# Optimization Methods of Compressively Sensed Image Reconstruction Based on Single-Pixel Imaging
## 4 Experiments
### Parameter \(p\)  
Setting in DWT Optimization  
In our proposed optimization method of wavelet decomposition basis, one can know that the size of \(p\) in Equation (10) directly determines the restraint degree of the restraint matrix on wavelet coefficients. The distribution of wavelet coefficients is regulated by adjusting the value of \(p\). If \(p\) is too large, the affect of constraint matrix constraining the wavelet coefficients is weakened, and the sparsity of wavelet coefficients cannot be effectively improved, which is not good for improving the quality of image reconstruction. However, if \(p\) is too small, the constraint degree of constraint matrix in DWT domain is higher, which would increase the computational difficulty of our algorithm operation and cause information loss, this is also not helpful to reconstruct images. Therefore, choosing a proper \(p\) value that performs well is very important for the optimization of wavelet decomposition basis, and directly affects the results of image reconstruction.  
In order to choose a relatively optimal \(p\) value, when \(m=64\) and \(n=256\), images _House_ and _Building_ were reconstructed separately with different \(p\) values under the 0-1 binary sensing matrix and the part Hadamard sensing matrix, then we determined an appropriate parameter \(p\) according to the simulation experimental results. We obtained the PSNR results of images reconstruction with different \(p\) values by our method optimizing DWT under 0-1 binary matrix and part Hadamard matrix, respectively. Since the 0-1 binary matrix and the part Hadamard matrix were generated randomly, in order to avoid experiments' randomness and make experimental results more stable and reliable, the averages of five experiments' results were calculated and the average experimental results are shown in Figure 7.  
Figure 7: Relationships between PSNR and \(p\) in reconstructing images _Building_ and _House_ for **(a)** 0-1 binary sensing matrix and **(b)** part Hadamard sensing matrix by our method optimizing wavelet decomposition basis.  
As shown in Figure 7, whether for the 0-1 binary sensing matrix or the part Hadamard sensing matrix, the PSNR of reconstructed images increases first and then decreases as \(p\) increases, and the optimal range of \(p\) value is about [0.995, 0.997]. Thus, after giving full consideration to the balance of images _Building_ and _House_ and the balance of two types of binary sensing matrices, \(p\) value is set to 0.996 in our actual imaging algorithm.  
As shown in Figure 7, after adding the constraint condition of exponential decay to the wavelet decomposition basis, the PSNRs of reconstructed images are effectively improved. For both of the 0-1 binary sensing matrix and the part Hadamard sensing matrix, the PSNRs of reconstructed image _Building_ are improved by about 1.5 dB, and the PSNRs of reconstructed image _House_ are improved by about 2 dB. When \(p\) is set to 0.996, the optimized DWT improves PSNRs more significantly by about 2 to 2.5 dB. The experiments of Figure 7 also implicitly demonstrate the effectiveness of our method in improving the quality of compressively sensed images reconstruction.