Extending depth of field for hybrid imaging systems via the use of both dark and dot point spread functions
L. V. NHU,1,2,* ZHIGANG FAN,2 SHOUQIAN CHEN,2 AND FANYANG DANG2
1Department of Optical Engineering, Le Quy Don Technical University, 236 Hoang Quoc Viet Street, Hanoi, Vietnam
2Harbin Institute of Technology, Heilongjiang, Harbin 150001, China
*Corresponding author: [email protected]
Received 21 June 2016; revised 7 August 2016; accepted 13 August 2016; posted 15 August 2016 (Doc. ID 268883); published 8 September 2016
In this paper, we propose one method based on the use of both dark and dot point spread functions (PSFs) to extend depth of field in hybrid imaging systems. Two different phase modulations of two phase masks are used to generate both dark and dot PSFs. The quartic phase mask (QPM) is used to generate the dot PSF. A combined phase mask between the QPM and the angle for generating the dark PSF is investigated. The simulation images show that the proposed method can produce superior imaging performance of hybrid imaging systems in extending the depth of field. © 2016 Optical Society of America
OCIS codes:(110.1758) Computational imaging; (110.0110) Imaging systems.
http://dx.doi.org/10.1364/AO.55.007345
1. INTRODUCTION
Wavefront coding is a powerful technique that can be used to extend depth of field. By placing a phase mask in the pupil plane, we can reduce the impact of a defocus error on the optical transfer function (OTF) or the point spread function (PSF) [1]. Phase masks can be divided into two types: asym- metrical phase mask and radially symmetrical phase mask.
Many kinds of asymmetrical phase mask to increase depth of field have been introduced; some examples are the cubic phase mask [1], the tangent phase mask [2], the non-integer order phase mask [3], the exponential phase mask [4], the sinusoidal phase mask [5], the logarithmic phase mask [6], the polynomial phase mask [7], and the square root phase mask [8,9]. Some types of radial phase masks to enhance depth of field have been suggested; some examples are the quartic phase mask (QPM) [10], the logarithmic axicon [11], the diffraction hybrid lens [12], and the logarithmic asphere [13]. The asym- metrical phase masks extend the depth of field greater than the radially symmetrical phase masks. For asymmetrical phase masks, a simple deconvolution filter needs to be used to restore sharp images, and image artifacts are one intrinsic problem [14–16], while the radially symmetrical phase masks can indi- rectly lead to sharp images [17].
Recently, wavefront coding systems with a phase mask pair are used to improve imaging performance in extending the depth of field. One conjugate asymmetric phase mask pair produces the PSF to have invariant imaging characteristics of an asymmetrical phase mask and symmetric imaging
characteristics through an orthogonal axis pair of radially symmetric phase masks, as shown in Ref. [18]; however, this method has a small frequency-offcut value. In Ref. [19], image artifacts on the restored images can be removed by using two asymmetrical phase masks, but this method requires more time. In Ref. [20], a conjugate radially symmetrical phase mask pair is used to obtain a defocused modulation transfer function (MTF) that is more invariant over a range of defocus. However, in order to achieve the defocused MTF near to the diffraction limited MTF of a traditional imaging system, digital processing needs be used. In this paper, we propose one method based on the use of both dark and dot PSFs to obtain a defocused MTF that is more invariant over a large range of defocus and to have a high magnitude value of the defocused MTF.
The organization of this paper is as follows: in Section 2, the imaging theory of both dark and dot PSFs is presented:
Section 3shows the simulation results and analysis based on the use of PSFs, MTFs. and simulation images; finally, several conclusions are given in Section4.
2. THEORY
First, we build a layout of the imaging system with two phase masks, which can give two images. An imaging system with two phase masks gives two recoded images, as shown in Fig.1.
The recorded image of the hybrid imaging system with a phase mask can be expressed by
gh⊗on; (1)
1559-128X/16/267345-06 Journal © 2016 Optical Society of America
wherehis the PSF of the hybrid imaging system with a phase mask, ois the object, and ndenotes the noise.
The PSF of the imaging system is can be presented by
h jFFTPρj2; (2)
where FFT denotes the fast Fourier transform.
