Due to the diffraction effect of light-field propagation, the resolution of imaging systems is limited. For over a century, Rayleigh’s criterion has been the most influential principle of the resolution limit for incoherent optical imaging systems [1]. Nonetheless, various technical methods to break Rayleigh’s criterion have been proposed [2]. In structured illumination microscopy (SIM) [3] and Fourier ptychographic microscopy [4], by utilizing multiple structured illuminating light fields, the diffraction limit imposed by the finite aperture of an imaging system can be broken through post-synthetic processing. By reducing the scanning spot size of illuminating or emitting light using the near-field or stimulated emission depletion effect, point-scanning super-resolution (SR) imaging methods, such as near-field scanning optical microscopy (NSOM) [5] and stimulated emission depletion microscopy (STED) [6], successfully surpass the Rayleigh diffraction limit. Another category of far-field SR imaging techniques utilizes the single-molecule localization, including stochastic optical reconstruction microscopy (STORM) [7] and photoactivated localization microscopy (PALM) [8]. These methods estimate the positions of individual fluorescent molecules from diffraction-limited image sequences and synthesize an SR image from thousands to tens of thousands of localization images. More recently, overcoming the “Rayleigh’s curse” is investigated by utilizing quantum properties of classical and quantum light fields [9,10]. In parallel with the above physics-based SR techniques, numerous SR algorithms [11] have been developed to achieve SR images from low-resolution images, such as spectrum extrapolation [12], multi-frame SR [13], SR image using deep learning [14], sparsity prior [15,16], and wavelength multiplexing [17,18].

Traditionally, almost all SR imaging systems have been designed for the direct point-by-point imaging mode, in which the emitted light from each two-dimensional (2-D) spatial point at the object plane is directly recorded on the corresponding pixel of the detector at the image plane. The SR capability of this imaging mode lies in the ability to enhance the precision of separating two adjacent points in the 2-D space. Essentially, light fields can be fully characterized by a multidimensional plenoptic function involving the temporal, spatial, spectral, and polarization dimensions [19]. Shannon’s channel capacity of an imaging system can be described as $\mathcal{H}=N_{\mathrm{DOF}}\log (1+\mathrm{SNR})$, which is determined by the detection signal-to-noise ratio (SNR) using the Gaussian noise model and multiple optical degrees of freedom (DOFs), where $N_{\mathrm{DOF}}=N_t·N_s·N_c·N_{\psi }$, with $N_t,N_s,N_c$, and $N_{\psi }$ denoting the temporal, spatial, spectral, and polarization DOFs, respectively [20,21]. However, the existing direct point-by-point imaging mode poses challenges when capturing image information in the high-dimensional (H-D) light-field space using a 2-D photon detector, and it will “degenerate” the DOFs of light fields in single-shot detection. Therefore, the utilization of H-D imaging channel capacity is limited, which makes it hard to achieve SR with a single-shot measurement even for existing state-of-the-art SR imaging techniques.

Different from the direct point-by-point imaging mode, ghost imaging (GI) acquires object’s image information through a “global random” encoding mode [22–25]. Specifically, GI utilizes fluctuating light fields to modulate the image information of an object, which is encoded into detectable lower-dimensional signals. Subsequently, the H-D light-field information is embedded in the high-order light-field correlation. To achieve the correlation, typical GI systems require multiple samplings in the temporal dimension [24]. In contrast, by utilizing the ergodicity of thermal light fields, the GI camera via sparsity constraints (GISC camera) [26] has been proposed to extract image information from light-field correlation in the spatial dimension rather than the temporal dimension, thereby enabling data acquisition in a single shot. So far, the GISC camera has been demonstrated to be able to simultaneously acquire objects’ information of spatial, spectral, and polarization dimensions in a single shot [27]. This implies that the GISC camera can transmit H-D image information and efficiently utilize the imaging channel capacity compared to conventional imaging systems based on the direct “point-by-point” imaging mode. Consequently, the GISC camera will provide new possibilities to break Rayleigh’s criterion using the H-D image information of objects.

