## Abstract

Digital cameras use detector arrays with regular geometry for optical sampling. Though regular arrangement was demonstrated to be optimal for two-dimensional sampling, it causes aliasing at high frequencies exceeding its Nyquist limit. Here, we proposed a randomization procedure to generate 2D hyperuniform patterns that can be used to suppress aliasing in image retrieval. Experiments are performed using a single-pixel camera, where the sampling patterns do not necessarily follow a fixed Cartesian geometry. Results demonstrate that the images reconstructed by hyperuniform patterns have a lower root mean squared error and exhibit less moiré fringes at high frequencies than the images reconstructed by regular square patterns do. Furthermore, the same conclusion can be applied to the production of conventional detector arrays, where manufacturing imperfection could be utilized to suppress frequency aliasing in image retrieval.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Optical imaging is an important tool for many applications, and even more so since the invention of digital cameras, which provide us an access to a numerous amount of images. A digital camera samples a scene with a detector array, the pixels of which are regularly fixed in a Cartesian geometry in most cases, or a triangular-lattice arrangement (i.e., a hexagonal array) in a rare case [1], because manufacturing hexagonal multipixel image sensors increases cost dramatically for mass-production, even though classical sampling theory [2] as well as recent works [3,4] demonstrated that a hexagonal array is optimal for two-dimensional (2D) imaging sampling. Nevertheless, species free from business rules have evolved this optimal arrangement of their retinal mosaic, such as compound eyes of the insect [5].

An earlier study suggested that any irregularity in optical sampling degrades image quality [6]. However, regular sampling has its own pros and cons. While capturing image details whose spatial bandwidth does not exceed its Nyquist limit, regular sampling also introduces aliasing, which converts frequencies above the limit into moiré fringes [cf. Fig. 1(a)] in the reconstructed images [7]. One possible solution to this dilemma is the fact that a modest degree of irregularity within the sampling mosaic can actually suppress aliasing introduced by the perfect regular array [8]. Random sampling has been used in detector array structures to increase bandwidth [9]. Interestingly, biologists recently found such structures within avian retina [10] and believed that this is one of the reasons that birds have the most sophisticated vision of any vertebrate. Avian retina consists of five types of cones (four sense colors and one detects luminance [11]), each of which independently exhibits a disorder on small length scales but a quasi-uniformity on large scales. The remarkable arrangement of correlated disorder is known as hyperuniformity [12].

The density fluctuation in a many-particle system, which is of great fundamental interest in astronomic [13], and biological sciences [14], contains crucial structural information about the system [15] and can be described using point patterns [12], whose points are the centroids of the structure shapes. Consider a 2D point pattern [cf. Fig. 1(b)] with Poisson point distribution (the concept can be expanded to $n$-dimensional Euclidean space); the variance of the number of points ${\sigma}_{\mathrm{pois}}^{2}(R)$ within a circular domain $S$ is proportional to the square of its radius $R$, i.e.,

Disordered hyperuniformity has been observed in galaxy systems, molecule structures, and biology organisms and exhibited exotic properties in all cases. It would be interesting to artificially generate hyperuniform patterns and investigate its effect on optical sampling.

Because a hexagonal array is optimal for imaging sampling, we generated a regular array with $64\times 64$ hexagonal blocks in a 2D grid, as shown in Fig. 2(a). For clarity, these hexagonal blocks are referred to as resolution cells henceforth. The 2D grid has an underlying Cartesian structure of $512\times 512\text{\hspace{0.17em}}\text{pixels}$ but contains $64\times 64=4096$ hexagonal cells, each of which consists of 64 pixels. Different colors are used to distinguish adjacent cells. The centroids of the cells are calculated [black dots in Fig. 2(a)] and form a regular triangular-lattice point pattern, as shown in Fig. 2(b), which describes the structure information of the sampling cell pattern. To introduce irregularity into the regular hexagonal pattern, a randomization procedure is proposed, as shown in Fig. 2(c). The protocol of the procedure is as follows:

- 1. Put the triangular-lattice point pattern on a blank $512\times 512\text{\hspace{0.17em}}\text{pixels}$ 2D grid. For each centroid, choose a square region [dashed pink square in Fig. 2(c)] that places the corresponding centroid at the center. The pixel size of the region is $a\times a$, and the irregularity to be introduced into the regular pattern can be determined by tuning $a$.
- 2. For each region, randomly place a resolution cell of one pixel size [‘$\mathbf{S}$’ in Fig. 2(c)] inside the region as the starting point of cell growth. For one iteration of the growth, randomly choose one side of the cell for it to grow a pixel. All sides of the cell have the same possibility to grow, except for those adjacent to other resolution cells and therefore have no possibility to grow.
- 3. During the iterations, the growth is performed on all resolution cells simultaneously. The growth iteration continues until the resolution cells fill the whole 2D grid, and there is no blank pixel left.

Figure 2(d) shows a randomized pattern with $a=6$. The centroids of the resolution cells demonstrated a moderate deviation from the regular triangular-lattice array [Fig. 2(b)].

To verify the randomized patterns for hyperuniformity and evaluate the irregularity induced by the randomization, we generated three patterns with $a=4$, 6, and 8, respectively. To calculate the variance ${\sigma}^{2}(R)$ of a point pattern, 1500 circular domains $S$ are randomly placed in the pattern without overlapping the system boundary for a specific $R$. The pattern size $L$ limits the maximum radius to be $R=L/2$. For comparison, the variance data of a regular hexagonal pattern were also calculated. Noting that the strict variance characteristics for a hexagonal pattern were provided in [12,17], and the randomized patterns were generated from the hexagonal one, we therefore constructed the fitting function of the computed ${\sigma}^{2}(R)$ data as

Figure 3 shows four groups of calculated variance data (color dots) at different $R$. Fitting curves (color dashed lines) of the data groups are also obtained, with their computed coefficients listed in Table 1. The variance data and coefficients of the fitting curves demonstrate that all three randomized patterns satisfy Eq. (2) and therefore qualify as hyperuniform. Furthermore, the value of coefficient $B$ becomes smaller as $a$ increases, showing that the induced irregularity can be controlled by tuning $a$. In short, it is demonstrated that hyperuniform patterns can be generated with a tunable irregularity.

A single-pixel camera encodes spatial information in the temporal dimension [18] and reconstructs images from a set of sequential measurements, each of which contains different spatial information and is sensed by a single-pixel detector [19]. Though not performing as well as commercial digital cameras, this strategy enables imaging in situations that are challenging or impossible with multipixel image sensors [20], such as x-ray and terahertz band detection [21,22], 3D depth mapping [23,24], adaptive imaging [25], and compressive sensing [26].

Our previous works [25,27] showed that without a fixed Cartesian pixel geometry, pixel resolution of a single-pixel camera can be spatially and temporally varying to adapt different applications. In this work, a single-pixel camera is used to perform investigation on hyperuniform optical sampling. We performed single-pixel imaging using hyperuniform sampling patterns, and without adding any imaging resources, the yielded images demonstrated enhancement in image quality as well as frequency aliasing suppression over conventional Cartesian sampling.

In a single-pixel camera, the pixel resolution of the acquired image is equivalent to the spatial resolution of the sampling patterns, which are usually displayed on a high-speed spatial light modulator (SLM), such as a digital micromirror device (DMD). The sampling patterns ${P}_{i}$ are either partially correlated [28] or orthonormal [29]. If the sampling patterns form an orthonormal basis, such as a Hadamard matrix [30], then an $N$ pixelated image of the scene can be reconstructed by $N$ measurement ${S}_{i}$ as

