P HYSICAL J OURNAL D
Regular Article
Image-contrast enhancement of wavefront coding systems by combining radially symmetrical phase and amplitude masks
Vannhu Le1,2,a, Xuanhoi Hoang2, Minhnghia Pham2, and Hoanghai Le2
1 Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam
2 Le Quy Don Technical University, Hanoi, Vietnam
Received 11 February 2019 / Received in final form 3 July 2019 Published online 27 August 2019
c EDP Sciences / Societ`a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2019
Abstract. The quartic phase mask is a radially symmetrical phase mask which is commonly used to analyze imaging performance and to apply in many practical applications. In this paper, we propose a simple, novel method to improve the image contrast for extending depth of field in wavefront coding system with radially symmetrical quartic phase mask. An amplitude mask, which improves image contrast in low spatial frequency region, is added on the wavefront coding system with quartic phase mask to obtain the remarkable improvement of imaging performance for extending depth of field. Some evaluation methods are used to show the effectiveness of imaging performance of the proposed method in comparison to previously phase mask. The results demonstrated that the proposed method yields better imaging property in extending depth of field.
1 Introduction
For traditional imaging systems, the sharpness of the images depends on the effect of the defocus which is lim- ited by the lens aperture. However, more plentiful object information can be obtained by using a large depth-of- field imaging system. The images of the object which are captured by an imaging system with the bigger depth of field are more similar with a real scene. In addition, there are many practical applications require a wide range of defocus for the acquisition of three-dimensional objec- tive information such as biomedical imaging [1], vision- based applications [2], the reductions of aberrations [3], and so on. Recently, to extending depth of field by using a phase mask placed in the pupil plane has been received much attention of researchers in the world. When a phase mask (PM) is placed in the pupil plane of an optical system, a long focusing line is acquired and the imag- ing characteristic is invariant to defocus. Some types of the PMs to increase the depth of field have been intro- duced. They can be separated into two kinds including:
the radially symmetrical PMs and the asymmetrical PMs [4]. Some types of asymmetrical PMs are suggested to extend the depth of field, such as the cubic phase mask [5], the logarithmic phase mask [6], the root square phase mask [7], the exponential phase mask [8], the sinusoidal phase mask [9], the tangent phase mask [10], the polyno- mial phase mask [11]. When the asymmetrical PMs are used to place in the pupil plane of the imaging systems,
ae-mail:levanhu [email protected]
the recorded images are blurred. The digital processing should be employed to restore the final high-quality image.
However, the image artifacts and SNR are both intrin- sic problems in wavefront coding system with the asym- metrical PMs [12–14]. Some kinds of the radially sym- metrical PMs to enhance the depth of field have been introduced, such as quartic phase mask (QPM) [15], loga- rithmic axicon [16], diffraction hybrid lens [17], and loga- rithmic asphere [18]. The radially symmetrical PMs have symmetrical and sharpness point spread function (PSF).
When the radially symmetrical PMs are used to place the pupil plane of the imaging systems, the sharpness image can directly be acquired, while the digital pro- cessing should not be used? Additionally, there are not image artifacts over images of the radially symmetrical PMs.
Among all the kinds of the radially symmetrical PMs, the QPM is the most phase mask which is used in prac- tical applications. The phase function of the QPM is pre- sented by,
f(x, y) =a(x2+y2)2+b(x2+y2). (1) When the QPM is placed in the pupil plane of the imaging system, the defocused MTFs of the imaging sys- tem are more invariant compared that of the conventional imaging system. However, the defocused MTFs of the QPM are the cut-off spatial frequency in the low spatial frequency region. Moreover, the defocused MTFs of the QPM are much lower in comparison with the diffraction- limited in-focus MTF of the conventional imaging system.
Therefore, the image contrast is not high. In this paper, we introduce a new method to improve the higher defo- cused MTFs of the QPM in the low spatial frequency region. Therefore, the image contrast is increased in this region. When the amplitude mask (Q(x,y) = 1−x2−y2) is added in the pupil plane, the image contrast in the low spatial frequency region is improved in comparison with the conventional imaging system [19]. Since the MTF of the amplitude mask in low spatial frequency region is higher than the diffraction-limited in-focus MTF. In here, we show that when the amplitude mask is used in the imaging system with the QPM, the defocused MTFs of the imaging system in the low spatial frequency region is improved. This means that the image contrast is enhanced.
The paper is organized as follows. Section 2 presents optimization and analysis of the QPM. The simulation and comparison results between the combination mask and the QPM are shown in Section3. Finally, the conclusions are presented in Section4.
2 Optimization and analysis of amplitude and quartic phase mask pair
The recorded image of imaging system can be pre- sented by,
g(x0, y0) =o(x0, y0)∗h(x0, y0) +n(x0, y0) (2) whereois the object,his the point spread function (PSF), nis the noise and∗denotes convolution operation;x0and y0 are coordinates in the imaging plane.