In this study, we propose a single-shot SR imaging scheme by leveraging the object’s discernibility in the H-D light-field information acquired by the GISC camera. Through theoretical and experimental studies, our results demonstrate that while the resolution in the H-D light-field space, characterized by the second-order correlation of light fields, remains limited by diffraction, the spatial resolution can be significantly improved by leveraging the discernibility in other dimensions, such as the spectral and polarization dimensions. We theoretically analyze the overall limitations of imaging resolution in the H-D light-field space and establish quantitative relationships among the imaging resolution, SNR, number of samplings, and object sparsity level. Moreover, the relationship between the spatial SR capability of the GISC camera and the discernibility in the H-D light-field space is experimentally investigated. Our experimental results demonstrate that the single-shot SR imaging scheme can achieve 5-fold spatial resolution improvement over the Rayleigh diffraction limit. It is worth mentioning that unlike those SR imaging approaches that sacrifice resolution in other dimensions to enhance spatial resolution [28], our proposed scheme efficiently exploits the channel capacity of imaging systems, enabling SR imaging in a single shot.

The SR scheme via discernibility in the H-D light-field space based on the GISC camera is shown as Fig. 1. Figure 1(a) shows the principle of the proposed SR imaging. Considering that the object consisting of two sources with H-D light-field information is imaged through a lens system, the two sources become diffraction-limited and cannot be resolved through the direct 2D point-by-point imaging mode due to the large size of the diffuse spots. In contrast, in the GISC camera, the object’s H-D information is encoded into a detectable 2D speckle pattern, which can be detected by an array detector in a single shot. By utilizing prior constraints, the H-D information with high resolution can be retrieved from the single-shot detection speckle image using the pre-calibrated system responses of the GISC camera when the system is set up. Furthermore, the SR image is achieved by synthesizing these H-D images.

Figure 1(b) shows the schematic diagram of the GISC camera for capturing H-D light-field information for SR imaging. Specifically, the proposed SR scheme based on the GISC camera involves three stages: calibration, detection, and SR. In the calibration stage, the system responses of the GISC camera are measured by scanning a point calibration source in the H-D light-field space. This process is performed only once when the GISC camera is set up. At the detection stage, the object with H-D information is modulated into a 2-D speckle pattern, which is captured in a single shot. Finally, the H-D image information of the object is reconstructed using the single-shot measurement and the pre-calibrated responses via an optimization algorithm, and the SR image is then achieved by a synthesis process.

As the core system of the proposed scheme, the apparatus of the GISC camera system is as follows. A conventional imaging system consisting of lens 1 with focal length $f_1$, an iris with aperture $D$, and lens 2 with focal length $f_2$ is used to project the object onto its focal plane. A spatial random phase modulator (SRPM) placed $z_1$ away from the focal plane of lens 2 is exploited to modulate the light fields of the captured H-D information, and finally the detector $z_2$ away from the SRPM records the modulated light signal. The SRPM plays a critical role by modulating thermal light into a spatially fluctuating pseudo-thermal light and acting as a random grating that generates uncorrelated speckles for different wavelengths. It facilitates the modulation of object’s H-D information into a detectable 2D plane, allowing the GISC camera to achieve H-D imaging in a single exposure. Basically, the H-D imaging information is contained in the second-order light-field fluctuation correlation [22–25] as $$\begin{gathered} \mathrm{\Delta }{G}^{(\mathrm{2,2})}({\mathbf{r}}_{\mathit{r}},k_l)={⟨\mathrm{\Delta }{I}_t({\mathbf{r}}_t)\mathrm{\Delta }{I}_r({\mathbf{r}}_t;{\mathbf{r}}_r,k_l)⟩}_{{\mathbf{r}}_t} \\ \propto \int \int T({{\mathbf{r}}^{\prime }}_{\mathit{r}},{k^{\prime }}_l)\mathrm{\Delta }{g}^{(2)}({\mathbf{r}}_{\mathit{r}},{{\mathbf{r}}^{\prime }}_{\mathit{r}};k_l,{k^{\prime }}_l)\mathrm{d}{{\mathbf{r}}^{\prime }}_{\mathit{r}}\mathrm{d}{k^{\prime }}_l \\ \propto T({\mathbf{r}}_r,k_l)*g^{(2)}({\mathbf{r}}_r,k_l), \end{gathered}$$(1)where $I_t({\mathbf{r}}_t)=\iint T({\mathbf{r}}_r,k_l)I_r({\mathbf{r}}_t;{\mathbf{r}}_r,k_l)\mathrm{d}{\mathbf{r}}_r\mathrm{d}{k}_l$ is the recorded signal of the H-D (spatial- and spectral-dimensional) image information $T({\mathbf{r}}_r,k_l)$ in the detection process, $I_r({\mathbf{r}}_t;{\mathbf{r}}_r,k_l)$ represents the system response of the GISC camera measured during the calibration process, ${⟨·⟩}_{{\mathbf{r}}_t}$ indicates that the ensemble average is calculated over the spatial dimension (this holds due to the ergodicity of thermal light fields), and * represents the convolution operation. This implies that the normalized second-order correlation function $g^{(2)}$ generally characterizes the capability of the GISC camera for acquiring H-D imaging information, which is further confirmed in Section 2.B. See Appendix B for a detailed derivation of Eq. (1).

