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Abstract
In this paper, we present edge detection schemes with specially designed superposed spiral phase plate (SSPP) filters in the Fourier domain both for intensity or phase objects. A special SSPP whose function is equivalent to Sobel operator in space domain is firstly designed by weighting different topological charge spiral phase plate (SPP) filters. Later, a SSPP with controllable direction parameters is then discussed to enhance the anisotropic edges by controlling the direction parameter. Numerical simulation and experimental results show that either isotropic or anisotropic edge information can be enhanced by using our proposed schemes. The signal-to-noise ratio and the root-mean-square-error performance are improved in comparison with those using traditional SPP filter. Importantly, it is the first time to present the special ways of superposing and the SSPP can be designed before the experiment so that a clear edge can be achieved at real time without the convolutional operation.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Edge detection is on the basis of the dramatic change in the transmittance or reflectance of an object, which is nowadays widely used in target recognition, security checks, earth observation and so on [1,2]. In many imaging applications, it is sufficient to detect the periphery of an object [3]. Commonly, the edge information can be archived by an edge operator convoluted on the images, for example, the Sobel operator, one classical first-order gradient edge detection operator. Edge information can also be detected based on the principle of ghost imaging (GI) with the advantages on good anti-disturbance imaging and direct edge detection of unknown objects [4–6].
Usually, these edge detection methods mentioned above are useful for the intensity objects. It is still a challenge for the phase objects’ edge detection, e.g., biological cells. In order to obtain the edge information of both intensity and phase objects, the spiral phase contrast imaging (SPC) was proposed [7], where a spiral phase plate (SPP) with a topological charge $\ell$ = 1 was served as a filter in designed 4f system. As a significant optical implementation for edge enhancement, there has been a series of related research works owing to its parallel processing, bioimaging and so on [8–14]. It was the first time to talk about the phase contrast microscope in 1942 [8]. Subsequently, Fürhapter et al. demonstrated a novel optical edge enhancement method in light microscopy worked by holographic Fourier plane filtering of the microscopic image with a SPP [9]. Recently, a new nonlinear filter was presented which was constructed by equivalently imprinting the SPP onto the potassium titanyl phosphate crystal using second harmonic generation (SHG), leading to a visible edge enhancement with invisible illumination [10]. There was also the work to use the single-pixel detection to directly acquire the Fourier spectrum of the edge-enhanced object by scanning spiral phase-encoded plane waves in k-space [11].
Moreover, owing to the symmetry of the SPP with integer topological charge, all edges of the object are enhanced isotropically, which is not suitable in the case where some direction edges are preferredly emphasized more than other directions. By introducing the Sine function in conventional spiral phase distribution [15] or by superposing two SPPs with opposite topological charges [16], several methods have been presented to obtain the anisotropic edge enhancement. Nevertheless, only one direction edge can be filtered out by using the existed methods.
In the paper, we propose edge detection schemes with specially designed SSPP filters both for isotropic or anisotropic edge enhancement. Firstly, we present the isotropic edge enhancement by designing a special SSPP with the equivalent functionality of Sobel operator in Fourier domain. Secondly, we present the anisotropic edge enhancement by constructing a special structured SSPP with some controllable direction parameters. With numerical simulations and experiment setup, we discuss the performance of the proposed edge detection schemes, and compare them with the existing edge schemes on SPC.
The organization of the paper is as follows. In Section 2, the presented scheme with two kinds of specially designed SSPP filters for isotropic and anisotropic edge enhancement are proposed. The performance of the proposed scheme is discussed by numerical simulations and experiments in Section 3 and Section 4, respectively. Finally, Section 5 concludes the paper.
2. Theory
The SPP was firstly used at millimetre-wave frequencies to transform a free-space, fundamental Hermite-Gaussian mode into a Laguerre-Gaussian(LG) mode with an orbital angular momentum(OAM) equivalent to $\ell \hbar$ per photon [17]. The transmission function of SPP is described as $\exp \left (i\ell \varphi \right )$, where $\varphi$ is the polar angle and $\ell$ is the topological charge. Then, SPP was found that it could be applied to SPC imaging for edge enhancement because of its sensitivity to intensity or phase gradients [14].
