interpolation process and demonstrates the relationship between the \(I_{\text{LR}}\) (\(i\), \(j\)) pixels and the \(I_{\text{HR}}\) (\(2i\), \(2j\)) pixels in the \(2\times 2\) image interpolation case. To obtain HR image pixel values, we duplicate the LR pixels at \((i,j)\) to \((2i-1,2j-1)\) in the HR image, as shown in Equation (1). Next, we need to construct unknown pixels around known pixel points in \(I_{\text{HR}}\) (\(2i\), \(2j\)). \[I_{\text{LR}}(i,j)=I_{\text{HR}}(2i-1,2j-1),1\leq\ i\ \leq M,1\ \leq\ j\ \leq\ N \tag{1}\]  
To evaluate the interpolation method's effectiveness, we first downsample the original image; then, we interpolate the downsampled image and, finally, compare the interpolated image to the original image. Currently, the most common downsampling methods are direct-extraction downsampling and function downsampling. Direct-extraction downsampling means keeping the odd rows and columns while deleting the remaining pixels from the HR image; this process yields an LR image, as shown in Figure 1a. Its pixelrelationship is shown in Equation (1). Function downsampling makes use of MATLAB's \"imresize\" function, which includes three methods: \"nearest\", \"linear\", and \"bicubic\". When acquiring low-resolution images, direct-extraction downsampling is frequently used in the literature [18; 19; 20; 21] on image interpolation methods. The advantage of direct-extraction downsampling is that the obtained image pixels are part of the original image, allowing the original image information to be preserved to the greatest extent possible. This is consistent with the central question of image interpolation, that is, how to create an image as close to the original as possible with only pixels of the original image? The data obtained by function downsampling are calculated from the original image's pixel values, and this process destroys the original image's information. We performed bicubic interpolation on LR images obtained through various downsampling methods and used PSNR [29] to objectively evaluate the interpolated images. Table 1 summarizes the findings. We can see from Table 1 that the LR image obtained by direct extraction displayed superior PSNR performance after interpolation compared to the others, demonstrating that this method can preserve the greatest amount of original image information. Figure 2 depicts the results of the interpolation of the four downsampling methods used on the same test image, labeled \"house\". The luminance information of direct-extraction downsampling and nearest downsampling can be seen to be relatively complete. In contrast, the luminance information of bilinear downsampling and bicubic downsampling suffered from some loss. In Section 4, the low-resolution image is obtained by direct-extraction downsampling. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Interlaced Row/Column** & **Nearest** & **Bilinear** & **Bicubic** \\ \hline PSNR & 31.13 & 27.56 & 26.09 & 25.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A comparison of different downsampling methods with respect to the PSNR (dB). Figure 1: Illustration of the interpolation. The dots denote the known pixels, and the circles denote the missing pixels. (**a**) low-resolution (LR) image; (**b**) high-resolution (HR) image. Figure 2: Interpolation results of different downsampling methods. (**a**) test image; (**b**) direct extraction; (**c**) nearest; (**d**) bilinear; (**e**) bicubic. Figure 3a shows a relatively simple image, and Figure 3b shows the edges considered by the standard nonlinear image interpolation method.