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Abstract
Diffraction calculation techniques based on Fourier transform, such as Fresnel diffraction, are essential in computational optics. Notably, zero padding is applied in diffraction calculations to manipulate sampling pitch and convolution calculations. However, zero padding can generate ringing artifacts due to sudden changes in value, which affect hologram reconstructions, etc. Several existing methods reduce the ringing artifact by decreasing the sudden changes in values. Therefore, in this study, we propose and validate a “ringing artifact extraction method” that focuses on the pattern of ringing artifacts, which depends on the conditions of diffraction calculation.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Diffraction calculation techniques based on Fourier transform, including Fresnel diffraction [1], are significant in computational optics, as observed from their application in computer-generated holograms [2], digital holography [3], and phase retrieval algorithms [4,5]. Diffraction calculations are usually classified into two groups: those based on a single Fourier transform and others based on convolution calculations.
In the method based on a single Fourier transform, the sampling pitch depends on the number of data points after diffraction calculation. Therefore, to maneuver the sampling pitch after the diffraction calculation, zero padding is often used to adjust the data size [6], which surrounds the original data with zero value data. Additionally, data size modification using zero padding is applied to conduct linear convolution with the fast Fourier transform, which is better in computational speed in methods based on convolutional calculations. Data size modification using zero padding is also applied when varying the sampling pitch before and after diffraction calculations [7–9] or when conducting diffraction calculations between nonparallel planes [10].
Zero padding is frequently used in diffraction calculations. However, when zero padding is applied, ringing artifacts occur, owing to unexpected alterations at the boundary between the original data portion and the portion increased by zero padding. Therefore, a technique to reduce ringing artifacts has been proposed. This technique smoothens sudden changes at the boundary by using a window function. Gaussian, Han, and Hamming windows are applied as window functions for smoothing, and their efficiency in reducing ringing artifacts has been established [11,12]. Notably, smoothing is applied to some of the diffraction calculation results. Although a technique using a flat top window function has also been proposed to prevent this challenge [13,14], it does not have much effect on reducing ringing artifacts. Other proposed techniques include high-pass Fourier filtering [14–16], boundary processing [17], subtraction [18], and average subtraction [19] methods, which are based on the concept of decreasing sudden changes at the boundary between the original data portion and the portion increased by zero padding. Additionally, the methods are based on reducing abrupt changes at the boundary between the original data and the portion increased by zero padding.
Alternatively, ringing artifacts appear as a lattice-like pattern on the complex amplitude distribution after diffraction computations, such as in the reconstructed image of a hologram. This pattern is determined by how zero padding is introduced using parameters in diffraction calculations, such as the diffraction distance, which is the change in sampling pitch before and after diffraction calculations [20], and in the situation off-axis systems [21], the amount of position shift. Moreover, the trend of ringing artifacts can be obtained by conducting diffraction calculations with the same parameters. Therefore, in this study, we propose a method to eliminate ringing artifacts generated by diffraction calculations using the pattern of ringing artifacts obtained in advance. Furthermore, we verified the effectiveness of this method by comparing it with other ringing artifact reduction methods.
2. Proposed method
The diffraction calculation method based on a single Fourier transform is given in Eq. (1):
The convolution-based diffraction calculation method is expressed as in Eq. (2):
Fig. 1. Procedure for convolution-based diffraction calculations and the effect of zero padding on multiple diffraction calculations.
An example of diffraction calculation results with ringing artifacts is presented in Fig. 2. In Fig. 2, two diffraction calculations are performed for the images (blank image, Pepper and Parrot), one in the forward propagation and the other in the back propagation. The parameters employed in the diffraction calculations are displayed in Table 1.
Fig. 2. Example of diffraction calculation results with ringing artifacts. (a) result of blank image when diffraction distance is 0.1 m; (b) result of “Pepper” image when diffraction distance is 0.1 m; (c) result of “Parrot” image when diffraction distance is 0.1 m; (d) result of blank image when diffraction distance is 0.2 m; (e) result of “Pepper” image when diffraction distance is 0.2 m; (f) result of “Parrot” image when diffraction distance is 0.2 m.
Table 1. Parameters used in the diffraction calculations.
Additionally, the diffraction distance was set to 0.1 m for Figs. 1(a)−(c) and 0.2 m for Figs. 1(d)−(f).