where ${S}_{i}$ is the intensity of the light modulated by the pattern ${P}_{i}$.Numerical simulations are performed on a group of 50 pictures. Each picture ${I}_{o}$ functions as the object and is sampled by a series of Hadamard patterns ${P}_{i}$, which are in a hyperuniform pattern with $a=6$. By associating ${P}_{i}$ with the corresponding signals ${S}_{i}$, using Eq. (4), an image ${I}_{r}$ can be reconstructed. For comparison, square geometry patterns and hexagonal geometry patterns are used during the sampling and reconstruction of each image. The original images are of $512\times 512\text{\hspace{0.17em}}\text{pixel}$ resolution, and the patterns are of $64\times 64$ cell resolution with an underlying Cartesian $512\times 512\text{\hspace{0.17em}}\text{pixel}$ structure. The qualities of reconstructed images are evaluated using the percentage root mean squared error (RMSE) as

where $m=n=512$ are the dimensions of the image, and all images are normalized to unity.Figure 4(a) shows the RMSEs of the reconstruction images using three different sampling patterns, which are square, hexagonal, and hyperuniform. The data are in descending order of square pattern RMSE values (blue square dots). Coincident with the conclusion that the triangular-lattice array is optimal for 2D optical sampling, hexagonal patterns yield the lowest RMSE (red triangular dots) in most cases and the lowest averaged RMSE value among the three groups’ data. Hyperuniform patterns reconstruct images with RMSEs (green circular dots) lower than those of square patterns but slightly higher than those of hexagonal patterns in most cases because the hyperuniform patterns we generated originated from triangular lattice, though the irregularity causes degradation in image quality. In some cases where moiré fringes are presented in the reconstructed images [Figs. 4(b) and 4(c)] due to regular sampling of periodical high frequency, the hyperuniform patterns suppress aliasing and therefore reconstruct images without severe moiré fringes. The RMSEs of the reconstructed images demonstrate the effectiveness of aliasing suppression [dotted groups labeled as “$\mathbf{b}$” and “$\mathbf{c}$” in Fig. 4(a), corresponding to the images shown in Figs. 4(b) and 4(c)].

Experiments are performed with a single-pixel camera [cf. Fig. 5(a)]. The objects are placed $\sim 0.5\text{\hspace{0.17em}}\mathrm{m}$ away from the camera. A lens ($f=50\text{\hspace{0.17em}}\mathrm{mm}$) images the object onto a DMD (Texas Instruments Discovery 4100, $1024\times 768$), which is placed at the focal plane of the lens. The predesigned patterns are displayed on the DMD to sample the image, and the reflected light intensities are measured by a bucket detector (Thorlabs PDA100A-EC with a collection lens) and transferred by a digitizer (National Instruments DAQ USB-6361) to computer for image reconstruction.

Figures 5(b) and 5(c) demonstrate that the reconstructed images using regular sampling exhibit moiré fringes at high frequencies while hyperuniform sampling suppresses frequency aliasing and therefore has no severe moiré fringes in the reconstructed images. The observation is coincident with that of the numerical simulation, though quality of the reconstructed images is degraded due to non-uniform illumination and noise presented in the experiment.

In conclusion, we proposed a randomization procedure to generate 2D hyperuniform patterns that can be used to suppress frequency aliasing in image retrieval. The randomized patterns are verified to be hyperuniform with a tunable irregularity. Numerical simulations and practical experiments are performed using a single-pixel camera. The results demonstrate that the images reconstructed by the generated hyperuniform patterns have a lower RMSE than conventional Cartesian patterns do, and exhibit much less moiré fringes at high frequencies than regular patterns do. It has been shown that with the same cell resolution, the image quality of single-pixel cameras can be improved by hyperuniform sampling. Furthermore, the same conclusion can be applied to the production of conventional detector arrays, where manufacturing imperfection of pixel shape and arrangement could be utilized to suppress frequency aliasing in image retrieval.

## Funding

National Natural Science Foundation of China (NSFC) (61307021, 61675016); Natural Science Foundation of Beijing Municipality (4172039).

## Acknowledgment

We thank M. J. Padgett for helpful discussions.

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