Figure 1 shows the procedure of SPC imaging which can be recognized as a 4f system, where the two lens (L1, L2) in the 4f system are with the same focal length f. The input object, mathematically described by a function as $s(r,\phi )$ in the real space, is placed in the object plane of the 4f system, and the SPP filter $H(\rho ,\varphi )$ in Fourier domain is at the back focal plane of L1 [18]. The input object is illuminated by a beam of light. The first lens L1 carries out the Fourier transform so that one can obtain the Fourier spectrum of the input object at its back focal plane,
where $F$ represents spatial Fourier transform. When the light field carrying the Fourier spectrum information of the input object passes through the filter, one can obtain the Fourier field of the filtered edge information by multiplying the Fourier spectrum of the input object and the filter function, namely,When the modulated light described by Eq. (2) passes through L2, mathematically equivalently a Fourier transform again, the output image can be obtained at the back focal plane of L2, that is,
where $h$($r$, $\phi$) = $F^{-1}$[$H$($\rho$, $\varphi$)], is the reverse Fourier transform of $H$($\rho ,\varphi$), $F^{-1}$ denotes the reverse spatial Fourier transform [9]. According to the convolution algorithm, every corresponding pixel of the input object is multiplied with the convolution kernel $h\left (r,\phi \right )$ and summed, resulting in an edge detection, that is, the edge regions with different phases or intensities are highlighted.
Fig. 1. Schematic illustration of spiral phase contrast imaging, where the object $s(r,\phi )$ in space domain is located at front focal plane of L1, and the SPP filter $H(\rho ,\varphi )$ in Fourier domain is at the back focal plane of L1. The edge information $e(r,\phi )$ in space domain is obtained at the image plane of the 4f system. All the coordinate systems are polar coordinate system.
2.1 Isotropic edge enhancement
The filter $H\left (\rho ,\varphi \right )$ plays an important role in SPC system. In previous works, the most commonly used SPP filter is LG mode with topological charge $\ell$ = 1, which is named as traditional SPP filter in the paper. In fact, the filters with other topological charge, such as, $\ell$ = 2, 3, $\cdots$, also have contributions to edge enhancement. And according to our work in [19], the SPP filters with odd topological charges have a greater contributions.
On the other hand, the edge information can also be obtained by the convolution operation of edge detection operator on the image, where the Sobel operator is one of the classical edge detection operators and has a better edge recovering effects. Figure 2 shows the two 3$\times$3 convolution kernels of Sobel operator for the horizontal and vertical direction [20–22].
As the integer OAM eigenstates form a complete, orthogonal and infinite basis, in principle, all the filters in Fourier domain can be expressed in terms of OAM eigenstates [23,24]. Different SPPs can produce different OAM eigenstates. Therefore, the Sobel operator in Fourier domain can be described by the superposition of SPPs. The proposed superposed spiral phase plate (SSPP) filter for isotropic edge enhancement can be described as,
where $H_{Sobel}$($\rho ,\varphi$) represents the Fourier spectrum of Sobel convolution kernel (here we employ the horizontal Sobel convolution kernel Gx, in fact, whether Gx or Gy is used, the result is identical) and $A_\ell$ = $\int \int H_{Sobel}$ ($\rho ,\varphi$) $\cdot \exp$($i\ell \varphi$)$\rho d\rho d\varphi$ presents the weight of SPP functions with topological charge $\ell$. It is noted that only the odd topological charge ($\ell$ = 2m-1, m = 1, 2,…, $n$) are employed in Eq. (4). When the isotropic SSPP filter is used, the edge detection results shown in Eq. (3) is changed to [25],2.2 Anisotropic edge enhancement
Generally, the edge enhancement is isotropic, particularly when the SPP is used as a radial Hilbert phase mask. But in some applications, it is preferable to have some directional edges emphasized more than others. The conventional anisotropic edge detection system can only filter the edge in the same direction. To overcome this issue and make a broader applications, an improved SSPP filter to implement multi-directional edge enhancement is presented here. By superposing multiple SPPs with opposite topological charges, and then changing the phase distribution between these SPPs with controllable direction parameters $\theta _j$, a SSPP filter for multi-directional edge enhancement is designed, that is,
Moreover, when the anisotropic SSPP filter is used, the result of edge detection is changed to,
3. Numerical simulations
In this section, we discuss the performance of the presented scheme with two kinds of specially designed SSPP filters by numerical simulations. The numerical simulations are done with CPU of AMD Core A10-7300(Lenovo with Radeon R6 core 1.9GHz and memory 8GB) by Labview 2012(32bits).
