# Single-Pixel Color Imaging Method with a Compressive Sensing Measurement Matrix
## 3 Design of Pseudo-Random Circulant Measurement Matrix Based on Chaotic Sequence
### Design of Pseudo-Random Circulant Measurement Matrix Based on Chaotic Sequence  
LCS-MM is combined with the Toeplitz matrix to construct a circulant measurement matrix (LCS-CMM), as indicated in Formula (10).  
\[A=\left[\begin{array}{cccc}a_{N}&a_{N-1}&\cdots&a_{2}&a_{1}\\ a_{N+1}&a_{N}&\cdots&a_{3}&a_{2}\\ \vdots&&&\vdots&\vdots\\ a_{N+M-1}&\cdots&\cdots&a_{M+1}&a_{M}\\ \end{array}\right]. \tag{10}\]  
\(A\) is an LCS-CMM, that consists of two elements, \(1\) and \(-1\), and is a deterministic measurement matrix of binarization.  
On the basis of the Toeplitz matrix structure, the RIP of order \(3k\) for every \(\delta_{3k}\in(0,1/3)\) is satisfied for an \(M\times N\) Bernoulli Toeplitz-structured matrix when \(M\geq c_{1}k^{3}\ln N/k\) with probability at least  
\[p=1-e^{-c_{2}M/k^{2}}, \tag{11}\]  
where constants \(c_{1},\ c_{2}>0\) exist that depend only on \(\delta_{3k}\)[28].  
After the combination of chaotic sequence with the Toeplitz matrix, simple number calculation is used to generate the first line, and a common cyclic shift is used to accomplish the generation of the entire measurement matrix. Compared with the method without combining circulant structure, the generated LCS-MM immensely reduces the pressure on hardware, and improved production speed of the measurement matrix. However, the recovery capability is relatively poor.  
LCS-CMM is improved to reduce matrix coherence. Compared with a matrix obtained from random exchange of lines and columns of an orthogonal matrix with any given matrix, incoherence is immensely improved [29]. Thus, the random circulant is introduced based on common Toeplitz matrix to maintain low coherence between the columns and between lines. Specifically, the elements in Line 2-M of the matrix are obtained through the random cycle of elements in the first line. However, the chaotic sequence is introduced in conducting the random circulant on deterministic matrix, to pseudo-randomly select the number of cycles each time. In this way, the pseudo-random circulant measurement matrix based on chaotic sequence (LCS-PRC-MM) is formed. The specific generation method is expressed as follows:  
Step 1: The required measurement matrix should be \(M\times N\) when the resolution ratio of the target recovered image is \(N\times N\), and the number of measurements is \(M\). The logistic chaos system used to generate sequence S1 in inverted order consists of \(N\) elements, and acts as the first line of measurement matrix, where \(\text{S1}=\{a_{N},a_{N-1},\cdots,a_{2},a_{1}\}\).  
Step 2: Circulate based on the Toeplitz matrix is conducted to generate line \(i\), with original value of \(2\), where \(1<i<N\).  
Step 3: Determine whether the vector of circulant line generated at Step 2 can act as Line h of the measurement matrix based on the value of the next element generated by logistic chaotic sequence(multiple cycles can be set). Line \(i\) generated through circulation is taken as line h of the measurement matrix, \(\text{h}=\text{h}+1\), when the condition is met, and the process moves to Step 4. It directly proceeds to Step 4 when the condition is not met. The original value of h is \(2\), \(2<\text{h}<M\);  
The measurement matrix utilizes the exterior determinacy and inherent randomness of chaotic sequence. In this way, the measurement matrix improves the image reconstruction capacity, and makes the camera adaptive in capturing images with a different sparsity by setting multiple of cycles.