## Abstract

In the paper, we propose a new edge detection schemes, based on a single-pixel imaging in the frequency domain. In SCHEME-I, special sinusoidal patterns for the *x*-direction edge and also *y*-direction edge of the unknown object are first designed. The frequency spectrum for the edge is then obtained using the a four-step phase-shifting technique with the designed sinusoidal patterns in a single-pixel imaging system. In SCHEME-II, the frequency spectrum of the unknown object is first obtained, then the frequency spectrum for the edge is obtained by calculations. The resulting edges are finally obtained by the inverse Fourier transform on their frequency spectrum. We have also verified the proposed schemes by experiments and numerical simulations. The results show that the proposed schemes can produce higher quality edges of character and also image objects. Comparing with SCHEME-II, the application of SCHEME-I to high frequency components has greatly improved signal-to-noise ratio of the received data in the bucket detector, resulting in better experimental results. Comparing with the edge detection scheme by speckle-shifting in ghost imaging systems, the proposed SCHEME-I shows an improvement in the signal-to-noise ratio. Since a single-pixel imaging system is used, the proposed schemes are capable of reconstructing edges from indirect measurements. The number of measurements required can be effectively reduced due to the sparsity of natural images in the Fourier domain and the conjugate symmetry of real-valued signals’ Fourier spectrum.

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

## 1. Introduction

*Ghost imaging* (GI), also termed *correlated imaging*, is currently a hot topic in quantum optics [1–16]. Normally, there are two optical beams in a GI system. One beam, termed a signal beam, crosses an unknown object and is detected by a bucket detector without any spatial resolution. The other beam, termed as idle beam, is detected by a spatially resolving detector. Based on the correlation of entanglement or field fluctuations, one can obtain a clear image of a non-localized object, at the idle path, by the intensity correlation between the signal beam and idle beam. GI has turned out to be an intriguing optical technique, allowing the imaging of objects to be located in various optically harsh or noisy environments. This techniques is also a promising candidate for distributed image processing, remote sensing and distributed communication systems.

The first implementation of a GI scheme was performed by an entangled-photon pair source [1], generated by spontaneous parametric down-conversion (SPDC) [2, 3], in 1995. Later, it was found that a thermal light or a pseudo-thermal light could also be used to realize GI [4, 5]. Thereafter, a new configuration, named *computational ghost imaging (CGI)*, was proposed to simplify GI implementations [6–10]. The spatial sweeping was used to increase the imaging speed of CGI [10]. Compressive sensing, an advanced signal processing technique, had been used in the new configuration to improve the quality of the imaging and to reduce the image reconstruction time [11–15]. Soon after, however, it was discovered that the single-pixel imaging is actually same as GI. As a consequence, some orthogonal phase patterns, such as sinusoidal patterns [16], Fourier patterns [17] and Hardmard patterns [18], were proposed to improve the quantity of single-pixel imaging. Because natural images are generally sparse in the Fourier domain, single-pixel imaging allows reconstruction of a recognizable image with a small number of measurements. Furthermore, it allows to produce a feasible placement of the detector, even under environmental illumination [16], since indirect measurements are adopted in the single-pixel imaging technique.

In many imaging applications, it is sufficient to detect the periphery of an unknown object (caled the edge of the object). This process is called edge detection [19]. Edge detection tests recognize the object edge on the basis of some dramatic change in the transmittance or reflectance of the object. They are nowadays widely used in the target recognition, earth observation, security checks [20, 21] and so on. In most of the previous edge detection approaches, the objects that are under consideration (targets) are imaged before the process of edge detection starts [22]. Hence, the poor imaging quality would severely limit the usage of edge detection, such as the target recognition and localization applications in remote sensing and biological imaging [20–23]. Recently, several reports have shown that edge detection could be directly achieved for unknown objects by using GI technique [24–27], which cannot use significantly addition of hardware or expensive computation. In [24], authors presented gradient of ghost imaging (GGI) for better edge detection, where a proper gradient angle had been chosen based on a prior knowledge of the object. In [25], several corresponding shifted groups of speckle patterns, instead of random speckle patterns, were used in a GI system to avoid the gradient angle or any other prior knowledge of the object. This technique was termed speckle-shifting ghost imaging (SSGI). In [26], the subpixel speckle patterns were adopted. The structured illuminations came from the interference of two phase masks were used in [27] to replace the speckle-shifting patterns.

