In this paper, we proposed a simple rational phase mask with two mask parameters to extend the depth of field. A series of performance comparison between the proposed phase mask with the cubic, logarithmic, sinusoidal, and exponential phase mask is presented.
Nghiên cứu khoa học công nghệ OPTIMIZED RATIONAL PHASE MASK TO EXTEND THE DEPTH OF FIELD IN IMAGING SYSTEMS Nguyen Phuong Nam1*, Le Van Nhu2 Abstract: Wavefront coding technique includes the optics and the digital processing step to the final high-quality images One of the most important parts of wavefront coding technique is the design of a suitable phase mask to obtain the defocus invariant characteristics In this paper, we proposed a simple rational phase mask with two mask parameters to extend the depth of field A series of performance comparison between the proposed phase mask with the cubic, logarithmic, sinusoidal, and exponential phase mask is presented The simulation results demonstrated that the proposed phase mask is superior to other compared methods relating to imaging performance in extending the depth of field of an imaging system Keywords: Wavefront coding; Phase mask; Depth of field INTRODUCTION In recent years, wavefront coding is well known as a powerful technique which can be employed to extend the depth of field of an imaging system Wavefront coding is a hybrid imaging technique and it includes two parts: the optics and the digital processing step By placing a phase mask in the exit pupil plane, the modulation transfer function (MTF) or the point spread function (PSF) reduces sensitivity to defocus error Then, middle images which are captured by a detector are the same blurred images to defocus error and become sharp and clear after using a simple deconvolution filter to decode them in the digital processing step Conversely, in traditional imaging systems, the aim of the optics is to result in an imaging as sharp and clear as possible After that, the image might be postprocessed to reveal information that are relevant to a specific application However, the optimizations of the optics and the postprocessing are implemented separately Moreover, in comparison between the wavefront coding system and traditional imaging system under the impact of defocus, the MTF of an wavefront coding system is lower than that of traditional imaging system It introduces no zeros in the MTF of an wavefront coding system Therefore, all frequencies can be restored to near diffraction-limited performance In wavefront coding system, there are two parts which are designed together However, the most important part of wavefront coding system lies in the design of suitable phase masks to obtain the defocus invariant characteristics As a result, many researchers keep focusing on the development of this technique in term of phase masks So far, many kinds of odd asymmetrical phase masks to increase the depth extension, such as cubic phase mask [1], logarithmic phase mask [2-5], exponential phase mask [6], sinusoidal phase mask, [7, 8], polynomial phase mask [9], free-form phase mask [10], rational phase mask [11], tangent phase mask [12], and high-order phase mask [13], have been reported All these phase masks can achieve the goal of the depth of field extension Although many phase masks have been mentioned above, there are two main methods to research a phase mask One is based on resolution analysis using the Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san Viện Điện tử, - 2020 41 Kỹ thuật điện tử stationary phase approximation method, as in the cubic phase mask [1] by minimizing the variation of the optical transfer function to the defocus, or in the given logarithmic phase mask by making the point spread function insensitive to the defocus [2] Other methods are based on the optimization of parameters of the phase mask By using this method, many phase masks with high performance in extending the depth of field were presented, as in [3-13] However, the second method should use an evaluation function Some evaluation functions have been widely applied to optimize parameters of the phase mask, such as the Fisher information, the mean square error of the MTF or the PSF With each evaluation function, two conditions should be considered The first one, the magnitude of the MTF should big enough to make the digital processing easier The second one, the optical transfer function (OTF) should be invariant enough to defocus According to [1, 4, 6, 7], the cubic, the improved logarithmic, the exponential, and the sinusoidal phase mask can be given by Eqs.