# Optimization Methods of Compressively Sensed Image Reconstruction Based on Single-Pixel Imaging
## 3 Methods
### Binary Sensing Matrix Optimization  
According to the introduction in Section 2, it is required that the elements of sensing matrix must be binary in single-pixel camera, and the traditional binary sensing matrices usually used in single-pixel camera are 0-1 binary sparse matrix and part Hadamard matrix. Thus, in this subsection, our proposed sensing matrix optimization method is based on the traditional random binary sensing matrices. We focus on seeking a more optimized binary sensing matrix that can minimize the mutual coherence between sensing matrix and sparsifying basis and further improve the accuracy of signal reconstruction. Since there may be both positive and negative elements in binary measurement matrices, such as the Hadamard matrix, we divide \(\Phi\) into the positive part \(U\) and the negative part \(V\), so \(\Phi\) is denoted as:  
\[\Phi=U-V,u_{i,j}\geq 0,v_{i,j}\geq 0, \tag{13}\]  
Figure 5: The wavelet coefficients constraint matrix. where \(u_{i,j}=\left(\phi_{i,j}\right)_{+}\) and \(v_{i,j}=\left(-\phi_{i,j}\right)_{+}\), \(i=1,2,\ldots,m\) and \(j=1,2,\ldots,n\). \(\left(\phi_{i,j}\right)_{+}\) is the positive-part operator, such as \(\left(\phi_{i,j}\right)_{+}=\max\left\{0,\phi_{i,j}\right\}\). \(u_{i,j}\), \(v_{i,j}\) and \(\phi_{i,j}\) are the elements in the matrices \(U\), \(V\) and \(\Phi\) respectively. From the model (8), one can know that, due to the coupling of sensing matrix and sparsifying matrix, it is very complicated to directly solve Equation (8) and also difficult to guarantee the construction accuracy of the sensing matrix. In order to construct a more optimized sensing matrix, we propose a new method of solving Equation (8) by adding a constraint term of approximate equivalent dictionary \((\text{matrix})\Theta\approx\hat{\Phi}\Psi\), for ensuring higher accuracy in constructing sensing matrix. Therefore, the following optimization problem is further obtained by updating Equation (8):  
\[\hat{\Phi}=\underset{\hat{\Phi}}{\text{argmin}}\left\|\Theta^{H}\Theta-I \right\|_{F}^{2}\text{, s.t. }\left\|\Theta-\hat{\Phi}\Psi\right\|_{F}^{2}\leq\xi, \tag{14}\]  
where \(\xi\) is the upper bound of the approximate sensing matrix term and \(\hat{\Phi}\) is the estimated solution of \(\Phi\). Equation (14) is a convex constrained optimization problem. We convert the constrained optimization problem to an unconstrained optimization problem by lagrangian method and propose a new mathematical model based on Equation (14), as expressed in Equation (15):  
\[\hat{\Phi}=\underset{\hat{\Phi}}{\text{argmin}}\Gamma\!\left(\hat{\Phi}\right) =\frac{1}{2}\left\|\Theta-\hat{\Phi}\Psi\right\|_{F}^{2}+\tau\left\|\left(\hat {\Phi}\Psi\right)^{T}\!\hat{\Phi}\Psi-I\right\|_{F}^{2}, \tag{15}\]  
so that the constraint becomes a penalty. For a proper choice of \(\tau\), the two problems (14) and (15) are equivalent, where \(\tau\) is the weighting factor chosen empirically for balancing the two items of Equation (15). \(F(\hat{\Phi})\) is the loss function (or objective function) that needs to be solved via seeking the minimum, so Equation (15) is an unconstrained convex optimization problem. The term \(\left\|\Theta-\hat{\Phi}\Psi\right\|_{F}^{2}\) is based on the approximate solution of \(\Theta=\Phi\Psi\) and makes it easier to solve \(\Phi\) by balancing \(\left\|\Theta-\hat{\Phi}\Psi\right\|_{F}^{2}\) and \(\left\|\left(\hat{\Phi}\Psi\right)^{T}\!\hat{\Phi}\Psi-I\right\|_{F}^{2}\). Our proposed new model (15) not only can ensure smaller mutual coherence between sensing matrix and decomposition matrix by the cost \(\left\|\left(\hat{\Phi}\Psi\right)^{T}\!\hat{\Phi}\Psi-I\right\|_{F}^{2}\), but also is the accuracy guarantee of \(\Phi\) by solving the penalty term \(\left\|\Theta-\hat{\Phi}\Psi\right\|_{F}^{2}\). For the new model (15) with the specific two terms, the descent direction of \(F(\hat{\Phi})\) is not only the gradient direction but also the direction of gradient projection, which better ensures that our method is able to approximate the optimal solution of \(\Phi\).