1. Introduction

Ghost imaging (GI), also known as correlated imaging, is one of the frontiers and hot spots in quantum optics [1, 2]. It is based on the correlation of quantum entanglement or field fluctuations of classic lights. Normally, there are two beams in a GI system, one is signal beam, which is received by a bucket detector that does not have spatial resolution. The other is reference beam, which is received by a point detector with spatial resolution. By intensity correlation between the signal and reference beam, GI could non-locally obtain an image of the object, and has provided a method to obtain a clear image in instance where conventional imaging techniques are not effective. Since first GI was realized by experiments [3], GI has been received much attention [4–16]. For example, differential ghost imaging [7, 26] and normalized ghost imaging [8, 26] were proposed to enhance the imaging quality. The reference beam can be obtained by a calculation rather than an experimental detection in computational ghost imaging (CGI) [5], which could make the GI system simpler.

Edge detection can get the edge of an object by finding dramatic changes of the boundaries. It has been already applied to image analysis, pattern recognition or earth observation [17–23]. Traditional edge detection methods should firstly get the image of the target object, and then obtain the edge information by edge detection algorithms. In spite of a lot of improvements have been achieved in edge detection algorithms, it is still essential to obtain a clear image of the object at first [17, 18]. However, in many real application occasions in which turbulence and scattering medium exist, the optical imaging is difficult to perform and then the edge detection algorithms cannot be implemented. The edge detection based on GI provides a way to directly obtain the edge information of an unknown object [19–22]. Furthermore, the edge information of most objects is sparse, which in turn improves the signal-to-noise ratio of the detected edge based on GI [19].

At present, the operators used in edge detection based on GI are all differential operators, of which Sobel operator is the best one [20]. Although Sobel operator has certain anti-noise ability, but its size is $3\times 3$, the coefficients in the template is small, and is only effective in both horizontal and vertical directions, which is insensitive in other directions. In the paper, we first design a variable size Sobel edge detection operator, named V-Sobel operator. The template size of the V-Sobel operator can be arbitrary, say, $N\times N$. Its coefficients are isotropic and their values are determined by the distance between current pixels and the central pixel. The minimum coefficient is 1, the factor between any adjacent coefficients is $\sqrt{2}$. When N=3, V-Sobel operator returns to Sobel operator. We later discuss its property on detecting the unknown objects’ edges, together with the method to keep the measurement times unchanged when the size of the template increases. With the ‘calculated speckles’ come from the random speckles and the used V-Sobel operator, the theory proves that there is no increase of measurement times with of the size of V-Sobel operator. Finally, we verify the performance of the proposed edge detection method by numerical simulations and experiments.

The structure of the paper is the following. In the second section, a variable size Sobel operator is first designed, and an improved edge detection method with this Sobel operator based on GI is later presented. In the third section, the experiment and numerical simulation results are presented to verify the proposed edge detection. In the fourth section, some conclusions are drawn.

2. Efficient edge detection based on ghost imaging

In this section, we first propose a variable size Sobel edge detection operator (named V-Sobel operator), and then present an edge detection method based on GI using a larger template size V-Sobel operator, meanwhile without the increase of measurement times.

