## Abstract

Far-field imaging beyond the Rayleigh limit is one of the most important challenges in optics, microwave, and ultrasonics. We propose a novel sparsity-promoted super-oscillation imaging scheme for reconstructing more universal objects in subwavelength scales, which solves a weighted optimization problem constrained by *l _{p}*-norm-based sparsity regularization ($\text{0}\le p\le \text{1}$). We demonstrate numerically that the proposed imaging technique improves the resolution related to existing approaches remarkably for the case of very high signal-to-noise ratio (SNR), including the traditional super-oscillation imaging and sparsity-based super-resolution imaging. The standard superoscillation based super-resolution imaging approach can be regarded as the first-iteration solution of the proposed scheme. Numerical results for one- and two-dimensional super-resolution imaging are presented for validation.

© 2014 Optical Society of America

## 1. Introduction

Since the discovery of the diffraction limit, i.e., the Rayleigh limit, by Lord Rayleigh in 1891, imaging with a resolution beyond such barrier has been one of the most important issues in microwave, optics, ultrasonics, etc. The near-field scanning technique and its various variants, which rely on capturing directly or indirectly the evanescent waves carrying the fine-scale information of an illuminated object, have been widely accepted to break the Rayleigh limit. All of these super-resolution imaging techniques require an imaging lens (whether the conventional lens or other metamaterial based super-/hyper-lens [e.g., 1, 2]) or a probe to be placed in the vicinity of the imaged samples, thus suffer from several important limitations such as time-consuming data acquisition, and the fabrication of subtle lens, etc.

Numerous efforts have been directed towards the super-resolution imaging without the usage of evanescent waves, and three representative strategies amongst them include the sparse reconstruction [3, 4], structured illumination [5], and super-oscillation imaging [6–9]. Here, we focus on the study of superoscillation imaging in the context of computational imaging. Superoscillations mean that a waveform can oscillate, over a finite interval, arbitrarily faster than its highest constitute frequency component. As such, the concept of superoscillations holds the promise for the subwavelength focusing at much longer imaging distances of several wavelengths and beyond. Aseries of super-oscillation microscopes have been invented for far-field optical imaging, which rely on fabricating a huge-size binary amplitude mask of optimized concentric rings with different widths and diameters [8]. Recently, to avoid creating such subtle mask, a computational super-oscillation strategy, instead of a hardware implementation, has also been proposed to perform sub-wavelength imaging [6, 7]. For instance, Amineh et al proposed a simple yet effective technique that constructs a super-oscillatory filter by adapting from the principle of super-directivity antenna, in combined with traditional back-propagation algorithm. Although the superoscillation based imaging technology in principle has no physical constraints on the size of smallest-achievable focal spot that can be created in the hardware or algorithmic manner, its specific implementations do suffer from an important limitation: to trade off the imaging resolution with the size of field of view (FOV). Therefore, such superoscillation imaging strategy is typically limited in imaging universal objects, unless incorporating an adaptive data acquisition. For example, a typical scenario is that the probed object consists of several groups of point sources separated in the subwavelength scale, while the centers of different groups are well separated from each other.

In this paper, we extend the standard super-oscillation imaging for narrow-FOV objects to a sparsity-promoted super-oscillation imaging for more universal objects. The proposed method consists of solving weighted optimization problems constrained by a -norm-based sparse prior ($\text{0}\le p\le \text{1}$), which fully exploits two super-resolution mechanisms of super-oscillation and sparse reconstruction. We demonstrate numerically that the proposed imaging technique outperforms the traditional super-oscillation imaging and sparsity-based super-resolution imaging approaches remarkably in the case of very high signal-to-noise ratio (SNR). The solution rendered by the methods in [6,7] can be viewed as the first-iteration solution of the proposed method.

