Edge enhancement in three-dimensional vortex imaging based on FINCH by Bessel-like spiral phase modulation

Jiuru He; Pengwei Zhang; Jianpo Su; Junqiao Wang; Yongzhi Tian; Yongsheng Hu; Yongsheng Hu; Fengying Ma; Fengying Ma

doi:10.1364/OE.511205

1. Introduction

Edge enhancement can emphasize the edge and detail information of the objects, which has been widely applied in the fields of feature recognition [1,2], fingerprint identification [3], bioimaging [4], and astronomical observation [5,6]. There are two common methods to achieve image edge enhancement: digital image processing and optical filtering [7]. Digital image processing is a post-processing technique for the obtained images, and the performance is subject to the imaging quality acquired in the earlier stage. By comparison, the method of optical filtering can directly obtain edge-enhanced images, which is more conducive to highlighting the edge details of the sample and achieving higher contrast of edge enhancement. Furthermore, achieving higher contrast of the edge-enhanced images and suppressing the noise during image acquisition can be extremely significant since it can improve the accuracy of image detection and will be greatly beneficial to image processing, including image segmentation, pattern recognition, and feature matching.

The optical vortex filtering based on Hilbert transform provides an effective method for edge enhancement [8–10]. However, the early operation of this spatial filtering technique is one-dimensional (1D) as it can enhance edges along only a single direction [11]. To address this issue, a radially symmetric version of the Hilbert transform is proposed to realize two-dimensional (2D) edge enhancement of the arbitrarily shaped objects [10]. But at the same time, the traditional vortex filtering causes background noise and contrast reduction due to the diffraction sidelobes caused by central singularities and sharp edges of the spiral phase plate (SPP) [12]. Several methods are used for diffraction noise suppression and imaging quality improvement, including annular phase structure [13], helical axicon [14], and different kinds of spiral phase filters [15,16]. Based on these works, the resolving capacity in edge enhancement of 2D vortex imaging is dramatically improved.

In recent years, edge enhancement in 3D vortex imaging has also been successfully implemented by Fresnel incoherent correlation holography (FINCH) combined with vortex filtering [17–20]. FINCH is a promising 3D imaging technique that shows significant advantages of scanning-free, high lateral resolution, and easy matching with existing mature optical systems [21–24]. The edge enhancement study in FINCH using vortex filtering was first demonstrated by P. Bouchal and Z. Bouchal [17], and the system was dubbed spiral-FINCH. Recently, our group has also done some work on the spiral-FINCH, including the mathematical model established [18] and the tunable edge enhancement [19]. In spiral-FINCH, one part of the light emitted from each object point is modulated by the spiral phase, and it interferes with the other part modulated by the quadratic phase to generate a point source hologram. When numerically reconstructed by Fresnel propagation, the recorded hologram generates the 3D edge enhancement images. Since the spiral phase mask (SPM) loaded on the spatial light modulator (SLM) functions as the spiral phase filter, the sidelobes in the generated optical vortices can also deteriorate the quality of the recorded holograms [15,16]. And this in turns leads to increased background noise and decreased contrast of the reconstructed edge-enhanced images, which seriously degrades the quality of the 3D vortex imaging.

Generally, the edge enhancement effect in vortex imaging is isotropic. However, in some practical applications, it is significant to achieve anisotropic edge enhancement since emphasizing the local features along certain orientations is necessary for image processing [25–27]. The 2D anisotropic edge enhancement has been extensively studied and achieved by the methods of fractional or shifted vortex filtering [28], vectorial vortex filtering [29], and superposed vortex filtering [30]. However, despite the 3D vortex imaging has been studied in several papers [17–20,31], most of them are focused on isotropic edge enhancement. Presently, there is only one study of the anisotropic edge enhancement in 3D vortex imaging, which was adopted by the sine-modulated spiral modulation [17].

