Improving the Resolution of Correlation Imaging via the Fluctuation Characteristics

Abstract

The resolution is an important factor in evaluating image quality. In general, the resolution of correlation imaging is taken to the full width at half maximum (FWHM) of the point spread function (PSF) produced by the second-order correlation function. In this paper, we show that the resolution of correlation imaging can be improved by the fluctuation characteristic of the second-order correlation function. It is demonstrated both experimentally and theoretically that the resolution of the system can be drastically improved. We also prove that the FWHM of the PSF can be narrowed by $\sqrt{2^n}$ by extracting the n-order fluctuation information of the second-order correlation function.

1. Introduction

Correlation imaging is a technique based on second-order intensity correlation [1,2,3,4,5,6,7,8,9,10,11,12]. In traditional correlation imaging, a beam emitted by a light source is split into two beams, which then travel along different optical paths, usually referred to as the signal and the reference beam. The signal beam passes through an object and the intensity of transmitted light is measured by a detector, whereas the reference beam does not interact with the object. The image of the object is retrieved by measuring the correlations between the intensities at the two detectors [1,2,3]. In 2008, a computational correlating imaging system was proposed by Shapiro [13]. In this system, the source is a programmable light source [8,13], which is realized using a programmable spatial light modulator (SLM) and a laser [13]. Later on, it was found that a programmable light source can also be achieved using a digital light projector (DLP) [14,15], which itself contains a digital micro-mirror device (DMD) and a light emitting diode, to produce structured digital light fields. Compared to SLM, DMD is characterized by a higher modulation speed [16]. The programmable light source creates deterministic illumination patterns at the object’s position, such that the reference beam is no longer needed, and only a detector is required for imaging [8,13]. This greatly simplifies the overall imaging system and makes its application much easier. Correlation imaging techniques have several practical advantages compared to traditional imaging, including higher sensitivity and robustness [4], which have been exploited in different fields, such as optical lithography [17], remote imaging [18], microscopy imaging [19], X-ray imaging [20] and imaging for occluded objects [7]. In all the above applications, the spatial resolution is a crucial factor [21] and methods to improve the resolution have received widespread attention in recent years [21,22,23,24,25].

In traditional imaging systems, the resolution is limited by the Rayleigh criterion [5], i.e., by the properties of the point spread function (PSF) [26,27]. The same happens for correlation imaging systems when the pixel size of the detector is much smaller than the average size of the speckles [21,22]. In general, the resolution is quantified by the full width at half-maximum (FWHM) of the PSF, which is approximately equal to the average size of the speckles [5,21,28]. Several schemes to improve the spatial resolution of correlation imaging have been proposed. They may be divided into two main classes according to whether they exploit physical principles or are based on image reconstruction. Among physical methods, Han et al. reported a two-arm microscope scheme that employs second-order intensity correlation imaging to narrow the PSF [29]. Shih et al. reported a super-resolution method that uses the spatial frequency filtered intensity fluctuation correlation to reduce the FWHM of the PSF [30]. The narrowing of the PSF by the higher-order correlation of non-Rayleigh speckle fields has been also reported [31]. A sub-Rayleigh resolution experiment has been performed via the spatial low-pass filtering of the instantaneous intensity to narrow the PSF [21]. Li et al. used localization and thresholding to reduce the effect of the PSF in a correlation imaging system [32]. Preconditioned deconvolution methods [33] and the use of the coefficient of skewness [34] have been also suggested. Among the schemes based on image reconstruction, compressed sensing correlation imaging reduced the effect of the PSF on the imaging quality, using sparsity constraints to improve the spatial resolution of correlation imaging [24,35,36,37]. The use of deep neural network constraints [38] and sped up robust features for a new sum of modified Laplacian (SURF-NSML) [39] have been also suggested, and high-resolution correlation imaging through complex scattering media has been obtained via temporal correction [40].

In this paper, we focus on how the fluctuation characteristics of the second-order correlation may be exploited to improve the resolution of correlation imaging. The paper is organized as follows. In Section 2, we review the concept of resolution for traditional correlation imaging techniques, whereas, in Section 3, we analyze the fluctuation characteristics of the second-order correlation function. It is shown that the PSF of correlation imaging can be narrowed via the fluctuation information. It is also shown that the grayscale information of a grayscale object can be extracted through this fluctuation information. Based on these characteristics, the high-resolution image of the object can be obtained by using the fluctuation information. In Section 4, we describe our experimental setup and results, which confirm the theoretical predictions. Section 5 closes the paper with some concluding remarks.

