https://doi.org/10.1140/epjp/s13360-020-00806-w R eg u l a r A r t i c l e
Extending the depth of field in hybrid imaging system by subtracting different asymmetrical phase masks
Vannhu Le1,2,a
1Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam 2Department of Optical Engineering, Le Quy Don Technical University, Hanoi, Vietnam Received: 8 July 2019 / Accepted: 24 September 2020
© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The subtraction method has been proposed to improve resolution and contrast in confocal scanning fluorescent microscopy. In this paper, we propose applying the subtraction method based on the use of rotating two different asymmetric phase masks along three directions to extend the depth of field. The combining optical transfer function is generated by using three subtraction optical transfer functions. The proposed method can produce high- quality image, in which the digital processing by a deconvolution filter absents. A square root phase mask is used to demonstrate the proposed concept. Compared with the traditional imaging system with the clear aperture, the subtraction method can be used to attain sharp and invariant image over a wide range of defocus.
1 Introduction
Wavefront coding can be used to extend the depth of field in incoherent optical systems.
Some successful applications of the wavefront coding have been introduced, such as optical aberration reduction [1–3], iris recognition [4,5], thermal imaging [6]. In the technique, the wavefront of the incident light is corrected by adding an asymmetrical phase mask (PM) in the pupil plane, so that the optical transfer function (OTF) or point spread function (PSF) is nearly invariant to defocus. These images sampled by the detector are a series of encoding images, which are intentionally blurred over a wide range of defocus but have a nearly uniform level of blur. These images can be subsequently decoded by using a simple deconvolution kernel, so that we can obtain the final high-quality images near to the diffraction-limited in-focus image of traditional imaging system.
In the wavefront-coding technique, the most important part depends on the design of suitable PMs to obtain the invariant imaging characteristics over a big range of defocus.
So far, kinds of asymmetrical PMs to extend the depth of field have been introduced, such as the cubic PM [7], the logarithmic PM [8], the sinusoidal PM [9], the tangent PM [10], the polynomial PM [11], the exponential PM [12], the square root PM [13,14], the high- order PM [15]. All these PMs can acquire the improvement of depth of field, but they would result in totally different-invariant imaging characteristics over a big range of defocus. In particular, the cubic PM is the most common PM and its accurate analytical solution in both spatial frequency and spatial domains has been successfully employed to reveal the favorable
ae-mail:[email protected](corresponding author)
Fig. 1Schematic diagram for the subtraction method using PMs (PM1 and PM2)
imaging characteristics. The logarithmic PM and the exponential PM are two popular PMs.
The tangent PM performs remarkably in the generation of defocus-invariant MTFs. Compared with other PMs, the square root PM has the best imaging performance in improving the depth of field.
According to Refs. 16-19, the subtraction method has been employed to enhance the resolution and contrast in confocal scanning fluorescence microscopy by subtracting two images obtained under different acquisition conditions. The subtraction method obtains the remarkable improvement of the higher modulation transfer function (MTF). Moreover, the advantage of the subtraction method is the simplification of laborious calculation. Several different imaging conditions have been introduced using subtraction method, such as two sizes of apertures [16,17], two polarizations [18], two signals [19], two detectors [20]. In this paper, the subtraction method is introduced in rotating two asymmetrical PMs with three directions to obtain high defocused MTFs over a wide range of defocus, resulting in clearer image in extending depth of field. In addition, the proposed method achieves the image with the reduction of the effect of the noise in comparison with single phase mask. Moreover, the proposed method of this manuscript is different in comparison with Ref. [21]. Ref. [21]
needs to use the inverse filter to restore high-quality image, while the proposed method does not need to use the inverse filer.
In this paper, we present the subtraction method based on the usage of rotating two different asymmetrical PMs with three directions to obtain the invariant and high-quality images over a wide range of defocus. Section2shows the imaging theory of the subtraction method of using two different asymmetrical PMs. Section3shows the simulation results and analysis.
Finally, the conclusions are presented in Sect.4.
2 Imaging theory of different asymmetrical phase masks
The two asymmetrical PMs have been placed in the optical system to obtain two images sequentially, as shown in Fig.1[21]. In the schematic diagram the two lenses combined with two different PMs (PM1 and PM2) face the same object, and the two encoded images are detected by the two separate detectors. These two encoded images are different in signal collection since two PMs introduce variable phase modulations. The high-quality image can be obtained by subtracting one image to other.
The recorded image of imaging system can be presented by
IPMh∗o+n (1)
wherehis the PSF,ois the object, * is the convolution operation,nis the noise.