In this paper, the Bessel-like spiral filter has been introduced into the spiral-FINCH instead of the traditional spiral filter. Comparing the conventional spiral-FINCH, we demonstrate that the edge enhancement of FINCH by employing Bessel-like spiral phase modulation can achieve high-quality 3D vortex imaging. Reconstruction images of the small circular aperture, resolution target, and the Drosophila melanogaster demonstrate its apparent advantages in background noise suppression and edge contrast enhancement. Furthermore, aiming at the lack of research on the anisotropic edge enhancement in 3D vortex imaging, we developed a new method by breaking the symmetry of the helical phase in the algorithmic model of isotropic edge enhancement. By controlling the constant phase factor of the spiral phase, the edges of the given objects can be selectively enhanced in any desired direction. The results of this work may find potential applications in the fields of edge detection and pattern recognition.

2. System analysis

The Bessel-like phase function is described as:

(1)$$B{S_l}({r,\varphi } )= circ({{r / R}_0})\textrm{exp} [{{J_l}(r) \times il\varphi } ]$$

where (r, φ) is the polar coordinate, J_l(r) is the Bessel function of the first kind, R₀ is the size of the aperture, and l is the angular quantum number.

In the Bessel-like spiral phase modulated FINCH, two diffractive lenses are spatially multiplexed on the SLM. One is a conventional lens, and the other is a spiral lens modulated by the Bessel phase. The modulation function of the SLM can be written as:

(2)$$R({x,y} )= {A_1}\textrm{exp} [\frac{{ - i\pi }}{{\lambda {f_{d1}}}}({x^2} + {y^2}) + i{\theta _j}] + {A_2}\textrm{exp} [\frac{{ - i\pi }}{{\lambda {f_{d2}}}}({x^2} + {y^2}) + iBS]$$

where A₁ and A₂ are constants, λ is the working wavelength, f_d1 and f_d2 are the focal lengths of two diffractive lenses. θ_j (j = 1, 2, 3) are phase constants, which can be used to eliminate the zero-order and twin image. BS is the angle of the BS_l.

Figure 1 presents the schematic of the spiral-FINCH system by employing Bessel-like spiral phase modulation. Suppose an ideal point source at P(0, 0, -z_s) induces a diverging spherical wave and propagates in the front surface of the collimating lens L (whose focal length is f₀). The complex amplitude in the front surface of the lens L can be expressed as exp[iπ(z_sλ)⁻¹(x² + y²)]. After passing through the lens L, propagating a distance d and omitting the constant coefficients, the complex amplitude on the SLM plane gets exp{iπ[(f_e + d)λ]⁻¹(x² + y²)}, here 1/f_e = 1/z_s - 1/f₀. Then reflected by the SLM, the output light is divided into two beams of A₁exp[iπ(λf₁)⁻¹(x² + y²) + iθ_j] and A₂exp[iπ(λf₂)⁻¹(x² + y²) + iBS], here 1/f_1,2 = 1/(f_e + d) – 1/f_d1,2. Finally, in the charge-coupled device (CCD) plane at a distance z_h from the SLM, the complex amplitude modulated by the lens with focal length f_d1 becomes:

(3)$${U_{fd1}}(x,y) = {A_1}\textrm{exp} \left[ {\frac{{i\pi ({x^2} + {y^2})}}{{\lambda ({f_1} + {z_h})}} + i{\theta_j}} \right]$$

The complex amplitude modulated by the spiral lens with focal length f_d2 is:

(4)$${U_{fd2}}(x,y) = {A_2}\textrm{exp} \left[ {\frac{{i\pi ({x^2} + {y^2})}}{{\lambda ({f_2} + {z_h})}} + iBS} \right]$$

Fig. 1. Schematic of the spiral-FINCH system by employing Bessel-like spiral phase modulation. Image1 and Image2 are the images of the objects converged by lenses with the focal lengths f_d1 and f_d2, respectively. L: collimating lens; SLM: spatial light modulator; CCD: charge-coupled device.