After the signal detection, the H-D imaging information is reconstructed using the compressed-sensing (CS) algorithm [29,30]. In particular, the detection process can be discretized into the matrix form as $$\mathbf{Y}=\mathbf{\Phi }\mathbf{x},$$(2)where $\mathbf{Y}\in {\mathbb{R}}^{\mathit{m}}$ represents the sampling signals composed of the speckle intensity $I_t({\mathbf{r}}_t)$, the sampling matrix $\mathbf{\Phi }\in {\mathbb{R}}^{\mathit{m}\times n}$ has columns consisting of the speckle patterns $I_r({\mathbf{r}}_t;{\mathbf{r}}_r,{\lambda }_l)$, $m$ is the sampling number, and $\mathbf{x}\in {\mathbb{R}}^n$ denotes the image of the object in the spatial and spectral dimensions. The $\mathbf{x}$ can be recovered by solving an inverse problem: $$\underset{\mathbf{x}}{\min }{\Vert [\mathbf{Y}-\overline{\mathbf{Y}}]-[(\mathbf{\Phi }-\overline{\mathbf{\Phi }})\mathbf{x}]\Vert }_2+\beta \mathit{R}(\mathbf{x}),$$(3)where $\overline{\mathbf{Y}}$ contains identical entries equal to the mean value of $\mathbf{Y}$, $\overline{\mathbf{\Phi }}$ consists of columns, each of which has identical entries equal to the mean value of the corresponding column of $\mathbf{\Phi }$, and $\mathit{R}(\mathbf{x})$ denotes the regularization on $\mathbf{x}$. To solve the ill-posed inverse problem Eq. (3), there are roughly three well-known categories of model-based algorithms: (1) optimization algorithms, (2) greedy algorithms, and (3) thresholding-based algorithms. In previous works, total variance regularization and rank minimization have been utilized for GISC spectral imaging in continuous scenes [26,27], since these priors represent a smooth property in both spatial and spectral dimensions. In contrast, our current study focuses on the super-resolution ability for sparse scenes in the H-D light-field space. Hence, we use a sparsity prior of the object in both spatial and spectral dimensions to solve Eq. (3). Specifically, we adopt $l_1$-regularization for image restoration using the gradient projection for sparse reconstruction (GPSR) algorithm [31], as it offers superior noise reduction and stable reconstruction performance compared to $l_0$-regularization.

After retrieving high-quality H-D image information, the SR image is finally achieved by a synthesis process, where the multiple images representing H-D information are merged and the high-resolution image is obtained. Along with this scheme, we further investigate a theory to illustrate that the spatial resolution of the GISC camera can be enhanced by utilizing the discernibility of objects in the H-D light-field space, such as the spectral space.

