Quantum-inspired superresolution for incoherent imaging

Xiao-Jie Tan; Luo Qi; Lianwei Chen; Aaron J. Danner; Pakorn Kanchanawong; Pakorn Kanchanawong; Mankei Tsang; Mankei Tsang

doi:10.1364/OPTICA.493227

1. INTRODUCTION

The resolution of incoherent sources with subdiffraction separations is a fundamental problem in optical imaging, with applications ranging from fluorescence microscopy to astronomy [1]. Quantum information theory [2] has recently lent a fresh perspective to the age-old problem [3,4]: the fundamental quantum limits turn out to be far away from what can be achieved by direct imaging, while a measurement of the photons in a judicious basis of spatial modes can approach the limits and offer substantial improvements. This quantum perspective has since inspired many follow-up studies—both theoretical and experimental—and grown into a field of research called quantum-inspired superresolution [4].

Numerous experiments on quantum-inspired superresolution have been reported in recent years [5–29], but most of them focus on the estimation of the separation between two point sources. This paper reports an experiment that uses spatial-mode demultiplexing (SPADE) in two dimensions (2D) to estimate both locations of two point sources, not just their separation, as well as the moments of a distribution of multiple point sources [7,30].

In theory, SPADE in the Hermite–Gaussian (HG) basis responds to the source locations in a symmetric manner, which introduces sign ambiguities to the estimation and prevents one from resolving the locations accurately without an adaptive scheme [3,31–33]. In practice, we discovered that the SPADE response in our experiment is asymmetrical because of experimental imperfections, and the asymmetry enables us to estimate both locations accurately. For a distribution of multiple point sources, we demonstrate that their moments can still be measured in the context of semiparametric estimation [7,30,34]. While our experiment is not close to the quantum limit, our findings pave the way towards the use of SPADE under realistic scenarios, where many parameters are unknown and adaptive measurements are difficult. We conjecture that, with shot-noise-limited sources and detection that are typical in optical astronomy [35] and fluorescence microscopy [36], our scheme can be close to the fundamental quantum limits while retaining the practical advantages observed in this experiment.

2. EXPERIMENT

Figure 1 illustrates our experimental setup. A collimated Gaussian beam is produced by a laser at wavelength 1550 nm (Thorlabs S1FC1550), a single-mode fiber, and a collimator (Thorlabs CFC2-C). The beam radius is given by 190 µm, or $\sigma = 95 \;{\unicode{x00B5}{\rm m}}$ in intensity standard deviation. We use the Gaussian beam to simulate the image-plane field produced by a point source and a diffraction-limited imaging system. SPADE is implemented by the multi-plane light conversion (MPLC) device (Cailabs PROTEUS-C), which is placed on a 2D translation stage ($2 \times {\rm Thorlabs}$ PT1/M) to produce 2D transverse displacements between the Gaussian beam and the MPLC device. A rotational stage (Zolix RSM82-1A) between the translation stage and the MPLC device provides additional alignment control. Each of the six MPLC output channels, corresponding to the first six ${{\rm HG}_{\textit{nm}}}$ modes with

(1)$$(n,m) \in {\cal N} \equiv \{(0,0),(0,1),(1,0),(0,2),(2,0),(1,1)\} ,$$

is coupled to a photodetector (Thorlabs PDA20CS2), and the electrical signals are fed into an oscilloscope (Tektronix MSO46) for digital data collection. By scanning in 2D, we record effectively the six mode outputs as a function of the point-source location. The scanning is performed over a $2 \;{\rm mm} \times 2\;{\rm mm} $ area; the step size is 50 µm in each direction and refined to 10 µm for displacements between ${-}{100}$ and 100 µm.

Fig. 1. Experimental setup. See the main text for details.

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Fig. 2. Sample mean of the photodetector readings $\{Y_{\textit{nm}}^{(j)}({\boldsymbol R})\}$ versus input beam displacement ${\boldsymbol R} \in {\mathbb{R}^2}$ for the six SPADE output channels denoted by $\{{{\rm HG}_{\textit{nm}}}:(n,m) \in {\cal N}\}$. The $x$ and $y$ axes are normalized with respect to the beam standard deviation $\sigma$, while the photodetector readings are proportional to the optical power and in an arbitrary unit.