(1)-(4), respectively, as follows: f x, y ax3 ay (1) f x, y ax exp bx ay exp by f x, y sgn x ax log x b sgn y ay log y b (2) (3) f x, y ax sin bx ay sin by (4) Where x and y are the pupil plane coordinates; both a and b denote the phase mask parameters to control the magnitude of phase deviation; sgn(·) describes the sign function which is defined as for x and -1 for x The high performance of the rational phase mask is demonstrated in [11] However, this rational phase mask has twenty-four mask parameters to control the magnitude of phase deviation It is relatively complex Hence, in this paper, the aim of work is to find a simple phase type of rational phase function to obtain the invariant modulation transfer function over a wide range of defocus Based on using optimization method, and from trial and error, we propose a simple rational phase mask with two mask parameters to control the magnitude of the phase deviation Nonetheless, this mask can obtain the high performance in extending the depth of field The proposed rational phase mask can be presented by, ax ay f x, y (5) b x b y2 where these symbols in Eq (5) have meaning the same as above, but b OPTIMIZATION OF PHASE MASK PARAMETERS To make performance comparison between five phase masks, these five masks should be determined under the same condition Firstly, the integral areas of the infocus MTF for the five phase masks are the same This condition ensures that the phase masks have the same ability to transfer signal In this paper, we choose the value of the integral area of the in-focus MTF for the five phase masks is equal 0.3 With this value, the digital processing can make certain to obtain the restored final high-quality images For the cubic phase mask, it has only one mask parameter to 42 N P Nam, L V Nhu, “Optimized rational phase mask … field in imaging systems.” Nghiên cứu khoa học công nghệ control the magnitude of phase deviation while the value of integral area of the infocus MTF is equal to 0.3 Therefore, it is easy to calculate the mask parameter a of the cubic phase mask as given in table For other phase masks, they have two mask parameters to control the magnitude of phase deviation Such, with the value of the integral area of the in-focus MTF is equal to 0.3, there are many values of (a, b) of the phase masks to satisfy this condition However, there is a value pair of (a, b) to give the best performance in extending the depth of field To determine this (a, b), we need to choose an evaluation function Here, we adopt mean square error of the MTF to choose the optimal parameters (a, b) of the phase mask According to [11], the optimization procedure based on mean square error of the MTF can be described by, MSE H , u H 0, u u 1 (6) 0 min MSE 0 Where H is the one-dimensional optical transfer function; is the defocus parameter which the maximum value is the parameter 0 and it can be presented by, L2 1 (7) 4 f d0 di Where L is the aperture diameter, is the wavelength of light, , d0, and di, are the focal length, the object distance, and the image sensor distance, respectively Fig Phase profiles of five phase masks Based on the optimization procedure as shown in Eq.(6) with 0 = 30, the corresponding optimum parameters (a, b) of the logarithmic, sinusoidal, exponential, and fractional phase mask are indicated in table From the data in table 1, the one-dimensional phase profiles of five phase masks are shown in fig As can be seen in the figure, the phase profile of the rational phase mask is clearly Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san Viện Điện tử, - 2020 43 Kỹ thuật điện tử separate to the other phase masks when the variation of height of the rational phase mask profile is faster than the other phase masks Table Optimum parameters of five phase masks a b Cubic 89.85 Logarithmic 139.99 -3.43 Sinusoidal 189.94 1.87 Exponential 42.99 1.37 Rational 229.68 1.94 PERFORMANCE COMPARISON FOR PHASE MASKS With the data in table 1, we can illustrate defocused MTF curves of the five phase masks at the in-focus condition = as well as at the out-focus condition, = 6, 12, 18, 24 and 30 as given in fig The defocused MTF curves of the cubic phase mask are indicated in fig 2(a) The defocused MTF curves of the logarithmic phase mask are shown in fig 2(b) The defocused MTF curves of the sinusoidal phase mask are given in fig 2(c) Fig 2(d) shows the defocused MTF curves of the exponential phase mask The defocused MTF curves of the rational phase mask are presented in fig 2(e) From fig 2, it is clear that the defocused MTF curves of the five phase masks are less sensitive to defocus However, the defocused MTF curves of the cubic, logarithmic, and sinusoidal phase mask are less invariant than that of the other phase masks because the defocused MTF curves of the cubic phase mask present a strong oscillations at high spatial frequency part, while the two logarithmic and sinusoidal phase masks have noticeable oscillations at low and high spatial frequency parts The defocused MTF curves