# Super-Resolution Reconstruction of an Array Lidar Range Profile
## 3 Experimental Results  
### Dataset Experiment and Result Analysis  
In this paper, the dataset (redwood-3 dscan) used in the literature [27] was used for quantitative evaluation, which was a range profile with a resolution of 640 \(\times\) 480 collected by PrimeSense Carmine camera (Made by PrimeSense, Tel Aviv, Israel). The experiments were verified on a computer with Intel(R) Core (TM) i7-7700 HQ and 16 GB memory, and MATLAB R2022a was used as the experimental simulation platform. In order to verify the reconstruction effect of the super-resolution method, the range profiles of the dataset were firstly downsampled by 1/2, 1/4, and 1/8 to simulate the low-resolution range profiles and then reconstructed by 200%, 400%, and 800%, respectively, and compared to bilinear interpolation, bicubic interpolation, POCS, and the POCS method reproduced in the literature [17]. The average gradient (\(AG\)) and edge strength (\(ES\)) were used to objectively evaluate the reconstructed range profile [28]. The \(AG\) is the average of the pixel gradient values in the image, and the gradient reflects the rate of change in the pixel intensity in the image. This indicator not only reveals the change trend of the entire image, but also captures the change in the contrast and  
Figure 2: Flowchart of super-resolution reconstruction based on POCS with an LK Gaussian pyramid. texture of tiny details, which is closely related to the clarity of the image, and the larger the value, the clearer the image, as in:  
\[AG=\frac{1}{M\times N}\sum_{i=1}^{M}\sum_{j=1}^{N}\sqrt{\frac{\left(\frac{\partial f }{\partial x}\right)^{2}+\left(\frac{\partial f}{\partial y}\right)^{2}}{2}} \tag{17}\]  
where \(M\) and \(N\) represent the number of horizontal and vertical pixels of the range profile, respectively. \(\frac{\partial f}{\partial x}\) is the horizontal gradient and \(\frac{\partial f}{\partial y}\) is the vertical gradient. \(ES\) is an important index to evaluate edge sharpness and intensity in images. An edge refers to an area where the strength of a pixel changes rapidly, often representing the outline or boundary of the target. The edge intensity is measured by measuring the gradient amplitude of the edge points in the image, and the larger the value, the better the retention effect of the edge information, as in:  
\[ES(i,j)=\sqrt{\
abla xf(i,j)^{2}+\
abla yf(i,j)^{2}} \tag{18}\]  
The first-order difference formula of the image along the \(x\) and \(y\) directions at point \((i,j)\) is as follows:  
\[\left\{ \begin{array}{l}\
abla xf(i,j)=f(i,j)-f(i-1,j)\\ \
abla yf(i,j)=f(i,j)-f(i,j-1)\end{array} \right. \tag{19}\]  
For the dataset experiment with the reference high-resolution range profile, we additionally introduced peak signal to noise ratio (\(PSNR\)) and structural similarity (\(SSIM\)) [21] to analyze the algorithm performance. The \(PSNR\) measures the quality between the original image and the enhanced image. A higher \(PSNR\) value indicates less degradation of the image. It is estimated by calculating the value of mean square error (\(MSE\)) as:  
\[MSE=\frac{1}{MN}\sum_{m=1}^{M}\sum_{n=1}^{N}\left(f^{(\alpha)}(m,n)-f(m,n) \right)^{2} \tag{20}\]  
\[PSNR=10\log\left(\frac{(L-1)^{2}}{MSE}\right) \tag{21}\]  
The \(SSIM\) is an indicator of the similarity between two images, which is defined as:  
\[SSIM(x,y)=\frac{\left(2\mu_{x}\mu_{y}+c_{1}\right)\left(2\sigma_{xy}+c_{2} \right)}{\left(\mu_{x}^{2}+\mu_{y}^{2}+c_{1}\right)\left(\sigma_{x}^{2}+ \sigma_{y}^{2}+c_{2}\right)} \tag{22}\]  
where \(\mu_{x}\) and \(\mu_{y}\) are the averages of \(x\) and \(y\), respectively; \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\) are the variances of \(x\) and \(y\), respectively; \(\sigma_{xy}\) is the covariance of \(x\) and \(y\); and \(c_{1}=(0.01L)^{2}\) and \(c_{2}=(0.03L)^{2}\) are constant numbers to stabilize the equation. The range of \(SSIM\) is \([0,1]\). When the two images are exactly the same, the value of \(SSIM\) is equal to \(1\).