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The intensity distribution of holograms recorded by the CCD is:

(5)$$\begin{aligned} {I_P}(x,y) &= {|{{U_{fd1}}(x,y) + {U_{fd2}}(x,y)} |^2}\\ &= {C_1} + {C_2}\textrm{exp} [\frac{{i\pi }}{{\lambda {z_r}}}({x^2} + {y^2}) + i{\theta _j} - iBS] + c.c. \end{aligned}$$

where C₁ and C₂ are complex functions, c.c. is the complex conjugate of the second term on the right, and z_r is in the form of

(6)$$\frac{1}{{{z_r}}} = \frac{1}{{{f_1} + {z_h}}} - \frac{1}{{{z_h}}}$$

Considering the FINCH system is shift-invariant, the point source hologram (PSH) of any point source located at (x_s, y_s, -z_s) can be expressed as:

(7)$${I_{PSH}}(x,y) = {C_1} + {C_2}\textrm{exp} \left\{ {\frac{{i\pi }}{{\lambda {z_r}}}[{{{(x - {M_T}{x_s})}^2} + {{(y - {M_T}{y_s})}^2}} ]+ i{\theta_j} - iBS} \right\} + c.c.$$

where M_T is the lateral magnification of the system.

(8)$${M_T} = \frac{{{z_h}{f_e}}}{{{z_s}({f_e} + d)}}$$

To remove the zero-order and twin images, three holograms with the phase of θ₁ = 0°, θ₂ = 120°, and θ₃ = 240° are captured, respectively.

(9)$$\begin{aligned} {I_{F1}}({x,y} )&= {I_{PSH1}}(x,y)[\textrm{exp} ({\pm} i{\theta _3}) - \textrm{exp} ({\pm} i{\theta _2})]\\ &+ {I_{PSH2}}(x,y)[\textrm{exp} ({\pm} i{\theta _1}) - \textrm{exp} ({\pm} i{\theta _3})]\\ &+ {I_{PSH3}}(x,y)[\textrm{exp} ({\pm} i{\theta _2}) - \textrm{exp} ({\pm} i{\theta _1})]\\ &= C\textrm{exp} \left\{ {\frac{{i\pi }}{{\lambda {z_r}}}[{{{(x - {M_T}{x_s})}^2} + {{(y - {M_T}{y_s})}^2}} ]- iBS} \right\} \end{aligned}$$

Here, C is a complex function. The point spread function (PSF) of the spiral-FINCH system by employing Bessel-like spiral phase modulation can be reconstructed from I_F₁(x, y) by Fresnel propagation as:

(10)$${I_{psf1}}({x,y} )= {I_{F1}}(x,y) \ast \textrm{exp} [\frac{{i\pi }}{{\lambda {z_r}}}({x^2} + {y^2})]$$

For a general 3D object, the reconstructed image is the convolution of overall object g(x, y, z) and the PSF as follows:

(11)$${S_1}({x,y,z} )= g(x,y,z)\ast {I_{psf1}}(x,y)$$

The analysis above presents the algorithmic model of isotropic edge enhancement. To implement the anisotropic edge enhancement, the symmetry of the helical phase in the algorithmic model of isotropic edge enhancement was tried to break. The constant phase factor α₀ was introduced to the Eq. (9) to achieve anisotropic edge enhancement imaging with a selectable direction.

(12)$${I_{F2}}({x,y} )= C\textrm{exp} \left\{ {\frac{{i\pi }}{{\lambda {z_r}}}[{{{(x - {M_T}{x_s})}^2} + {{(y - {M_T}{y_s})}^2}} ]- iBS - i{\alpha_0}} \right\}$$

Similar to the Eq. (10), we simulate Fresnel propagation in the computer.

(13)$${I_{psf2}}({x,y} )= {I_{F2}}(x,y) \ast \textrm{exp} [\frac{{i\pi }}{{\lambda {z_r}}}({x^2} + {y^2})]$$

To break the symmetry of the spiral phase, take the real or imaginary part of I_psf₂ and then take the intensity.