From Eq. (1), it can be found that $g^{(2)}$ actually characterizes the resolving ability within the framework of the second-order correlation ensemble average, where an infinite number of samplings are assumed. However, in practical imaging, the samplings are always finite, and the relation shown in Eq. (1) will be inaccurate. Besides, algorithms other than light-field correlation, such as CS algorithms that further exploit various kinds of sparsity prior, have been applied for H-D information retrieval in the GISC camera. Therefore, we conduct statistical resolution and algorithmic resolution analyses for the GISC camera rather than the description with $g^{(2)}$.

Statistical resolution analysis: The theoretical analysis is conducted first from the perspective of statistical resolution [32,33]. Considering that the GISC camera does not satisfy the spatial invariance of point spread functions (PSFs), performing a statistical resolution study for the case of imaging only two-point sources (like what statistical resolution analyses on conventional imaging systems with spatial-invariant PSFs commonly do) is insufficient. Thus, we propose to consider imaging $K$ monochromatic point sources located at different positions in the H-D space with illumination $a$$({\{a\delta (r_i,k_i)\}}_{i=1}^K)$ and focus on the resolving condition between two of them, treating the response signals associated with other point sources as disturbances. We derive the spatial statistic resolution limit of the GISC camera by pursuing the Cramér–Rao lower bound (CRLB) of estimating the spatial distance $\mathrm{\Delta }r=r_2-r_1$. Given the detection model of the GISC camera represented by the probability density function $p(\mathbf{y}|\mathit{\theta };\mathit{\eta })$, where $\mathbf{y}$ consists of the detection signals in multiple pixels, $\mathit{\theta }=(r_1,r_2)$ is the parameter to be estimated, $\mathit{\eta }=(k_1,k_2)$ is used to represent the discernibility in the spectral information and supposed to be known, and the Fisher information matrix (FIM) and CRLB for estimating parameters $\mathit{\theta }$ can be calculated via ${\mathbf{J}}_{\mathit{ij}}(\mathit{\theta };\mathit{\eta })={\mathbb{E}}_{\mathbf{y}}\{-\frac{{\partial }^2[\ln p(\mathbf{y}|\mathit{\theta };\mathit{\eta })]}{\partial {\theta }_i\partial {\theta }_j}\}$ and $\mathrm{CRLB}({\theta }_i;\mathit{\eta })={\mathbf{J}}_{\mathit{ii}}^{-1}(\mathit{\theta };\mathit{\eta })$, respectively. By performing CRLB transformation and combining it with the statistical resolution principle $\mathrm{\Delta }r\ge \mathrm{CRLB}(\mathrm{\Delta }r;\mathit{\eta })$, we achieve the expression for spatial statistical resolution $\mathrm{\Delta }{\mathbf{r}}_{\text{statis}}$ of the GISC camera as $$\frac{2}{m}(\frac{K^2}{\mathrm{DSNR}}+K-2)={\alpha }^2\mathrm{\Delta }{\mathbf{r}}_{\text{statis}}^2[1-g^{(2)}(\mathrm{\Delta }{\mathbf{r}}_{\text{statis}},\mathrm{\Delta }k)(1-{\alpha }^2\mathrm{\Delta }{\mathbf{r}}_{\text{statis}}^2)],$$(4)where $m$ is the sampling number, DSNR denotes the detection SNR, which is defined as $\mathrm{DSNR}={\overline{I}}_{\text{signal}}^2/{\sigma }_{\text{noise}}^2$, $\alpha =\overline{k}D/2f_1$, and $\mathrm{\Delta }k$ is the measure of discernibility of the spectral space. The above expression highlights that the statistical limit of the resolving ability of the GISC camera does relate to sampling number $m$, DSNR, sparsity level $K$, and discernibility in the spectral information of the object, as well as the normalized second-order correlation function in the H-D light-field space. See Appendix C for more details about the statistical resolution. It is worth noting that the resolution limit determined by the CRLB is the best-case bound independent of the specific algorithm.