The PSF of the imaging system with the generalized pupil function,P(x,y), can be described by
h |FFT[P(x,y)]|2, (2) The generalized pupil function,P(x,y), with a PM, ƒ(x,y), and a defocus parameter,ψ, can be represented as
P(x,y) √12exp
if(x,y)+iψ
x2+y2
,ifx2+y2≤1
0 other (3)
and
ψ πL2 4λ
1 f − 1
d − 1 d0
2π
λ W20, (4)
whereLis the pupil plane dimension, ƒ,d0,d, are the focal length, the image distance, and the object distance, respectively,λis the length of light wave.W20is the defocus coefficient associated with the wave aberration and is in unit of light wavelength.
Using two different PMs, we can obtain a high-quality image by subtracting the two images which can be presented by
I IPM1−δIPM2
(PSFPM1−δPSFPM2)∗o+n1−δn2
PSFsub∗o+n1−δn2 (5) whereδis the subtraction coefficient andδ> 0.
In Fourier domain, the image, as shown in Eq. (5), can be rewritten by
I¯iFFT−1(Hsub×O+N1−δN2), (6) whereH is the OTF,Ois the Fourier transform of objecto,N is the Fourier transform of noise,n.
The OTF,H, is equal to Fourier transform of PSF and, therefore, can be obtained by HsubFFT(PSFsub)HPM1−δHPM2 (7) As mentioned earlier, many asymmetrical PMs for the improvement of depth of field have been introduced. Among them, the square root PM can increase the depth of field relatively better, especially in acquiring defocus-invariant MTFs.13Therefore, in this paper, we choose the square root PM to demonstrate the proposed concept. According to Ref. 13, the phase function of the square root PM can be described by
fPM1(x,y)αx
β− β−x2
+αy
β− β−y2
, (8)
where bothyandxare the pupil plane coordinates in the vertical and horizontal direction, respectively, andαandβare the PM parameters to control magnitude of phase deviation, forβ≥1. According to these definitions,x,yandβhave no unit, butαis in units of radians because this parameter indicates the magnitude of phase deviation, suggested by phase mask through the relationα2πξ/λ, whereξis the corresponding optical path different (OPD).
Fig. 2Phase profiles ofaPM1 with parameters of (α42.15,β1) andbPM2 with parameters of (α 42.15,β1,η4.52)
As can be clearly seen from simulation and experimental results shown in Fig.2a, b and e, f in Ref. 22, both images of Gaussian and doughnut beams are shown, respectively. There is the displacement between two images, indicating that the image of Gaussian beam and the image of doughnut beam will not cover exactly the same area. The effectiveness of the subtraction method is better when the displacement between two images is proper. All the asymmetrical PMs generate the displacement of PSF along one direction, indicating that the encoded image and the original scene will not cover exactly the same area. This displacement of the PSF would relate the different defocus magnitude, different PM parameters, and the different PMs. Furthermore, the displacement of PSF of asymmetrical PMs is along a line y0 x0, wherey0andx0 are normalized coordinates of the imaging plane. When the PM parameter groups for most asymmetric PMs are chosen closely, the different displacement of the PSFs is small. In order to achieve the better result of the subtraction method, the two encoded images would require significant relatively displacement between two PSFs of two asymmetrical PMs. Therefore, in this paper, the second PMs are achieved by adding a linear function to the first PMs, which can control the relative displacement between two PSFs and can be presented by
fPM2(x,y)αx
β− β−x2
+αy
β− β−y2
+η(x+y), (9) whereηis the PM parameter which controls the displacement of the PSF along a liney0 x0.
3 Simulation results
The subtraction method can improve resolution and contrast in optical imaging system. It is crystal clear from Eq.5that the result of subtraction method is solely controlled by three parameters (both PSFs and value ofδ). In this paper, the square root PM is adopted to study the effectiveness of the subtraction method for depth of field extension. Primarily, two sets of PM parameters are chosen to generate two different square root PMs. Generally, in order to evaluate and analyze imaging performance of wavefront-coded imaging system using PMs, then the PM parameters should first be optimized, because the randomly selected PM
parameters do not reflect true imaging performance of it. Some optimization functions are used to generate the best PM parameters, such as the OTF, the MTF, and the PSF or the phase transfer function (PTF) [23,24]. An optimization function should satisfy two conditions: the imaging performance of the PMs should be invariant over one big range of defocus, and the magnitude of the MTF makes sure that the final image has the good quality. The mean square error (MSE) of the MTF is a best way to evaluate the level of defocused MTF changes over a wide range of defocus. This would lead to choose optimal PM parameters for PMs. The MSE optimization procedure based on the usage of the MTF to obtain optimal parameters can be presented by
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩ min
⎡
⎣ψmax
ψ0 u1
u−1
|MTFsubtraction(u, ψ)−MTFsubtraction - mean(u, ψ)|2
⎤
⎦
subject :
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎩ 1 ψmax
ψmax
ψ0 u1
u−1
MTFsubtraction(u, ψ)dudψ≥T H1
1 ψmax
ψmax
ψ0 u1
u−1
MTFP M1(u, ψ)dudψ≥T H2
(9)
where MTFsubtraction is the normalized subtraction MTF in the subtraction method and MTFsubtraction-mean is the mean value of all normalized subtraction defocused MTFs. The maximum value and normalized space frequency are represented byψmax andu, respec- tively.TH1is the magnitude of the MTF which is chosen to obtain the better image without using the digital processing.TH2is the magnification of the MTF which is chosen to obtain accepted signal-to-noise ratio.