The PSF of the imaging system is expressed by the square modulus of fast Fourier transform, FFT, of the complex amplitude-phase pupil function, P(x, y), where xandy are coordinates in the pupil plane,
h(x0, y0) =|FFT [P(x, y)]|2 (3) where
P(x, y) = (√1
2Q(x, y)exp{i[f(x, y) +ψ(x2+y2)]} ifx2+y2≤1
0 other
(4) and
ψ= πL2 4λ
1 f −1
d− 1 d0
(5) whereLis the pupil plane dimension;λis the wave length of light;f,d, d0 are the focal length, the object distance, and the image distance, respectively.
The OTF,H,is equal to Fourier transform of PSF, and therefore,H can be given by,
H(fx, fy) = FFT [h(x0, y0)] (6) wherefxandfy are the normalized spatial frequency.
The first, we find amplitude mask which can produce the higher image contrast in the low spatial frequency region at the focal plane. According to reference [19], the
Fig. 1. The MTFs of traditional imaging system and the amplitude mask.
amplitude mask as shown in equation (7) can produce the value of MTF at the low spatial frequency region higher than the value of the diffraction-limited in-focus MTF of traditional imaging system. The MTFs of the ampli- tude mask and traditional imaging system are shown in Figure 1. In Figure 1, the cut-off spatial frequency for clear pupil has been normalized to one. As Figure1indi- cates, it is not difficult to see that the amplitude mask is better in the low spatial frequency region in comparison with the diffraction-limited MTF of traditional imaging system.
Q(x, y) = 1−x2−y2. (7) To realize imaging property comparison between phase masks, the phase mask parameters should be optimized because the imaging performance of the phase masks can be degraded with the mask parameters selected randomly.
Several optimized methods for selection of mask parame- ters have been suggested, such as the fisher information, mean square error (MSE) of the MTF [12], or the PSF [20], PTF [21,22], and OTF [23]. These methods should satisfy two conditions: (1) the imaging properties of the imaging systems with PMs should be invariant over a wide range of defocus and (2) the minimum acceptable magni- tude of MTF makes sure that the recorded images can be restored. Since the QPM is the radially symmetrical phase mask, the PTF has no effect to the imaging quality and, while, the MTF has the effect to the imaging qual- ity. In order to obtain the invariant performance over a wide range of defocus, the MTF should be invariant over a wide range of defocus. In this paper, we adopted MSE of the defocused MTFs with the in-focus MTF to opti- mize the phase mask parameters (aandb) of the PQM. If the MSE is equal to zero, all defocused MTFs within the designed range of defocus are the same, indicating that the MTF is absolutely insensitive to focus error. In order to obtain the acceptable signal noise ratio, a threshold, TH, is used to determine the minimum acceptable value of the MTF of the QPM. The optimized function based on MSE of the defocused MTFs with the in-focus MTF
Fig. 2.The defocused PSFs of traditional imaging system in top and the QPM in bottom at different defocus values of (a, a1) ψ= 0, (b, b1)ψ= 5, (c, c1)ψ= 10 and (d, d1)ψ= 15.
Fig. 3.The defocused MTFs of (a) traditional imaging system and (b) the QPM at different defocus values ofψ= 0, ψ= 5, ψ= 10 andψ= 15.
can be presented by,
mina,b
ψ=ψmax
P
ψ=0 fx=1
P
fx=0
|MTF(fx, ψ)−MTF(fx, ψ= 0)|2
subject : ψ1
max
ψmax
R
0 1
R
0
MTF(fx, ψ)dfxdψ >TH
(8) where ψis the defocus parameter,ψmax is the maximum defocus value,fxis the normalized spatial frequency.
To perform optimization of phase mask parameters of the QPM, the maximum defocus value ψmax is set to 15 and the minimum acceptable average magnitude of MTF is set to TH= 0.242. Based on the optimized function shown in equation (8), the optimal mask parameters for the QPM are equal toa= 16.95 and b=−15.1.
There are some different quality criteria were introduced to analyze imaging property of an optical imaging system.
Among them, the PSF is commonly used to that purpose.
The defocused PSFs of the QPM at different defocus val- ues are depicted in Figure2, where defocus value is equal to ψ= 0, ψ= 5, ψ= 10 and ψ= 15. For comparing with a traditional imaging system with a clear aperture, the corresponding defocused PSFs are also shown in Figure2.
For traditional imaging system, the size and shape of defo- cused PSFs rapidly are with the change of defocus. For wavefront coding system with QPM, the size and shape of defocused PSFs have the low variation to defocus. This means that imaging system with the QPM can produce a more stable defocus imaging property, i.e., the QPM can produce a better imaging performance in extending the depth-of-field imaging.