The phase pupil function with a phase mask and defocus error can be presented by
Pρ
p1ffiffi2 expfifρ ψρ2g if ρ≤1
0 other ; (3)
where ρ denotes a range from 0 to 1, and ψ is the defocus parameter and can be described by
ψ2π
λ W20; (4)
whereλis the light wavelength;W20is the defocus coefficient associated with wave aberrations.
As shown in Fig. 1, this imaging system includes two branches; the above branch of the imaging system produces the dot PSF, and the other branch produces the dark PSF.
The combination of images of the two recorded images given from the imaging system in Fig.1can be presented by
gcombination gdot−γgdark hdot−γhdark⊗o 1−γn;
(5) wheregdotis the image of the hybrid imaging system with the dot PSF,gdark is the image of the hybrid imaging system with the dark PSF, andγ is the ratio factor (γ>0).
There are some phase masks generating the dot PSF, such as the QPM, the logarithmic phase mask in Ref. [10], the circularly symmetric phase mask in Ref. [17], and so on. In this paper, we use the QPM to prove our concept. The phase function of the QPM can be presented by
fQPMρ aρ2−0.52; (6) wherea is the phase mask parameters controlling the magni- tude of the phase deviation.
In order to generate the dark PSF, in this paper, we propose a phase mask based on the combination of the QPM and an angle, and this phase mask can be presented by
fCombinedρ aρ2−0.52mφ; (7) where mis an integer, andφis a range from 0 to 2π.
3. SIMULATION RESULTS
Generally speaking, in order to analyze and evaluate the imag- ing performance of the phase mask, the phase mask parameters should be optimized first. In wavefront coding systems, the optimal parameter value of the phase mask should be satisfied under two conditions: (1) the imaging properties of the imag- ing systems with phase masks should be invariant over a wide range of defocus, and (2) the magnitude of the MTF makes sure that the final images have good quality. For the QPM, there are several optimal parameters that have been shown in several previous papers. For a range of the depth of field Fig. 1. Imaging system with two phase masks for leading two
recoded images. PM1 and PM2 are phase masks.
Fig. 2. Defocused PSFs of (top) a traditional imaging system with clear aperture and (bottom) an imaging system with the QPM. Left to right,ψ0,ψ3,ψ6,ψ9, andψ12.
[−12, 12], the optimized parameter in Ref. [10] will be used to perform simulations and evaluations. According to Ref. [10], the optimal parameter of the QPM is set to a1.5π2. With this optimal mask parameter, the defocused PSFs of the QPM are shown in Fig.2. For comparison, the correspond- ing defocused PSFs of a traditional imaging system with a clear aperture are also indicated in Fig. 2. As Fig. 2 shows, the defocused PSFs of a traditional imaging system with a clear aperture spread fast with an increase of defocus value, whereas the defocused PSFs of the QPM display stability over a wide range of defocus.
Next, we consider the defocused PSFs of the combined phase mask, the phase mask parameter of which is still equal to a1.5π2. The defocused PSFs of the combined phase mask for the three parameters of m1, 2, and 3 are shown in Fig.3. It can be seen that the defocused PSFs of the com- bined phase mask are more invariant to defocus compared to the defocused PSFs of a traditional imaging system with a clear aperture. As Fig.3shows, when parametermis increas- ing, the dark region in the center of the defocused PSFs of the combined phase mask is wider. For each parameterm, when the defocus value is increasing, the dark region in the center of the defocused PSFs of the combined phase mask has a bigger area.
Figure 4shows defocused MTFs of a traditional imaging system with a clear aperture and the QPM. It can be seen that the defocused MTFs of a traditional imaging system have a strong change to the defocus, and when the defocus value is increasing, the magnitude of the defocused MTFs reduces very fast. The magnitude of the defocused MTFs of the QPM is lower compared to the in-focus MTF of a traditional imaging system, but the defocused MTFs of the QPM have more stability for defocusing.
Figure5shows the defocused MTFs of both dark and dot PSFs for three differentmparameters andγ0.5. As Fig.6 indicates, the defocused MTFs of both dark and dot PSFs are more invariant to defocus compared to one of a traditional imaging system with a clear aperture. It can be clearly seen that the magnitude of the defocused MTFs of both dark and dot PSFs is a larger value than that of the QPM.
