Image edge enhancement using Airy spiral phase filter

Yi Zhou; Shaotong Feng; Shouping Nie; Jun Ma; Caojin Yuan

doi:10.1364/OE.24.025258

1. Introduction

Techniques of edge enhancement have been widely used in image processing [1–3], especially in bioimaging [4], astronomical observation [5, 6] and fingerprint identification [7]. Radial Hilbert transform is one of the methods to achieve edge enhancement [8,9], which highlights the gradient of the complex refractive index of an object. Therefore, it is able to enhance the edges of amplitude-contrast and phase-contrast objects. The spiral phase plate (SPP) is a practical element as a common spiral phase filter (SPF) to implement Radial Hilbert transform in 4f system [9]. When the local features at the certain direction are required to enhance, special designed SPFs based on SPP have been proposed, such as fractional SPP [10, 11], shifting SPP [12, 13] and vectorial vortex filter [14].

It has to be mentioned that the superfluous side lobes near the main lobe in the coherent spread function (CSF) of the traditional SPP worsen the resolution and the image quality in the coherent imaging system. These side lobes are caused by the diffraction light from the central singularity of the SPP and the sharp edge of the aperture. To solve this problem, Laguerre-Gaussian spatial filter (LGSF) has been proposed to modify the amplitude of the singularity and the radius of the central area. Since the diffraction light from the singularity is blocked, LGSF can achieve higher contrast of edge enhancement than the conventional SPP [15]. Bessel-like spiral filter (BLSF) modulates the amplitude distributions not only in the central singularity but also near the edge of filter to become gradually dark. The latter process can avoid amplitude jump between the inside and outside of the filter edge. BLSF can partly eliminate superfluous side lobes by adjusting, namely, it can further reduce the imaging diffraction noise compared with LGSF [16].

Berry et al. theoretically predicted the existence of a unique wave packet solution in the form of a non-spreading Airy function from Schrodinger equation and paraxial wave equation [17]. Siviloglou G et al. firstly demonstrated that Airy beam which contains Airy function is able to be realized in optics [18, 19]. Since then, Airy beam has attracted more and more attention due to its peculiar properties of non-diffracting, self-healing and self-acceleration, such as the plasma guiding, the light bullet generation, the optical micromanipulation [20–24]. In image processing, it has been demonstrated by us that the edge enhancement can be also completed by adopting Airy function filter [25]. Although the amplitude of side lobes by adjusting the parameter $w_{0}$ in the Airy function filter is weaker than its in LGSF and BLSF, the infinite energy of the Airy function results in a low contrast.

A method is proposed to enhance the object edge by using novel Airy spiral phase filter (AiSF), which is formed by introducing a decaying item and a controlling parameter into an Airy function. The edge enhancement process is described in details in this paper. The CSF, as an image quality determined function, varying with the parameters $w_{0}$ and $ρ_{0}$ is numerically analyzed. Meanwhile, we compare the resolution and the contrast of the proposed methods with the conventional filters according to the CSFs in theoretical analysis section. Furthermore, to implement the edge enhancement in a certain orientation, the characteristics of the edge enhancement by using off-axis AiSF are analyzed on considering the center positions of the filter. The experiments are carried out to verify the proposed isotropic and anisotropic methods.

2. System principle analysis

To analyze the imaging system, the coordinates are given in Fig. 1(a). The imaging system is built on a 4f system. The object plane is located at the front focal plane of the first lens L1. The Fourier plane is at the rear focal plane of the first lens L1. The novel AiSF is located at the Fourier plane and the pattern is given in Fig. 1(b). The image plane is located at the rear focal plane of the second lens L2. $(r_{0}, θ_{0})$ , $(ρ, ϕ)$ and $(r_{1}, θ_{1})$ denote the polar coordinates of the object plane, the Fourier plane and the image plane, respectively.

Fig. 1 (a) The diagram of the imaging system. L1 and L2 represent lenses, $f_{1}$ and $f_{2}$ are focal length of L1 and L2, respectively. (b) The pattern of the AiSF.

