Parametric blind - deconvolution method to remove image artifacts in wavefront coding imaging systems

 includes a phase mask of asymmetric 
 phase mask kind in the pupil plane to extend the depth of field of an imaging system 
 and the digital processing step to obtain the restored final high-quality image. 
 However, the main drawback of wavefront coding technique is image artifacts on 
 the restored final images. In this paper, we proposed a parameter blind-
 deconvolution method based on maximizing of the variance of the histogram of 
 restored final images that enables to obtain the restored final image with artifact-
 free over a large range of defocus. 
Keywords: Imaging systems; Computational imaging; Deconvolution. 
 1. INTRODUCTION 
 Since 1995, an asymmetrical cubic phase mask was firstly reported by E. R. 
Dowski and W. T. Cathey as mean for increasing depth of field for imaging 
systems [1]. Util now, many phase masks of asymmetrical phase mask kind to 
increase the depth of field have been reported, such as the cubic phase mask [1], 
the logarithmic phase mask [2-5], the sinusoidal phase mask [6, 7], the exponential 
phase mask [8], the polynomal phase mask [9], the rational phase mask [10], the 
tangent phase mask [11], and the high-order phase mask [12]. All these phase 
masks can obtain the aim for large depth extention. Because the modulation 
transfer function (MTF) is nearly invariant to defocus and involves no zeros, the 
digital processing using a single deconvolution kernel can be used to obtain the 
final image near diffation-limited image of a traditional imaging system. However, 
as in [13] shown that the restored final images are damaged by the image artifacts, 
which come from non-linear phase transfer function (PTF) between the PTF of the 
imaging optical transfer function (OTF) and the PTF of the restored OTF. Only 
when the restored OTF perfectly matches the imaging OTF, the restored final 
image has the best image quality. Recently, a technique to remove image artifacts 
was introduced to obtain the restored final image with artifacts-free based on 
minimizing of the high frequency content of the restored image [14]. However, this 
way need more computation times. Nearly, some methods by using some phase 
masks have been introduced to remove image artifacts [15, 16]. However, these 
methods are complex and expersive. To decrease ring artifacts, some iterative 
restoration techniques have been introduced [17]. The main problem of the 
iterative restoration techniques is their computational speed. Recently, another fast 
non-iterative technique, which employs Fourier transform based deconvolution 
followed by wavelet de-noising, has been presented [18]. Iterative restoration 
techniques for ringing artifacts decrease have been introduced for WFC imaging 
system with phase masks [19]. 
 In this paper, we proposed simple image artifacts metric to obtain the restored 
final image with artifacts-free. The image artifacs metric is based on maximizing 
62 L. V. Nhu, H. V. Dang, D. C. Toan, “Parametric blind-deconvolution  imaging systems.” 
Research 
of variance of the histogram of restored final image, which is known as parameter 
blind-deconvolution. The image artifact metric enables to determine the restored 
deconvolution kernel in order to obtain the restored final image with artifacts-free. 
 2. METHOD 
 We consider the performance of the proposed metric with the most common 
cubic phase mask. Acoording to Ref. [1], the cubic phase mask can be presented by: 
 f x, y ax33 ay (1) 
 Where: a is the phase mask parameter to control the magnitude of phase 
deviation; x and y are the coordinates in the pupil plane. 
 A defocused or middle image with added noise for wavefront coding imaging 
system in the Fourier domain, IMilddle(u, v; 20), where u and v are the spatial 
frequencies, can be presented by: 
 IMiddle uv, ;,2020 ; ,, Huv OuvNuv (2) 
 Where: H(u, v; 20) is the OTF of wavefront coding imaging system at a given 
defocus parameter 20; O(u, v) and N(u, v) refer the Fourier transfer of a given 
scene and noise, respectively. The defocus parameter 20 can be presented by: 
 L2 1 1 1
  20 (3) 
 4 f d0 di
 r
 The restored final image in the spatial domain, IFinal(x0, y0; 20 ), with a restored 
deconvolution kernel, F(u, v;  r ), can be described as: 
 20
 I xy,;,;,;rr iFFTI 1 xy  Fxy  (4) 
 Final 0020 Middle 0020 0020 
 -1
 Where: x0 and y0 are the coordinates in image plane; iFFT is the inverse 
Fourier transfer. The deconvolution kernel is a Wiener filter given by Eq. (5) as: 
 H* u,; v  r
 r 20 
 F u,; v  20 2 (5) 
 * r
 H u,; v 20 K
 Where: * is the complex conjugate; K is a regularization parameter which 
optimally is the inverse of the signal-noise-ratio [14]. 