Algorithmic resolution analysis: Nevertheless, during the practical imaging process, some algorithms are employed to reconstruct the image, thereby influencing the achieved resolution. Therefore, we investigate an algorithmic resolution analysis. In this analysis, the case of imaging $K$ monochromatic point sources is considered. The analysis is conducted by combining the relationship between $g^{(2)}(\mathrm{\Delta }\mathbf{r},\mathrm{\Delta }k)$ and the mutual coherence $\mu$ of the system’s transmission matrix with the exact recovery condition in the noisy scenario [34] in the CS paradigm [29,30]. The derived expression for the algorithmic resolution is formulated as below: as long as the H-D distance $\{\mathrm{\Delta }\mathbf{r},\mathrm{\Delta }k\}$ between each pair of point sources among those $K$ points in the H-D space satisfies $$g^{(2)}(\mathrm{\Delta }{\mathbf{r}}_{\mathrm{alg}},\mathrm{\Delta }k)+\sqrt{\frac{1}{m}\{1+4g^{(2)}(\mathrm{\Delta }{\mathbf{r}}_{\mathrm{alg}},\mathrm{\Delta }k)+3{[g^{(2)}(\mathrm{\Delta }{\mathbf{r}}_{\mathrm{alg}},\mathrm{\Delta }k)]}^2\}}=\frac{1-\frac{2K}{\sqrt{\mathrm{DSNR}}}}{2K-1},$$(5)the information of those point sources can be well retrieved and, thus, can be resolved. Please refer to Appendix D for more details. Similar to the statistical resolution, the algorithmic resolution also depends on the sampling number $m$, DSNR, and sparsity level $K$. Because the exact recovery condition on $\mu$ is stringent to ensure signal retrieval in the worst case, the algorithmic resolution given by Eq. (5) describes the worst-case resolution, which hardly happens in practical experiments.

Though the expressions for both statistical and algorithmic resolutions have been obtained, due to the complexity of both the expressions and the involved Bessel function in $g^{(2)}$, a direct analytic expression for the imaging resolution similar to the Rayleigh criterion seems hard to give. Nevertheless, it can be conveniently obtained by a numerical solution once the parameters are given. To show that, the two imaging resolution bounds are visually illustrated by plotting them under different conditions including the sparsity level, sampling number, and DSNR in Fig. 2. As a reference, the Rayleigh diffraction resolution is plotted in Fig. 2(b) with a dark-red color. It clearly shows that the algorithmic resolution bounds are uniformly worse than the corresponding statistical resolution bounds, while both bounds are superior to the Rayleigh resolution. Also, both spatial resolution bounds improve as the discernibility in the spectral space increases, demonstrating the feasibility of the GISC camera in improving the spatial resolution by utilizing the discernibility in the H-D light-field space. Moreover, both resolution bounds get better when the sampling number or DSNR gets larger, and they get worse as the sparsity $K$ increases, indicating that the super-resolution ability is related to the object’s sparsity. Especially, when the object is not sparse, namely the sparsity $K$ of the object is very large, both resolution bounds of the GISC camera without exploiting the H-D space discernibility will approach the classical Rayleigh’s limit in the noise-free case (see Appendix C.2 and Appendix D for detailed analysis). Our theoretical results also reveal that, for objects with the same wavelength, spatial resolution highly depends on the DSNR, the object’s sparsity ($K$), and the sampling number. Both a smaller $K$ and a higher SNR can lead to an improved spatial resolution, which is consistent with previous work [35].