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Assuming that the complex amplitude of the object is described in the object plane by

o (r_{0}, θ_{0}) = | o (r_{0}, θ_{0}) | \exp [i Φ (r_{0}, θ_{0})],

where

| o (r_{0}, θ_{0}) |

and

Φ (r_{0}, θ_{0})

denote the amplitude and the phase of the object, respectively. If the item

\exp [Φ (r_{0}, θ_{0})]

is expanded by the Taylor formula expansion, the complex amplitude of object can be rewritten as

o (r_{0}, θ_{0}) = | o (r_{0}, θ_{0}) | [1 + i Φ (r_{0}, θ_{0}) + \frac{{[i Φ (r_{0}, θ_{0})]}^{2}}{2!} + \frac{{[i Φ (r_{0}, θ_{0})]}^{3}}{3!} + \dots] .

At Fourier plane, the Fourier spectrum of the object is written as

F [o (r_{0}, θ_{0})]

, the symbol

F []

denotes the Fourier transform. The designed SPF is located at the Fourier plane to modulate the object spectrum. After passing through the filter, the field can be written by multiplying the object spectrum with the SPF transmittance function

H (ρ, ϕ)

:

O (ρ, ϕ) = F [o (r_{0}, θ_{0})] \times H (ρ, ϕ) .

The transmittance function of AiSF with topological charge l = 1 is:

H (ρ, ϕ) = A i (\frac{ρ_{0} - ρ}{w_{0}}) exp (\frac{ρ_{0} - ρ}{w_{0}}) c i r c l (\frac{ρ}{R_{0}}) \exp (i ϕ),

where

A i ()

is the Airy function which is a solution of Helmholtz equation and its expression is

A i (ρ) = \frac{1}{π} \int_{0}^{+ \infty} \cos (\frac{t^{3}}{3} + ρ t) d t

.

w_{0}

is the scaled parameter in Airy function,

ρ_{0}

is the radius of the main ring of the Airy function, and

exp [(ρ_{0} - ρ) / w_{0}]

is a decaying item.

c i r c l (ρ / R_{0})

describes a circular filtering aperture with radius

R_{0}

. When

| ρ |

is larger than

R_{0}

, the value of the circular aperture function equals to zero which means the light beyond the aperture cannot pass through the filter. Therefore, we only consider the situation with

| ρ | \leq R_{0}

and

c i r c l (ρ / R_{0}) = 1

. Through the Fourier transform, the CSF of the filter can be approximately expressed as

h (r_{1}, θ_{1}) = - \frac{2 π}{λ f_{2}} \exp (i θ_{1}) \int_{0}^{R_{0}} A i (\frac{ρ_{0} - ρ}{w_{0}}) exp (\frac{ρ_{0} - ρ}{w_{0}}) J_{1} (\frac{2 π r_{1} ρ}{λ f_{2}}) ρ d ρ,

where

J_{1}

is the first-order Bessel function of the first kind. The output distribution at the image plane can be written as

\begin{matrix} \tilde{o} (r_{1}, θ_{1}) = o (r_{0}, θ_{0}) * h (r_{1}, θ_{1}) \\ = exp [i 4 π k (f_{1} + f_{2})] \int_{0}^{R_{0}} \int_{0}^{2 π} h (r_{1} - r_{0}, θ_{1} - θ_{0}) | o (r_{0}, θ_{0}) | \\ [1 + i Φ (r_{0}, θ_{0}) + \frac{{[i Φ (r_{0}, θ_{0})]}^{2}}{2!} + \frac{{[i Φ (r_{0}, θ_{0})]}^{3}}{3!} + \dots] d r_{0} d θ_{0}, \end{matrix}

where

k

is wave number,

exp [i 4 π k (f_{1} + f_{2})]

[26] corresponds to a fixed phase that can be discarded since it plays no role in imaging. Therefore, we only consider the integral term in Eq. (6). If specimen is an amplitude-contrast object,