## 2. Adaptive super-oscillation imaging beyond the Rayleigh limit

To illustrate the operational principle of the proposed methodology, we start our formulations assuming a linear scanned aperture and a linear imaged domain (both along the $x$ direction). With reference to Fig. 1(a) for the one-dimensional (1D) holographic imaging setup, the single-frequency coherent data $S(x)$ is acquired on a 1D scanned line of $-{L}_{x}\le x\le {L}_{x}$ at $z={z}_{0}$, and the sources under investigation $J({x}^{\prime})$ are located on the imaged line $z=0$. Following the guideline adopted in [7], the governing equation of coherent data $S(x)$ in relation to the probed sources $J({x}^{\prime})$ reads

where $J()$ denotes the current density of the line source aligned infinitely along*y*-direction, $G(x-{x}^{\prime})=\frac{j}{4}{H}_{0}^{(1)}\left(\sqrt{{z}_{0}^{2}+{(x-{x}^{\prime})}^{2}}\right)$ is the two-dimensional (2D) Green’s function, and ${H}_{0}^{(1)}()$ is a Hankel function of the first-kind and zero-order.

Recall the spectral representation of 2D Green’s function:

First, we discuss briefly the principle of standard superoscillation based super-resolution imaging method. Similar to the well-known filtered back-propagation algorithm, the super-oscillation imaging formula reads formally:

It is worth remarking that the superoscillation based subwavelength imaging technique relies heavily on the use of an important prior: the objects under investigation are confined into a *known* finite interval of [$-{D}_{x}$,${D}_{x}$]. More specifically, the larger ${D}_{x}$ is, the higher the resolution is. To take automatically such prior into account, we consider the discrete counterpart of Eq. (3), and construct the following constrained optimization problem:

Initializing $\widehat{J}$ to be a zero vector, and applying the standard Newton’s iterative approach, we obtain immediately the first-iteration solution to Eq. (6) as

Note that Eq. (7) is exactly the discrete form of traditional superoscillation imaging formula represented by Eq. (4). For fixed $W$ the closed-form solution to Eq. (6) can be obtained asComparing Eq. (8) with Eq. (7), we can notice there is an extra term, i.e., ${\left({F}^{\prime}WF\right)}^{-1}$, which is responsible for suppressing the high-level side lobes involved in the traditional super-oscillation imaging, as demonstrated numerically below. For notable convenience, we refer to the imaging scheme by Eq. (8) as the improved super-oscillation imaging. Since the imaging routine by Eq. (6) is capable of taking automatically FOVs into account, it is flexible in treating the case when the objects under investigation fall into several finite disjoint sub-regions, as demonstrated in the next section.In a nutshell of sparse reconstruction, a high-dimensional sparse signal can be retrieved from its non-adaptive low-dimensional projections by solving a tractable convex program. In this way, the locations of a train of sparse well-resolved spikes can be determined in the super-resolution manner [4]. However, it is not guaranteed in resolving quantitatively two objects separated in the subwavelength distance by using the sparse reconstruction alone, as demonstrated theoretically in [4]. To improve the imaging quality rendered by solving Eq. (6), we suggest penalizing Eq. (6) by introducing the sparse regularization of l_{p}-norm ($0\le p\le 1$). As a result, we have

## 3. Results and discussions

We hereby present several numerical examples to demonstrate the performance of the proposed methodology for the 1D and 2Dsuper-resolution imaging applications. We conduct the study for 1D imaging, but the conclusions can be extended for 2D and 3D imaging. The MATLAB 7.3 code for reproducing the results in this paper can be freely obtained by sending a request email to lianlin.li@pku.edu.cn.

#### 3.1 *1D imaging*

In this subsection, we investigate the performance of proposed method for 1D super-resolution imaging from far-field observations, where the operational wavelength is chosen to be unit without loss of generality. Referring to Fig. 1(a), the simulation parameters are set as follows: $p=0.5$, ${L}_{x}=10$, ${D}_{x}=2$, and ${z}_{0}=50$; as a result, the corresponding Rayleigh limit is around 5.0. An additive Gaussian noise with SNR being 60dB is corrupted to the simulated data acquired over the scanned aperture. Moreover, $\gamma $ is set to be for this noise level throughout this paper.