In order to validate the analytical results, we perform simulations. In the simulation, the parameters of the GISC camera are set as $f_1=50\mathrm{mm}$, $D=1.48\mathrm{mm}$, $n=1.53$, $\omega =3.5\mathrm{\mu m}$, $\eta =12\mathrm{\mu m}$, $z_1=5\mathrm{mm}$, and $z_2=40\mathrm{mm}$, and the working waveband is $\lambda =530-560\mathrm{nm}$, so $\overline{\lambda }=545\mathrm{nm}$. First, the sampling matrix whose columns consist of speckle patterns is created, where speckle patterns are simulatively achieved for the $64\mathrm{\mu m}\times 64\mathrm{\mu m}$ field of view (FoV) and wavelength ranging from 530 nm to 560 nm with interval 15 nm; see Appendix E for more details about the simulation process. Second, a resolution test target is built [Fig. 3(A)], and the spatial distances between each slit are, respectively, 6 μm,5 μm,4 μm, and 3 μm. To imitate the diffraction limit of the conventional imaging system, we convolve the image of the resolution test target with a PSF = $(2J_1(\pi D|x|/\lambda f_1)/(\pi D|x|/\lambda f_1{)}^2$, to get the diffraction-limited image [Fig. 3(B)], where the full width at half-maximum (FWHM) of the PSF is about 8 μm at wavelength 530 nm. The sampling signals $\mathbf{Y}$ for the resolution test target with or without spectral difference are created via Eq. (2). By solving Eq. (3) using the GPSR algorithm, the reconstructed result of the resolution test target without a spectral difference shows that the spatial distance below 5 μm is hard to resolve [Fig. 3(D)]. But when the resolution test target has a spectral difference, where the labels 1, 2, 3 on the resolution test target have different wavelengths [Fig. 3(C)], the spatial distance as low as 3 μm in the reconstructed result can be recognized [Fig. 3(E)]. Specifically, Fig. 3(F) reveals that the spatial resolution of the GISC camera is obviously improved with the spectral difference.

The influence of the detection noise on improving the spatial resolution of the GISC camera is also investigated through simulations. To this end, the recovery performances of the object with and without spectral difference are compared under different noise levels. The detection noise is assumed as Gaussian white noise, and the noise level is measured by the DSNR in dB, defined as ${\mathrm{DSNR}}_{\mathrm{dB}}=20{\log }_{10}(\overline{\mathbf{Y}}/\sigma )\mathrm{dB}$, where $\overline{\mathbf{Y}}$ is the mean value of the sampling signals $\mathbf{Y}$ and $\sigma$ is the standard variance of Gaussian white noise. In the simulation, the ${\mathrm{DSNR}}_{\mathrm{dB}}$ is set from 5 dB to 30 dB with an increasing step of 5 dB. The reconstructed images of the object with and without spectral difference through the GPSR algorithm are shown in Figs. 4(B) and 4(C). We adopt the quantitative evaluation metrics, peak signal-to-noise ratio (PSNR), and structural similarity (SSIM) [36], to evaluate the imaging quality. One can see that the spatial resolution as well as the imaging quality of the object with spectral difference is much better than that of the object without spectral difference, especially for the low DSNR, which clearly demonstrates the advantage of the object with the spectral difference in improving the spatial resolution and image quality of recovered images.

Figure 5 shows the imaging results of the GISC camera in improving the spatial resolution for the dot objects, which investigates the potential for fluorescent microscopy. The dot object consists of different orientation’s dots [Fig. 5(A)], and their adjacent dots emit lights of the same wavelength or lights with different wavelengths (530 nm and 545 nm, respectively). It can be observed from Fig. 5(D) that the dots with spectral difference can be resolved (see blue rectangles) and the dots without spectral difference cannot be resolved (see yellow rectangles) in the reconstructed image, which clearly demonstrates the advantage of the GISC camera in improving the spatial resolution for the application of fluorescent microscopy by exploiting the spectral difference of fluorescent emitters.

The experimental setup of the GISC camera is outlined schematically in Fig. 6. Ahead of practical imaging, the impulse responses are measured through a calibration process, which involves employing artificially generated monochromatic point sources. After calibration, the imaging object is introduced, and the resultant response signal is captured by detector 1. The captured signal is used as input for subsequent image information retrieval. For comparison, the diffraction-limited image of the conventional imaging system is recorded on detector 2 at another optical path. The classical Rayleigh’s criterion of the conventional imaging systems at the object plane is about 22.2 μm at the incident wavelength 540 nm. See Appendix F for details about the parameter settings for the GISC camera.