2.1. Variable size Sobel operator

Among all the edge detection operators, Sobel operator performs a better performance on gray scale and noisy images. Some of the difficulties for edge detection are caused by noise. The visually distinct edges sometimes cannot be discriminated within a small image subarea belongs to the homogeneously textured domain of an image. It is apparent to solve this problem by increasing the size of the subarea [24]. The bigger the subarea size is, the larger the template size of the edge detection operator would have. After that, there are more pixels can be processed within one edge detection operation, resulting in the lower noise in edge detection. Meanwhile, the more precise the template coefficient is, the better edge the operator could obtain. Besides that, the edge detection operator with isotropic property may have a higher anti-noise ability [25]. Therefore, we redesign Sobel operator with a large template size, a more precise coefficient, and an isotropic structure, named variable size Sobel operator (V-Sobel operator). The template size of V-Sobel operator is N$\times$ N, where N is an integer greater than $3$. The isotropic coefficients are adopted in V-Sobel operator, where the coefficient is determined by the distance between the current point and the central point of the template. All the coefficients are same for all the points in the template with equal distances to the central point. The minimum coefficient is setup to 1, and the factor between any adjacent coefficients is setup to $\sqrt{2}$. Hence, the template of a V-Sobel operator can be shown as Eq. (1) and Eq. (2). Here, $G_x$ is the template for x-direction, and $G_y$ is for y-direction. For symmetry, $N$ may be an odd integer. Therefore, we are able to obtain a vertical and horizontal edge with Eq. (1) and Eq. (2). As an example, say, if $N=5$, the V-Sobel operator can be expressed as respectively Eq. (3) and Eq. (4).

2.2. Efficient edge detection using V-Sobel operator

An edge information is a differential of object information, and is equivalent to the difference of an object image and its shifted image [19]. Since the object can not be shifted in a GI system, the speckles should be shifted so as to generate the edge information in the edge detection method based on GI.

The procedure of edge detection based on GI can be described as follows. Firstly, a group of random speckles, such as $M$ random speckles, are generated at the source. Then, the shifted-speckles are obtained for each speckle in the group by shifting the speckle in x-direction and in y-direction according to the selected V-Sobel operator. Here, the shifted number is determined by the size of the used V-Sobel operator, and there exist the shifted-speckles in $x$-direction and in $y$-direction, respectively. After that, these shifted-speckles are illustrated on an unknown object as it usually does in a normal GI system. Note that these shifted-speckles are used in a sequence, that is all $x$-directional shifted-speckles are first used, followed by the $y$-directional shifted-speckles. After the detection results obtaining from the bucket detector, the edge information of the unknown object can be directly achieved by the second-order intensity correlation.

For a $N\times N$ V-Sobel operator, both $x$-direction and $y$-direction matrices of the V-Sobel operator can be written as a matrix as,

Assume that the $k^{th}$ ($k\in \left\{1,2,\cdots ,M\right\}$) speckle in the group is denoted as $I_k\left(x_i,y_j\right)$, where $x_i,y_j$ are the Cartesian coordinates for spatial domain. Normally, $x_i,y_j$ vary the same size of the unknown object. Its shifted-speckles, with $a$ pixel in $x$-direction and $b$ pixel in $y$-direction, can then be expressed as $I_k\left(x_{i+a},y_{j+b}\right)$, where $a,b$ belong to $\left[-w,w\right]$. When the $k^{th}$ shifted-speckle is projected onto an unknown object $T\left(x_i,y_j\right)$, the bucket detector result ${\mathrm{y}}_{a,b}^k$ is,

Here, $T\left(x_i,y_j\right)$ represents the transmission function (or the reflection function) of the unknown object. Then, the total record for the $k^{th}$ measurement (${\mathrm{y}}^k$) can be obtained as,

It is shown that the total detection result is combination of the V-Sobel template coefficients and bucket detector values at different positions. Eq. (7) can also be rewritten as,

On the other hand, the edge of the unknown object $T\left(x_i,y_j\right)$ is the differential of object information, which is operated by the V-Sobel operator as,

With Eq. (9), Eq. (8) can be rewritten as,

According to Eq. (10), the edge image can be reconstructed by calculating the second-order intensity correlation in the principle of GI system,

However, it is shown that one need to measure the bucket detector $N\times N$ times for the $k^{th}$ measurement in the edge detection according to Eq. (7). Therefore, the total measurement times would be $N\times N\times M$ for the edge detection. Here, we present a method to reduce the measurement times, and the total measurement times does not increase with the size of the selected V-Sobel operator. The deduction is as follows. Let us rewrite Eq. (7) again as,