2. The Resolution of a Correlation Imaging System

A schematic diagram of the experiment is shown in Figure 1a. It is similar to the traditional ghost imaging (GI) scheme [21], but each beam is now detected by a spatially resolving charge-coupled device (CCD) detector $CC{D}_i$ (i = 1, 2). The quantities x, $\alpha$ and $\beta$ denote the transverse coordinates on the source plane, object plane and imaging plane, respectively. The distances from the source to the object and image planes are $s_o$ and $s_i$, respectively.

We assume that the source is a monochromatic light beam with a wavelength $\lambda$. The light field PSFs from source to object and from source to $CC{D}_2$, respectively, may be written as [3,41]

$$h_t(x,\alpha )=\frac{e^{-ik{s}_o}}{i\lambda s_o}exp(\frac{-i\pi }{\lambda s_o}{(x-\alpha )}^2),$$

(1)

$$h_r(x,\beta )=\frac{e^{-ik{s}_i}}{i\lambda s_i}exp(\frac{-i\pi }{\lambda s_i}{(x-\beta )}^2),$$

(2)

where $k=\frac{2\pi }{\lambda }$. The light intensity distributions on the object and $CC{D}_2$ plane are denoted by $I\left(\alpha \right)$ and $I\left(\beta \right)$, respectively, and the light field transmission function of the object is denoted by $t\left(\alpha \right)$. The correlation function of the intensity fluctuations at the two detectors is [3]

$$\begin{array}{cc}\hfill & \Delta G^{\left(2\right)}(\alpha ,\beta )={\langle \left[I\left(\alpha \right)\right|t\left(\alpha \right)|}^2-{\langle I\left(\alpha \right)\left|t\left(\alpha \right)\right|}^2\rangle ][I\left(\beta \right)-\langle I\left(\beta \right)\rangle ]\rangle \hfill \\ \hfill & =\langle \Delta I\left(\alpha \right)|t\left(\alpha \right){|}^2\Delta I\left(\beta \right)\rangle \hfill \\ \hfill & =|\int G^{\left(1\right)}(x,x\prime )t\left(\alpha \right)h_t(x,\alpha )h_r^{*}(x\prime ,\beta )dxdx\prime {|}^2,\hfill \end{array}$$

(3)

where $\langle \dots \rangle$ represents the ensemble average. We assume that the light source is spatially incoherent and the intensity distribution is uniform, i.e.,

$$G^{\left(1\right)}(x,x\prime )=I_0\delta (x-x\prime ),$$

(4)

where $I_0$ is the intensity of the source and $\delta \left(x\right)$ is the Dirac delta function. We further assume that $s_o=s_i=z$. Then, upon substituting Equations (1) and (2) and Equation (4) in Equation (3), we have

$$\begin{array}{cc}\hfill & \Delta G^{\left(2\right)}(\alpha ,\beta )={\langle \Delta I\left(\alpha \right)\left|t\left(\alpha \right)\right|}^2\Delta I\left(\beta \right)\rangle \hfill \\ \hfill & =\frac{I_0^2}{{\lambda }^4{z}^4}\times {\left|t\left(\alpha \right)\right|}^2{\sin \mathrm{c}}^2(\frac{2\pi R}{\lambda z}(\alpha -\beta )),\hfill \end{array}$$

(5)

where R is the radius of the light source. By integration over the $\alpha$-variable, we retrieve the correlation image as

$$\begin{array}{cc}\hfill & \Delta G^{\left(2\right)}\left(\beta \right)=\langle \Delta I\left(\beta \right)\Delta B_t\rangle \hfill \\ \hfill & =\frac{I_0^2}{{\lambda }^4{z}^4}\int {\left|t\left(\alpha \right)\right|}^2{\sin \mathrm{c}}^2(\frac{2\pi R}{\lambda z}(\alpha -\beta ))d\alpha ,\hfill \end{array}$$

(6)

where $\Delta B_t=\int \Delta I\left(\alpha \right){\left|t\left(\alpha \right)\right|}^{2}d\alpha$ is a quantity equivalent to the intensity fluctuation at the bucket detector of a ghost imaging system. The image resolution is constrained by this PSF, and the resolution is determined by the FWHM of the ${\sin \mathrm{c}}^2$ function in Equation (6).

The imaging system can be simplified by using a DLP as a programmable light source, as shown in Figure 1. The DLP includes a DMD and a diode illumination source, which can generate deterministic illumination patterns at the object position [14,15]. In this way, the $CC{D}_2$ in the reference arm is no longer required, as shown in Figure 1b. For simplicity, our subsequent experimental system uses the single-arm setup.