for the two exponential and rational phase masks have high stability However, it can be seen that the defocused MTF curves of the rational phase mask is more invariant than that of the exponential phase mask Therefore, the rational phase mask could perform better in extending the depth of field of an imaging system Besides the defocused MTF, another approach to evaluate the defocus invariant characteristic of an imaging system with a phase mask is the mean square error of the MTF (MSE) as given in Eq (6) The MSE reveals the stable level of the defocused MTF The lower the MSE is, the higher invariant to defocus the phase mask Fig shows that MSE curves of the five phase masks corresponding to their optimum parameters which are listed in table From fig 2, it is clear that the MSE of the sinusoidal phase mask is bigger among all phase masks, while the MSE of the exponential-rational mask pair is lower than that of the cubiclogarithmic mask pair The MSE of the rational phase mask is lower than that of the exponential phase mask The value of the MSE of the rational phase mask is lower approximately 32% in comparison with the exponential phase mask at the defocus parameter = 30 So the rational phase mask is less sensitive to defocus 44 N P Nam, L V Nhu, “Optimized rational phase mask … field in imaging systems.” Nghiên cứu khoa học công nghệ (a) Cubic phase mask (b) Logarithmic phase mask (c) Sinusoidal phase mask (d) Exponential phase mask (e) Rational phase mask Fig Defocused MTF curves of five phase masks Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san Viện Điện tử, - 2020 45 Kỹ thuật điện tử Fig MSE curves of five phase masks Fig Hilbert space angle curves of five phase masks Another powerful tool to evaluate defocus invariant imaging property is the Hilbert space angle The Hilbert space angle indicates the similar level between two functions The smaller the Hilbert space angle, the more similar the two functions will be Here we use the Hilbert space angle between the in-focus MTF and the out-focus MTF to evaluate the imaging performance of the phase mask The Hilbert space angle, (), between the in-focus MTF and the out-focus MTF can be presented by, H , u H 0, u (8) cos 1 H , u H 0, u Where the symbol · is the inner product, the symbol · denotes the norm By using Eq (8) and the data in table 1, Hilbert space angle curves of the five phase masks are illustrated in fig It can be seen that when defocus is small, the Hilbert space angle of the cubic phase mask is bigger among all phase masks, and when defocus increase, the Hilbert space angle of the sinusoidal phase mask increase and bigger than the other masks While the Hilbert space angles of the exponential-rational mask pair are lower than that of the other phase masks In which, the Hilbert space angle of the rational phase mask is lower than that of the exponential phase mask at any defocus parameter value When defocus parameter is set to 30, the value of the Hilbert space angle of the rational phase mask is lower approximately 22% in comparing with the exponential phase mask This means that the rational phase mask can perform better in enlarging the depth of field of an imaging system Finally, we consider images of these phase masks with a cameraman target with the size of 256256 pixels This is a direct way to evaluate imaging performance of the phase masks in extending the depth of field Fig shows the images of the five phase masks at the in-focus condition = as well as at the out-focus condition = 15, and = 30 The top row in fig is a set of final images of the wavefront coding system with the cubic phase mask The second row in fig is a set of final images of the wavefront coding system with the logarithmic phase mask 46 N P Nam, L V Nhu, “Optimized rational phase mask … field in imaging systems.” Nghiên cứu khoa học công nghệ Fig Simulation images of cameraman target (rows: cubic, logarithmic, sinusoidal, exponential, and rational; columns: =0, = 15, and = 30) Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san Viện Điện tử, - 2020 47 Kỹ thuật điện tử The third row in fig is a set of final images of the wavefront coding system with the sinusoidal phase mask The fourth row in fig is a set of final images of the wavefront coding system with the exponential phase mask The bottom row in fig is a set of final images of the wavefornt coding system with a rational phase mask As can be seen in fig 5, the image with the cubic phase mask is more degraded during the digital processing, such as artifacts on images, especially when defocus parameter is bigger The image degradation of the cubic phase mask comes from the oscillations and the spatial frequency cutoff of its defocused MTF as shown in fig (a) The quality of restored final images from the logarithmicsinusoidal