In order to optimize the parameters of PM, we introduce the initial conditions of the subtraction method in extending depth of field as follows: the range of defocus is considered to be from 0 to 30,TH11.92 andTH20.48, whereTH1is near equal to the magnitude of the diffraction-limited MTF onuaxis andTH2 can obtain the high-quality image for the single PM due to the post processing. Based on the optimization procedure as shown in Eq. (9), the optimal parameters of the square root PMs and the subtraction parameter are obtained such that the first PM parameters areα42.15 andβ 1; the second PM parameters areα42.15,β1, andη4.52; the subtraction parameter isδ0.76. With the above sets of PM parameters, phase profiles of two square root PMs are shown in Fig.2.
Based on the above optimal parameters, the defocused MTFs of the square root PMs (PM1 and PM2) for some different defocus values along axis are depicted in Fig.3, where defocus value is set toψ0, 6, 12, 18, 24, and 30. The defocused MTFs of PM1 and PM2 are the same, but the defocused PTFs of PM1 and PM2 are different. It is obvious from Fig.3that the defocused MTFs of PM1 and PM2 are invariant to defocus compared to the defocused MTFs of traditional imaging system. In addition, it can be seen that as the defocus parameter increases, the defocused MTFs of PM1 and PM2 are decreased.
Next, we evaluate the stable level for the normalized defocused MTF of the subtraction method using two different square root PMs. For the PM parameters described above, Fig.4 shows the normalized defocused MTFs of the subtraction method for some different defocus values on axis, where defocus value is set toψ0, 6, 12, 18, 24, and 30. For comparison, the corresponding defocused MTFs of a traditional imaging system are shown in Fig.4. As Fig.4indicates that the defocused MTF of a traditional imaging system is very sensitive to
Fig. 3The defocused MTFs of aPM andbPM2. The defocus value is set toψ0, 6, 12, 18, 24, and 30
Fig. 4Defocused MTFs on axis for:atraditional imaging system,bthe subtraction method with two different square root PMs. The defocus value is set toψ0, 6, 12, 18, 24, and 30
defocus, the magnitude of the defocused MTF decreases rapidly with increasing defocus.
Additionally, the higher the defocus value, the smaller the frequency-cutoff value. Whereas the normalized defocused MTFs of the subtraction method are highly invariant to defocus, the frequency-cutoff value is always large enough. In addition, the normalized defocused MTFs of subtraction method depend on the defocused MTFs of original PMs. The subtraction method has almost the same ability to extend depth of field as the original PMs which is clear from Figs.3and4b. As well known to us, the ability to extend depth of field of asymmetrical PM can be ten times compared to traditional imaging system. However, from Figs.3and 4b, it is not difficult to see that the defocused MTFs of the original PM are lower than the diffraction-limited in-focus MTF of traditional imaging system, while the normalized defocused MTFs of the subtraction method are close to the diffraction-limited in-focus MTF of traditional imaging system.
Another reliable method to evaluate imaging performance of optical imaging system with a PM is to perform imaging with the spokes target. The images of the subtraction method and a tradition imaging system for some different defocus values are given in Fig.5, where defocus value is equal toψ0, 10, 20, and 30. For a traditional imaging system, the edge of the spokes becomes more blurred as defocus value increases; the information of image with large defocus value at the certain high spatial frequency is lost. It is obvious that the image
Fig. 5Simulation images for left:
a traditional imaging system, right: the subtraction method with two square root PMs.
Defocus value is set toa,a1ψ 0,b,b1ψ10,c,c1ψ20,d, d1ψ30
of the subtraction method is more invariant to defocus; the edge of some parts of the spokes is sharper; the contrast of the parts is higher. This means that the proposed method can be used to perform the effectiveness in extending depth of field.