3. Enhance the Resolution via the Fluctuation Information

If we consider that the power of a light source cannot be kept stable, $\Delta G^{\left(2\right)}(\alpha ,\beta )$ in Equation (5) and $\Delta G^{\left(2\right)}\left(\beta \right)$ in Equation (6) should be substituted with $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )$ and $\Delta G^{\left(2\right)}(I_0,\beta )$, respectively. Fluctuations of $I_0$ lead to fluctuations of $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )$ and $\Delta G^{\left(2\right)}(I_0,\beta )$ [41]. In other words, the two quantities $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )$ and $\Delta G^{\left(2\right)}(I_0,\beta )$ are statistically fluctuating functions. In order to account for this effect, we employ the cumulant-generating function $K(s,\alpha ,\beta )$, defined as

$$\begin{array}{cc}\hfill K(s,\alpha ,\beta )& =ln(\langle exp(s\Delta G^{\left(2\right)}(I_0,\alpha ,\beta ))\rangle )=\sum _{n=1}^{\infty }{\kappa }_n(\alpha ,\beta )\frac{s^n}{n!}\hfill \\ \hfill & =\mu (\alpha ,\beta )\times s+{\sigma }^2(\alpha ,\beta )\times \frac{s^2}{2}+\dots ,\hfill \end{array}$$

(7)

where ${\kappa }_n(\alpha ,\beta )$ is the nth-order cumulant of $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )$, $\mu (\alpha ,\beta )=\langle \Delta G^{\left(2\right)}(I_0,\alpha ,\beta )\rangle$, and ${\sigma }^2(\alpha ,\beta )=\langle {[\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )-\mu (\alpha ,\beta )]}^2\rangle$. The nth-order cumulant is given by

$${\kappa }_n(\alpha ,\beta )=\frac{d^{\left(n\right)}K(s,\alpha ,\beta )}{d{s}^{\left(n\right)}}{|}_{s=0}.$$

(8)

From Equations (7) and (8), we see that the second-order cumulant of $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )$ is given by

$$\begin{array}{cc}\hfill & {\kappa }_2(\alpha ,\beta )=\langle {[\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )-\langle \Delta G^{\left(2\right)}(I_0,\alpha ,\beta )\rangle ]}^2\rangle \hfill \\ \hfill & =\frac{\langle {(I_0^2-\langle I_0^2\rangle )}^2\rangle }{{\lambda }^8{z}^8}\times {\left|t\left(\alpha \right)\right|}^4{\sin \mathrm{c}}^4(\frac{2\pi R}{\lambda z}(\alpha -\beta )).\hfill \end{array}$$

(9)

According to Equation (9), ${\kappa }_2(\alpha ,\beta )$ contains information about the fluctuations of $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )$, due to the fluctuations of the light source. Upon the comparison of Equation (5) with Equation (9), we also see that this quantity has a narrower PSF than that of $\Delta G^{\left(2\right)}(\alpha ,\beta )$. Then, we integrate over $\alpha$ and obtain the new imaging information as

$$\begin{array}{cc}\hfill & {\kappa }_2^{\left(1\right)}\left(\beta \right)=\int {\kappa }_2(\alpha ,\beta )d\alpha \hfill \\ \hfill & =\frac{\langle {(I_0^2-\langle I_0^2\rangle )}^2\rangle }{{\lambda }^8{z}^8}\times \int {\left|t\left(\alpha \right)\right|}^4{\sin \mathrm{c}}^4(\frac{2\pi R}{\lambda z}(\alpha -\beta ))d\alpha .\hfill \end{array}$$

(10)

In other words, using ${\kappa }_2^{\left(1\right)}\left(\beta \right)$ instead of $\Delta G^{\left(2\right)}(I_0,\beta )$ to image the object, one has a narrower PSF. From Equation (9), we also see that the fluctuations of $I_0$ lead to fluctuations of ${\kappa }_2(\alpha ,\beta )$ and, in turn, after integrating over $\alpha$, we obtain an overall narrower PSF of the imaging system

$$\begin{array}{cc}\hfill & {\kappa }_2^{\left(2\right)}\left(\beta \right)=\int \frac{d^{\left(2\right)}ln\left(\langle exp\left(s{\kappa }_2(\alpha ,\beta )\right)\rangle \right)}{d{s}^{\left(2\right)}}{|}_{s=0}d\alpha \hfill \\ \hfill & \int |t\left(\alpha \right){|}^8{\sin \mathrm{c}}^8(\frac{2\pi R}{\lambda z}(\alpha -\beta ))d\alpha .\hfill \end{array}$$

(11)