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For each displacement ${\boldsymbol R} \in {\mathbb{R}^2}$ and each ${{\rm HG}_{\textit{nm}}}$ channel, a photodetector reading is a sample $Y_{\textit{nm}}^{(j)}({\boldsymbol R})$ of a random variable ${Y_{\textit{nm}}}({\boldsymbol R})$, the mean ${\bar Y_{\textit{nm}}}({\boldsymbol R})$ of which is theoretically given by

(2)$${\bar Y_{\textit{nm}}}({\boldsymbol R}) \propto {\left| {\iint \phi _{\textit{nm}}^*({\boldsymbol r})\psi ({\boldsymbol r} - {\boldsymbol R}){{\rm d}^2}{\boldsymbol r}} \right|^2},$$

where ${\phi _{\textit{nm}}}({\boldsymbol r}) \propto {{\rm He}_m}(u/\sigma){{\rm He}_n}(v/\sigma)\exp [- ({u^2} + {v^2})/(4{\sigma ^2})]$ with $u \equiv (x + y)/\sqrt 2$ and $v \equiv (x - y)/\sqrt 2$ is the wavefunction of the ${{\rm HG}_{\textit{nm}}}$ mode, ${{\rm He}_n}$ is a Hermite polynomial [37], and $\psi ({\boldsymbol r}) = {\phi _{00}}({\boldsymbol r})$ is the wavefunction of the input beam at zero displacement. For each ${\boldsymbol R}$ and each ${{\rm HG}_{\textit{nm}}}$ output, we record a set of $J$ samples denoted as

(3)$$\{Y_{\textit{nm}}^{(j)}({\boldsymbol R}):j = 1, \ldots ,J\} ,$$

where $J \approx 10,000$. Figure 2 plots the mean of these samples, which we take to be the impulse-response function ${\bar Y_{\textit{nm}}}({\boldsymbol R})$ of our system in our subsequent data analysis. For microscopy, remote sensing, or astronomy, we envision the use of a fluorescence bead, a beacon, or a guide star to measure such a function in a calibration procedure before the imaging system is used in practice. We also note that our data analysis takes the displacement of our translation stage to be the true displacement and does not take into account its imprecision. This systematic error can be reduced by using a better stage and does not affect the principles demonstrated by our experiment.

Because of experimental imperfections, the outputs in Fig. 2 exhibit asymmetric patterns with respect to the diagonal axes, unlike the ideal case indicated by Eq. (2). Moreover, because the wavefunction $\psi ({\boldsymbol r})$ of the beam in our experiment is not exactly equal to the mode wavefunction ${\phi _{00}}({\boldsymbol r})$ of the SPADE device, the beam can couple into the ${{\rm HG}_{02}}$ and ${{\rm HG}_{20}}$ outputs at zero beam displacement, leading to lobes at the origin. We also note that the variances of our data are nowhere near the shot-noise limit because of excess laser noise, excess detector noise, and mechanical noise. We find that these noise sources are time-varying and the variances of our data exhibit irregular behavior over time, so we do not assume that the variances are known in our data analysis.

3. RESULTS

A. Localization of Two Point Sources

We use the experimental data set to simulate various imaging scenarios. The first scenario is the localization of two point sources. For each trial of the simulated experiment, we randomly draw $K = 100$ sets of samples $\{(Y_{\textit{nm}}^{(k)}({{\boldsymbol R}_1}),Y_{\textit{nm}}^{(k)}({{\boldsymbol R}_2})):k = 1, \ldots ,K;(n,m) \in {\cal N}\}$ at two displacements ${{\boldsymbol R}_1}$ and ${{\boldsymbol R}_2}$ from our data set. Then

(4)$$Z_{\textit{nm}}^{(k)}({{\boldsymbol R}_1},{{\boldsymbol R}_2}) = Y_{\textit{nm}}^{(k)}({{\boldsymbol R}_1}) + Y_{\textit{nm}}^{(k)}({{\boldsymbol R}_2})$$

simulates the ${{\rm HG}_{\textit{nm}}}$ output given two incoherent point sources at locations ${{\boldsymbol R}_1}$ and ${{\boldsymbol R}_2}$. To estimate the locations from the $K$ outputs, we use the least-squares method