mask pair is almost the same and is better than that of the cubic phase mask Whereas the exponential-rational mask pair provide a better image quality among all phase masks Additionally, the visual perception of the rational phase mask is likely better than that of the exponential phase mask Hence, the rational phase mask could be better in extending the depth of field of an imaging system CONCLUSION In the paper, the simple rational phase mask to reduce the impact of defocus error in the modulation transfer function has proposed Based on the evaluation of different methods such as: evaluation ways of the defocused MTF, the mean square error of the MTF, and Hilbert space angle between the in-focus MTF, the out-focus MTF and image restoration, it can be concluded that the proposed phase mask has remarkable improvement in imaging performance in extending the depth of field of an imaging system Practically, it is feasible that a series of rational phase masks can be employed to enlarge the depth of field Additionally, noise effect to image quality will be also investigated This issue will be further studied in the future REFERENCES [1] E R Dowski, Jr and W T Cathey, “Extended depth of field through wavefront coding,” Appl Opt 34, 1859-1866 (1995) [2] S S Sherif, E R Dowski, and W T Cathey, “A logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems,” Proc SPIE 4471, 272 -279 (2001) [3] H Zhao and Y Li, “Performance of an improved logarithmic phase mask with optimized parameters in a wavefront-coding system,” Appl Opt 49, 229-238 (2010) [4] H Zhao and Y Li, “Optimized logarithmic phase masks used to generate defocus invariant modulation transfer function for wavefront coding system,” Opt Lett 35, 2630 -2632 (2010) [5] H Zhao, Q Li and H Feng, “Improved logarithmic phase mask to extend the depth of field of an incoherent imaging system,” Opt Lett 33, 1171-1173 (2008) [6] Q Yang, L Liu, and J Sun, “Optimized phase pupil masks for extended depth of field,” Opt Commun 272, 56-66 (2007) [7] H Zhao and Y Li, “Optimized sinusoidal phase mask to extend the depth of field of an incoherent imaging system,” Opt Lett 35, 267-669 (2010) 48 N P Nam, L V Nhu, “Optimized rational phase mask … field in imaging systems.” Nghiên cứu khoa học công nghệ [8] J Wang, J Bu, M Wang, Y Yang and X C Yuan, “Improved sinusoidal phase plate to extend depth of field in incoherent hybrid imaging systems,” Opt Lett 37, 4534-4535 (2012) [9] N Caron and Y Sheng, “Polynomial phase mask for extending depth-of-field optimized by simulated annealing,” Proc SPIE 6832, 68321G-1-10 (2007) [10] Y Takahashi, and S Komatsu, “Optimized free-form phase mask for extension of depth of field in wavefront-coded imaging,” Opt Lett 33, 15151517 (2008) [11] F Zhou, G Li, H Zhang, and D Wang, “Rational phase mask to extend the depth of field in optical-digital hybrid imaging systems,” Opt Lett 34, 380382 (2009) [12] V N Le, Z Fan, S Chen, “Optimized asymmetrical tangent phase mask to obtain defocus invariant modulation transfer function in incoherent imaging system,” Opt Lett 39, 2171-2174 (2014) [13] A Sauceda and J O eda-Casta n eda, “High focal depth with fractionalpower wave fronts,” Opt Lett 29, 560-562 (2004) TÓM TẮT MẶT NẠ PHA HÀM PHÂN THỨC TỐI ƯU MỞ RỘNG ĐỘ SÂU TRƯỜNG TRONG HỆ THỐNG TẠO ẢNH Công nghệ mã hóa mặt sóng bao gồm bước tạo ảnh quang học trình xử lý số cho nhận ảnh chất lượng cao Phần quan trọng công nghệ mã hóa mặt sóng nằm thiết kế mặt nạ pha phù hợp để nhận đặt tính bất biến lệch tiêu Ở báo này, đề xuất mặt nạ pha hàm phân thức với hai tham số cho mở rộng độ sâu trường Một loạt so sánh tính tốn mặt nạ pha đề xuất với mặt nạ pha hàm bậc ba, hàm logarit, hàm sin, hàm mũ trình bày Các kết mô chứng minh rằng, mặt nạ pha đề xuất có khả cao cho mở rộng độ sâu trường hệ thống tạo ảnh Keywords: Wavefront coding; Phase mask; Depth of field Nhận ngày 26 tháng năm 2020 Hoàn thiện ngày 28 tháng năm 2020 Chấp nhận đăng ngày 28 tháng năm 2020 Địa chỉ: Institute of Electronics, Academy of Military Science and Technology; Le Quy Don Technical University * Email: ngnamaus41@gmail.com Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san Viện Điện tử, - 2020 49 ... exponential phase mask Hence, the rational phase mask could be better in extending the depth of field of an imaging system CONCLUSION In the paper, the simple rational phase mask to reduce the impact of. .. mask pair is lower than that of the cubiclogarithmic mask pair The MSE of the rational phase mask is lower than that of the exponential phase mask The value of the MSE of the rational phase mask. .. that the defocused MTF curves of the rational phase mask is more invariant than that of the exponential phase mask Therefore, the rational phase mask could perform better in extending the depth of