# Optimization Methods of Compressively Sensed Image Reconstruction Based on Single-Pixel Imaging
## 3 Methods
### Imaging Method in Our Single-Pixel Camera  
We have built a CS-based single-pixel imaging system (single-pixel camera) in our laboratory and reconstructed two-dimensional imaging scenes by the methods of CS-based signal processing. Figure 1a,b respectively show the schematic diagram and a physical view of our single-pixel imaging system. As shown in Figure 1, our single-pixel imaging system consists of: the imaging lens 1, the Digital Mirror Device (DMD, V-7001, ViAIUX, Chemnitz, Germany, operating at 22.7 kHz/s), the mirror, the light collection lens 2, the Silicon Photomultiplier (SPM) Detector (C13369, HAMAMASTSU, Shizuoka, Japan, 320 nm to 900 nm, photoelectric sensitivity at \(1.0\times 10^{9}\) V/W), the Analog Digital Converter (ADC, PCI50641, TDEC, Chengdu, China, sampling frequency at 50 MHz) and the computer. The lens of our single-pixel camera are designed for distant scenes, and the imaging objects are distant buildings. The SPM detector is very sensitive to optical signals with detection ability of photon level,which can detect valid signals quickly and accurately and it is beneficial to improve the signal-to-noise ratio of signal detection. Our single-pixel camera's specific imaging process is: firstly, the lens 1 collects the light of the distant scene (equivalent to the original signal \(x\)) and projects the light onto DMD, which means that the image plane of the imaging scene falls exactly on the DMD surface; then the computer encodes the row masks of sensing matrix \(\Phi\) to the DMD scanning sequence, and DMD performs spatial light modulation on the image plane, which means that DMD picks the necessary pixel light and filters out the unnecessary pixel light (equivalent to the signal sensing measurement process of \(\Phi\) in CS) as shown in Figure 2; then the sampled pixel lights are reflected by the mirror and collected by the lens 2 into the photodetector (SPM); the SPM converts optical signals into electric signals, and then ADC converts analog electric signals to digital signals (the digital signals are equivalent to the measured signals \(y\)); finally our computer reconstructs the imaging scene (equivalent to \(\hat{x}\)) from the digital signals \(y\) through CS reconstruction algorithms, which means that the imaging scene is recovered from the small sampled data by single-pixel imaging. It should be noted that DMD is a spatial light modulator (SLM) with digital micromirror arrays and facilitates the hardware implementation and engineering application of the sensing matrix in the CS-based imaging system. The DMD sequence indicates the elements distribution status in binary sensing matrix. Due to the hardware limitation, as shown in Figure 2, every micromirror unit of DMD arrays only has binary states with the two flipping angles of +12 degrees and \(-\)12 degrees, and the two flipping angles are corresponding to the \"1\" and \"0\" (or \"\(-\)1\") binary states, respectively, in the sensing matrix, so only deterministic binary sensing matrices can be used in the single-pixel camera, such as the 0-1 binary sparse matrix with the elements of only \"0\" and \"1\" and the part Hadamard matrix with the elements of only \"+1\" and \"-1\". Figure 1: (a) Schematic diagram, (b) experimental set-up of our single-pixel imaging system. Figure 2: DMD modulates the spatial light and and the flipping micromirrors on DMD arrays (this picture has been modified according to our content on the basis of the original one, Figure 3 of [60]). In the 0-1 binary sensing matrix, the element \"0\" means that the corresponding pixel micromirror of DMD is off (\(-12\) degrees) removing the unnecessary pixel light, and the element \"1\" indicates the corresponding pixel of DMD is on (+12 degrees) sampling the necessary pixel light. The sampled signal value Y is detected by the photodetector, as shown in Figure 1a. However, for the part Hadamard matrix, it is very hard for DMD to physically realize the \"-1\" state of the Hadamard matrix directly.