 When the restored deconvolution kernel, F(u, v;  r ), is not driven from the 
 20
imaging OTF, H(u, v; 20), there are image artifacts on the restored final image, 
while the restored deconvolution kernel, F(u, v;  r ), perfectly matches the imaging 
 20
OTF, H(u, v; 20), the restored final image has no image artifacts. Therefore, the aim 
 r
of optimization is to find the value of the optimal filter defocus  20 that is equal to 
20, yielding artifacts-free image. In order to resolve this problem, we proposed an 
 r
evaluated metric to obtain the optimal value of defocus  20 . The optimal value of 
Journal of Military Science and Technology, Special Issue, No.72A, 5 - 2021 63 
 Physics 
 r
defocus  20 is determined based on maximizing of a parameter proportional to the 
variance, VAR, of the histogram of the restored final images. The proposed metric, 
as a function of defocus parameter, can be presented by: 
 hh min
 rr 
 Metric VAR 20 20 (6) 
 maxhhrr min
 20 20 
 r
 Where: h r is the histogram of the restored image, IFinal(x0, y0;  20 ). 
  20 
 3. SIMULATION RESULTS 
 In order to evaluate the performance of the proposed metric, first, we describe 
numerical simulations with a spoke target shown in Fig. 1 and the cubic phase 
mask with mask parameter of a = 30. 20 is computed for numerical simulations of 
the five arbitrary, 20 = 5, 20 = 10, 20 = 15, 20 = 20, and 20 = 25 with a 
defocus parameter range from 0 to 30 in 101 steps. For each restored final image, 
we add an amount of white Gausses noise for SNR= 80 dB, 60 dB, and 40 dB in 
the middle image. By using Eq. (6), the result is shown in Fig. 2, where there is a 
convention that the maximum value of Metric is equal to 1. Fig. 2 (a) shows the 
result with added noise for SNR = 80 dB. Fig. 2(b) shows the result with added 
noise for SNR = 60 dB. Fig. 2(c) shows the result with added noise for SNR = 40 
 r
dB. The estimation of the optimal defocus of value 20 is corresponding to the 
maximum value of these curves. As Fig. 1 shows, it can be seen in all cases that
 r
 20 is approximate to the imaging defocus 20. In the case with added noise for 
SNR = 80 dB, the mean error for five curves is set to 0.12. In the case with added 
noise for SNR = 60 dB, the mean error for 5 five curves is equal to 0.10. In the 
case with added noise for SNR = 40 dB, the mean error for five curves is set to 
0.42. With two case SNR = 80 dB and 60 dB, theses curves are smooth, while SNR 
= 40 dB, these curves have oscillations and the value of the mean error is bigger. 
 Figure 1. Spoke target. 
64 L. V. Nhu, H. V. Dang, D. C. Toan, “Parametric blind-deconvolution  imaging systems.” 
Research 
 (a) SNR = 80 dB 
 (b) SNR = 60 dB 
 (c) SNR = 40 dB 
 Figure 2. Variation of Metric applied to the spoke target with defocus range 
 from 0 to 30. The blue, red, green, magenta, and cyan plots are correspond to 
 20 = 5, 20 = 10, 20 = 15, 20 = 20, and 20 = 25, respectively. 
 We show the restored images of the proposed method and the traditional 
method for Lena target at the defocused value for 20 = 20. The restored image of 
the proposed method is shown in Fig. 3(a). While the restored image of the 
traditional image is indicated in Fig. 3(b). It is not difficult to see that there are not 
image artifacts on the restored image of the proposed method. While, there are 
image artifacts on the restored image of the traditional image. This means that the 
proposed method can be used to remove image artifacts on the restored image in 
wavefornt coding imaging system. 
Journal of Military Science and Technology, Special Issue, No.72A, 5 - 2021 65 
 Physics 
 (a) (b) 
 Figure 3. (a) The restored image of the proposed method and (b) the restored 
 image of traditional method for Lena target at the defocused value for 20 = 20. 
