# Single-Pixel Color Imaging Method with a Compressive Sensing Measurement Matrix
## 4 Experimental Results and Analysis
### Performance Comparison of Measurement Matrices  
A number of measurement matrices have been proposed. These matrices can be classified into two main categories: random and deterministic [30; 31; 32]. On the one hand, Random matrices are generated by identical or independent distributions such as Fourier matrix, Bernoulli and Hadamard sparse random matrices. Random matrices are easy to satisfy the RIP with high probability. However, because of the randomness, they present some drawbacks such as high computation and costly hardware implementation. On the other hand, deterministic matrices are constructed deterministically to have a small mutual coherence or satisfy the RIP. In this study, we designed a pseudo-random circulant measurement matrix based on chaotic sequence. It belongs to the deterministic measurement matrix. The advantage of the measurement matrix easily generates hardware on the basis of ensuring better reconstruction effect and is suitable for the single-pixel imaging system. In this study, two image quality models, namely, peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) [33], are used to conduct the analysis and comparison of results. PSNR is defined as following. PSNR = 10 \(\log_{10}\) ((\(2^{k}-1\))\({}^{2}\)/MSE),  
\[\text{MSE}\ =\ \frac{1}{mn}\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}||I(i,j)-K(i,j)||^{2} \tag{12}\]  
where \(m\times n\) is the image size, \(I\), \(K\) represent the target image and reconstructed image respectively, MSE represents the mean square error between two images, \(i\) and \(j\) represent the image row and  
Figure 1: Single-pixel color imaging system with color wheel. Figure 2: Single-pixel color imaging system with chromatic dispersion theory. column pixel position respectively, \(k\) is the number of bits of each sample value. The value of PSNR is inversely proportional to the degree of image distortion obtained by reconstruction. The magnitude of PSNR value is positively correlated with the quality of image reconstruction. PSNR is widely used objective measure of quality evaluation. SSIM (structural similarity index) is a measure of the similarity between two images. It is defined as following. \[l(x,y)=(2\mu_{x}\,\mu_{y}+c_{1})/({\mu_{x}}^{2}+{\mu_{y}}^{2}+c_{1}),\]  
\[c(x,y)=(2\sigma_{x}\,\sigma_{y}+c_{2})/({\sigma_{x}}^{2}+{\sigma_{y}}^{2}+c_{2}),\]  
\[s(x,y)=(\sigma_{xy}+c_{3})/(\sigma_{x}\,\sigma_{y}+c_{3}), \tag{13}\]  
\[SSIM=l(x,y)\times c(x,y)\times s(x,y), \tag{14}\]  
where \(x\) and \(y\) represent the target and reconstructed images respectively, \(\mu_{x}\) and \(\mu_{y}\) represent the mean value of images \(x\) and \(y\) respectively, \(\sigma_{x}\) and \(\sigma_{y}\) represent the variance of images \(x\) and \(y\) respectively, \(\sigma_{xy}\) is the covariance of images \(x\) and \(y\), and \(c_{1}\), \(c_{2}\), and \(c_{3}\) are constants as shown in Formula (15). To avoid the case where the denominator is 0, \(K_{1}=0.01\), \(K_{2}=0.03\), and \(L=255\) are generally used. The SSIM value range is [0, 1], and the larger the value is, the smaller the image distortion becomes. \[c_{1}=(K_{1}\times L)^{2},c_{2}=(K_{2}\times L)^{2},c_{3}=c_{2}/2, \tag{15}\]  
The last process aims to recover the original signal from a small set of measurement matrices [34; 35; 36; 37; 38; 39]. Because the TVAL3 algorithm has a higher reconstruction rate and better reconstruction effect, it is widely used as a recovery algorithm in the single-pixel imaging system [40]. In the comparison experiment, TVAL3 is therefore used as the recovery algorithm, and the resolution ratio of target pictures is \(128\times 128\). Bernoulli matrix, sparse random matrix, Walsh-Hadamard measurement matrix, LCS-MM, LCS-CMM and LCS-PRC-MM are compared. The results are shown in Figures 3 and 4.