Firstly, the imaging of double point-sources centered at in free space is studied. Figures 2(a)-2(d) show, respectively, the images reconstructed by the traditional super-oscillation imaging technique [7], the improved super-oscillation imaging approach by Eq. (8), the standard back propagation method [11], the proposed sparsity-promoted super-oscillation approach shown in Table 1, and the conventional sparsity-promoted reconstruction approach corresponding to the case of $W=I$ in Table 1. For comparison, the ground truth, denoted by the red circle, has been added in these figures. It is observed that the almost perfect reconstruction can be obtained by the proposed sparsity-promoted superoscillation approach (Fig. 2(d)) and the conventional sparsity-promoted reconstruction approach (Fig. 2(e)), which means that the use of sparse prior is very helpful in resolving two points separated in the 0.18 times Rayleigh limit. From Fig. 2(a), one can immediately observe the well-known features of traditional super-oscillation imaging that multiple point sources can be focused in the subwavelength scale, 0.18 times the Rayleigh limit for this example, and that the focus spot is surrounded by sidebands with remarkably high level. Comparing Fig. 2(b) with Fig. 2(a), one can deduce another interesting conclusion that the improved super-oscillation imaging technique from Eq. (8) is capable of resolving two sub-wavelength point sources with suppressed sidebands.

Secondly, we consider the reconstruction of double point sources located at (−0.15,0) and (0.15,0), which corresponds to the study of imaging with the resolution of 0.06 times Rayleigh limit. The corresponding results are shown in Fig. 3. From these figures, we can see that only the joint use of superoscillation and sparse processing can produce correct imaging of two point sources separated in the 0.06 times the Rayleigh limit, as shown in Fig. 3(d). More specifically, two point sources separated at the distance of 0.06 times Rayleigh limit can still be almost perfectly reconstructed by the proposed sparsity-promoted super-oscillation method, which benefits from the joint use of super-oscillation and sparse reconstruction. From Fig. 3(e) we notice that two misplacement point sources are obtained, which are mainly due to the shrinkage property of sparsity-promoted reconstruction technique. In addition, it is demonstrated from Figs. 3(a) and 3(b) that the use of superoscillation alone fails to do this task.

To be more realistic, we consider the performance of proposed super-resolution imaging method for reconstructing two groups of double point sources, which are located at (−1.3, 0), (−0.8, 0), (0.7,0) and (1.2,0). Figures 4(a)-4(d) show, respectively, images reconstructed by the traditional superoscillation imaging technique [7], the improved super-oscillation imaging approach by Eq. (8), the standard back propagation algorithm, the proposed sparsity-promoted super-oscillation approach in Table 1, and the conventional sparsity-promoted approach. As illustrated in Fig. 4(a), the traditional super-oscillation imaging technique completely fails to tackle this task due to the relatively big size of FOV. Moreover, the traditional sparsity-promoted reconstruction can efficiently determine centers of two groups, however, fails to restore finer structures inside each group. Once again, it is observed from Fig. 4(d) that the four point sources can be almost perfectly reconstructed by performing the proposed sparsity-promoted super-oscillation method. Now, we can safely conclude that the proposed method is capable of producing super-resolution imaging of more universal objects from the far-field observation, and lifting up the limitation of the traditional super-oscillation imaging method, i.e., sidebands with very high amplitude significantly larger than that of hotspot intensity.

Finally, we investigate the effect by adding WGN to the acquired field on the scanned aperture. The simulation parameters are the same as those in Fig. 2. We have added WGN with signal-to-noise ratio values of 40dB (see results in Fig. 5), and 30dB (see results in Fig. 6) to the simulated fields acquired over the scanned aperture. It is observed that the proposed sparsity-promoted superoscillation imaging approach performs very well with the reasonable SNR value of 30dB. As expected, both false objects and undesirable displacement will appear due to the use of strong sparse operation with decreased SNR values, partly owing to the use of too strong sparse penalty of p = 0.5. In order to impose relatively weaker sparse constraint, we consider furthermore the situation of *p* = 1 in our algorithm in Table 1. The results reconstructed by the proposed sparsity-promoted super-oscillation algorithm and the conventional sparsity-promoted reconstruction algorithm with *p* = 1 are shown in Figs. 7(a) and 7(b), respectively. From these two figures, we can observe that false objects disappear; however, the reconstruction of two point sources is remarkably expanded. More or less, we can deduce from Figs. 5-7 that for the case of relatively low SNR, the sparsity-promoted super-oscillation approach behaves similarly to the sparse reconstruction.