(5)$$({{\check {\boldsymbol R}}_1},{{\check {\boldsymbol R}}_2}) = \mathop {\rm argmin}\limits _{{({{\boldsymbol R}_1},{{\boldsymbol R}_2})}}\sum\limits_{k = 1}^K \sum\limits_{(n,m)} {[Z_{\textit{nm}}^{(k)} - {\bar Y_{\textit{nm}}}({{\boldsymbol R}_1}) - {\bar Y_{\textit{nm}}}({{\boldsymbol R}_2})]^2},$$

where ${\bar Y_{\textit{nm}}}({\boldsymbol R})$ as a function of ${\boldsymbol R}$ is determined by a cubic-spline interpolation of the sample means $\sum\nolimits_j Y_{\textit{nm}}^{(j)}({\boldsymbol R})/J$ of our data. For each set of true locations $({{\boldsymbol R}_1},{{\boldsymbol R}_2})$, we repeat the sampling and estimation procedure for $L = 100$ trials to obtain $L$ estimates $\{({\check {\boldsymbol R}}_1^{(l)},{\check {\boldsymbol R}}_2^{(l)}):l = 1, \ldots ,L\}$ and compute the root-mean-square error

(6)$${\rm RMSE}({{\boldsymbol R}_1},{{\boldsymbol R}_2}) \equiv \sqrt {\frac{1}{L}\sum\limits_{l = 1}^L |{\check {\boldsymbol R}}_1^{(l)} - {{\boldsymbol R}_1}{|^2} + |{\check {\boldsymbol R}}_2^{(l)} - {{\boldsymbol R}_2}{|^2}} .$$

The preceding experiment is then repeated for $M = 200$ randomly selected sets of true locations $\{({\boldsymbol R}_1^{(m)},{\boldsymbol R}_2^{(m)}):m = 1, \ldots ,M\}$, resulting in a set of RMSEs given by

(7)$$\{{\rm RMSE}({\boldsymbol R}_1^{(m)},{\boldsymbol R}_2^{(m)}):m = 1, \ldots ,M\} .$$

To summarize the four-dimensional ${\rm RMSE}({{\boldsymbol R}_1},{{\boldsymbol R}_2})$ in one plot, we divide the $M$ RMSEs into bins according to the object size $\Delta ({{\boldsymbol R}_1},{{\boldsymbol R}_2})$ defined as

(8)$$\Delta ({{\boldsymbol R}_1},{{\boldsymbol R}_2}) \equiv \sqrt {\frac{{|{{\boldsymbol R}_1}{|^2} + |{{\boldsymbol R}_2}{|^2}}}{2}} ,$$

such that the set of RMSEs in each object-size bin is given by

(9)$${\{{\rm RMSE}\} _j} \equiv \{{\rm RMSE}({{\boldsymbol R}_1},{{\boldsymbol R}_2}):{\Delta _j} \le \Delta ({{\boldsymbol R}_1},{{\boldsymbol R}_2}) \lt {\Delta _{j + 1}}\} ,$$

(10)$${\Delta _j} = j \times (0.1\sigma),\quad j = 0,1,2, \ldots$$

Figure 3(a) plots the mean of ${\{{\rm RMSE}\} _j}$ for each object-size bin, with each error bar denoting the minimum and maximum of ${\{{\rm RMSE}\} _j}$ within the bin, while Fig. 3(b) plots the same errors normalized by the true separation between the two point sources $\delta ({{\boldsymbol R}_1},{{\boldsymbol R}_2}) \equiv |{{\boldsymbol R}_1} - {{\boldsymbol R}_2}|$. Figure 3(c) plots three representative examples of the point-source locations, the estimates, as well as the hypothetical direct images of the point sources to illustrate the subdiffraction resolution.

Fig. 3. (a) Summary of the root-mean-square errors ${\rm RMSE}({{\boldsymbol R}_1},{{\boldsymbol R}_2})$ in estimating the locations $({{\boldsymbol R}_1},{{\boldsymbol R}_2})$ of two point sources via SPADE. The horizontal axis has the object sizes $\Delta$ in linear scale grouped into multiple bins, while the vertical axis is the RMSE in log scale. Each bar (blue) is the average RMSE for sources with object sizes $\Delta ({{\boldsymbol R}_1},{{\boldsymbol R}_2})$ within the bin. Each error bar (red) gives the maximum and minimum RMSEs in that bin. Both axes are normalized with respect to the beam standard deviation $\sigma$. (b) Relative error ${\rm RMSE}({{\boldsymbol R}_1},{{\boldsymbol R}_2})/\delta ({{\boldsymbol R}_1},{{\boldsymbol R}_2})$ of the estimation, where $\delta ({{\boldsymbol R}_1},{{\boldsymbol R}_2}) \equiv |{{\boldsymbol R}_1} - {{\boldsymbol R}_2}|$ is the true separation between the two point sources. (c) Three representative examples of the point-source locations (black circles), the estimates (red crosses), and the hypothetical direct images of the two point sources (background).