If we use ${\kappa }_2^{\left(0\right)}(\alpha ,\beta )$, ${\kappa }_2^{\left(0\right)}\left(\beta \right)$ and ${\kappa }_2^{\left(1\right)}(\alpha ,\beta )$ to denote $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )$, $\Delta G^{\left(2\right)}(I_0,\beta )$ and ${\kappa }_2(\alpha ,\beta )$, respectively, we may write, according to Equation (6) and Equations (10) and (11),

$$\begin{array}{cc}\hfill & {\kappa }_2^{\left(n\right)}\left(\beta \right)=\int {\kappa }_2^{\left(n\right)}(\alpha ,\beta )d\alpha \hfill \\ \hfill & =\int \frac{d^{\left(2\right)}ln(\langle exp(s{\kappa }_2^{(n-1)}(\alpha ,\beta ))\rangle )}{d{s}^{\left(2\right)}}{|}_{s=0}\hfill \\ \hfill & +\delta \left(n\right)[{\kappa }_2^{\left(0\right)}(\alpha ,\beta )-\frac{d^{\left(2\right)}ln(\langle exp(s{\kappa }_2^{(n-1)}(\alpha ,\beta ))\rangle )}{d{s}^{\left(2\right)}}{|}_{s=0}]d\alpha ,\hfill \end{array}$$

(12)

where n is the number of iterations. The meaning of each iteration is that we go one step further in considering cumulants to extract the fluctuation information. From Equation (12), we find that ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ is the information consisting of all ${\kappa }_2^{\left(n\right)}(\alpha ,\beta )$. ${\kappa }_2^{\left(n\right)}(\alpha ,\beta )$ is the fluctuation information of ${\kappa }_2^{(n-1)}(\alpha ,\beta )$ when n ⩾ 1, which is equivalent to $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )$ for n = 0.

In Ref. [41], the authors have presented a scheme to enhance the resolution of correlation imaging, referred to as second-order cumulant ghost imaging (SCGI). The imaging information of the SCGI can be written as [41]

$$\begin{array}{c}\hfill {\kappa }_2\left(\beta \right)=\int {\kappa }_2(\alpha ,\beta )d\alpha +L\left(\beta \right),\end{array}$$

(13)

where ${\kappa }_2\left(\beta \right)$ is the fluctuation information of $\Delta G^{\left(2\right)}(I_0,\beta )$. $L\left(\beta \right)$ is cross-information, which reduces the resolution of ${\kappa }_2\left(\beta \right)$ [41]. From Equation (10) and Equation (13), we see that ${\kappa }_2\left(\beta \right)$ includes ${\kappa }_2^{\left(1\right)}\left(\beta \right)$ and cross-information $L\left(\beta \right)$, and that ${\kappa }_2^{\left(1\right)}\left(\beta \right)$ greatly enhances the resolution but $L\left(\beta \right)$ decreases the resolution. In the framework of SCGI, $K(s,\beta )$ is the cumulant-generating function of $\Delta G^{\left(2\right)}(I_0,\beta )$, and ${\kappa }_2\left(\beta \right)$ is the second-order cumulant obtained from $K(s,\beta )$. Thus, ${\kappa }_2\left(\beta \right)$ includes not only the fluctuation information of $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )$ for different $\alpha$ points, but also the cross-information generated correlating $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )-\langle \Delta G^{\left(2\right)}(I_0,\alpha ,\beta )\rangle$ and $\Delta G^{\left(2\right)}(I_0,{\alpha }^{{}^{\prime }},\beta )-\langle \Delta G^{\left(2\right)}(I_0,{\alpha }^{{}^{\prime }},\beta )\rangle$ for different $\alpha$ and ${\alpha }^{{}^{\prime }}$ ($\alpha \ne {\alpha }^{{}^{\prime }}$) [41]. This is the reason for $L\left(\beta \right)$ appearing. In our case, we have a spatially resolving charge-coupled detector $CC{D}_1$ instead of the bucket detector of [41]. If the distance between the object and the $CC{D}_1$ is smaller than their longitudinal coherence length, we consider $K(s,\alpha ,\beta )$, i.e., the cumulant-generating function of $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )$. The second-order cumulant obtained by $K(s,\alpha ,\beta )$ is the fluctuation information of $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )$. In other words, the term $L\left(\beta \right)$ disappears in our scheme, and this leads to a better resolution limit than that of SCGI.