#### 3.2 2D imaging

Now, we investigate the performance of proposed the method for 2D super-resolution imaging from far-field observations. Referring to Fig. 1(b), the simulation parameters are set as follows: ${L}_{x}={L}_{y}=10$, ${D}_{x}={D}_{y}=2$, and ${z}_{0}=50$. First, the considered sources to be reconstructed consist of four point sources located at (−0.3,-0.3, 0), (−0.3,0.2,0), (0.2,-0.3,0) and (0.2,0.2,0). For this case, we add the WGN with SNR = 60dB to the simulated field. Figures 8(a)-8(d) show, respectively, the images reconstructed by the traditional super-oscillation imaging technique, the improved super-oscillation imaging approach by Eq. (8), the standard back propagation algorithm, the proposed sparsity-promoted super-oscillation approach, and the conventional sparsity-promoted approach. From these figures, we notice that these four closely separated point sources can be well-resolved only by using the proposed sparsity-promoted superoscillation approach. Numerical tests conducted for the relatively lower SNR = 30dB case show that these four point cannot be resolved any more, which are not provided here due to the limited space. Next, we consider the reconstruction of four point sources located at (−0.6,-0.6, 0), (−0.6, 0.4, 0), (0.4,-0.6,0) and (0.4,0.4,0), respectively. For this set of numerical investigations, we set SNR to be 30dB, and p = 1 for imposing relatively weaker sparse constraint, as discussed previously. The reconstructed results are provided in Fig. 9. From Fig. (8) and Fig. (9), previous conclusions through 1D imaging tests are verified again. Besides, we have normalized these results with respect to their own maximums for visual purpose.

## 5. Conclusion

We have studied a super-oscillation approach to overcome the diffraction limit for imaging sources distanced from a scanned aperture. We propose to use the super-oscillation imaging to solve weighted optimization problems constrained by *l _{p}*-norm-based sparsity regularization, leading to a novel sparsity-promoted super-oscillation imaging schemefor reconstructing more universal objects in subwavelength scales. The two super-resolution imaging mechanisms of super-oscillation and sparse reconstruction are simultaneously taken into account. We demonstrate that the proposed imaging technique outperforms remarkably both the traditional super-oscillation imaging and the sparsity-based super-resolution imaging for the case of very high SNR. The existing methods in [6, 7] can be regarded as the first-iteration solution of the proposed scheme.

We remark that there is big space for improving the proposed scheme by considering more specialized methods to design the super-oscillation filters [6, 12] or the weighting matrix in Eqs. (6)-(9), which is beyond the scope of this study. It is expected that the proposed method will find applications in the microwave, optical, and ultrasonic super-resolution imaging.

## References and links

**1. **J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**(18), 3966–3969 (2000). [CrossRef] [PubMed]

**2. **X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. **7**(6), 435–441 (2008). [CrossRef] [PubMed]

**3. **S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express **17**(26), 23920–23946 (2009). [CrossRef] [PubMed]

**4. **E. J. Candes and C. Fernandez-Granda, “Towards a mathematical theory of super-resolution,” (submitted) (2014).

**5. **L. Li and F. Li, “Beating the Rayleigh limit: orbital-angular-momentum-based super-resolution diffraction tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **88**(3), 033205 (2013). [CrossRef] [PubMed]

**6. **R. K. Amineh and G. V. Eleftheriades, “2D and 3D sub-diffraction source imaging with a superoscillatory filter,” Opt. Express **21**(7), 8142–8156 (2013). [CrossRef] [PubMed]

**7. **L. Li, X. Xu, and F. Li, “Towards super-resolution microwave imaging: general framework,” 10th International Symposium On Antenna, Propagation & EM Theory (2012). [CrossRef]

**8. **E. T. F. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. **15**(9), 094008 (2013). [CrossRef]

**9. **F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Lett. **9**(3), 1249–1254 (2009). [CrossRef] [PubMed]

**10. **L. Li and B. Jafarpour, “Effective solution of nonlinear subsurface flow inverse problems in sparse bases,” Inverse Probl. **26**(10), 105016 (2010). [CrossRef]

**11. **A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging **4**(4), 336–350 (1982). [PubMed]

**12. **A. M. H. Wong and G. V. Eleftheriades, “Adaptation of Schelkunoff’s superdirective antenna theory for the realization of superoscillatory antenna arrays,” IEEE Antennas Wirel. Propag. Lett. **9**, 315–318 (2010). [CrossRef]