 We consider the effectiveness of proposed metric to remove image artifacts for 
varying with added noise SNR = 80 dB, 60 dB, and 40 dB. A mask parameter 
range is a range from 1 to 60. The result is shown in Fig. 4. As Fig. 4 indicates, the 
mean error is dependent to value of mask parameter and value of the signal-noise-
ratio. With added noise SNR =80 dB, the value of the mean error is relatively 
stable with mask parameter range [20, 60]. For SNR = 60 dB, the value of the 
mean error is relatively stable with phase mask parameter range [20, 45]. By using 
the proposed way, we can optimize mask parameter, , based on minimizing of 
value of the mean error. From Fig. 4, the optimal phase mask parameter is equal to 
10, 20, and 20 for SNR = 40 dB, 60 dB, and 80 dB, respectively. Such at SNR = 60 
dB and 80 dB, the optimal phase mask parameter is the same and this value is 
identical to optimization results in [20] based on full-reference image measure of 
restored image (in [20], the optimal phase mask parameter = 18). In comparison 
to the optimization result in [1] (as in [1], the phase mask parameter a is set to 90), 
the optimal phase mask parameter in here is reduced by an approximate factor 4.5. 
It is well-known that the clear advantage of the lowest is reduction of the noise 
amplification in the restored final image. When SNR = 40 dB, the optimal phase 
mask parameter is equal to about half of the optimal phase mask parameter with 
SNR = 60 dB and 80 dB, this is advantageous for a reduction of the noise 
amplification in the restored final image. 
 Figure 4. Mean errors of five arbitrary 20 and the spoke target as a function of 
 phase mask parameter. 
66 L. V. Nhu, H. V. Dang, D. C. Toan, “Parametric blind-deconvolution  imaging systems.” 
Research 
 Figure 5. Mean errors for five arbitrary 20 and the spoke target 
 as a function of SNR. 
 Finally, we consider the performance of the proposed method for varying SNR 
from 10 to 80 with an interval of 10, where the mask parameter a is still equal to 
30. The result is shown in Fig. 5. As Fig. 5 shows that the proposed method can 
obtain the small mean error with the high SNR. When SNR > 60 dB, the mean 
error is nearly stable. When SNR > 30 dB, the most maximum value of the mean 
error is less than 1.23 and at these low defocus deviations, we can ignore image 
artifacts on the restored final image such that the image artifacts cannot be 
remarked in high SNR. Whereas, for lower SNR, the mean error increase fast and 
remarkably, and therefore image artifacts on the restored final images can visible 
by eye. When SNR<15 dB, the mean error is bigger than 1.5, the effectiveness of 
the proposed method is not high. 
 4. CONCLUSIONS 
 We proposed a novel parameter blind-deconvolution method to obtain the 
restored final image with artifact-free for wavefront coding imaging system. By 
using the proposed way, we can to determine the accuracy defocus 20 on a large 
range of defocus, which can be obtained the restored final image without image 
artifacts. Additionally, we have indicated that the proposed way enables the 
reduction of the phase mask strength and obtain the optimal mask parameter of the 
cubic phase mask that is identical to the previous optimization result based on full-
reference measure. 
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 TÓM TẮT 
 PHƯƠNG PHÁP CUỘC ĐẢO NGƯỢC MỜ CHO LOẠI BỎ TẠP CHẤT 
 Ở HỆ THỐNG MÃ HÓA MẶT SÓNG 
 Công nghệ mã hóa mặt sóng bao gồm đồng thời phần quang học và xử 
 lý ảnh. Phần quang học là một mặt nạ pha đặt tại mặt phẳng đồng tử ra 
 của hệ thống quang học truyền thống đưa đến ảnh mờ đồng đều trên một 
 độ sâu trường lớn. Phần xử lý ảnh truyền thống được thực hiện bằng sử 
 dụng một phin lọc nghịch đảo để nhận ảnh sắc nét trên độ sâu trường lớn. 
 Tuy nhiên, nhược điểm chính của kỹ thuật xử lý ảnh này là làm xuất hiện 
 tạp chất trên ảnh khôi phục. Trong bài báo này, chúng tôi đề xuất phương 
 pháp cuộn nghịch đảo mờ trên cơ sở tối ưu hóa hàm histogram của tập ảnh 
 khôi phục với các độ sâu trường để nhận được ảnh khôi phục chất lượng 
 cao mà không bị xuất hiện tạp chất trên ảnh. 
Từ khóa: Các hệ thống tạo ảnh; Tạo ảnh tính toán; Cuộc nghịch đảo. 
 Received 2nd March 2021 
 Revised 27th April 2021 
 Accepted 10th May 2021 
Author affiliations: 
 1Học viện Kỹ thuật quân sự; 
 2Viện Vũ khí, Tổng cục Công nghiệp Quốc phòng. 
 *Corresponding author: levannhuktq@gmail.com. 
68 L. V. Nhu, H. V. Dang, D. C. Toan, “Parametric blind-deconvolution  imaging systems.”