Here, ${I^{\prime }}_k\left(x_i,y_j\right)$, named ‘calculated speckle’, is defined as,

For a computational GI structure, the random speckle $I_k\left(x_i,y_j\right)$ ($k\in \left\{1,2,\cdots ,M\right\}$) could be generated by computer before the experiment. With Eq. (13), one can obtain all the ‘calculated speckles’ with these random speckles and the used V-Sobel operator. With the ‘calculated speckles’, one can do GI experiment for $M$ times to obtain enough data, and obtain the edge image by calculating the second-order intensity correlation. Here, the total measurement times for the experiment is $2\times M$, it will no increase with of the size of the used V-Sobel operator.

3. Results and analysis

In this section, we present the experiment and simulation results to verify the proposed method. Numerical simulation is calculated by LabVIEW software which is operated on intel i7-4790 CPU, 8G RAM, and 64 bit Windows 7 system.

 

Fig. 1 The experimental setup for the edge detection with V-Sobel operator based on GI. DLP: Digital Light Procession.

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Fig. 1 is the experiment setup for the edge detection based on GI using V-Sobel operator. The binary character ‘NUPT’ and figure ‘Ghost’ with $128\times 128$ pixel, $18mm\times 18mm$ are adopted as the objects. At first, we obtain $2\times M=32768$ ‘calculated speckles’ according to Eq. (13), where $M=16384$ is the measurement times using in the GI system for the selected objects. The group of speckles are Hadamard matrices which are generated by computer, and the V-Sobel operator matrix are listed as Eq. (3) and Eq. (4). Then we use a Digital Light Procession (Digital Light Crafter 4500) to produce the ‘calculated speckles’, and let the ‘calculated speckles’ illustrate on the objects after a projecting lens (the focal length is 250 mm). After the light reflected from the object, we use a collecting lens (focal length is 250 mm) to collect the beam, and detect the beam by a detector (Thorlabs S120C and PM100USB). Finally, we reconstruct the edge by calculating the second-order intensity correlation.

 

Fig. 2 The experiment and simulation results for edge detection based on ghost imaging with V-Sobel operator, where the original edges are obtained by Sobel operators on the original objects. The size of the V-Sobel template is $5\times 5$.

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Fig. 2 shows the experiment and simulation results for edge detection based on GI with V-Sobel operator. Here, the original edges are obtained by Sobel operators on the original objects. The size of the V-Sobel template is $5\times 5$. It can be seen that the edge information of the unknown objects can directly be obtained without imaging first by the proposed method. Both the experimental and simulation edges are clear and continuous. Additionally, the simulation results look the same as the original edges, but there are a little noise in the experimental edges due to the environmental noise.

In order to compare the quality of the reconstructed edge quantitatively, signal-to-noise ratio (SNR) is used as an objective evaluation [19], which is defined as,

 

Fig. 3 The experiment and simulation comparison results for the same objects with Sobel operator and V-Sobel operator. The size of the V-Sobel template is $5\times 5$.

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Fig. 3 shows the experiment and simulation comparison results for the same objects with Sobel operator and V-Sobel operator. An additive Gaussian white noise (AGWN) with zero mean is adopted as the simulation environment. The SNR of the bucket detector $SN{R}_{\text{BD}}$ is defined as

In order to further testify the anti-noise characteristics of the proposed V-Sobel operator, we present the edge information of the same object with different template size V-Sobel operator. Here, we increase the strength of the noise, and $SN{R}_{\text{BD}}$ is setup to 18.10dB.

 

Fig. 4 The anti-noise property of different template size Sobel operator, where $SN{R}_{\mathbf{BD}}$ is setup to 18.10dB.