Because the asymmetrical phase mask generates the point spread function which is not circular, the subtraction MTF is not circular. The subtraction MTF is shown in Fig.6b. It can be seen that the subtraction MTF is not circular. The subtraction MTF near center and along axis is high and is near to the diffraction-limited MTF as shown in Fig.6c, while other part is low. This leads the subtraction image which has some shaper parts (as parts 1 in Fig.6a) and has some blurred parts (as shown in parts 2 in Fig.6b). In order to deal with this problem, we use to rotate the phase mask. When the phase mask is rotated with an angle, the MTF of it also is rotated with the same angle. This will be presented in the following.
Assuming that the PM rotates with an angleϕ about the optical axis and obtains the coordinate in the source system [x,y]Tand the transformed coordinate in the target system is [xr,yr]T, the transformation formula is
xr
yr
cosϕ sinϕ
−sinϕcosϕ x
y
(10)
Fig. 6 aThe image of the subtraction method.bThe 2D MTF for subtraction method.cThe 2D MTF of traditional imaging system
Fig. 7Profiles with different rotating angles,aϕ0,bϕπ/6,cϕ −π/6
When the PM is rotated to the new the transformed coordinate in the target system is [xr, yTr], Eq. (8) can be presented by
fPM1(xr,yr)αxr
β− β−xr2
+αyr
β− β−yr2
, (11)
Profiles of phase mask of Eq. (11) with three angles ofϕ0,ϕπ/6,ϕ −π6 are shown in Fig.7. The subtraction MTFs of three phase masks are shown in Fig.8. It can be seen that the subtraction MTFs of the phase masks with angles ofϕπ/6,ϕ −π/6 also are rotated with the same angles as indicated in Fig.8b and c. In order to obtain high subtraction OTF, we combine three subtraction OTFs. The subtraction OTF can be presented by
Hsub
⎧⎨
⎩
Hsub,ϕ0i f abs(Hsub,ϕ0)≥ abs(Hsub,ϕπ/6) andabs(Hsub,ϕ−π/6) Hsub,ϕπ/6i f abs(Hsub,ϕπ/6)≥ abs(Hsub,ϕ0) andabs(Hsub,ϕ−π/6) Hsub,ϕ−π/6i f abs(Hsub,ϕ−π/6)≥ abs(Hsub,ϕπ/6) andabs(Hsub,ϕ0)
(12) By using Eq. (12), the high subtraction MTF is shown in Fig.9. As Figs.9and6c indicate, the subtraction MTF is near the diffraction-limited MTF. This means that the high-quality image can be obtained by proposed method.
Figure10shows the images of the subtraction method from combining three rotating directions and a tradition imaging system for some different defocus values. It can be seen
Fig. 82D MTFs with different rotating angles,aϕ0,bϕπ/6,cϕ −π/6 Fig. 9The 2D combination MTF
that the edge of the spokes is sharper; the contrast of image is higher. This means that the proposed method can be used to perform the effectiveness in extending depth of field.
In order to show the imaging effectiveness, we perform the comparison of the proposed method with traditional wavefront coding. In the traditional wavefront-coding method, the single phase mask is placed in the pupil plane of imaging system and the digital processing is used to obtain high-quality image. The simulated images of traditional wavefront-coding method are generated by the following two steps: the convolution of the original standard image with the PSF at different levels of defocus is used to generate the encoded images, and (2) all the encoded images for the different levels of defocus are deblurred by applying only one in-focus inverse filter kernel. The simulation results for comparing both methods are presented at the in-focus(ψ 0). In addition, white Gaussian noise is added to make the method more realistic. Figure11shows the images with the white Gaussian noise of the proposed method and traditional wavefront-coding method. Figure 11a and b shows the simulation results with the noise density of 0.005 for subtraction method and traditional wavefront-coding method, respectively. From the visual comparison of Fig.11, it is clear that the combination image of the subtraction method has better quality than that of traditional wavefront-coding method. This would guarantee that the proposed method can be used to further improve the imaging effectiveness in extending the depth of field.
4 Conclusion
In this paper, we have proposed the subtraction method based on the introduction of rotating two different asymmetrical PMs with three directions to get invariant and high-quality images over a wide range of the depth of field. The calculation of proposed method is simple. The
Fig. 10The
combination–subtraction images and traditional images. Defocus value is set toa,a1ψ0,b,b1 ψ10,c,c1ψ20,d,d1ψ 30
Fig. 11Simulation images at the in-focus position (ψ0) forathe subtraction methodbthe traditional wavefront-coding method
normalized defocused MTFs of the subtraction method have higher magnitude; therefore, the digital processing with a deconvolution filter is not required. Simulation images authen- ticated the ability of the subtraction method for wavefront-coding technique in depth of field expansion.
Acknowledgements This work is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number (103.03-2018.08).
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