For the problem of poor reconstruction quality and resolution degradation of underwater ghost imaging, an underwater ghost imaging method based on speckle degradation compensation was proposed to recover the target image degraded by the water body. Compared with ghost imaging in air medium, underwater ghost imaging has been studied by scholars in many aspects, such as the absorption effect of the water body, signal-to-noise ratio detection of the system, backward scattering noise, underwater illumination spot, and underwater turbulence. Image degradation and recovery methods based on underwater optical transmission models have been used in array detector optical imaging. However, there is no relevant study to analyze and solve the problem of degradation of underwater ghost imaging starting from the inherent optical properties of the water body. The scattering effect of the water body on the beam reduces the contrast of the speckles shining on the surface of the target and degrades the resolution, which deteriorates the intensity fluctuation characteristics of the target obtained by bucket detection, thus affecting the reconstruction quality of ghost imaging. Therefore, we hope to recover the underwater ghost imaging results affected by the water body through a method similar to deconvolution by introducing a point spread function (PSF) related to the intrinsic optical parameters of the water body.

In this research, the water body scattering degradation model was introduced into the ghost imaging image reconstruction to improve the image quality. First, the S-S (Sahu-Shanmugam) scattering phase function was linearly approximated in logarithmic coordinates in a small angular range (0.1°-5°). Then, the Hankel transform of the scattering phase function was substituted into the Wells modulation transfer function (MTF) model to obtain the modulation transfer function for image propagation in water. The MTF is a frequency domain expression of the PSF, which can better describe the degradation effect of real seawater on the scattering spot. The intrinsic optical parameters of the water body (such as scattering coefficient, absorption coefficient, and scattering phase function) and the imaging distance were used to construct this MTF. Finally, the obtained MTF was used for correcting the reference arm speckles, so as to compensate for the object arm speckle degradation of the target surface. Meanwhile, we represented the process of convolution PSF of speckle patterns as matrix multiplication. In addition, the role of the speckle degradation compensation method in the image reconstruction process was studied theoretically by the second-order correlation algorithm and the pseudo-inverse algorithm, respectively. Finally, the usefulness of the speckle degradation compensation method and the correctness of the theoretical model were verified by simulation and experiment.

In this study, the matrix form of the speckle degradation compensation method is derived theoretically. The mathematical nature of the correction compensation of the reference arm speckle before reconstruction by the second-order correlation algorithm or the pseudo-inverse algorithm is analyzed. Equation (16) shows that in the second-order correlation calculation, the reference arm speckle is convolved with the PSF of the water body equivalent to the image convolved with the PSF of the water body obtained by second-order correlation for the original non-degradation compensation. Therefore, this method, for second-order correlation reconstruction, will make the reconstruction effect doubly degrade. As shown in Eq. (17), the pseudo-inverse ghost imaging with speckle degradation compensation is essentially a method of deconvolution by obtaining the convolution kernel of the PSF of the water body from the optical parameters of the water body. If the correction compensation of reference arm speckles is consistent with the actual degradation of the object arm, the degradation of the water body can be better removed. The simulation results and experimental validation results are shown in Fig. (4) and Fig. (6), respectively. The reconstruction results of the second-order correlation algorithm with speckle degradation compensation deteriorate the image quality compared with the original second-order correlation algorithm. The image quality and resolution of the reconstruction results of the pseudo-inverse algorithm with speckle degradation compensation are significantly improved compared with the original pseudo-inverse algorithm.

In this study, an MTF of the water body that can describe the underwater speckle transmission is derived, and the reference arm speckle is corrected with the same degree of degradation as the object arm speckle, so as to compensate for the degradation of the object arm speckle. The method restores the congruence between the object arm speckle and the reference arm speckle and then performs the reconstruction calculation of the target image. Through theoretical analysis, simulation, and experiments, it is proved that the spot degradation compensation will aggravate the image degradation for the second-order correlated image reconstruction, while it can improve the image resolution and imaging quality for the pseudo-inverse reconstruction. The method has some degradation removal effect for pseudo-inverse algorithm and greedy algorithm based on least squares in underwater target image reconstruction. Unlike blind deconvolution, the accuracy of the method depends on the accuracy of the MTF or PSF of the water body, and the improvement of the image reconstruction quality characterizes the correctness of the derived MTF. The method is essentially a deconvolution method based on the scattering model of the water body, which generates ringing artifacts and noise amplification in the case of the low signal-to-noise ratio of bucket detection, making the reconstruction quality worse, which is also an important direction for subsequent research.