(14)$${I_{PSF2}}({x,y} )= {|{\textrm{Re} ({I_{psf2}})} |^2} = {|{abs({I_{psf2}})\cos (\varphi + {\alpha_0})} |^2}$$

(15)$${I_{PSF3}}({x,y} )= {|{{\mathop{\rm Im}\nolimits} ({I_{psf2}})} |^2} = {|{abs({I_{psf2}})\sin (\varphi + {\alpha_0})} |^2}$$

Here, φ is the phase variation induced by the propagation of the vortex beam. By analyzing the above equations, we can find the α₀ in the cosine function of the real part and the sine function of the imaginary part. This asymmetry will cause the 3D vortex imaging to no longer achieve isotropic edge enhancement, instead the anisotropic edge enhancement with controllable direction. From the perspective of similarity and complementarity, the angle difference between the real and imaginary parts is 90 degrees, which also induces the direction of final imaging to differ by 90 degrees. Taking the strength value as the sum of the squares of the real and imaginary parts, isotropic edge enhancement is the superposition of the two directions.

For a general 3D object, the reconstructed image of the anisotropic edge enhancement for the 3D vortex imaging can also be referred to by the Eq. (11). By controlling the constant phase factor α₀, we can selectively enhance the edges of the objects in any desired direction.

3. Experimental results

Figure 2 shows the experimental setup of the spiral-FINCH system by employing Bessel-like spiral phase modulation. A solid-state plasma lamp (Thorlabs HPLS-30-04, 230 W) is used as the incoherent light source. BF is a bandpass filter with a peak wavelength of 633 nm and a bandwidth of 20 nm. To focus and restrain the beam emitted from the incoherent light source, a convex lens L₁ with a focal length of 125 mm is positioned after it. The SLM is phase only (Holoeye Pluto, 1920 × 1080 pixels, and 8 µm pixel pitch). BS₁ and BS₂ are beam splitter cubes. The distance between the collimating lens L₂ (with a focal length of 250 mm) and the SLM is d = 140 mm. P is a polarizer whose polarization is adjusted to the direction specified for the SLM. The two focal lengths on the SLM are f_d1 = 245 mm and f_d2 = 255 mm, respectively. To facilitate subsequent processing, the CCD (QIMAGING digital camera RETIGA 6000, pixel size 4.54 µm, 2750 × 2200 pixels) uses only 2048 × 2048 pixels.

Fig. 2. Experimental setup of the spiral-FINCH system by employing Bessel-like spiral phase modulation. SLM: spatial light modulator; CCD: charge-coupled device; BS₁ and BS₂: beam splitter; P: polarizer; L₁ and L₂: collimating lens; BF: bandpass filter.

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3.1 2D isotropic edge enhancement

Based on the experimental setup in Fig. 2, a small circular aperture with a diameter of 25 µm is selected as an ideal point source and imaged by a 20 × 0.4 NA microscope objective with a working distance of 5.9 mm. The recorded PSHs for the conventional spiral-FINCH and the Bessel-like spiral phase modulated FINCH with a phase shift of 0°, 120° and 240° are shown in Fig. 3(a)-(c) and 3(e)-(g), respectively. Figure 3(d) and 3(h) show the corresponding PSFs. It can be found that strong superfluous noise exits in the PSHs and PSF of the conventional spiral-FINCH in Figs. 3(a)-(d). In Figs. 3(e)-(h), the Bessel-like spiral phase modulated FINCH makes the energy concentrated within the main lobe, and the sidelobes almost completely disappear. The noise suppression and imaging quality improvement are mainly attributed to the Bessel-like spiral filter, which can effectively suppress the sidelobes in the generated optical vortices since the diffraction light originates from the central and the outer ring areas of the phase mask can be efficiently eliminated [32,33]. This sidelobe suppression process will cause more energy to be filtered out, resulting in the brightness and contrast of images by Bessel-like spiral phase modification worse than images by conventional spiral phase modification. However, we aim to suppress the sidelobes generated by the vortex, which could improve the signal-to-noise ratio of the reconstructed images. Figure 3(i) and 3(j) show the normalized intensity across the center of Fig. 3(d) and 3(h). Through the comparison, we can find that effective sidelobes suppression is also demonstrated in the PSF. The above results verified the validity and feasibility of the Bessel-like spiral phase modulation technique combined with the FINCH in suppressing the diffraction noise and improving the imaging contrast.