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Fig. 4 shows the anti-noise property of different template size Sobel operator, where $SN{R}_{\text{BD}}=18.10dB$ is adopted as the simulation environment. The sizes of the two V-Sobel operators are $5\times 5$, $7\times 7$, respectively. The edge detection results show that there is no edge information by using Sobel operator, but there are some edge information by using V-Sobel operators when $SN{R}_{\text{BD}}=18.10dB$. For ‘NUPT’, the SNR of the edge using Sobel operator is $0.16$, while they are $1.67$ for V-Sobel operator with $5\times 5$ size and $2.35$ for V-Sobel operator with $7\times 7$ size. For the binary figure ‘Ghost’, the SNR of the edge using Sobel operator is $0.22$, while it has been improved to $1.42$ by using V-Sobel operator with $5\times 5$ size, and has been improved to $2.71$ by using V-Sobel operator with $7\times 7$ size. The results also indicate that the bigger template size of the V-Sobel operator, the higher anti-noise property the edge operator has.

Finally, we change the $SN{R}_{\text{BD}}$, and get the SNR performance of edge information against $SN{R}_{\text{BD}}$ both using Sobel operator and the V-Sobel operator with $5\times 5$ template size.

 

Fig. 5 The signal-to-noise ratio performance of edge information against $SN{R}_{\mathbf{BD}}$ both using Sobel operator and the V-Sobel operator. The size of the V-Sobel template is $5\times 5$.

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Fig. 5 shows the SNR performance of edge information against $SN{R}_{\text{BD}}$ both using Sobel operator and the V-Sobel operator. The size of the V-Sobel template is $5\times 5$. The results show that the SNRs performance of edge information using Sobel operator decrease greatly with the increase of the $SN{R}_{\text{BD}}$, while the SNRs varies a very little by using the proposed V-Sobel operator. When the $SN{R}_{\text{BD}}$ is infinity, the SNR performance for the edge obtained by V-Sobel operator is $2.82$, while it is $2.26$ when $SN{R}_{\text{BD}}=11.89dB$. The results show that the V-Sobel operator has a good anti-noise characteristics. In addition, there is a cross between the two curves, and a more high SNRs performance can be obtained by using V-Sobel operator when $SN{R}_{\text{BD}}$ is smaller. The reason is that there would be some noise produced by the edge detection method based on GI, and the noise increase with the template size of the V-Sobel operator. Simultaneously, there would be other interference come from the simulation environment, and the interference is greatly increased with the decrease of $SN{R}_{\text{BD}}$. When $SN{R}_{\text{BD}}$ is smaller, the later interference is larger than that noise caused by the V-Sobel operator.

4. Conclusion

In the paper, we have designed a variable size Sobel operator (V-Sobel) at first, and then we have proposed an efficient edge detection method based on GI by using the designed V-Sobel operator. The template coefficients of V-Sobel are isotropic, and their values are determined by the distances between current pixels and the central pixel. The template size of V-Sobel is variable, and V-Sobel operator returns to Sobel operator when the template size is 3. The experiment and simulation results have shown that we have directly obtained the edge information before imaging the objects by using the proposed edge detection method. Meanwhile, the measurement times has not increased with the increase of V-Sobel template size. The SNRs performance are greatly improved by using V-Sobel operator in comparison with those results using Sobel operator for the same unknown objects. The bigger template size of V-Sobel operator has, the higher anti-noise property the proposed edge detection method has. When there is no edge information by using Sobel operator for a bigger noise, say, $SN{R}_{\text{BD}}=11.89dB$

, there is still a clear edge by using V-Sobel operator.

In summary, the proposed edge detection method has some advantages. Compared with the edge detection method in Ref.[19], our proposed method could directly achieve the edges of an unknown object without choosing the gradient angle or any other prior knowledge of the object. Compared with the edge detection method in Ref.[20], our proposed method has more large template size and more precise coefficient, resulting in the improvement of SNRs performance and anti-noise property. Compared with the method in Ref.[22], the edge image using our proposed method has a better quality even under a noisy environment, since the proposed V-Sobel operator has an ability to resist noise interference. Furthermore, our edge detection method has a simper imaging system, and be easier to implement than that in Ref.[22].