# Optimization Methods of Compressively Sensed Image Reconstruction Based on Single-Pixel Imaging
## 4 Experiments  
In this section, we describe experiments conducted to test the performance of our optimized wavelet decomposition basis and optimized binary sensing matrix for compressively sensed image reconstruction, compared with the traditional wavelet decomposition basis and the conventional binary measurement matrices that are usually used in single-pixel imaging respectively. Finally our proposed methods were applied to improve the traditional imaging methods of our single-pixel camera by conducting practical imaging experiments. The imaging resolution of our single-pixel camera is \(256\times 256\). In order to better balance the sampling speed and the image reconstruction accuracy, we set the usually-used sampling rate to 0.25 in our imaging system with \(m=64\) and \(n=256\). Because our single-pixel camera used the imaging method of column-by-column scanning via DMD, we measured the each column of images or imaging scenes separately for 64 times (64 rows of the sensing matrix) by the binary sensing matrix of size \(64\times 256\), and then reconstructed images' columns of size \(256\times 1\) one by one, finally added the 256 reconstructed columns back to the full image of size \(256\times 256\). Our DMD worked at 22.7 kHz/s, so the imaging time of column-by-column was \(\frac{1\times 64\times 256}{22700}\approx 0.72\) seconds. The imaging time of scanning was very short, which could reconstruct a full 2D image quickly and well. Since our single-pixel camera experiment system was designed to take images of distant scenes such as buildings, the imaging field of view is some limited. Before the practical imaging experiments, in order to determine the proper empirical parameters in our algorithm, we conducted the simulation experiments in reconstructing images that are similar to our real imaging scenes. The images used in our simulation experiments are images _Building_ and _House_ that are similar to the actual imaging scenes, as shown in Figure 6, and the two images can well simulate the real imaging scenes. In the experiments, 0-1 binary sparse sensing matrix and part Hadamard sensing matrix are used, and the image reconstruction algorithm is OMP algorithm in which the iteration number is \(m\), as mentioned in Sections 2.1 and 3.1. The 0-1 sparse binary sensing matrix was produced by the function \"SparseRandomMtx(m, n, m/2)\" of Matlab, in which m denotes the row number, n denotes the column number and m/2 denotes the number of \"1\" in every column, so the elements \"0\" and \"1\" are equally weighted with the same 50% respectively in the 0-1 binary matrix. The Hadamard matrix sequence was generated by the function \"hadamard(n)\" of Matlab, and the Hadamard matrix \(H_{a}\) is a square matrix and \({H_{a}}^{T}H_{a}=nI\), where [\(n\)\(n\)]=size(\(H_{a}\)) and \(I=\) eye(\(n\),\(n\)), and the part Hadamard matrix was constructed by extracting the corresponding rows or columns from Hadamard square matrix. The algorithms operated in the environment of MATLAB 2017b on a standard PC with a 2.20 GHz Intel Core i5 CPU and 4 GB memory, running on the Windows 10 operating system. Figure 6: (**a**) Original image _Building;_ (**b**) original image _House._  
In the simulation experiments, we used the peak signal to noise ratio (PSNR) and the structural similarity (SSIM) to evaluate the accuracy and quality of image reconstruction, PSNR and SSIM are defined as:  
\[PSNR(dB)=20\log\frac{255}{\frac{1}{HW}\sum_{i=1}^{H}\sum_{j=1}^{W}\left(\hat{x} \left(i,j\right)-x\left(i,j\right)\right)^{2}}, \tag{31}\]  
\[SSIM=\frac{(2\mu_{x}\mu_{\hat{x}}+c_{1})(2\alpha_{xt}+c_{2})}{\left(\mu_{x}^{2 }+\mu_{\hat{x}}^{2}+c_{1}\right)\left(\sigma_{x}^{2}+\sigma_{\hat{x}}^{2}+c_{2 }\right)}, \tag{32}\]  
where \(x\) is the original image, and \(\hat{x}\) is the reconstructed image. \(H\) and \(W\) are the height and the width of the image respectively. \(\mu_{x}\) and \(\mu_{\hat{x}}\) are respectively the mean values of \(x\) and \(\hat{x}\).