In this paper, we propose novel edge detection schemes in the frequency domain based on single-pixel imaging by means of Fourier spectrum acquisition, named SCHEME-I and SCHEME-II. When the usual definition of edge detection is considered, we can deduce that the edges (their images) can be directly obtained by applying the Inverse Fourier Transform to the edges’ frequency spectrum, which is produced by the phase-shifting sinusoidal structured illuminations on the unknown object in a single-pixel imaging system. In SCHEME-I, two corresponding sinusoidal patterns are designed, while only one type of sinusoidal patterns are used in SCHEME-II. However, the illustration with SCHEME-I of the high frequency components has greatly improved the signal-to-noise ratio (SNR) of the received data in the bucket detector in experiments. Experiments and numerical simulations are presented to illuminate and analyze the usefulness of the proposed schemes.

Because the proposed schemes are based on single-pixel imaging, and the illustrations with the special sinusoidal patterns enhance the SNR of data from the bucket detector, the proposed schemes have some benefits: (1) In this paper new proposed edge detection schemes are feasible for any unknown objects without a prior imaging of the object under consideration. (2) Because any object edge is usually sparser than the object itself, the SNR of the proposed edge detection schemes is greatly enhanced in comparison with those methods where the object has to be imaged before the edge is to be extracted. (3) Our new schemes take advantages from the single-pixel imaging technique, enabling the reconstruct of edges from indirect measurements which lead to a feasible placement of the detector. The schemes also allow operation under any environmental illumination. (4) Here introduced edge detection methods are different from the existing edge detection schemes based on GI. Indeed, the proposed edge detection schemes operate in the frequency domain, and efficiently represent the differential of the gray-scale functions. (5) Comparing with the existing edge detection schemes based on GI, the proposed schemes have higher SNR values, and the number of measurements can be greatly reduced by the sparsity of natural images in the Fourier domain and the conjugate symmetry of real-valued signals’ Fourier spectrums.

The organization of the paper is as follows. In Section 2, new edge detection schemes based on single-pixel imaging by means of Fourier spectrum acquisition are proposed, and the corresponding theoretical deductions are presented. In Section 3, the experimental and numerical simulation results are presented. Finally, several conclusions are drawn in Section 4.

## 2. Edge detection based on single-pixel imaging

In a digital image, so-called edge is a collection of pixels whose gray values have step or roof changes. Hence, the edge of an object is the indication or reflection of the discontinuity of the gray values. Edge detection process is therefore an analysis of changes of a single image pixels in a gray area. Variations in the edge area, representing the first order or the second-order of gray function differential, is then often used to detect the edge. So called Sobel operator is one of the usually used gradient operators for edge detection, and it is expressed as [19],

*f*(

*x*,

*y*) is the gray representing function of an object,$\frac{\partial f(x,y)}{\partial x}$ is the differential of

*f*(

*x*,

*y*) with respect to

*x*, and $\frac{\partial f(x,y)}{\partial y}$ is the differential of

*f*(

*x*,

*y*) with respect to

*y*. One can get also other edge detection operators, such as the Laplace operator, Laplacian of the Gaussian operator [19], by the first order or the second-order of the differential of

*f*(

*x*,

*y*).

Comparing with other edge operators, the Sobel operator has two main advantages: (1) It causes a smoothing effect on random noise of the image due to the introduction of the averaging factor. (2) It causes an enhancements of edge elements on both sides, because of the differential of two rows and two columns in the Sobel operator matrix. Here, we are considering the Sobel operator only. Of course other operators can also be used for edge detections that are based on a single-pixel imaging in the frequency domain.