Fig. 3. PSHs and PSFs for different spiral phase modifications. (a)-(c) recorded PSHs for the conventional spiral phase modification; (e)-(g) recorded PSH for the Bessel-like spiral phase modification; (d) and (h) PSFs from the corresponding recorded data; (i) and (j) normalized intensity profiles in lateral direction of (d) and (h), respectively. The scale bar is 20 µm.

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To compare the imaging characteristics of the conventional spiral-FINCH and the Bessel-like spiral phase modulated FINCH, we carried out two comparative experiments. The experimental results of the USAF 1951 resolution target are shown in Fig. 4. Figures 4(a)-4(c) show the reconstructed images of the dual-lens FINCH, conventional spiral-FINCH, and spiral-FINCH by employing Bessel-like spiral phase modification, respectively. As can be seen from the partially enlarged images of the red box area (Figs. 4(a)-4(c)) in Figs. 4(d)-4(f), the conventional spiral-FINCH and the Bessel-like spiral phase modulated FINCH all show obvious edge contrast enhancement. But different from the conventional spiral-FINCH, the Bessel-like spiral phase modulated FINCH shows lower background noise and higher image contrast. To obtain quantitative comparisons among these images, the normalized intensity profiles along the red lines in Figs. 4(d)-(f) are plotted and shown in Figs. 4(g)-(i). It can be found that the edges enhanced by the Bessel-like spiral phase modification are sharper than those that use the conventional spiral phase modification. Meanwhile, the relative intensity of the former is much higher than the latter.

Fig. 4. Experimental results of the USAF 1951 resolution targets. (a)-(c) reconstructed images of the dual-lens FINCH, conventional spiral-FINCH, and the spiral-FINCH by employing Bessel-like spiral phase modification, respectively; (d)-(f) partially enlarged images of the red box area in (a)-(c), respectively; (g)-(i) normalized intensity profiles of the identical area from (d)-(f) depicted by the red lines, respectively.

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To further investigate the superiority of the Bessel-like spiral phase modulated FINCH in the imaging resolution, the images from the yellow box area in Fig. 4(b) and 4(c) are intercepted and enlarged, presented in Figs. 5(a) and 5(b). As can be seen, the resolution of the conventional spiral-FINCH and the Bessel-like spiral phase modulated FINCH reach 40.3 lp/mm (Element 3 of Group 5) and 45.25 lp/mm (Element 4 of Group 5), respectively, indicating a distinct promotion for the Bessel-like spiral phase modulated FINCH. Since the visibility is widely used to characterize the imaging system’s resolution, we also calculated and compared each unit of group 5 for these images. Visibility refers to the degree to which the human eye or detection equipment can observe an object under specific conditions, which is defined as (I_max - I_min) / (I_max + I_min) [24]. As shown in Fig. 5(c), the visibility of the Bessel-like spiral phase modulated FINCH is increased for both the horizontal and the vertical elements compared with the conventional spiral-FINCH.

Fig. 5. (a) and (b) are partially enlarged images of the yellow box area in Fig. 4(b) and 4(c), respectively; (c) the visibility of each unit of group 5 for the images in (a) and (b).