Two-point super-resolution: In the first experiment, two-point sources with different spatial distances $\mathrm{\Delta }r$ and different spectral gaps $\mathrm{\Delta }\lambda$ are constructed. Specifically, two-point sources with three different spatial distances 4.0 μm, 4.6 μm, 5.1 μm corresponding to spectral gaps 23 nm,17 nm, 9 nm around the central wavelength 540 nm, respectively, are constructed. As shown in Figs. 7(A) and 7(E), the classic Richardson–Lucy deconvolution method [37] cannot resolve the two-point sources through the conventional imaging system, but the GISC camera can resolve the two-point sources. Moreover, the resolved spatial distance of the GISC camera is improved when the spectral discernibility of the H-D information of the object increases. Besides the resolved spatial distance, the localization error for the point source, which is estimated by the width of the reconstructed spots, reduces as the spectral-space difference increases, as shown in Fig. 7(B). To evaluate the achieved resolution more accurately, we analyze the obtained images in Fig. 7(A) with the Fourier rolling correlation (FRC) method [38]. The corresponding FRC curves with the $2-\sigma$ curve are shown in the three subplots in Fig. 7(C), respectively, from which the critical resolutions are obtained.

We compare the experimental resolution with the developed resolution bounds, as shown in Fig. 7(D), where the “Experimental” relationship between the resolving distance and the spectral-space difference is from Fig. 7(A), the error bar denotes the localization error from Fig. 7(B), the “Experimental-FRC” with the orange-star marker denotes the evaluated resolution from the FRC method, and the two theoretical resolution bound curves are given by Eqs. (4) and (5), with experimental parameters as follows: the independent sampling number $m=1600,{\mathrm{DSNR}}_{\mathrm{dB}}=24\mathrm{dB}$. It can be observed that the experimental imaging resolution is bounded by both the statistical resolution and algorithmic resolution. This is because the algorithmic resolution defined by Eq. (5) builds on the worst case in CS [34] and the statistical resolution given by Eq. (4) represents a best-case bound for resolution estimation, which is irrelevant to specific reconstruction algorithms but relies on the sampling number and DSNR. We would like to mention that the fabrication of two-point sources in our experiments results is not merely two isolated pixels in the object plane, but rather two clusters. Therefore, for the statistical resolution curve plotted in Fig. 7(D), the sparsity parameter $K$ is set as 18, where the two point-like sources are approximately two spots with $3\times 3$ pixels determined from the reference image captured by detector 2. In contrast, when considering the algorithmic resolution, which represents the minimum distance of any two points among $K$ resolvable point sources in the case of exact recovery, since in the experiment the object consists of two clusters and points inside each cluster that are not resolved, $K$ is set to 2 in this instance.

Super-resolution results of three slits: Figure 8 presents the experimental result of three slits. The spatial distance and spectral gap between each slit are 5.0 μm and 15 nm, respectively, and the spectral width of each slit is about 4.5 nm, as shown in Fig. 8(D). Figures 8(B), 8(C), and 8(E) are the corresponding diffraction-limited image of the conventional imaging system, the deconvolution image from the diffraction-limited image, and the reconstructed super-resolution image, respectively. It can be observed that, in the case of 5.0 μm spatial distance between each slit, the imaging result of the conventional imaging system is a largely diffused spot, which is limited by Rayleigh’s criterion, and the three slits cannot be resolved from that as well as its deconvolution image. In contrast, they are clearly resolved by utilizing the H-D space discernibility. It implies that exceeding four-folds over the classical Rayleigh’s criterion, is achieved via the discernibility in the H-D $(r,\lambda )$ space in this experiment.

In a nutshell from the results, the best resolved spatial resolution of the proposed SR scheme based on the GISC camera can achieve 4 μm at the spectral gap 23 nm, which achieves 5-fold enhancement beyond the classical Rayleigh’s criterion ($\sim 22.2\mathrm{\mu m}$). While current imaging results still face challenges in achieving lower-bounded statistical resolution, the relationship between the resolved spatial distance and the discernibility in the $(r,\lambda )$ space highly correlates with the statistical resolution result. Moreover, the improvement of spatial resolution through the discernibility in the spatial-polarization space has been investigated experimentally, and a similar relationship between the resolved spatial distance and the discernibility in the spatial-polarization space is presented in Appendix H.