By explicitly evaluating Equation (12), we see that the PSF at the n-th iteration is the 2ⁿ-th power function of sinc². Since the PSF of $\Delta G^{\left(2\right)}(I_0,\beta )$ is the 2-th power function of sinc (see Equation (6)), the FWHM of the PSF for ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ is narrower by a factor of $\sqrt{2^n}$ than that of $\Delta G^{\left(2\right)}(I_0,\beta )$. For a binary object, the resolution of the correlation imaging can be thus enhanced by a factor $\sqrt{2^n}$. In order to address a concrete example, a schematic diagram of our numerical simulations is shown in Figure 1a. We take 20,000 measurements for every simulation. The wavelength of the source is $\lambda$ = 500 nm, and the radius is R = 1.28 × ${10}^{-4}$ m, whereas $s_o$ = $s_i$ = 0.022 m. The object consists of two squares, as shown in Figure 2a. The side length of both squares is r = 1.6 × ${10}^{-5}$ m. The distance between the centers of the two squares is d = 4 × ${10}^{-5}$ m. The distance between the object and the $CC{D}_1$ is $s_d$ = 0.002 m. According to Equations (6)–(8) in Ref. [42], we have that $s_d$ is less than the spatial longitudinal coherence length between the object plane and the $CC{D}_1$.

At first, we reconstruct the image by the $CC{D}_1$ and $\Delta G^{\left(2\right)}(I_0,\beta )$, respectively. Results are shown in Figure 2b and Figure 2c, respectively. From Figure 2b, we see that imaging of the object is not possible using the $CC{D}_1$ due to the interference effect of light. From Figure 2c, we find that also $\Delta G^{\left(2\right)}(I_0,\beta )$ fails to provide acceptable imaging. This is because the distance between the square centers is smaller than allowed by the Rayleigh criterion, i.e, $d<\frac{\lambda s_o}{2R}$. Using SCGI (i.e., ${\kappa }_2\left(\beta \right)$), we obtain Figure 2d, which shows a better resolution, since the PSF of ${\kappa }_2\left(\beta \right)$ is narrower than that of $\Delta G^{\left(2\right)}(I_0,\beta )$. Finally, we have reconstructed the image using ${\kappa }_2^{\left(1\right)}\left(\beta \right)$ and ${\kappa }_2^{\left(2\right)}\left(\beta \right)$, respectively. The results are shown in Figure 2e,f. We see that the resolution provided by ${\kappa }_2^{\left(1\right)}\left(\beta \right)$ is better than ${\kappa }_2\left(\beta \right)$. This is due to the disappearance of the $L\left(\beta \right)$ term. Finally, if we then compare Figure 2e to Figure 2f, we see that the resolution provided by ${\kappa }_2^{\left(2\right)}\left(\beta \right)$ is better than that of ${\kappa }_2^{\left(1\right)}\left(\beta \right)$, since the FWHM of the PSF of ${\kappa }_2^{(n+1)}\left(\beta \right)$ is smaller than that in ${\kappa }_2^{\left(n\right)}\left(\beta \right)$. For a binary object, the resolution becomes better and better as n grows.

In order to further compare the quality of the reconstructed image, we evaluate the Peak Signal-to-Noise Ratio (PSNR), which is defined as [43]

$$\begin{array}{c}\hfill PSNR=10\times {log}_{10}[\frac{maxVa{l}^2}{MSE}],\end{array}$$

(14)

where $MSE=\frac{1}{N}\sum {\left(G_{array}-G_O\right)}^2$, $G_{array}$ and $G_O$ denote the reconstructed image and the original object, respectively; N is the total number of pixel points for each of $G_{array}$; and $maxVal$ is the maximum gray value of $G_O$. By using Equation (14), we calculate the PSNRs of the normalized images in Figure 2b–f, obtaining PSNR = 15.86, 17.19, 20.65, 23.41 and 24.17, as shown in

Table 1. The PSNRs obtained by different methods in the two-square object imaging.

Method	${\mathbf{CCD}}_1$	$\mathsf{\Delta }{\mathit{G}}^{\left(2\right)}({\mathit{I}}_0,\mathit{\beta })$	${\mathit{\kappa }}_2\left(\mathit{\beta }\right)$	${\mathit{\kappa }}_2^{\left(1\right)}\left(\mathit{\beta }\right)$	${\mathit{\kappa }}_2^{\left(2\right)}\left(\mathit{\beta }\right)$
PSNR	15.86	17.19	20.65	23.41	24.17

. This is consistent with our previous analysis and further confirms the reliability and accuracy of the new protocol.