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We observe that the RMSEs are all substantially below the diffraction limit defined by $\sigma$ and remain so for subdiffraction object sizes. Such results would be implausible if the experiment were perfect. In theory, the ideal mean outputs ${\bar Y_{\textit{nm}}}({\boldsymbol R})$ are all symmetric along the ${\hat{\boldsymbol u}} = ({\hat{\boldsymbol x}} + {\hat{\boldsymbol y}})/\sqrt 2$ and ${\hat{\boldsymbol v}} = ({\hat{\boldsymbol x}} - {\hat{\boldsymbol y}})/\sqrt 2$ directions, where ${\hat{\boldsymbol x}}$ and ${\hat{\boldsymbol y}}$ are unit vectors of the horizontal and vertical axes. Writing each source location as ${{\boldsymbol R}_j} = {u_j}{\hat{\boldsymbol u}} + {v_j}{\hat{\boldsymbol v}}$ in terms of the coordinates $({u_j},{v_j})$ along ${\hat{\boldsymbol u}}$ and ${\hat{\boldsymbol v}}$, the symmetry implies

(11)$${\bar Y_{\textit{nm}}}({{\boldsymbol R}_j}) = {\bar Y_{\textit{nm}}}({u_j}{\hat{\boldsymbol u}} + {v_j}{\hat{\boldsymbol v}}) = {\bar Y_{\textit{nm}}}(|{u_j}|{\hat{\boldsymbol u}} + |{v_j}|{\hat{\boldsymbol v}}).$$

If the probability distribution of the data depends only on such symmetric functions, such as the Poisson case, then the data are fundamentally insensitive to the signs of $({u_j},{v_j})$, and the maximum-likelihood estimator may converge to a solution with the wrong signs, leading to large errors. Our experimental outputs depend on ${\boldsymbol R}$ asymmetrically, however, as shown in Fig. 2, and the asymmetry together with the six output channels turn out to resolve the sign ambiguities and give low errors in estimating both locations of the sources, as demonstrated empirically by Fig. 3.

To further study the sign problem, we simulate the outputs of a SPADE device that responds symmetrically along the ${\hat{\boldsymbol u}}$ and ${\hat{\boldsymbol v}}$ directions by replicating one quadrant of our data set $\{Y_{\textit{nm}}^{(j)}({\boldsymbol R}):j = 1, \ldots ,J;(n,m) \in {\cal N};u \lt 0\; {\rm and}\; v \lt 0\}$ to all four quadrants. We then repeat the two-point-localization analysis with the new data set. The yellow bars in Fig. 4 plot the resulting RMSEs in the same manner as Fig. 3(a), demonstrating the substantially increased localization errors in the symmetric case. To confirm that the increased errors are due to the sign problem, we take each location estimate ${{\check {\boldsymbol R}}_j}$ in the symmetric case, correct the signs of its $({u_j},{v_j})$ coordinates by comparing them with the true location, and compute the localization errors again. This new set of errors, with the sign problem artificially removed, is plotted as the blue bars in Fig. 4. The blue bars are comparable with Fig. 3(a), thus demonstrating that the sign problem is indeed the culprit for the increased errors in the symmetric case shown by the yellow bars. The similarity between Fig. 3(a) and the blue bars in Fig. 4 also demonstrates that our asymmetric system is able to both resolve the sign problem and maintain the nominal performance of a symmetric device.

Fig. 4. Yellow bars: localization errors with a hypothetical SPADE device that responds symmetrically to the point-source displacement ${\boldsymbol R}$ along the ${\hat{\boldsymbol u}}$ and ${\hat{\boldsymbol v}}$ directions. The outputs of such a device are generated by replicating one quadrant of the experimental data as a function of ${\boldsymbol R}$ to all four quadrants. Blue bars: localization errors of estimates with the sign problem artificially removed. The plots and the error bars otherwise follow the same format as that of Fig. 3(a).