3.1 Isotropic edge enhancement
Firstly, we study the edge detection performance of SPP filter with $\ell$ from -8 to 8 by simulation. Here a binary image of a flower, gray value of 255 for flower part and gray value of 0 for background part, is used as object image which is shown in Fig. 3(a). The simulation results are shown in Fig. 3(b) and (c) and the image size is 500 $\times$ 500 pixels. The results show that the edge patterns look the same when the topological charge value of SPP is the same. The edge obtained by SPC is independent of the sign of the topological charge. From the figure, we can also find that the edges with different topological charges SPP have a cycle pattern for 4. The edge with $\ell =1$ looks like the edge with $\ell =5$. The edge with $\ell =2$ looks like the edge with $\ell =6$. The edge with an even topological charge has a bright field, such as $\ell =2$ and $\ell =4$, where $\ell$ = 2 has only curved edges, while $\ell$ = 4 has all bright information regions. This is because the azimuthal is doubled the spiral phase for $\ell$ = 4 [26]. Hence, only the odd SPPs are used in Eq. (4).
Fig. 3. The edge information between SPPs with different topological charges (SPP spectra). (a)original image, (b1 - b8) the results of SPC imaging by SPP filter, for $\ell$ from -1 to -8, (c1 - c8) for $\ell$ from 1 to 8.
In order to compare the quality of the reconstructed edge quantitatively, signal-to-noise ratio (SNR) and root-mean-square-error (RMSE) are used as objective evaluations [27]. The SNR is defined as,
Figure 4 demonstrates the proposed SSPP filter in Eq. (4) for several binary intensity objects with the same size of 500 $\times$ 500 pixels. And the last one is transformed into a pure phase object by a phase modulation (0 $\sim \pi$) according to the intensity (0 $\sim$ 255). For comparison, we also present the edges with the traditional SPP filter and the original edge achieved by Sobel operators. The first and second row are original images and original edges, respectively. And the results with traditional SPP filter are displayed at the third row. From the fourth to the fifth row, the results are demonstrated with the SSPP filter when $n$ = 2 and 10 (’$n$’ represents the superposition number of SPPs, i.e., $\ell$ = 2m-1, m = 1, 2,…, $n$. For example, when $n$ = 4, the SSPP filter is obtained by the superposition of SPP with $\ell$ equal to 1, 3, 5, 7). From the simulation results, we can find the background noise near the edges has been effectively suppressed by using SSPP filter, in comparison to that using traditional SPP filter. The sharp and clear edges mean the spatial resolution of the edge image is greatly improved when our SSPP is adopted. And it is not difficult to find that with the increase of $n$, the proposed scheme shows a better edge enhanced results.
Fig. 4. The demonstration of the proposed scheme with some binary images by isotropic SSPP filter, together with the results by traditional SPP filter. The first row: original images; the second row: original edges; the third row: results of conventional SPC imaging with traditional SPP filter; the fourth and fifth row: results of SPC imaging with the SSPP filter for $n$ = 2, 10 respectively.
Figure 5 shows the performance of edge information by both traditional SPP filter and the proposed SSPP filter($n$ from 2 to 10), Fig. 5(a) for SNR performance and Fig. 5(b) for RMSE performance. The results in Fig. 5(a) show that the SNR performance has a significant increase when the SSPP filter is adopted. The SNR performance of image ’1’ obtained by the conventional SPC imaging scheme is 0.75, while it is 1.55 when the proposed SSPP filter with $n$ = 4 is used, that is, there is a 106.7$\%$ improvement in SNR. Meanwhile, the results in Fig. 5(b) show that there is a steep decline of RMSE when the superposition number is increased, that is consistent with the results in Fig. 5(a). The RMSE performance of image Geometry obtained by the conventional SPC imaging scheme is 0.191, while it is 0.092 when the SSPP filter with $n$ = 7 is used, that is, there is a 51.8$\%$ improvement in RMSE.
Fig. 5. The performance of edge information with different superposition number $n$. (a) for signal-to-noise ratio(SNR), (b) for root-mean-square-error(RMSE).
According to Eq. (1), we list the weight $A_\ell$ in Table 1 about the SSPP filters presented above. One can see that there is a rapid decline of the weight with the increase of $\ell$. It means that the SPP with a smaller topological charge has a larger contribution to edge enhancement. This result is also consistent with the SNR performance above, that is, the enhancement of SNR is increased slowly after $n\geq$ 7.
Table 1. The SPP Weight $A_\ell$ for Different Topological Charges $\ell$
3.2 Anisotropic edge enhancement
Subsequently, we employ the filter shown in [16] to realize conventional directional edge enhancement. Here a circular pattern is used as the object shown in Fig. 6(a) with the size of 500 $\times$ 500 pixels. We use the $\theta$ = $0^\circ$, $45^\circ$, $90^\circ$, $135^\circ$, which are corresponding to the horizontal, diagonal, vertical and anti-diagonal direction, respectively. The simulation results are illustrated in Fig. 6(b)-(e). It can be found there is an obvious gap in each ring with expected direction which coincides with the adding $\theta$. Here, we define the horizontal gap direction is $0^\circ$.