Moreover, for a grayscale object, the resolution may be enhanced using ${\kappa }_2^{\left(n\right)}\left(\beta \right)$. In particular, according to Equation (12), we see that the information about the higher grayscale regions of the object can be gradually extracted from ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ with increasing n. As a consequence, we can extract high-resolution images of the different grayscale regions by combining ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ with some centroid localization algorithms. Then, the full high-resolution image of the grayscale object can be constructed by combining the images of the different regions, as in single-molecule localization microscopy (SMLM) [44]. Concerning the centroid localization, several algorithms may be used, such as thresholding [45] and DAOSTORM [46]. For the sake of simplicity, we employ a simple thresholding method and obtain the maximum value of ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ by considering

$$\begin{array}{c}\hfill N_{max}=max\left[{\kappa }_2^{\left(n\right)}\left(\beta \right)\right].\end{array}$$

(15)

and setting the threshold

$$\begin{array}{c}\hfill N=P\times N_{max},\end{array}$$

(16)

where 0 < P < 1. The smaller is P, the higher is the accuracy and the slower is the overall speed of the imaging technique, i.e., a trade-off level has to be set according to the specific application. Here, we set P = 0.5 and obtain the threshold by substituting P into Equation (16). Only the values of ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ larger than threshold are accepted to set the centroid of the image. The overall imaging process thus proceeds as follows. At first, we use ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ to reconstruct the high grayscale regions of the object. Then, we obtain the centroids of these regions by thresholding and eliminate the information about the centroid by blocking its position or using some specific algorithms. Then, we again use ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ to image the low grayscale regions. As an example, we consider an object composed of three squares with sides equal to 2.56 × ${10}^{-5}$ m and a center distance 4 × ${10}^{-5}$ m, as shown in Figure 3a. The transmittances of the squares are set to 1, 0.75 and 0.5, respectively, from left to right. The other parameters are set to the same values as in the previous simulations. The reconstructions obtained by ${\kappa }_2^{\left(0\right)}\left(\beta \right)$, ${\kappa }_2^{\left(1\right)}\left(\beta \right)$ and ${\kappa }_2^{\left(2\right)}\left(\beta \right)$ are shown in Figure 3b–d. As is apparent from the plot, the information in the high grayscale region can be extracted and improves for increasing n.

Since there is a grayscale difference between $\Delta G^{\left(2\right)}(I_0,\beta )$ and ${\kappa }_2^{\left(2\right)}\left(\beta \right)$, the peak of Figure 3d is reduced by a factor of

$$\begin{array}{c}\hfill C\left({\beta }_c\right)\approx \frac{\Delta G^{\left(2\right)}(I_0,{\beta }_c)}{{\kappa }_2^{\left(2\right)}\left({\beta }_c\right)},\end{array}$$

(17)

where ${\beta }_c$ is the peak coordinates of Figure 3d. To better recover the grayscale information, we multiply Figure 3d by this compensation factor $C\left({\beta }_c\right)$, and, by this procedure, the image of the left square can be retrieved. We use thresholding to obtain the centroid of Figure 3d and eliminate the information about the centroid from the imaging information. Then, we use the steps described above to obtain the image of the middle square, as shown in Figure 3e. Similarly, we obtain the image of the right square, as shown in Figure 3f. After eliminating the information of the three centroids, we obtain the image in Figure 3g. Upon the comparison of Figure 3d with Figure 3g, we see that Figure 3g contains only a fraction of the information of Figure 3d. This is because Figure 3g is already included in Figure 3d, but its intensity is negligible compared to Figure 3d. At this point, we assume that the imaging of all the regions has been completed and the full image may be obtained by combining the images of the different grayscale regions, i.e., combining the images of the left square from Figure 3d multiplied by the factor $C\left({\beta }_c\right)$, the middle square of Figure 3e and the right square of Figure 3f. We then normalize the image, as shown in Figure 3h. We also compare the quality of the reconstructed images quantitatively by using Equation (14). The PSNRs of the normalized images of Figure 3b, Figure 3c and Figure 3h are 16.43, 22.52 and 25.18, respectively, as shown in

Table 2. The PSNRs of the different methods.

Method	${\mathit{\kappa }}_2^{\left(0\right)}\left(\mathit{\beta }\right)$	${\mathit{\kappa }}_2^{\left(1\right)}\left(\mathit{\beta }\right)$	${\mathit{\kappa }}_2^{\left(2\right)}\left(\mathit{\beta }\right)$ with the Thresholding Method
PSNR	16.43	22.52	25.18

. The results show that our protocol can be used for grayscale object imaging and has a better resolution than traditional correlation imaging.

4. Experimental Results and Discussion

To verify our theoretical results, experiments are carried out. We use the simple single-arm setup shown in Figure 4a (which is equivalent to Figure 1a [14,21], because the intensity distribution on the $CC{D}_2$ ($I\left(\beta \right)$) can be obtained by calculations). The light source is a projector (XE11F), and there is a digital mirror device (DMD) in the source. The object is a double-slit object with width a = 0.223 mm, a distance between the centers of the two slits b = 0.445 mm and a slit height equal to g = 1.114 mm.