\int_{0}^{R_{0}} \int_{0}^{2 π} h (r_{1} - r_{0}, θ_{1} - θ_{0}) | o (r_{0}, θ_{0}) | d r_{0} d θ_{0}

plays a leading role in Eq. (6). If specimen is a phase-contrast object, the above equation becomes a uniform background. Then, Eq. (6) can be simplified to

\begin{matrix} \tilde{o} (r_{1}, θ_{1}) =C+ \int_{0}^{R_{0}} \int_{0}^{2 π} h (r_{1} - r_{0}, θ_{1} - θ_{0}) | o (r_{0}, θ_{0}) | \\ {i Φ (r_{0}, θ_{0}) + \frac{{[i Φ (r_{0}, θ_{0})]}^{2}}{2!} + \frac{{[i Φ (r_{0}, θ_{0})]}^{3}}{3!} + \dots} d r_{0} d θ_{0}, \end{matrix}

where C is the background. It can be clearly seen that the SPF can modulate not only the amplitude object but also the phase object. The intensity at the image plane is evaluated to be

I (r_{1}, θ_{1}) = {| \tilde{o} (r_{1}, θ_{1}) |}^{2} .

The image distribution given in Eq. (8) depends on the CSF in Eq. (5), which can be used to describe and analyze the system characteristics.

In Eq. (5), there are two parameters which are $ρ_{0}$ and $w_{0}$ , so the optimization process is carried out according to them. We set the value of $R_{0} = 3 m m$ in the filter, which is in accord with the physical size in the experiment. In order to show the characters more intuitively, we exhibit three typical values of $ρ_{0}$ with $w_{0} = 0. 5 R_{0}$ and plot their CSFs. Meanwhile, horizontal and vertical coordinates are normalized. As shown in Fig. 2(a), the green dotted line represents the amplitude distribution of the CSF when the value of $ρ_{0}$ equals to $0.2 R_{0}$ . In this situation, it is found that the CSF just contains the main lobe in the observable region, but the main lobe is wide which leads to a low resolution. When the value of $ρ_{0}$ gets increasing, the resolution is improved. However, it is worth noting that the contrast is decreased because of the increased amplitude of side lobes in the CSF. When $ρ_{0} > 0.6 R_{0}$ , a part of AiSF exceeds the boundary of the circular aperture. The intensity jump between the inside and outside filter is obvious, and it becomes a straight edge near the edge of the aperture. The diffraction from the straight edge generates stronger side lobes which further worsens the contrast.

Fig. 2 The CSF of AiSF with (a) the different values of $ρ_{0}$ when $w_{0} = 0. 5 R_{0}$ , (b) the different values of $w_{0}$ when $ρ_{0} = 0.6 R_{0}$ .

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Fortunately, it is able to suppress the side lobes while keeping the width of main lobe equal to $0.6 R_{0}$ by changing the value of $w_{0}$ . When $w_{0} \geq 0.5 R_{0}$ , the side lobes can be found in the blue dotted line of Fig. 2(b), which is generated by the diffraction of the straight edge of the aperture. With $w_{0} \leq 0.1 R_{0}$ , the side lobes increase again because the filter become a thin ring in this time. When the light passes through this thin ring, the diffraction phenomenon is similar to the slit diffraction. We use the green dotted line to denote it in the numerical simulation. It is found that the side lobes are successfully suppressed and the width of main lobe become small when $ρ_{0} = 0.6 R_{0}$ and $w_{0} = 0. 2 R_{0}$ .

To compare the amplitude characteristics of the CSFs of SPP, LGSF, BLSF and AiSF, the CSFs of these filters are simulated and given in Fig. 3(a)-(d), respectively. The plots across the center of Fig. 3(a)-(d) are normalized and shown in Fig. 3(e)-(h). It can be seen that strong superfluous noise exists in the CSF of the SPP in Figs. 3(a) and 3(e). In Figs. 3(b) and 3(f), the noise around the main lobe is partially reduced. In Figs. 3(c) and 3(g), the diffraction noise is further suppressed, but noise still exists around the main lobe. In Figs. 3(d) and 3(h), the AiSF makes the energy concentrate within the main lobe and side lobes are almost vanished. Comparing with the CSF of the Airy function filter [25], it can be found that there is less diffraction noise in CSF of the AiSF and its imaging resolution and the contrast are improved.

Fig. 3 CSF of: (a) SPP, (b) LGSF (c) BLSF and (d) AiSF, (e)-(h) are amplitude profile in radial section of (a)-(d), respectively.