Fig. 6. Simulation results of anisotropic edge enhancement. (a) original image, (b)-(e) conventional directional edge enhancement with $\theta$ = $0^\circ$, $45^\circ$, $90^\circ$, $135^\circ$, (f) 2-directional edge enhancement with $\theta _1$ = $45^\circ$, $\theta _2$ = $135^\circ$, (g) 3-directional edge enhancement with $\theta _1$ = $0^\circ$, $\theta _2$ = $45^\circ$, $\theta _3$ = $135^\circ$, (h) 3-directional edge enhancement with $\theta _1$ = $45^\circ$, $\theta _2$ = $90^\circ$, $\theta _3$ = $135^\circ$, (i) 4-directional edge enhancement with $\theta _1$ = $0^\circ$, $\theta _2$ = $45^\circ$, $\theta _3$ = $90^\circ$, $\theta _4$ = $135^\circ$.
Then we employ the improved SSPP filter shown in Eq. (6) to realize multi-directional edge enhancement. The superposition number $j$ and the corresponding direction parameter $\theta _j$ can be defined flexibility according to actual demands. Here we still use the same circular pattern as the object and choose $j$ = 2, 3, 4. The simulation results are illustrated in Fig. 6(f)-(i), Fig. 6(f) for $j$ = 2 with $\theta _1$ = $45^\circ$, $\theta _2$ = $135^\circ$, Fig. 6(g) for $j$ = 3 with $\theta _1$ = $0^\circ$, $\theta _2$ = $45^\circ$, $\theta _3$ = $135^\circ$, Fig. 6(h) for $j$ = 3 with $\theta _1$ = $45^\circ$, $\theta _2$ = $90^\circ$, $\theta _3$ = $135^\circ$, Fig. 6(i) for $j$ = 4 with $\theta _1$ = $0^\circ$, $\theta _2$ = $45^\circ$, $\theta _3$ = $90^\circ$, $\theta _4$ = $135^\circ$. It can be found several clearly gaps are well observed in each image and the directions of the gaps are coincides with corresponding $\theta _j$. And with the increase of superposition number $j$, our multi-directional edge enhancement scheme can theoretically filter out edges in any number of directions.
4. Experiments
In this section, we discuss the performance of the presented scheme by experiments.
The experimental system of the proposed SPC imaging system is shown in Fig. 7. Lens L3 and L4 with the same focal length of f = 50cm form a 4f system. The object used as input image and the CCD (Microvision, MV-VS142FM) are placed in the object plane and the image plane of the 4f system, respectively. While the SLM (Hamamatsu, X10468-07) [28] used as SSPP filter is positioned at the back focal plane of L3 which can freely modulate light phases by the Liquid Crystal on Silicon (LCOS) chip. The input light is a linear fundamental mode Gaussian beam derived from a 5-mW, 633-nm He-Ne laser (Thorlabs, HRS015). An attenuator (ATT) after the laser is used to adjust the beam power, and the half-wave plate (HWP) ensures the suitable polarization direction of the input light to match SLM. After being expanded and collimated by a telescope ($L_{1}$ and $L_{2}$), the beam passes through the object which are a portion of the United States Air Force (USAF) resolution target with an actual size of 40 $\times$ 40 mm (about 1280 $\times$ 1280 pixels, 913 pixels per inch (ppi)) and a circular aperture with an actual size of about 5 $\times$ 5 mm (about 160 $\times$ 160 pixels, 913 ppi). Then, the transmitted light carrying the information of the object is incident on the computer-controlled SLM which displays the SSPP filter function. Here we use the superposition of phase distribution of SPP, $\exp \left (i\ell \varphi \right )$, as holographic grating to realize SSPP filter in experiment. For example, when n = 2, we implement the SSPP filter by superposing the phase function $\exp \left (i\varphi \right )$ and $\exp \left (i3\varphi \right )$ with a certain weight as holographic grating to load on SLM. Finally, the filtered light beam reflected by the SLM is recorded by the CCD, namely, the edge information of the object is obtained. Of key importance in this system is the set of Fourier lens ($L_{3}$ and $L_{4}$) used to perform the spatial Fourier transform of the object in the plane of the SLM and CCD, as are described by Eq. (1) and Eq. (3) respectively.
Fig. 7. The sketch of the experimental system to implement the proposed spiral phase contrast imaging. ATT: attenuator, HWP: half-wave plate, L: lens, SLM: spatial light modulator, CCD: charge coupled device. (a) holographic grating of SSPP with $\ell$ = 1, 3, (b) holographic grating of SSPP for multi-directional edge enhancement.