For an unknown object the gray function *f*(*x*, *y*) is expressed as

*R*represents the frequency domain, the coefficient

*C*(

*f*,

_{x}*f*), also named spectrum, is defined as $\int {\int}_{\mathrm{\Omega}}f(x,y){e}^{-j2\pi ({f}_{x}\cdot x+{f}_{y}\cdot y)}dxdy$, and Ω denotes the space domain. Therefore, the differential of

_{y}*f*(

*x*,

*y*) with respect to

*x*, $\frac{\partial f}{\partial x}$, can be derived from

*F*

^{−1}represents the Inverse Fourier Transform (IFT). Using the same procedure as in Eq. (3) and Eq. (4), the differential of

*f*(

*x*,

*y*) with respect to

*y*can be described as

Motivated by the high-quality of single-pixel imaging by means of Fourier spectrum [16], we can use four phase-shifting sinusoidal structured illuminations to obtain the spectrum of the *x*-edge and *y*-edge, and we can then apply the Inverse Fourier Transform to obtain the differential of *f*(*x*, *y*).

At first, we design a special sinusoidal pattern *P _{ϕ}* with the initial phase

*ϕ*and the spatial frequency (

*f*,

_{x}*f*) as

_{y}*x*,

*y*) are Cartesian coordinates in the space domain,

*a*is the direct current (DC) component, which is equal to the average intensity of the image, and

*b*represents the contrast.

With the sinusoidal pattern *P _{ϕ}* illuminating the unknown object, the total intensity of the bucket detector [28–30] in the single-pixel imaging system is in theory

*f*(

*x*,

*y*) represents the unknown object. Since the bucket detector can not represent all intensity of the light transmitted or reflected by the unknown object, together with the environment illuminations, the resultant intensity in the bucket detector should be expressed as where

*k*is a scalar factor which depends on the location of the bucket detector, and

*I*is the response of environmental illuminations. Usually, the environmental illuminations are determined during experiments, and it could be regarded as a special constant in any experiment. Substitution of Eq. (8) to Eq. (7) and Eq. (6), Eq. (8) turns to

_{en}Using the four-step phase-shifting approach [16], where all four patterns have the same spatial frequency (*f _{x}*,

*f*) and with a constant phase shift $\mathrm{\Delta}\psi =\frac{\pi}{2}$ between the two adjacent patterns, the resultant is

_{y}*f*(

*x*,

*y*) with respect

*x*, $\frac{\partial f}{\partial x}$, can be obtained by the Inverse Fourier transform of the left side of Eq. (10),

Using the same considerations, one can design a phase-shifting sinusoidal pattern for obtaining the differential of *f*(*x*, *y*) with respect to *y*, $\frac{\partial f}{\partial y}$. Here, the designed sinusoidally structured pattern *Q _{ϕ}* is

At last, we can get an accurate description of edges of the unknown object by

*I′*(

_{ϕ}*f*,

_{x}*f*), the resultant intensity in the bucket detector, uses the sinusoidal pattern

_{y}*P*and

_{ϕ}*I″*(

_{ϕ}*f*,

_{x}*f*) uses the sinusoidally structured pattern

_{y}*Q*. It has been shown that one can get an accurate expression of the edge with the help of a single-pixel imaging by means of frequency domain acquisition. This scheme can directly obtain edges of the unknown objects without obtaining images of the objects. At the same time, the phase-shifting approach can not only assemble desired Fourier coefficients, but also effectively eliminate errors that are statistically the same. We call it SCHEME-I.

_{ϕ}In principle, there is an alternative approach to obtain object’s edge based on single-pixel imaging technique, that is, getting the spectrum of the object using the single-pixel imaging at first, and then extracting the spectrum of *x*-edge by multiplying with *j*2*πf _{x}* and the spectrum of

*y*-edge by multiplying with

*j*2

*πf*. We call it SCHEME-II. Here, the sinusoidal pattern

_{y}*S*with the initial phase

_{ϕ}*ϕ*and spatial frequency (

*f*,

_{x}*f*) should be

_{y}*x*,

*y*),

*a*and

*b*are the same as above. With the same four-step phase-shifting approach, the coefficient

*C*(

*f*,

_{x}*f*) is given s follows:

_{y}*I′″*(

_{ϕ}*f*,

_{x}*f*) in the bucket detector uses the sinusoidal pattern

_{y}*S*, and the spectrum of the edge could be obtained as

_{ϕ}It is also known that the edges are mainly on the high frequency domain, the variation on the small frequency spectrum has less effect on the resulting edge. Therefore, at the same time, the illustration with special sinusoidal patterns *P _{ϕ}*(

*x*,

*y*;

*f*,

_{x}*f*) or

_{y}*Q*(

_{ϕ}*x*,

*y*;

*f*,

_{x}*f*), especially for the high frequency components, will enhance the signal-to-noise ratio of the bucket detector’s received data in comparison with the sinusoidal patterns

_{y}*S*(

_{ϕ}*x*,

*y*;

*f*,

_{x}*f*).