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3. Anisotropic edge enhancement

Afore-mentioned filters are designed to complete isotropic edge enhancement. However, when the local features at a certain direction are more attractive to be enhanced than other directions, the symmetric AiSF is no longer appropriate. To obtain anisotropic edge enhancement, the center of the AiSF is shifted from the origin to $(ρ_{1}, ϕ_{1})$ in polar coordinates. This modified AiSF can be obtained by using coordinates transformation

H_{o f f} (ρ, ϕ, ρ_{1}, ϕ {}_{1}) = A i (\frac{B}{w}) \times \exp (\frac{B}{w}) \times \frac{[ρ \exp (i ϕ) - ρ_{1} \exp (i ϕ_{1})]}{\sqrt{C}},

where

\begin{array}{l} B = {(ρ_{0} - ρ)}^{2} + ρ_{1}^{2} + 2 (ρ_{0} - ρ) ρ_{1} \cos (ϕ - ϕ_{1}) \\ C = ρ^{2} + ρ_{1}^{2} + 2 ρ ρ_{1} \cos (ϕ - ϕ_{1}) . \end{array}

The subscript ‘off ’ stands for off-axis. Since the filter is decentered from the origin, we call it off-axis AiSF. The corresponding CSF is

h_{o f f} (r, φ, ρ_{1}, ϕ_{1}) = F [H_{o f f} (ρ, ϕ, ρ_{1}, ϕ_{1})] .

Numerical simulations are used to show the characteristics of off-axis AiSF with different values of

(ρ_{1}, ϕ_{1})

.

In order to observe the CSF clearly, we keep the azimuthal angle $ϕ_{1}$ be zero and choose different values of displacement distance $ρ_{1}$ . In Fig. 4 (a), the main lobe of the CSF become slightly non-uniform when the filter center is shifted to $(R_{0} / 8, 0)$ . The amplitude of the lower part of the main lobe accounts for almost eighty-three per cent of the amplitude of the upper part, which is measured from Fig. 4(e). When the center is removed from the origin, the non-uniform amplitude distribution is more and more obvious in Fig. 4(b)-(d). In Fig. 4(d), when $ρ_{1}$ increases to $R_{0}$ , the amplitude of the lower part of the main lobe accounts for merely twenty per cent of the amplitude of the upper part. Once $ρ_{1}$ is larger than $R_{0}$ , the singularity will be no more exist.

Fig. 4 CSF with $(ρ_{1}, ϕ_{1})$ equal to: (a) $(R_{0} / 8, 0)$ , (b) $(R_{0} / 4, 0)$ , (c) $(R_{0} / 2, 0)$ , (d) $(R_{0}, 0)$ , (e) - (h) are amplitude profile in radial section of (a)-(d), respectively.

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For a determined value of $ϕ_{1}$ in the AiSF, the direction where the main lobe is located in CSF has a phase shift of $π / 2$ after the direction of $ϕ_{1}$ [27, 28]. To set $ϕ_{1}$ at a determined angle, the enhancement direction will rotate with it. The edge enhancement is found along the white dashed line in in Fig. 5(a) and 5(b).

Fig. 5 CSF of AiSF with $(ρ_{1}, ϕ_{1})$ equal to (a) $(R_{0} / 2, 0)$ , (b) $(R_{0} / 2, π / 4)$ , respectively.

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A letter ‘E’ of which the stripes lie in horizontal and vertical directions is used to verify our approach. The positions of filter patterns, the anisotropic edge enhancement images and their intensity plots are given in Fig. 6. The yellow crosses denote the centers of the filters, while the red dotted line crosses represent the origins of the Fourier plane. In Fig. 6(a1), the center is shifted downward at $(R_{0} / 8, 3 π / 2)$ . The intensity of the right side of the sample edges is slightly stronger than the left side in Fig. 6(a2) and 6(a3). When the filter center is moved to the edge of the aperture $(ρ_{1} = R_{0})$ , half of the object spectrum is lost. Therefore, only the left side of edges can be seen in Fig. 6(b2). There is a strong intensity contrast between the right and left edges in Fig. 6(b3). The upper and bottom of edges are highlighted by shifting the filter centers to the left and right sides, respectively, as shown in Fig. 6(c) and 6(d). The anisotropic effect is more obvious with increasing $ρ_{1}$ , which is demonstrated with three typical values in the simulations. It also can be found that the shifting direction of the filter determined by the azimuthal angle is in vertical orientation of the edge enhancement.