In Fig. 2, a grid pattern along the edges of the image can be seen. This is the ringing artifact. Comparing Figs. 2(a)−(c) and 2(d)−(f), wherein the diffraction distances are equal, it can be observed that the ringing artifacts have almost the same pattern, despite the image content. This result shows that the parameters of the diffraction calculation more strongly impact the pattern of ringing artifacts than the content of the data for which the diffraction calculation is performed. Additionally, Figs. 2(a) and 2(d), in which the diffraction calculations are performed on images in which all pixels have the same value (blank image), can be considered to extract only the pattern of ringing artifacts from the respective diffraction calculations. Therefore, this study proposes a method to remove ringing artifacts by performing diffraction calculations on blank images in advance to obtain patterns of ringing artifacts.
The details of the proposed technique are explained, introducing ${b_{\textrm{source}}}({{x_1},{y_1}} )$ and ${b_{\textrm{target}}}({{x_2},{y_2}} )$ as the diffraction source and destination planes, respectively, and ${b_{\textrm{source}}}({{x_1},{y_1}} )$ as the blank image. All pixels in ${b_{\textrm{source}}}({{x_1},{y_1}} )$ have the same value. Since convolutions using FFT and FFT are linear, diffraction calculations using those calculations are also linear. Therefore, the value in ${b_{\textrm{source}}}({{x_1},{y_1}} )$ is not affect the pattern of ringing artifacts obtained or the final reconstructed image. However, the value is set to 1 to account for the division process in Eq. (4).
Thus, ${b_{\textrm{target}}}({{x_2},{y_2}} )$ and ${u_{\textrm{target}}}({{x_2},{y_2}} )\; $ have the same pattern of ringing artifacts. Thus, by dividing ${u_{\textrm{target}}}({{x_2},{y_2}} )$ by the ${b_{\textrm{target}}}({{x_2},{y_2}} )$ pixel, the ringing artifact can be removed. Let $u_{\textrm{target}}^{\prime}({{x_2},{y_2}} )$ be the diffraction destination plane after the ringing artifact is removed, and then $u_{\textrm{target}}^{\prime}({{x_2},{y_2}} )$ can be expressed as in Eq. (4):
If ${b_{\textrm{target}}}({{x_2},{y_2}} )$ is 0, replace it with a value sufficiently small compared to the value of ${b_{\textrm{source}}}({{x_1},{y_1}} )$ to avoid division by 0.
The proposed method is named the ringing artifact extraction method.
3. Result
In this section, the proposed method was evaluated and its efficiency in reducing ringing artifacts was verified. Comparisons with other methods for lowering ringing artifacts were also performed.
In this study, CWO++, a computational wave optics library, was used for the diffraction calculations [22].
3.1 Image quality evaluation using PSNR and SSIM
The effect of ringing artifact reduction was evaluated in this section by conducting diffraction calculations with the application of the ringing artifact reduction method and comparing the image quality of the resulting back propagation image. Figure 3 displays the original image, the backpropagation image devoid of ringing artifact reduction, the backpropagation image with the flat top window, and the backpropagation image with the ringing artifact extraction method.
Fig. 3. (a) Original image (Parrot); (b) Backpropagation image without reduction of ringing artifacts (Parrot); (c) Backpropagation image with flat top window (Parrot); and (d) Backpropagation image with ringing artifact extraction method (Parrot); (e) Original image (Pepper); (f) Backpropagation image without reduction of ringing artifacts (Pepper); (g) Backpropagation image with flat top window (Pepper); and (h) Backpropagation image with ringing artifact extraction method (Pepper).
The parameters used in the diffraction calculations are listed in Table 1. Additionally, the diffraction distance was set to 0.1 [m].
The flat top window has a function, as shown in Eq. (5):
The parameters in Eq. (5) are $\alpha = 5,\; L = 512,\; {L_0} = 432,{\; }$ $a ={-} \pi ({L - {L_0}} )/({2({{L_0} + 2 - L} )} ),$ and $b = \pi ({L + {L_0} + 2} )/({2({L - {L_0} - 2} )} )$, respectively.