The object is part of the resolution target (USAF1-228 LP/MM), as shown in Figure 4b. The distance between the source plane and the object is 25 cm. A camera (MV-VEM033SM) is taken, used as $CC{D}_1$. The distance between the object and the $CC{D}_1$ is 2 cm. In this experiment, we employ both the traditional correlation imaging and our method based on ${\kappa }_2^{\left(n\right)}\left(\beta \right)$. All the results are obtained by averaging over 20,000 exposure frames. For ${\kappa }_2^{\left(n\right)}\left(\beta \right)$, we extract $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )$ every 2000 steps. The FWHM of the different methods can be obtained by imaging a single pixel [31], and for $\Delta G^{\left(2\right)}(I_0,\beta )$, ${\kappa }_2^{\left(1\right)}\left(\beta \right)$ and ${\kappa }_2^{\left(2\right)}\left(\beta \right)$, we obtain 69 pixels, 49 pixels and 35 pixels, respectively, as summarized in

Table 3. Comparing the imaging results of the different methods for a binary double-slit object.

Method	$\mathsf{\Delta }{\mathit{G}}^{\left(2\right)}({\mathit{I}}_0,\mathit{\beta })$	${\mathit{\kappa }}_2^{\left(1\right)}\left(\mathit{\beta }\right)$	${\mathit{\kappa }}_2^{\left(2\right)}\left(\mathit{\beta }\right)$
The FWHM of the PSF	69 (pixels)	49 (pixels)	35 (pixels)
The normalized horizontal sections of the image	Figure 5d	Figure 5e	Figure 5f
The normalized intensity at ${\beta }_0$	0.95 (>0.81)	0.62 (<0.81)	0.32 (<0.81)

. Then, we obtain the image of the object by substituting $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )$ into Equation (12). The results are shown in Figure 5.

The image of the object, as obtained by traditional correlation imaging, is reported in Figure 5a, which shows that the two slits cannot be separated. Figure 5b shows the image of the object obtained using ${\kappa }_2^{\left(1\right)}\left(\beta \right)$, where the two slits are now visible, i.e., ${\kappa }_2^{\left(1\right)}\left(\beta \right)$ provides a better resolution than the traditional correlation imaging. This is because the resolution of the system is limited by the PSF, and ${\kappa }_2^{\left(1\right)}\left(\beta \right)$ has a narrower PSF than $\Delta G^{\left(2\right)}(I_0,\beta )$. Figure 5c shows the image of the object as obtained by ${\kappa }_2^{\left(2\right)}\left(\beta \right)$. The image is apparently clearer than those in Figure 5a,b. A quantitative comparison can be obtained from the normalized horizontal marginals, shown in Figure 5d–f. The PSF of ${\kappa }_2^{\left(2\right)}\left(\beta \right)$ is indeed the narrowest among the three models. The above experimental results show that the resolution obtained from ${\kappa }_2^{\left(2\right)}\left(\beta \right)$ is the best, and the resolution of ${\kappa }_2^{\left(1\right)}\left(\beta \right)$ is better than that of ${\kappa }_2^{\left(0\right)}\left(\beta \right)$ ($\Delta G^{\left(2\right)}(I_0,\beta )$). This means that the resolution provided by ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ improves for increasing n, which is consistent with our theoretical analysis.

Since the double-slit object is a real object, we cannot obtain perfect data for $G_O$, and therefore we are unable to calculate the PSNR for the images in Figure 5a–c by Equation (14). In order to compare quantitatively the quality of the reconstructed images, we evaluate the resolution using the intensity at saddle point position ${\beta }_0$ for the images in Figure 5d–f, respectively. The value of ${\beta }_0$ can be extracted from $\frac{{\beta }_2-{\beta }_1}{2}$, where ${\beta }_1$ is the center of the left slit in the object, and ${\beta }_2$ is the center of the right slit. According to Ref. [47], for a binary object, the double slit cannot be separated when the normalized intensity at ${\beta }_0$ is greater than 0.81. The normalized intensity at ${\beta }_0$ for the different methods is summarized in Table 3.

Table 3 shows that the double slit cannot be separated by traditional correlation imaging. However, it can be separated by ${\kappa }_2^{\left(1\right)}\left(\beta \right)$ and ${\kappa }_2^{\left(2\right)}\left(\beta \right)$. From the FWHM of the PSF in Table 3, we see that the resolution of ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ is improved by a factor of $\sqrt{2^n}$ compared to that of $\Delta G^{\left(2\right)}(I_0,\beta )$. This is consistent with our theoretical analysis.