Fig. 6 The first row is the off-axis AiSF with different $(ρ_{1}, ϕ_{1})$ . The values of $(ρ_{1}, ϕ_{1})$ are (a1) $(R_{0} / 8, 3 π / 2)$ , (b1) $(R_{0}, π / 2)$ , (c1) $(R_{0} / 4, 0)$ , (d1) $(R_{0} / 4, π)$ . The second row (a2)-(d2) are the output modified images which are modulated by the off-axis filters with corresponding $(ρ_{1}, ϕ_{1})$ in (a1)-(d1), respectively. The third row (a3)-(d3) are the intensity section distributions of (a2)-(d2) along the red arrow, respectively.

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4. Optical experimental results

The modified 4f optical setup for edge enhancement is shown in Fig. 7(a). The light emitting from a diode pump laser (500mW) is used as the light source. Then, it passes through a beam expander BE and a collimating lens L1 to correct its parallelism. After illuminating the specimen, the light carrying the information of the specimen is magnified by a microscopic objective (NA: 0.4, $20^{\times}$ ) and lens L2 (focal length is 75mm). The 4f system contains Fourier lens L3 and Fourier lens L4 (the focal lengths of the two lenses are 150mm and 200mm, respectively). A phase-only spatial light modulator (SLM, Holoeye Pluto VIS, $1920 \times 1080$ pixels, each pixel pitch is 8μm) is placed at the Fourier plane to load the Computer Generated Holograms (CGHs) which generate the designed filters. Since the filter is a complex amplitude one, GS iterative algorithm and CGH encoding method are required to generate the AiSF expressing in a phase function before being loaded. The SLM is tilted with a small angle deliberately to make the first order of output light to get into the CCD camera (Qimaging micropublisher 3.3 RTV, $2048 \times 1536$ pixels, each pixel pitch is 3.45μm). In order to avoid all diffraction orders of output light overlapping, the aperture is used to block unwanted orders of diffracted light. Since this filter is similar to a high-pass filter, the low frequency signals which contain lots of energy are blocked. The diffraction efficiency of the filter is 1.31% while $ρ_{0} = 0.6 R_{0}$ and $w_{0} = 0.2 R_{0}$ . Additionally, the CGHs used as a filter is complex-valued. Two third of the power will be lost if we only select the first order. To obtain a high visibility in the experiments, a high-power (500mW) laser is used in the experimental setup.

Fig. 7 (a) The experimental setup for edge enhancement. L2, L3 and L4 are the Fourier lens; BE is beam expander; BS is the beam splitter; SLM is the spatial light modulator; CCD is the Charge Coupled Device camera, (b) is the CGH based on AiSF.

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The 1951 USAF Glass Slide Resolution Target which is used as an amplitude-contrast object to, compare differences among LGSF, BLSF and AiSF. Figure 8(a) shows the output image of the specimen in a conventional bright-field microscope. Figures 8(b)-(d) show the microscopic images after filtering by LGSF, BLSF and AiSF, respectively. To obtain further quantitative comparisons among these images, we plot the intensity distributions along the red dotted lines and show them in Figs. 8(e)-(h). From these images, it is found that the image edges can be enhanced by all these SPFs. However, background noise is obvious and the width of enhanced edge is large in Fig. 8(f). It can be seen that the edges enhanced by AiSF are much sharper than those which use BLSF, meanwhile, the intensity of the former is higher and more uniform than the latter.

Fig. 8 The recorded images of an amplitude-contrast object by using (a) conventional bright field, (b) LGSF,(c) BLSF, (d) AiSF, (e)-(h) are the intensity section distribution of (a)-(d), respectively.

