# Optimization Methods of Compressively Sensed Image Reconstruction Based on Single-Pixel Imaging
## 1 Introduction  
According to information theory, one can know that the traditional signal sampling and signal recovery must meet the Shannon-Nyquist theorem [1], which states that to recover the original signal without distortion from a sampled discrete signal, it is required that the signal sampling frequency must be at least twice the bandwidth of the original signal. However, with the rapid development of information technology, the efficient transmission and effective processing of massive amounts of data have put more and more increasing requirements on signals' sampling compression rate (SCR), processing speed and signals' recoverability [2; 3; 4]. In 2006, compressive sensing (compressed sensing, CS) theory was proposed by [PERSON] et al. [1; 4; 5; 6]. They invented an effective way of economically converting analog signals into digital forms to compress the original signals. Compressivesensing is based on the premise that the original signal can be recovered and be sampled with fewer measurements than Nyquist sampling, and a compressed representation of the original signal is directly acquired by a compressed sensing measurement matrix without the intermediate process of Nyquist sampling [3, 4, 5, 6, 7]. The combination of signal sampling and signal compression is thus completed in one measurement process, which saves transmission and storage costs and reduces energy consumption and computational complexity [1, 6, 8, 9]. Finally, the original signal is reconstructed from the linear measurement data through corresponding reconstruction algorithms [1, 6]. Thus, compressive sensing greatly reduces the data amount of signal processing and breaks the sampling frequency limitation of the Nyquist sampling theorem. Because of the excellent performance of CS in the field of signal processing and image processing, compressive sensing has been widely applied in many fields since it was first proposed [7]. Some typical application research areas include radar imaging [10, 11, 12], images' recovery and wireless image sensor networks [13, 14], applications in classification problems [15, 16], biomedical signal processing [8, 17, 18, 19], such as nuclear magnetic resonance imaging (MRI) [20, 21], electrocardiographic (ECG) signal processing [22, 23] and single-pixel imaging [24, 25, 26, 27, 28]. For the single-pixel imaging, in 2008, Rice University invented a single-pixel imaging system (single-pixel camera) based on CS theory [25], which pioneered the application field of CS-based single-pixel imaging. In their research, a photodiode was used as the optical signal detector to measure the encoded information of an imaging scene, and they achieved CS-based single-pixel imaging for the first time. After Rice University, other researchers have contributed some works based on single-pixel imaging [26, 27, 28, 29]. In [26], [PERSON] et al., improved the quality of single-pixel imaging using digital microscanning. In [27], degradation factors added into the algorithm were used to avoid image blurring and achieve single exposure super-resolution compressive imaging. In [28], the authors used a dynamic single-pixel CS system to acquire super-resolution images. In [29], the performance of an imaging approach using two detectors simultaneously was evaluated for real-time video. Although a lot of works have contributed to optimize single-pixel imaging in image reconstruction algorithms and some important technologies, there still exist some problems in CS-based single-pixel imaging. On the one hand, for the image sparsifying basis, in order to be able to reconstruct images using compressive sensing, the imaging scene needs to be represented as a sparse combination of elements. Hence, if the scene is not sparse, it must first undergo some sparsifying transform, and discrete wavelet transform (DWT) is usually used as the signal decomposition basis [30, 31].