_{y}## 3. Experimental and simulation results

In this section, we will describe verification of the proposed edge detection schemes (SCHEME-I and SCHEME-II) by both numerical simulations and experiments. The numerical simulations are done with CPU of Intel Core i7-4790(Dell Optiplex 920 with intel core 3.6GHz and memory 24GB) by Labview 2012(32bits).

The experimental setup of the proposed scheme is indicated in Fig. 1. A digital light projector (DLP) (TI Digital lightCrafter 4500) is used to produce sinusoidal patterns which are generated from the designed patterns *P _{ϕ}* in Eq. (6) and the designed patterns

*Q*in Eq. (12) for SCHEME-I, and

_{ϕ}*S*in Eq. (14) for SCHEME-II. Then, the beam with the sinusoidal patterns is illuminated on an unknown object, which is printed on a white paper with 128 × 128 pixels. Note that For SCHEME-I, the sinusoidal patterns

_{ϕ}*P*and

_{ϕ}*Q*. They are then used in a sequence, that is all sinusoidal patterns

_{ϕ}*P*are first used to illuminate, followed by all patterns

_{ϕ}*Q*. A part of the beam reflected from the the unknown object is then detected by a bucket detector (Thorlabs PMM02-1), and subsequently transferred to a number by an analogue-digital convertor (NI USB-6341). In the experiment, the four-step phase-shifting sinusoidal illumination technique is adopted for Fourier spectrum acquisition. Hence, each Fourier coefficient is obtained by one of sinusoidal pattern with four-step phase shifting, according to Eq. (10). Actually, only half of the Fourier coefficients are needed because the Fourier spectrum of a real-valued signal is conjugate and symmetric. For SCHEME-I, this operation is repeated

_{ϕ}*M*times for

*M*different phase sinusoidal patterns

*P*and phase sinusoidal patterns

_{ϕ}*Q*. For SCHEME-II, all Fourier coefficients should be obtained using the sinusoidal patterns

_{ϕ}*S*(

_{ϕ}*M*times). Afterwards, modulations on Fourier coefficients are operated to get spectrum of the edge in

*x*and

*y*directions. Once the frequency spectrums for the unknown object’s edge are obtained, the horizontal and vertical edges of the unknown object are finally obtained by the Inverse Fourier Transform on spectrums, and the edge of the unknown object is obtained according to Eq. (1).

Fig. 2 shows the experimental and simulation results, with the proposed edge detection schemes, where two gray-scale character images(128 × 128 pixels) and two figure images (128 × 128 pixels) are used for numerical simulations and experiments. Here, all the target objects are printed on papers with dimensions 52*mm* × 52*mm*. The original edges are obtained when Sobel operators are applied on the original images. Both experimental and simulation measurements for SCHEME-I were performed 131072 times, and the measurements for SCHEME-II were done half as many times. The results show that we can get images of good edges of the unknown objects by using both the proposed edge detection schemes without prior imaging of the unknown objects. Concerning simulation results, the edges achieved with SCHEME-I and SCHEME-II were perfect. The images of edges obtained with SCHEME-I are better than those with SCHEME-II, and even seem sharper that the original edges. However, there may be some imperfections in the output of our systems. In particular, only partially reflections from the unknown objects may be detected in some experiments, resulting in a reduction of the contrast in the experimental results obtained. One imperfection can be caused by the quantization noise introduced when digital light projector operates in the 8-bit mode. The illumination patterns are needed for digitalization in the experiment, which leads to the periodic noise in the experimental results. Fortunately, locations of such periodic noise have some regularity so they can be picked out with a common spectral mask and reasonably replaced by their neighbour means [16]. Comparing with SCHEME-II, the illustration with SCHEME-I for the high frequency components was greatly improved in the signal-to-noise ratio of the received data in the bucket detector, while the noise was enlarged directly for SCHEME-II. Therefore, the experimental results were better for SCHEME-I. Therefore, we use SCHEME-I for the later comparisons following experiments..