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Then, the contrast experiments are conducted on the Drosophila melanogaster. Figure 6(a) shows the output image of the Drosophila melanogaster by direct optical imaging. Figures 6(b)-(d) show the reconstructed images by the dual-lens FINCH, conventional spiral-FINCH, and spiral-FINCH by employing Bessel-like spiral phase modification, respectively. Compared to the result of direct optical imaging, it is known that the FINCH has the advantage of higher lateral resolution [34,35]. As can be seen in Fig. 6(c), the edges of the D. melanogaster can be distinguished, but it is difficult to observe the detailed information of this sample, and there exists a large number of background noises. The result in Fig. 6(d) showed that the Bessel-like spiral phase modification has significant advantages in improving imaging quality. Not only the edges of the D. melanogaster are sharp and clearly distinguished, but also the noise is significantly reduced. Compared to the results shown in Fig. 6(a) and Fig. 6(b), we can obtain a clear edge profile of the D. melanogaster, as the edge profile of its leg (marked with a red ellipse) and wing (marked with a blue ellipse) of these small parts can be distinguished more clearly. Moreover, the result in Fig. 6(d) also shows a higher signal-to-noise ratio. All the results above demonstrated the superiority of the Bessel-like spiral phase modulated FINCH in enhancing the contrast, reducing the noise, and improving the resolution of the edge-enhanced images.

Fig. 6. Experimental results of the Drosophila melanogaster. (a) direct optical imaging; (b) dual-lens FINCH; (c) conventional spiral-FINCH; (d) spiral-FINCH by employing Bessel-like spiral phase modification. The red and blue ellipses are markers of some parts of the D. melanogaster for better comparison. The scale bar is 1 mm.

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3.2 3D isotropic edge enhancement

To further demonstrate the edge enhancement in 3D vortex imaging, experiments of the two USAF1951 resolution targets, which are non-coplanar-placed with a spacing of 10 mm, are carried out. Figure 7(a) and 7(b) show the reconstructed images at the best focal plane of the front resolution target and the rear resolution target, respectively. It can be found that the Bessel-like spiral phase modulated FINCH not only can obtain high contrast and low noise edge enhanced images but also has the ability to achieve 3D vortex imaging in which any plane of the 3D objects can be focused digitally by digital reconstruction. Meanwhile, we also carried out the experiment of 3D edge imaging for Drosophila melanogaster. The experimental results are presented in Fig. 8. The longitudinal distance from the leg to the bristle is about 1 mm. Figures 8(a) and 8(b) show the reconstructed images at the best focal plane of the leg and the bristle of the D. melanogaster, respectively. Figures 8(c)-(d) and 8(e)-(f) are the magnified parts in blue rectangular boxes and red rectangular boxes shown in Figs. 8(a)-(b), respectively. It can be seen that the edge features of the objects can be clearly distinguished. These results also demonstrate the superiority of the Bessel-like spiral phase modulated FINCH in 3D vortex imaging with edge detection. In this article, we implement the experiment of edge enhancement in 3D vortex imaging as the angular quantum number l = 1. However, as the angular quantum number increases, the optical path difference of this system will exceed the coherent length of the incoherent light source, which will seriously deteriorate the imaging quality.

Fig. 7. Experimental results of two USAF1951 resolution targets, which are non-coplanar-placed with a spacing of 10 mm. Spiral reconstruction with (a) the front resolution target (blue dashed box) in focus and (b) the rear resolution target (red dashed box) in focus.

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Fig. 8. Experimental results of the Drosophila melanogaster. Spiral reconstruction with (a) the leg and (b) the bristle in focus; Note that the longitudinal distance from the leg to the bristle is about 1 mm. (c) and (d) partially enlarged images of the blue box area in (a) and (b), respectively; (e) and (f) partially enlarged images of the red box area in (a) and (b), respectively. The scale bar is 0.5 mm.