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B. Moment Estimation

The second scenario we study is semiparametric moment estimation [30,34]. Instead of assuming that the number of point sources is known and estimating their locations, the goal of semiparametric estimation is to estimate only certain summary statistics about the source distribution without the need to know the source number. Each SPADE channel measures the generalized moment

(12)$${\beta _{\textit{nm}}} \equiv \iint {\bar Y_{\textit{nm}}}({\boldsymbol R})F({\boldsymbol R}){{\rm d}^2}{\boldsymbol R},$$

where $F({\boldsymbol R})$ is the source distribution, and here we take $\{{\beta _{\textit{nm}}}\}$ as the parameters of interest. ${\beta _{01}}$ and ${\beta _{10}}$, for example, are close to the second-order moments, which can quantify the object sizes along the two axes ${\hat{\boldsymbol u}}$ and ${\hat{\boldsymbol v}}$ [30], while combinations of the moments can form generalized Fourier coefficients for Fourier analysis and image reconstruction [38]. To demonstrate the estimation of each ${\beta _{\textit{nm}}}$ with our experimental data, we randomly draw $S$ locations $\{{{\boldsymbol R}_s}\} \equiv \{{{\boldsymbol R}_1}, \ldots ,{{\boldsymbol R}_S}\}$ for each trial of the simulated experiment, draw one sample $Y_{\textit{nm}}^{({l_s})}({{\boldsymbol R}_s})$ for each location, and take

(13)$${\check \beta} _{\textit{nm}}^{(l)} = \sum\limits_{s = 1}^S Y_{\textit{nm}}^{({l_s})}({{\boldsymbol R}_s})$$

as an estimator of the moment ${\beta _{\textit{nm}}}$, the true value of which is taken as

(14)$${\beta _{\textit{nm}}}(\{{{\boldsymbol R}_s}\}) = \sum\limits_{s = 1}^S {\bar Y_{\textit{nm}}}({{\boldsymbol R}_s}).$$

For each set of locations $\{{\boldsymbol R}_s\}$, we repeat the sampling and estimation procedure for $L = 100$ trials to obtain $L$ estimates $\{{\check \beta} _{\textit{nm}}^{(l)}:l = 1, \ldots ,L\}$ and compute the mean-square error (MSE) and the signal-to-noise ratio (SNR) given by

(15)$${{\rm MSE}_{\textit{nm}}}(\{{{\boldsymbol R}_s}\}) \equiv \frac{1}{L}\sum\limits_{l = 1}^L {[{\check \beta} _{\textit{nm}}^{(l)} - {\beta _{\textit{nm}}}(\{{{\boldsymbol R}_s}\})]^2},$$

(16)$${{\rm SNR}_{\textit{nm}}}(\{{{\boldsymbol R}_s}\}) \equiv \frac{{{{[{\beta _{\textit{nm}}}(\{{{\boldsymbol R}_s}\})]}^2}}}{{{{{\rm MSE}}_{\textit{nm}}}(\{{{\boldsymbol R}_s}\})}}.$$

We repeat the experiment $M = 10,000$ times by drawing $M$ sets of locations $\{\{{\boldsymbol R}_s^{(m)}\} :m = 1, \ldots ,M\}$, with a random source number $1 \le S \le 10$ for each set, and computing the resulting SNRs $\{{\rm SNR}(\{{\boldsymbol R}_s^{(m)}\}):m = 1, \ldots ,M\}$. To summarize these results in one plot, we divide the $M$ SNRs into bins according to the object size defined as

(17)$$\Delta (\{{{\boldsymbol R}_s}\}) \equiv \sqrt {\frac{1}{S}\sum\limits_{s = 1}^S |{{\boldsymbol R}_s}{|^2}} ,$$

such that the set of SNRs in each object-size bin is given by

(18)$${\{{{\rm SNR}_{\textit{nm}}}\} _j} \equiv \{{{\rm SNR}_{\textit{nm}}}(\{{{\boldsymbol R}_s}\}):{\Delta _j} \le \Delta (\{{{\boldsymbol R}_s}\}) \lt {\Delta _{j + 1}}\} .$$