Figure 3 indicates that the ringing artifacts appearing at the edges of the image are reduced when the ringing artifact extraction method was used compared to that in the case without ringing artifact reduction and the case with the flat top window. Comparing Fig. 3(d) with Fig. 3(a) and Fig. 3(h) with Fig. 3(e), a faint stripe pattern is seen in Fig. 3(d) and Fig. 2(h), which is considered to be a ringing artifact caused by sudden changes in luminance in the image. Furthermore, The results of image quality evaluation using PSNR (peak signal-to-noise ratio) and SSIM (structural similarity index measure) [23] based on Fig. 3(a) for Figs. 3(b)-(d) and Fig. 3(e) for Figs. 3(f)-(h) are shown in Table 2.
Table 2. Results of image quality evaluation.
Comparing the ringing artifact extraction method and the flat-top window method, when the center of the image (256 × 256) is compared, the PSNR and SSIM are superior to those obtained using the flat-top window. However, when the entire image is compared, the PSNR and SSIM are superior to those obtained using the ringing artifact extraction method.
These results are attributed to the fact that the flat-top window reduces ringing artifacts by reducing the luminance at the edges of the image. These results also quantitatively show that the proposed method can eliminate ringing artifacts without sacrificing image quality at the edges of the image.
In the next section, we will examine the use of the ringing artifact reduction method on holograms captured with actual optics.
3.3 Reconstruction of holograms captured by digital holography
To verify its effectiveness, the ringing artifact reduction method was applied to holograms captured with actual optics and reconstructed. Figure 4 illustrates the optical system used to capture the holograms. The conditions of the experiment are presented in Table 3. The optical system in Fig. 4 is an in-line transmission (Gabor-type) digital holography optical system. Figure 5 shows the holograms taken.
Table 3. Experimental conditions.
The parameters for the flat top window are $k = 5,\; {L_x} = 3264,\; {L_y} = 2448,\; {L_{0x}} = {L_x} - 80,{\; }{L_{0y}} = {L_y} - 80,{\; }a ={-} \pi ({L - {L_0}} )/({2({{L_0} + 2 - L} )} ),\; b = \pi ({L + {L_0} + 2} )/({2({L - {L_0} - 2} )} )$.
Figure 6 shows the outcomes of applying the ringing artifact reduction technique to holograms taken with actual optics and reconstructed. No processing (e.g., phase recovery) other than diffraction calculation and ringing artifact removal methods is performed during hologram reconstruction.
Fig. 6. Reconstructed images of holograms taken by the optical system with the ringing artifact reduction method applied: (a) Reconstructed image without ringing artifact reduction; (b) Reconstructed image with flat top window; and (c) Reconstructed image with the ringing artifact extraction method
Figure 6 indicates that the ringing artifact was reduced in the case of the ringing artifact extraction method compared to that in the situation without ringing artifact reduction and the case with a flat top window. This observation validates the effectiveness of the proposed technique in reducing ringing artifacts, even for holograms captured with actual optics.
4. Conclusion
In this paper, we verified that the pattern of ringing artifacts produced by diffraction calculations strongly depends on the conditions of the diffraction calculations. We proposed a technique to remove ringing artifacts by extracting the pattern of ringing artifacts using a “blank image,” which is an image in which all pixels have equal value.
The effectiveness of the proposed method was confirmed by performing diffraction calculations on the images, applying the ringing artifact reduction method, and the ensuing back propagation images were compared for image quality. The results indicated an increase of 9.34 [dB] in the PSNR and 0.1360 in the SSIM between the case without ringing artifact reduction and the case with the proposed technique. We also applied the ringing artifact reduction technique to holograms captured with actual optics and reconstructed them to verify their effectiveness. Therefore, the effectiveness of the proposed method in removing ringing artifacts even for holograms taken with real optics was confirmed.
The proposed method extracts patterns of ringing artifacts caused by sudden changes in values at the inner and outer boundaries of the original data portion. Therefore, it is highly effective for those ringing artifacts. On the other hand, it has the limitation that it cannot deal with ringing artifacts caused by sudden changes in values inside the original data portion.
The pattern of ringing artifacts can be reused, given that the pattern generated remains the same when the diffraction calculations are carried out under the same conditions, although the proposed method necessitates additional diffraction calculations to extract the pattern of ringing artifacts besides the diffraction calculations initially performed. Therefore, this method is more advantageous in the case of repeated diffraction calculations under the same conditions.
Funding
Japan Society for the Promotion of Science (JSPS) (21K17760).
Acknowledgments
The authors would like to thank Maruzen-Yushodo Co., Ltd. (https://kw.maruzen.co.jp/kousei-honyaku/) for the English language editing.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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