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3.3 2D anisotropic edge enhancement

To implement anisotropic edge enhancement, a circular aperture with a diameter of 300 µm is tested. As the spiral phase plate in the vortex imaging is sensitive to the wavelengths, we also designed phase masks whose focal lengths don’t change with wavelengths [36] and investigated its tolerance to the vortex imaging with different light illumination. Figure 9(a), 9(f), and 9(k) show the isotropic Bessel-like spiral phase modulated FINCH under illumination with wavelengths of 632.8 nm, 532 nm, and 405 nm, respectively. By changing the constant phase factor of the spiral phase, orientation-selective edge-enhanced images with different wavelengths of 632.8 nm, 532 nm, and 405 nm can be obtained and shown in Figs. 9(b)-(e), 9(g)-(j) and 9(l)-(o), respectively. To better illustrate the edge enhancement with selective orientation, we also potted the intensity profile along the arrowed line on the lower-right corners of each figure of Figs. 9(b)-(e), Figs. 9(g)-(j), and Figs. 9(l)-(o), respectively. These results proved the proposed method not only can achieve selective edge enhancement of 3D vortex imaging effectively but also demonstrates its robust characteristics with different light illumination.

Fig. 9. Experimental results of the circular aperture under visible light illumination with different wavelengths. (a), (f) and (k) isotropic spiral-FINCH under illumination with the wavelengths of λ₁ = 632.8 nm, λ₂ = 532 nm, and λ₃ = 405 nm, respectively; (b)-(e) anisotropic spiral-FINCH under illumination with the wavelengths of λ₁ = 632.8 nm and the constant phase factors of α = 0, α = π/2, α = π/4, and α = 3π/4, respectively; (g)-(j) anisotropic spiral-FINCH under illumination with the wavelengths of λ₂ = 532 nm and the constant phase factors of α = 0, α = π/2, α = π/4, and α = 3π/4, respectively; (l)-(o) anisotropic spiral-FINCH under illumination with the wavelengths of λ₁= 405 nm and the constant phase factors of α = 0, α = π/2, α = π/4, and α = 3π/4, respectively. On the lower-right corners of each figure of (b)-(e), (g)-(j), and (l)-(o), the intensity profile along the arrowed line is illustrated, respectively. The scale bar is 150 µm.

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3.4 3D anisotropic edge enhancement

We also performed the 3D anisotropic (orientation-selective) edge imaging for two non-coplanar placed circular apertures, whose longitudinal distance is 0.5 mm. Figures 10(a) and 10(b) show the reconstructed images at the best focal plane of the front and the rear circular apertures, respectively. These results demonstrate the feasibility of this system in 3D anisotropic edge imaging for a 3D object.

Fig. 10. Experimental results of two non-coplanar placed circular apertures with the constant phase factor of α = 0. Note that the longitudinal distance between the two apertures is 0.5 mm. Spiral reconstruction with (a) the front aperture (marked with a blue dashed box) and (b) the rear aperture (marked with a red dashed box) in focus. The scale bar is 3 mm.

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4. Conclusion

In conclusion, we have proposed the Bessel-like spiral phase modulated FINCH to realize the isotropic and anisotropic edge enhancement in 3D vortex imaging. In the isotropic edge enhancement in 3D vortex imaging, we found that the Bessel-like spiral phase modulated FINCH is effective in suppressing the background noise, improving the resolution, and enhancing the contrast of the edge-enhanced images compared with the conventional spiral-FINCH. The comparative experimental results of the small circular aperture, standard resolution target, and Drosophila melanogaster can support this conclusion. Moreover, by breaking the symmetry of the spiral phase in the algorithmic model, orientation-selective edge enhancement can be easily realized by controlling the constant phase factor of the spiral phase. We also successfully implemented this method for the anisotropic edge enhancement of a circular aperture. The results of this work could provide a promising route to detect objects with high resolution and contrast and will have great application potential in fields such as fingerprint recognition, biological imaging, and astronomical observation.

Funding

National Natural Science Foundation of China (11904323); Research Funds of Zhengzhou University (32213581, 32340305, 32410543).

Disclosures

The authors declare no conflicts of interest.

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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Edge enhancement in three-dimensional vortex imaging based on FINCH by Bessel-like spiral phase modulation

Abstract

1. Introduction

2. System analysis

3. Experimental results

3.1 2D isotropic edge enhancement

3.2 3D isotropic edge enhancement

3.3 2D anisotropic edge enhancement

3.4 3D anisotropic edge enhancement

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Equations (15)

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