# Optimization Methods of Compressively Sensed Image Reconstruction Based on Single-Pixel Imaging
## 3 Methods
### DWT-Based Decomposition Optimization  
The sparsity of signals is directly related to the signal sampling compression rate (SCR) and the signal reconstruction quality. This wouldn't mean that there must be K non-zero elements in sparse signals and all the remaining elements are 0 [61]. If all the coefficients of this signal were arranged from large to small in the sparse domain and the arranged coefficients were approximately an exponential decay distribution of approaching to zero, the signal would be also called an approximately sparse signal or a compressible signal [61]. DWT is usually used to compress signals, the coefficients distribution of non-sparse signals in DWT domain is very close to the exponential decay distribution, as shown in Figure 3. Figure 3a is a random non-sparse column signal of length 256, and the non-sparsecolumn signal is transformed to the sparse coefficients distribution by DWT (the wavelet decomposition basis), as shown in the blue curve of Figure 3b. One can see that the wavelet transform can make the non-sparse signal become more sparse from Figure 3a to Figure 3b. As shown in the blue and red curves respectively of Figure 3b, the wavelet coefficients are gradually decreasing and are very close to the exponential decay distribution. Based on the above analysis, we propose a new constraint condition of the positive exponential decay distribution of which the first term is 1, and this new constraint condition is applied to constrain the existing wavelet coefficients. The curve of this constraint function and the corresponding discrete histogram of this exponential decay are shown in Figure 4, in which the red solid curve is an exponential constraint distribution, and the blue columns are the constraint factors of wavelet coefficients of the corresponding sequence position. The constraint factors of blue columns are all positive real number less than 1 except the first term being 1. In other words, the wavelet coefficients are multiplied by the discrete exponential decay distribution with the first term being 1, and this constraint condition can make the wavelet coefficients distribution get more sparse and closer to the actual and real exponential decay. Figure 4: The exponential decay function and the corresponding discrete histogram. Figure 3: (**a**) The original random non-sparse column signal, (**b**) the coefficients distribution in DWT domain of the original column signal. Because the coefficients in DWT domain show a sparse distribution of approximately exponential decay, we use DWT to sparsify the non-sparse signal. In order to make wavelet coefficients more sparse and closer to a true exponential decay distribution, we design a diagonal constraint matrix with its diagonal elements exponentially decreasing as the constraint condition, as shown in Figure 5. This constraint matrix is denoted as \(W\) and the diagonal elements of \(W\) are \((p)^{i-1}\) in which \(p\) is a parameter that can adjust the constraint degree, \(0<p<1\), \(i\) is the \(i\)-th diagonal element and \(n\) is the length of the original signal. Therefore, the constraint matrix can be expressed by the following equation as:  
\[W=diag(p^{0},p^{1},\ldots,p^{i-1},\ldots,p^{n}),0<p<1,i=1,2,\ldots,n. \tag{10}\]  
In Equation (10), the first term is 1, and all the remaining elements are positive and less than 1 with the exponential decay distribution, so this constraint matrix can perform a constraint role of forcing the decaying wavelet coefficients to distribute more exponentially and more sparsely.