A ghost imaging (GI) system, as a new-type imaging system different from traditional imaging systems, has gained much attention and shown its imaging capabilities in related fields. GI is an imaging mode that requires computational reconstruction to obtain image information. For this type of imaging mode, the uncertainty of the retrieved image information cannot be obtained by direct measurement and is related to the specific reconstruction algorithm. Therefore, relevant research on error uncertainty analysis is required. Existing studies mostly give reconstruction error distributions corresponding to the intensity correlation algorithms. For algorithms of other types, it is hard to obtain a theoretical result since there is no explicit formula for the reconstructed information, and studies on related estimation methods have not been reported. Although there are some theoretical results about the reconstruction error of compressed sensing (CS) algorithms, they cannot be directly applied to the uncertainty estimation of GI, since there are differences between the speckle patterns in GI and the measurement matrix used in the CS theoretical derivation. In this study, we use the Bootstrap method to estimate the uncertainty of image information reconstruction. Bootstrap technology can effectively give the uncertainty estimation map of the entire reconstructed image in practical imaging scenarios where the theoretical expression cannot be given, and there is no original image for reference. It is expected that the proposed method can contribute to estimating the uncertainty map of image reconstruction with different reconstruction algorithms in practical GI systems.

In GI, detection signals Iti in the object arm and Iri(rr) in the reference arm are used together to retrieve the image information T(rr). There are various reconstruction algorithms, and they can be all expressed as Eq. (1). In this study, the uncertainty of the reconstructed T^(rr) is estimated, and the Bootstrap method is applied. The Bootstrap method is a technique in statistical analysis, and its core idea is to use the original sample of size n as a pseudo-population for resampling. Its steps are as follows. First, the computer is used to generate random numbers, and Bootstrap samples (y1*,y2*,?,yB*) of size n are independently drawn. Second, for each Bootstrap sample yb*, the corresponding parameter estimation is calculated, which results in a set of estimators T^b*,b=1,2,?,B. Third, the standard deviation of T^b* is calculated by using Eq. (3) to approximately estimate the standard error of T^. Finally, the approximate confidence interval at a confidence level of 1-α can be obtained by using the Bootstrap-t method. In thermal-source GI, the observed signals obtained by each sampling are mutually independent and identically distributed due to the random fluctuation characteristics of thermal light. Therefore, if each sampled signal Iri,Iti is considered as a sample, the obtained data Iri,Iti (i is from 1 to n) from a total number of n actual samplings are independent and identically distributed samples. Taking the data as the original sample and considering the reconstruction results T^(rr)=fIr(rr),It as the estimator, the uncertainty and confidence interval of the reconstructed image can be described by the non-parametric Bootstrap method.

To verify the effectiveness of the Bootstrap method, we first apply it to estimate the standard error of the reconstructed image from the intensity correlation algorithm, and a theoretical formula has been derived. The imaging target is selected as a hole with Gaussian distribution transmittance [Fig. 2(a)]. The result is shown in Fig. 3. It can be seen that the standard error map estimated by the Bootstrap method is highly consistent with the theoretical one. They have backgrounds of almost the same level. The difference in the region around the peak is mostly due to the statistical fluctuation in the estimation, but the estimated error map still has characteristics that the error of the central region is higher than that of the surrounding regions. Then, we use Bootstrap to estimate the result reconstructed from the CS algorithm. Specifically, the two-step iterative shrinkage thresholding (TwIST) algorithm with total variation regularization is used for reconstruction here. The result is shown in Fig. 4. It can be seen that the estimated standard error map is well correlated with the true absolute error map, and they have similar characteristics. In addition, a map that shows a confidence interval of 99% is shown in Fig. 4(b). In order to more accurately and quantitatively give the error distribution map, the bias and absolute error of the CS algorithm are estimated via the corresponding Bootstrap variants. Specifically, quantities in Eqs. (11) and (12) are estimated, and the result is shown in Fig. (5). It can be seen that the estimated bias and absolute error maps are consistent with the true ones in both the shape and the specific values. This shows that the proposed estimation can give a quantitative description of the reconstructed error map.

We propose to apply the non-parametric Bootstrap method to estimate the uncertainty of GI image information reconstruction. Simulation results show that the estimated standard error of the intensity correlation algorithm by Bootstrap is consistent with the theoretical one, which illustrates the reliability of the Bootstrap method in estimating the standard error of the GI algorithm. Furthermore, the Bootstrap method has been applied to estimate the uncertainty of the CS-based reconstruction. The TwIST algorithm is taken as an example, and the standard error map of the reconstructed image information is estimated, which has well demonstrated the confidence level of the reconstructed result in different positions of the target image. In addition, the bias and absolute error of the reconstructed T^ are estimated with the corresponding Bootstrap variant, which demonstrates that the Bootstrap results can quantitatively describe the image information error of the original sample reconstruction. Our research provides an effective solution for estimating the uncertainty of the reconstruction algorithms of GI, which is especially suitable for cases where there is no real reference image, and an explicit expression cannot be given. Subsequent work is applying the Bootstrap method to estimate the uncertainty of other reconstruction algorithms (e.g., deep neural network algorithm) of GI.