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For biological cells without dye which are phase-contrast objects, the contrast of images under bright-field microscopes are usually quite low. A transparent thyroid cell is adopted as the sample, which contains complex and irregular edges and lots of stripes structures. Figure 9(a) shows a microscopic image of a thyroid cell obtained by a phase contrast microscope, the cell is difficult to be distinguished in this image. To be clearly displayed, we implement image enhancement by loading different SPFs on SLM. In Fig. 9(b), the edges of this cell can be highlighted by using LGSF, but it is hard to see the details of this cell clearly in the figure and there is strong noise around the enhanced edges. In Fig. 9(c), it is noteworthy that the stripes can be observed more clearly than those in Fig. 9(b), but the noise still exists. In Fig. 9(d), not only the edges of the cell, but also its internal structures are sharp and clearly recognized. On the details, there are double stripes in its internal structures instead of single in the same position of the other images. Meanwhile, the noise is reduced significantly.

Fig. 9 The images of a phase-contrast object by using (a) phase contrast microscope, (b) LGSF, (c) BLSF, (d) AiSF, respectively.

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To implement anisotropic edge enhancement, a series of CGHs of off-axis AiSF with different center positions are loaded on the SLM. By changing the position parameters of center, edge enhancement images at different directions can be achieved and shown in Fig. 10. The thyroid cell is still used to carry out this experiment. To observe the details more clearly, we magnify the sections included in the red squares of the images and plot the magnified images along the red arrowed line in the inserted images of Fig. 10. With $ρ_{1}$ moving away from the origin, the AiSF becomes anisotropic. It can be found from Fig. 10(a) that the intensity of two edges are different and the right one is higher than the other. When the value of $ρ_{1}$ is further increased, the center of the AiSF is deviating from the origin of the Fourier plane and the main ring is closed to the origin, more and more low frequency information of the object cannot be suppressed. Therefore, background information of the thyroid cell will appear in Fig. 10(b) and background intensity blurs the image, which cause a poor image contrast. To verify the theory that changing the angle $ϕ_{1}$ can lead to the direction of the edge enhancement, we make $ρ_{1}$ equal to $R / 4$ firstly. The intensity distribution of the upper edges in each stripe pair is stronger than the other, as shown in Fig. 10(c). Then, we maintain $ρ_{1}$ but rotate $ϕ_{1}$ to $π$ , which leads to the direction of the edge enhancement opposite. The lower edges of stripe pairs are stronger than the upper edges in Fig. 10(d).

Fig. 10 The image of a phase-contrast object by using AiSF with different $(ρ_{1}, ϕ_{1})$ . The values of $(ρ_{1}, ϕ_{1})$ are (a) $(R_{0} / 8, 3 π / 2)$ , (b) $(R_{0}, π / 2)$ , (c) $(R_{0} / 4, 0)$ , (d) $(R_{0} / 4, π)$ , respectively. In the below left hand corner, the subfigure is the magnified image of the red box part image. In the below right hand corner, the subfigure shows the cross sections of magnified image along the red arrows.

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5. Conclusion

In this paper, the isotropic and anisotropic image edge enhancements method by employing AiSF is investigated. In the isotropic method, we deduce this filter transmittance function and analyze the CSF determined by the parameter $ρ_{0}$ and $w_{0}$ in this filter. When $ρ_{0} = 0.6 R_{0}$ and $w_{0} = 0.2 R_{0}$ , the main lobe in CSF can keep slim and the side lobes are fully suppressed. Compared with LGSF, BLSF and Airy function filter, it can be seen that AiSF is more effective in enhancing the contrast of the image and improving the resolution. Moreover, the off-axis AiSF is adopted to highlight the local features of some edges at a certain direction, we deduce the CSF of off-axis AiSF and analyze the primary factors of it. When $ρ_{1}$ equals to $R_{0} / 4$ , the contrast of edges are obvious and low frequency information of the object is well suppressed. Optical experiments by using amplitude-contrast and phase-contrast objects are carried out. This work can be used to detect the living cells with higher resolution and contrast. The interested sections of the living cells can be enhanced by using off-axis AiSF.

Funding

National Natural Science Foundation of China (NSFC) (61377003, 61575095); Key grant project of Jiangsu Provincial Department of Education, China (14KJA140001).

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Image edge enhancement using Airy spiral phase filter

Abstract

1. Introduction

2. System principle analysis

3. Anisotropic edge enhancement

4. Optical experimental results

5. Conclusion

Funding

References and Links

Cited By

Figures (10)

Equations (11)

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