Figure 5 plots the mean of ${\{{{\rm SNR}_{\textit{nm}}}\} _j}$ for each bin, with each error bar denoting the minimum and maximum of ${\{{{\rm SNR}_{\textit{nm}}}\} _j}$ within the bin. We observe high SNRs for all object sizes, including the subdiffraction regime. These SNRs can be translated to the SNRs for parameters related to the moments by error propagation. For example, ${\beta _{01}}$ can be related to the object size $\theta$ along the ${\hat{\boldsymbol u}}$ axis by ${\beta _{01}} = C{\theta ^2}$, where $C$ is a constant. Then error propagation implies that the MSE for $\theta$ is roughly ${{\rm MSE}_\theta} \approx {{\rm MSE}_{01}}/(\partial {\beta _{01}}\partial \theta {)^2} = {{\rm MSE}_{01}}/(2C\theta {)^2}$, and the SNR for $\theta$ is roughly ${\theta ^2}/{{\rm MSE}_\theta} \approx 4{{\rm SNR}_{01}}$.

Fig. 5. Summary of the signal-to-noise ratios ${\rm SNR}(\{{{\boldsymbol R}_s}\})$ in estimating the moments ${\beta _{\textit{nm}}}$ of multiple point sources via SPADE. In each plot, the title indicates the output channel that measures one of the moments, the horizontal axis has the object sizes $\Delta$ in linear scale grouped into multiple bins, and the vertical axis is the SNR in decibels (dB, $10\mathop {\log}\nolimits_{10} {\rm SNR}$). Each bar (blue) is the average SNR for sources with object sizes $\Delta (\{{{\boldsymbol R}_s}\})$ within the bin, while each error bar (orange) gives the maximum and minimum SNRs in that bin. Each horizontal axis is normalized with respect to the beam standard deviation $\sigma$.

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While SPADE can, in principle, offer substantial improvements over direct imaging in moment estimation when only photon shot noise is considered [30,34], we find it impossible to fairly compare our results here with direct imaging, since our experiment is dominated by excess noise and it is difficult to translate the noise level of our experiment to a per-pixel noise level for direct imaging in a fair manner. Even if excess noise is dominant, SPADE still offers the practical advantage of requiring only one photodetector for each moment, whereas direct imaging would require many pixels as well as more complicated data processing for moment estimation. We also note that the object moments naturally measured by our SPADE setup are close to the moments of even orders, and measurements in other optical mode bases, with the addition of controlled displacements [7] or interferometers [30,34], can be used to estimate other types of moments.

4. DISCUSSION AND CONCLUSION

In closing, we emphasize the novelties of our work relative to earlier experiments [5–28]. Most prior experiments considered only the estimation of the separation between two point sources. While our experimental setup is similar to that of Boucher et al. [16], our serendipitous use of the asymmetric SPADE response to estimate both locations of the two sources, not just their separation, as well as our demonstration of semiparametric moment estimation, are new to our knowledge. Brecht et al. have recently demonstrated quantum-limited simultaneous estimation of the centroid, offset, intensity imbalance for two pulses [19], but their demonstration was for pulses in the time domain, whereas ours is for imaging in the two-dimensional space domain. Another impressive recent experiment by Pushkina et al. demonstrated general superresolution imaging via SPADE [23]. It was for coherent sources, however, and the heterodyne detection they used would become much noisier with weak incoherent sources [39]. Our setup is envisioned to be translatable to more practical imaging scenarios in fluorescence microscopy, remote sensing, and astronomy, where photon shot noise may become dominant and SPADE can offer a fundamental advantage. The use of an asymmetric response to enable two-source localization is a practical alternative to adaptive measurements [33]; the asymmetry may be present naturally in an experiment or introduced intentionally by misalignment or another spatial-mode modulator. Semiparametric moment estimation, on the other hand, can offer important information about the object and contribute to Fourier analysis and general image reconstruction [38]. While it remains an open question whether these approaches can exactly attain the fundamental quantum limits to two-source localization and image reconstruction, our results demonstrate that the quantum-inspired superresolution techniques can be both experimentally feasible and applicable to general objects, thus paving the way towards their use in real applications.

Funding

National Research Foundation Singapore (QEP-P7).

Acknowledgments

We thank Wang Wei for useful discussions and Kenneth Y. W. Ng for his help with the early stage of the experiment.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Quantum-inspired superresolution for incoherent imaging

Abstract

1. INTRODUCTION

2. EXPERIMENT

3. RESULTS

A. Localization of Two Point Sources

B. Moment Estimation

4. DISCUSSION AND CONCLUSION

Funding

Acknowledgments

Disclosures

Data Availability

REFERENCES

Data Availability

Cited By

Figures (5)

Equations (18)

Optica