4.1 Isotropic edge enhancement
Figure 8 shows the experimental results of isotropic edge enhancement. Here we use two parts of USAF as the objects. For comparison, we first turn off the SLM, and the SLM acts as a mirror. The whole system just performs as a simple 4f system, and Eq. (6) can be simplified to $e_{out}\left (r,\phi \right ) = s\left (r,\phi \right )$, then the object images are obtained in first column. After the SLM is turned on, we display the traditional SPP filter and the SSPP filter with $\ell$ = 1, 3 (n = 2) by addressing suitable holographic grating on SLM, respectively. And the corresponding edge information is shown in second and third column, respectively. Moreover, we present the corresponding intensity profiles of each images in the white dotted lines of the image. Compare with traditional SPP edge detection results, the improved ones have more bright and sharp edges.
Fig. 8. The experimental results of the filtered image of part of USAF. The first column: original images; the second column: edge information with traditional SPP filter; the third column: edge information with the SSPP filter by using $\ell$ = 1, 3. The curve below each image is the corresponding intensity section distributions of the dotted line.
To analyze the image quality quantitatively, the contrast ratio is introduced and is defined as $V=\frac {I_{\max }-I_{\min }}{I_{\max }+I_{\min }}$. Generally, the higher V is, the better quality edge detection has. For the binary character object ’1’, V for our scheme is 87.9$\%$, while it is 76.8$\%$ using traditional SPP filter and 89.9$\%$ for the ’original image’. Besides, SNR for our scheme is 0.89, while it is 0.72 using traditional SPP filter. Obviously, the proposed edge detection scheme outperforms the conventional one and improves the image contrast ratio and SNR.
4.2 Anisotropic edge enhancement
We further perform experiment to verify anisotropic edge enhancement. Firstly, we turn off SLM and obtain the original image in Fig. 9(a). Then, we turn on the SLM with suitable holographic grating. The experimental results are illustrated in Fig. 9(b)-(e) and corresponding holographic gratings loaded on SLM are shown at the top left of the images with $\theta$ = $0^\circ$, $45^\circ$, $90^\circ$, $135^\circ$ respectively. One can see that the experimental results are consistent with the simulation results in Fig. 6, the uniform rounded edge enhancement is broken, and a gap in each ring is shown in $\theta$ direction.
Fig. 9. The experimental results of traditional directional edge enhancement. (a) original image, (b)-(e) filtered image with $\theta$ = $0^\circ$, $45^\circ$, $90^\circ$, $135^\circ$ respectively.
Then, we employ the improved filter in Eq. (6) to verify multi-directional edge enhancement. Here we use the same object and choose the filters of $j$ = 2 with $\theta _1$ = $0^\circ$, $\theta _2$ = $90^\circ$ and $\theta _1$ = $45^\circ$, $\theta _2$ = $135^\circ$. The experimental results are illustrated in Fig. 10. Moreover, the intensity profiles of the dotted lines are depicted ( the same color with the dotted lines). From the intensity profiles, it can be found the intensity value of the ring nearby $\theta _1$ and $\theta _2$ is far smaller than that one perpendicular to $\theta _1$ and $\theta _2$, it is indicated that the multi-directional edge enhancement has been successfully realized.
Fig. 10. The experimental results for multi-directional edge enhancement. The first row: original image, filtered image with $\theta _1$ = $0^\circ$, $\theta _2$ = $90^\circ$ and filtered image with $\theta _1$ = $45^\circ$, $\theta _2$ = $135^\circ$; the second row: corresponding intensity distribution.
5. Conclusion
In this paper, we have proposed an edge detection scheme with two types of specially designed SSPP filters to realize isotropic and anisotropic edge enhancement respectively. For isotropic edge enhancement, comparing with conventional SPC imaging system, our scheme improves the edge detection performance by introducing Sobel operator using SSPP in the Fourier domain. For anisotropic edge enhancement, our scheme expands directional edge enhancement to multi-directional edge enhancement. The numerical and experimental results have shown that the SPC system with the proposed isotropic SSPP filter has had better edge information in comparison with those results with the conventional one and the proposed anisotropic SSPP filter has been proven to filtered out the edges from different directions. It is the first time to present the special ways of superposing, like the decomposition of Sobel operator in Fourier domain and the any numbers controlled parameters superposition. Since all those SSPP filters mentioned above can be designed before the experiment, the proposed scheme can archive clear edge at real time. Furthermore, the proposed method also provides a new way to realize conventional spatial operations in optical Fourier domain.
Funding
National Natural Science Foundation of China (61871234, 62001249); Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University (KF201909).
Disclosures
The authors declare that there are no conflicts of interest related to this paper.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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