# Optimization Methods of Compressively Sensed Image Reconstruction Based on Single-Pixel Imaging
## 5 Analysis and Discussion  
These above experiments have well tested the superior performance of our proposed methods in optimizing traditional wavelet decomposition basis and constructing new binary measurement matrices based on the conventional binary sensing matrix. According to the experimental results, one can know that our methods have better performance in extracting information from the original signals and restoring the original signals from the sampled information. Our proposed methods of optimizing the traditional wavelet decomposition basis and the conventional binary sensing matrices can favorably suppress the noise of reconstructed images better compared with before optimization, as shown in Figures 8 and 9. In addition, the following four points are worth being analyzed and discussed:  
* Firstly, because we applied the method of column-by-column scanning to reconstruct the image, there are some column visual artifacts in the reconstructed image, which is the most obvious in Figure 8a,b and Figure 9a,b before optimization. Our optimization methods have effectively  
Figure 10: The evolution relationship between the imaging SNRs of reconstructed scenes and different compression sampling rates (SCR) for **(a)** 0-1 binary sensing matrix and **(b)** part Hadamard binary sensing matrix (”00” denotes the traditional imaging method before optimization; “01” denotes the imaging method after only optimizing wavelet decomposition basis; “10” denotes the imaging method after only optimizing binary sensing matrices; “11” denotes the imaging method after optimizing both wavelet decomposition basis and binary sensing matrices simultaneously).  
reduced the generation of column artifacts and reconstructed the images with more balanced structure and clearer texture, as shown in Figures 8c-h and 9c-h.
* Y1, it is required that Y1 is measured only once at the beginning when all the effective micromirror pixel units of DMD are set to on. This means that the measurement times of Hadamard matrix with the added measurement Y1 is only once more than that of 0-1 binary matrix. DMD works at 22.7 kHz/s, which makes a full 2D image be reconstructed quickly and well for about 0.72 s. Thus, the calculation method of Y = 2Y2
- Y1 will not extra increase more additional noise.
* Thirdly, in the process of signal sampling, the amount of information extracted from the original signal increases with the rise of sampling compression rate (SCR), which makes it easier to recover the original signal from more sampled information. This is why the improvement degree of imaging SNR in our single-pixel camera tends to decrease slowly as sampling rate increases, as shown in Figure 10. Thus, we have to consider the trade-off between SCR and recovery accuracy in our imaging experiments, and 0.25 is an appropriate middle sampling rate that can give full consideration to balancing the imaging speed and the imaging accuracy.
* Fourth, one can generally know that, in image reconstruction, the higher SNR is, the more difficult it is to further improve SNR. In the comprehensive experiments, we think adding the constraint matrix could affect the coherence between sensing matrix and wavelet decomposition basis to some extent, because the sensing matrix and the wavelet decomposition basis may be coupled from each other. Consequently, the simultaneous optimization (\"11\") of both wavelet decomposition basis and sensing matrix is a little superior than either one optimization method alone (\"01\" or \"10\"). However, compared with the traditional imaging method (\"00\") before optimization, the imaging SNRs have significantly been enhanced by our methods of whether separately (\"01\" or \"10\") or simultaneously (\"11\").