Recently, GISC nanoscopy with 80 nm spatial resolution in a single frame has been demonstrated by utilizing the sparsity of fluorescence emitters [35]. In the case that conventional Rayleigh’s criterion of the microscopy lens is about 200 nm, its spatial resolution is improved over 2-fold. Different from this work, our proposed scheme utilizes the H-D information, including not only the spatial dimension but also other dimensions like spectrum and polarization, to enhance the spatial resolution. In specific, our proof-of-principle experimental results achieve an imaging resolution exceeding 5-fold improvement over the classical Rayleigh’s criterion. It demonstrates that, besides utilizing the spatial sparsity, the capability of the GISC camera to break the Rayleigh’s criterion could be further improved by capturing H-D light-field information and exploiting the discernibility in the H-D space. Also, it is worth noting that the proposed SR imaging scheme, by utilizing the discernibility of H-D light fields, is not independent of existing SR methods. Therefore, rather than serving as an alternative to existing SR techniques, it can be incorporated into other existing SR methods to assist the resolution enhancement.

Currently, there are a few single-shot H-D imaging systems. By light-field modulation, H-D light-field images can be mapped onto a 2-D detectable plane through a compressed encoding manner and retrieved through digital decoding. However, not all of them are suitable for super-resolution imaging through the proposed scheme. Specifically, currently used modulation schemes can be sorted into two categories. One is the phase modulation, like the one used in the presented GISC camera. The other is the amplitude modulation on the image plane. For example, in coded aperture snapshot spectral imaging (CASSI) [39,40], spectral images are modulated using coded mask composed binary values (0 and 1) and are reconstructed by exploiting similarities in the H-D space. Those single-shot H-D imaging systems based on amplitude modulation, however, make it hard to apply to super-resolution imaging due to the following two points. First, the amplitude modulation sampling scheme highly relies on the assumption of image continuity so that the missed H-D information during the sampling process can be filled. However, this assumption does not hold in the field of super-resolution microscopy like fluorescence microscopic imaging. Second, to guarantee a sufficient detection SNR, the amplitude-modulation-based scheme requires the spatial size of sampling units on the image plane to significantly exceed the diffraction limit of the imaging system. Namely, the spatial resolution is mainly limited by the sampling resolution, which is far from the diffraction limit, making it unsuitable for super-resolution applications.

In summary, we present a single-shot super-resolution imaging scheme by leveraging the discernibility in the H-D light-field information acquired by the GISC camera. We demonstrate both theoretically and experimentally that, while the “Rayleigh’s curse” induced by the diffraction effect still exists when resolving two closely located points in the H-D light-field space, it can be avoided in the spatial dimension as long as the two points are distinguishable in other dimensions of the H-D light-field space, such as the spectral and polarization dimensions. We theoretically investigate the overall limits of imaging resolution in the H-D light-field space, establishing quantitative relationships among imaging resolution, SNR, number of samplings, and object sparsity level, and quantify the spatial-resolution improvements by leveraging discernibility in the H-D light-field space. The proof-of-principle experimental results well match the theoretical resolution bounds and show that exceeding 5-fold improvement beyond the classical Rayleigh’s criterion was achieved, while the conventional deconvolution approach cannot resolve the fine structure of the object, emphasizing the necessity and effectiveness of our proposed scheme.

With the rapid development in the fields of signal processing for sparse recovery [41,42], light-field optimization based on metasurfaces [43], megapixels and polarized detection technologies [44], and the time-of-flight detection technology [45], our work offers huge potential applications for sparse signals in their native domain, such as multi-color fluorescence microscopy [46], stimulated Raman scattering microscopy [47], polarization microscopy [48], and wavelength and polarization astronomy [49]. On the other hand, for broadband spectra or continuous scenes, the idea of realizing spatial super-resolution from the discernibility in the H-D space can be generalized. For example, with the help of distance measures in the field of manifold and information geometry [50,51], it is possible to pursue a generalized representation of H-D discernibility of broadband spectra or continuous scenes to be exploited for spatial super-resolution. Our future work will be directed toward this avenue.