The property of the measurement matrix has a great influence on the image reconstruction quality of single-pixel compressive imaging. Optimizing the measurement matrices is a core and crucial technology for single-pixel imaging. However, current optimization methods for measurement matrices often face the problems of local optimization and limited applicability. Additionally, existing analytical theories and methods based on the measurement matrix often fail to explain or predict the image reconstruction quality in many scenarios, and the quantitative relationship among measurement matrix characteristics, target properties, and image reconstruction results is unclear. For example, the reconstruction results vary obviously among different kinds of Hadamard encoding measurement matrices. Therefore, after combining optical imaging systems with compressive sensing theory, it has become an urgent issue for single-pixel compressive imaging to construct a characteristic function that can predict image reconstruction quality. We propose a characteristic function of high-quality image reconstruction for single-pixel compressive imaging to predict the imaging quality of targets with different sparsity, which is helpful for the optimal design of measurement matrices in single-pixel imaging systems.

Under the same sampling rate, the image reconstruction quality is significantly different for various kinds of Hadamard encoding measurement matrices, which can not be explained by existing compressive sensing theories. By combining compressive sensing theory with the characteristic parameters described in Ref. [23], the Gram matrix is obtained from the measurement matrix and then the relationship between the Gram matrix and the system's point spread function is clarified. Next, according to the point spread function and compressive sensing theory, four characteristic parameters are proposed, including the peak value of the strongest sidelobe, overlapped sidelobe peak value, spatial distance, and spectral cosine similarity. Based on these parameters, an image reconstruction characteristic function F(η) adopted for high-quality single-pixel compressive imaging is constructed. Meanwhile, by calculating the F(η) values of the random Hadamard encoding matrix in different sampling rates η and conducting data fitting, the relationship between the target's sparsity and the characteristic function is established. Furthermore, by changing the target's sparsity, sampling rate, and the type of encoding measurement matrices, the validity of the proposed characteristic function is verified by numerical simulations and experiments.

To demonstrate the validity of the proposed characteristic function, we conduct both numerical simulations and experiments based on the scheme in Fig. 1. Firstly, when the sampling rate η=0.6 is fixed, the sparsity thresholds for Natural, CC, RD, Random, and MP Hadamard encoding matrices are obtained and random grayscale point targets can be stably reconstructed at their respective sparsity thresholds Sε [Fig. 7(a)]. However, the sparsity threshold Sε for the Random Hadamard encoding matrix is much larger than that of the other four Hadamard encoding matrices. What's more, under S>Sε, Natural, CC, RD, and MP Hadamard encoding matrices can not recover the image of the slit shaped target [Figs. 7(b) and 7(c)]. Secondly, according to Fig. 6, under Sε=0.25000, the corresponding sampling rates η for the five kinds of Hadamard encoding matrices above are 0.89100, 0.88600, 0.86600, 0.72800, and 0.89100 respectively. Numerical simulations and experimental results demonstrate that random grayscale point targets can be perfectly reconstructed by all the five kinds of Hadamard encoding matrices when the target's sparsity S=0.25000 (Fig. 8). Additionally, when the sampling rate is η=0.728, only the random sequence Hadamard encoding matrix can accurately restore the radial target with the sparsity S=0.25000. Finally, the universality of the proposed characteristic function is further verified by Bernoulli random encoding matrices, Gaussian random encoding matrices, and Gaussian orthogonal encoding matrices in different representation bases (Tables 2 and 3, and Fig. 9). Meanwhile, Fig. 9 demonstrates that the relationship described by Equation (8) is valid for other common random encoding matrices, which means that the characteristic function can be employed as the objective function in optimizing measurement matrices for single-pixel compressive imaging systems.

Combined with the compressed sensing theory, four characteristic parameters based on the point spread function are proposed, including the peak value of the strongest sidelobe, overlapped sidelobe peak value, spatial distance, and spectral cosine similarity. A high-quality image reconstruction characteristic function of single-pixel compressive imaging is constructed and its validity is verified by numerical simulations and experiments. Both numerical simulation and experimental results demonstrate that the proposed characteristic function can not only explain the differences in single-pixel compressive imaging quality for Hadamard coding matrices with different sorting methods but also predict the image reconstruction results of a given measurement matrix. Additionally, the relationship between the proposed characteristic function and the target sparsity in high-quality image reconstruction is established. The characteristic function can serve as a criterion during the optimization of measurement matrices for single-pixel imaging.