Single-pixel imaging by means of Fourier spectrum acquisition is a compressive sampling-like approach. In general, most information of an edge concentrates on high-frequency components. Since natural images are very sparse in the high frequency domain, the Fourier spectrums can reconstruct the faithful-quality edges by performing sub-Nyquist measurements.

In order to express the quality of the edge detection quantitatively, the signal-to-noise ratio (SNR) is used as an objective quantitative evaluation of recognition quality for the edge [25], which is defined as

*I*and

_{edge}*I*are the intensity values of the edge detection results in the object’s edge and also background region, respectively.

_{background}*mean*(·) represents the average value, and

*Var*(·) denotes the variance value. Generally, the higher SNR is, the better quality edge detection has.

We have afterwords compared performance of the proposed edge detection scheme SCHEME-I with SSGI, together with original edge images, which were achieved by Sobel operators on reconstruction of the object using the single-pixel imaging. Here we call it as an ‘original edge image’. All edges were hen obtained using he Sobel edge detection operator. In that process, both SSGI and SCHEME-I obtained the edge without prior imaging unknown objects, in which SSGI uses speckle-shifting in the framework of computational ghost imaging, while SCHEME-I employs phase-shifting in single-pixel imaging by means of Fourier spectrum acquisition. The ’original edge image’ should get the image before the edge is extracted. Fig. 3 shows experimental results of comparisons. Again, all target objects were printed on papers of sizes 128 × 128 pixels and dimension 52*mm* × 52*mm*. The total number of measurements for SSGI was 128 × 128 × 8, the measurement times for SCHEME-I was 128 × 128 × 4 × 2, and the measurement times was 128 × 128 × 4 for the original edge image. The results therefore show that both SSGI and SCHEME-I could achieve the edges of unknown objects without obtaining the imaging. Moreover, the proposed scheme SCHEME-I has a better SNR performance in comparison with those with SSGI and ’original edge image’. For the binary character object ’NUPT’, SNR for our scheme is 2.26, while it is 0.63 using SSGI and 0.88 for the ’original edge image’. There are 259% and 157% improvements in the SNR performance. For the box figure, SNR for our scheme is 1.49, while it was 0.22 using SSGI and 0.25 for the ’original edge image’, there are therefore 577% and 496% improvements in the SNR performance. Compared with the original edge image, not only could our scheme obtain the edge without any prior imaging, but also has a better SNR performance. The reason is that the noise generated in the imaging would greatly impair the quality of the edge.

For all experimental results, we found that there exists a central bright line in the reconstructions and we adopted a simple method to remove it. Fig. 4 shows the results when the fixed central is removed line with a filter. When doing that, we used a paper without an image as a sample object, and cutted the noise when the proposed scheme was used for detecting the unknown object. The results show that the simple filtering could mitigate the central line, and the SNR performance has been improved.

## 4. Conclusion

In this paper, we have proposed new edge detection schemes based on a single-pixel imaging in the frequency domain without any prior imaging of the original object. In SCHEME-I, the frequency spectrum in *x* and *y* directions of the edge are at first obtained by the four-step phase-shifting method used in a single-pixel imaging system with special sinusoidal patterns. In SCHEME-II, the frequency spectrum of the unknown object is first obtained, then the frequency spectrum for the edge is obtained by calculations. The edge is obtained by the Inverse Fourier Transform on the frequency spectrum in *x* and *y* directions, respectively. Experiments and numerical simulations show that the proposed edge detection schemes are feasible, and obtained edges have high quality. The edges with SCHEME-I are better than those with SCHEME-II. Compared with the original edge image, not only could our scheme SCHEME-I obtain the edge without prior imaging, but also has a better SNR performance. In comparison with SSGI edge detection scheme, SCHEME-I exhibits an improvement in SNR. In addition, the proposed schemes are capable of reconstructing edges from indirect measurements, and can use any iterative or minimization algorithm to enhance the computational efficiency. The number of measurements required could be greatly reduced by the sparsity of natural images in the Fourier domain and the conjugate symmetry of a real-valued signal’s Fourier spectrum

## Funding

National Natural Science Foundation of China (Grant No. 61475075, 61271238); The open research fund of Key Lab of Broadband Wireless Communication and Sensor Network Technology, Ministry of Education (Grant No. JZNY201710).

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