Fig. 3. Defocused PSFs of the imaging system with the combined phase mask. Top,m1; middle,m2; bottom,m3. Defocus parameter is arranged as in Fig.2.
Fig. 4. Defocused MTFs of (a) a traditional imaging system with a clear aperture, and (b) an imaging system with the QPM.
Figure6shows the defocused MTFs of both dark and dot PSFs for differentγvalues. It can be seen that the magnitude of the defocused MTFs of both dark and dot PSFs has stability over a wide range of defocus compared to a traditional imaging
system with a clear aperture. It is not difficult to see that when theγvalue is larger, the magnitude of the defocused MTFs of both dark and dot PSFs is increased. The MTFs of both dark and dot PSFs withγ0.2are more invariant to defocus, but
Fig. 5. Defocused MTFs of the imaging system with both dark and dot PSFs for three different m parameters: (a)m1, (b) m2, (c)m3, andγ0.5.
Fig. 6. Defocused MTFs of the imaging system with both dark and dot PSFs for m2 and different γ parameters: (a) γ0.2, (b)γ0.5, and (c)γ0.8.
the magnitude value of the defocused MTFs is smaller.
Whereas the magnitude of the defocused MTFs of both dark and dot PSFs withγ0.8has very high value, the defocused MTFs of both dark and dot PSFs withγ0.8have a stronger oscillation. We can use theγvalue to control the magnitude of the defocused MTF of both dark and dot PSFs.
One method to evaluate imaging performance of imaging systems is to consider imaging with the spokes. Figure7indi- cates the images of both dark and dot PSFs form2,γ0.8, and the Gaussian white noise of a signal-to-noise ratio (SNR) equal to 10. For comparison, the corresponding recorded images of a traditional imaging system with a clear aperture and the QPM are also shown in Fig. 7. For a traditional imaging system with a clear aperture, when the defocus value is increasing, the edge of the spokes is more blurred and at the defocus valueψ12, the phase shift at a high frequency part
appears. The edges of the spokes of both dark and dot PSFs and the QPM over a large range of defocus are sharper compared to a traditional imaging system with a clear aperture. However, it is straightforward to find that the images of both dark and dot PSFs are sharper over a large range of defocus. This means that the proposed method can be used to obtain remarkable improvement in imaging performance of wavefront coding sys- tems for extending the depth of field.
According to Ref. [21], structural similarity (MSSIM) is a criterion which can be used to assess the difference between the original image and the distorted image. The MSSIM value has a range from 0 to 1; the bigger the MSSIM value, the smaller difference between the original image and the distorted image. Based on the images in Fig.7, the MSSIM values of the QPM and both dot and dark PSFs are shown in Table1. As Table1shows, it is obvious that the MSSIM of both dot and
Fig. 7. Simulation images with the spokes for different imaging systems with aSNR10; top, a traditional imaging system with a clear aperture;
middle, the QPM; bottom, both dark and dot PSFs withγ0.8andm2. Left,ψ0; middle,ψ6; right,ψ12.
dark PSFs is bigger than one of the QPM for the defocus values ofψ 0, 6, and 12. This means that our method can be used to obtain excellent performance in extending the depth of field.
4. CONCLUSION
In this paper, we have proposed one method based on the use of both dark and dot PSFs to improve the imaging performance of the hybrid imaging system in extending the depth of field. The imaging performance of combining the QPM and the angle has been analyzed and evaluated. The magnitude value of the de- focused MTFs of both dark and dot PSFs has a high value. The factor ratioγcan be used to control the magnitude value of the defocused MTFs of both dark and dot PSFs. The simulation images have proven that our method can be used to obtain a remarkable enhancement of the imaging performance of a wavefront coding system for extending the depth of field.
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Table 1. MSSIM of Different Types at Different Defocus Positions
ψ0 ψ6 ψ12
QPM 0.6444 0.5585 0.4017
Both dot and dark PSFs 0.8080 0.7832 0.6623