With above phase mask parameters of QPM, the defocused MTFs of the QPM are shown in Figure3b. For comparison, the corresponding defocused MTFs of tradi- tional imaging system are also depicted in Figure 3a. As Figure3a shows, it is not difficult to see that the defocused MTFs of traditional imaging system are sensitive to defo- cus, while the defocused MTFs of the QPM are invariant
Fig. 4. The defocused MTFs of the QPM and the combination mask at different defocus values ofψ= 0,ψ= 5,ψ= 10 and ψ= 15.
Fig. 5. MSEs of the QPM and the combination mask.
to defocus. As Figure3b indicates, the values of defocused MTFs of the QPM in the low spatial frequency region reduce from one to zeros, while the values of defocused MTFs of the high spatial frequency region are set near to
zeros. So the values of defocused MTFs of the QPM in the low spatial frequency region are used to obtain image. As can be seen from Figures1and3b, it can be seen that the values of defocused MTFs of the QPM are bigger zeros that lie in the low spatial frequency region, in this region the amplitude mask has the value of MTF higher than the value of the diffraction-limited in-focus MTF of tra- ditional imaging system. This means that this amplitude mask can be used to improve the higher MTF in the low spatial frequency region. We suggest the use of the ampli- tude mask added the imaging system with the QPM for improving the higher defocused MTFs in the low spatial frequency region, so the image contrast is improved. The combination between the amplitude mask and the QPM is called the combination mask.
3 Imaging enhancement by combining phase and amplitude masks
The defocused MTFs of the QPM and the combination mask at different defocus values are shown in Figure 4, where defocus value is set to ψ= 0, 5, 10, and 15. As Figure 4 indicates, the combination mask is not sensi- tive to defocus in comparison with traditional imaging
Fig. 6. Simulation images with spokes target for the QPM in the left and the combination mask in the right at different defocus values of (a, a1) ψ= 0, (b, b1)ψ= 5, (c, c1) ψ= 10 and (d, d1)ψ= 15.
system. It is clear that the defocused MTFs of the com- bination mask in all the values of low spatial frequency region are higher than the defocused MTFs of the QPM.
This means that the proposed combination mask can be used to improve the image contrast in extending depth of field. This means that the combination mask can produce
Table 1. MSSIM values of the QPM and the combination mask.
ψ 0 5 10 15
The QPM 0.7605 0.7019 0.5936 0.4735
The combination mask 0.8792 0.8997 0.8635 0.7831
a better imaging performance in extending the depth-of- field imaging.
In order to show the remarkable effectiveness of the proposed method, we use MSE to evaluate the difference between the defocused MTFs and the diffraction-limited in-focus MTF of traditional imaging system. The smaller the MSE value, the better defocused MTFs will be. Based on the above optimal mask parameters, the curves of MSE of the QPM and the combination mask to defocus are shown in Figure 5. It is clear that the curve of MSE of the combination mask at all defocus values is lower.
This means that the proposed method has a significant enhancement of performance in extending the depth of field.
A more direct approach to evaluate imaging perfor- mance of optical systems is to simulate the recorded images. In this paper, we use spokes target to simulate the recorded images. The images of wavefront coding sys- tems with the QPM and the combination mask are shown in Figure6. As Figure6shows, it is clear that the images of wavefront coding system with the combination mask at all defocus values are shaper than one of wavefront coding system with the QPM. In other words, the image con- trast of the combination mask is improved. This means that the combination mask can be used to produce better image quality in extending depth of field.
According to reference [24], structural similarity (MSSIM) is a criterion which can be used to evaluate the difference between the original image and the distorted image. The MSSIM value has a range from 0 to 1; the big- ger the MSSIM value, the smaller the difference between the original image and the distorted image will be. Based on the images in Figure6, the MSSIM values of the QPM and combination mask are shown in Table 1. As Table 1 shows, it is not difficult to see that the MSSIM values of the combination mask are bigger than one of the QPM at all the defocus values ofψ= 0, 5, 10 and 15. Additionally, the MSSIM values of the combination mask to defocus reduce more slowly than one of the QPM. These mean that our method can be used to obtain excellent perfor- mance in extending the depth of field.
4 Conclusion
In this paper, we have presented a simple method based on the addition of an amplitude mask in optical system with a radially symmetrical phase mask to improve con- trast of image in extending depth of field. An amplitude mask which improves the image contrast in the low spa- tial frequency region has been used to add in the wavefront coding system with the QPM. The wavefront coding sys- tem with combination mask has the remarkable improve- ment for image contrast in extending depth of field. By
using evaluation approaches of the MTF, MSE, simulation images and MSSIM, indicating that the proposed method can be used to obtain better image quality in extending depth of field. Specially, this method can open application in light sheet microscopy to extend field of view.
This work is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number (103.03-2018.08, 102.01-2017.04).
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