Finally, in order to show that the proposed method can be applied also to the imaging of grayscale objects, we take the double-slit object considered above and cover the left slit with a film. Results are shown in Figure 6. As is apparent from the plots, the image cannot be obtained by traditional correlation imaging (see Figure 6a), but it is well retrieved by ${\kappa }_2^{\left(1\right)}\left(\beta \right)$ (see Figure 6b). This is because the PSF of ${\kappa }_2^{\left(1\right)}\left(\beta \right)$ is narrower than that of $\Delta G^{\left(2\right)}(I_0,\beta )$. As a matter of fact, low grayscale regions are discarded if n in ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ is set too high. However, even without using centroid localization algorithms, the resolution of the imaging may be improved by setting the appropriate n according to the grayscale distribution of the object. We then obtain the image by combining ${\kappa }_2^{\left(2\right)}\left(\beta \right)$ with a thresholding method. The detailed imaging process is the same as that used for the simulations, and the result is shown in Figure 6c. The improvement in the resolution is apparent and is confirmed by the normalized horizontal sections shown in Figure 6d–f. Since the object is a grayscale one, the intensity at ${\beta }_1$ is not equal to the intensity at ${\beta }_2$. Compared to the image of the right slit (higher grayscale area), the image of the left slit (lower grayscale area) is more difficult to observe, and thus we further assess the resolution of the images in Figure 6d–f by calculating the ratio between the intensity at ${\beta }_0$ and at ${\beta }_1$. Results are shown in

Table 4. Comparison among the imaging results of the different models on the grayscale object.

Method	$\mathsf{\Delta }{\mathit{G}}^{\left(2\right)}({\mathit{I}}_0,\mathit{\beta })$	${\mathit{\kappa }}_2^{\left(1\right)}\left(\mathit{\beta }\right)$	${\mathit{\kappa }}_2^{\left(2\right)}\left(\mathit{\beta }\right)$ with Thresholding Method
The FWHM of the PSF	69 (pixels)	49 (pixels)	35 (pixels)
The normalized horizontal sections of the image	Figure 6d	Figure 6e	Figure 6f
The radio between the intensity at ${\beta }_0$ and ${\beta }_1$	1.21 (>0.81)	0.713 (<0.81)	0.25 (<0.81)

.

Table 4 shows that the double slit cannot be separated by the traditional correlation imaging, whereas it can be separated by ${\kappa }_2^{\left(1\right)}\left(\beta \right)$ and ${\kappa }_2^{\left(2\right)}\left(\beta \right)$ with the thresholding method. From Table 4, we also see that the resolution of ${\kappa }_2^{\left(2\right)}\left(\beta \right)$ with the thresholding method is the best one. This is because the PSF of ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ narrows as n grows, and the high-resolution image of the different grayscale regions can be obtained by thresholding. The image of the grayscale object can be thus obtained by combining the images of the different grayscale regions. The experimental results are in excellent agreement with our theoretical analysis.

Since ${\kappa }_2^{\left(1\right)}(\alpha ,\beta )$ contains information about the fluctuations of $\Delta G^{\left(2\right)}(I_0,\alpha ,\beta )$ and ${\kappa }_2^{\left(n\right)}(\alpha ,\beta )$ those of ${\kappa }_2^{(n-1)}(\alpha ,\beta )$ when n ≥ 1, the number of measurements required for imaging using ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ increases as n grows. We thus conclude that traditional correlation imaging is, in general, faster than our scheme, whereas our scheme shows a better resolution than traditional correlation imaging.

5. Conclusions

In conclusion, we have demonstrated that the resolution of correlation imaging schemes can be improved by exploiting the knowledge of the fluctuation characteristics of the second-order correlation function. Our results show that, using the fluctuation, we can obtain a narrower PSF. In particular, the PSF of ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ is narrower by a factor of $\sqrt{2^n}$ compared to the original PSF. We have also shown that the same method may be applied to the imaging of grayscale objects. In fact, information of the high grayscale regions can be extracted by ${\kappa }_2^{\left(n\right)}\left(\beta \right)$ by increasing n and then high-resolution images of the different grayscale regions are obtained by combining this fluctuations information with a centroid localization algorithm. Finally, a full high-resolution image of the grayscale object can be constructed by combining the images of the different regions. We have experimentally confirmed the resolution enhancement of our scheme, and the results show excellent agreement with the theoretical predictions. Although our scheme requires more measurements compared to traditional correlation imaging, it shows a better resolution. Other localization algorithms can be also used to obtain centroid information, such as DAOSTORM or Bayesian algorithms, to improve the overall resolution. Work along this line is in progress and results will be reported elsewhere.