Journal of information and communication convergence engineering 2023; 21(1): 90-97
Published online March 31, 2023
https://doi.org/10.56977/jicce.2023.21.1.90
© Korea Institute of Information and Communication Engineering
Correspondence to : Myungjin Cho^{1} (E-mail: mjcho@hknu.ac.kr), Min-chul Lee^{2} (E-mail: lee@csn.kyutech.ac.jp, Tel: +81-948-29-7699)
^{1}School of ICT, Robotics, and Mechanical Engineering, IITC, Hankyong National University, Kyonggi-do 17579, Republic of Korea
^{2}Department of Computer Science and Networks, Kyushu Institute of Technology, Fukuoka 820-8502, Japan
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Digital holographic microscopy (DHM) is a three-dimensional (3D) imaging technique that uses the phase information of coherent light. In the reconstruction process of DHM, a narrow region around the positive or negative sideband from the Fourier domain is windowed to avoid noise due to the DC spectrum of the hologram spectrum. However, the limited size of the window also degrades the high-frequency information of the 3D object profile. Although a large window can have more detailed information of the 3D object shape, the noise is increased. To solve this trade-off, we propose phase difference averaging (PDA). The proposed method yields high-frequency information of the specimen while reducing the DC noise. In this paper, we explain the reconstruction algorithm for this method and compare it to various conventional filtering methods including Gaussian, Wiener, average, median, and bilateral filtering methods.
Keywords Digital holographic microscopy, Noise reduction, Phase error, Three-dimensional imaging
Light sensors, including the human eye, and digital arrays cannot detect phase information directly, but can only detect the intensity of light. For this reason, the interference characteristics of light have been used for obtaining phase information. In particular, holography is a technology that uses interference to detect the phase of light. It utilizes the phase difference between a reference beam and an object beam to record three-dimensional (3D) information for objects. This technique has been studied extensively since it was first proposed in 1948 [1]. Digital holography (DH) uses the same basic principles as holography, but uses image sensors instead of films for recording. Unlike films that cannot be reused after recording, DH can record images without storage limitations. Recently, DH has been applied to quantitative phase imaging (QPI) [2,3], microscopy [2-10], 3D image encryption [11], and 3D object recognition [12]. Among these applications, as digital holographic microscopy (DHM) can obtain the thickness profile of an object with high resolution, it is used for disease diagnosis [13,14], 3D profiling of microstructures [15-17], and microbial research [18, 19]. In DHM, one of the sidebands (corresponding to object information) in the Fourier domain is windowed to obtain the depth information. In this process, when a wide window is used, high frequency information of the object is obtained. However, the information in the DC spectrum, as well as the noise, also increases, leading to phase error. In contrast, when a narrow phase region is masked, the phase error decreases, but high frequency information about the object is lost. Furthermore, in interferometers, it is difficult to adjust the interference fringe density and set the size of the window used to mask the sideband. In this paper, we propose a phase error filtering method for DHM, which can address this ambiguity. We refer to this filtering method as phase differences averaging (PDA).
Fig. 1(a) and (b) show the 2D and 1D Fourier spectra of the recorded hologram, respectively. To obtain depth information from the recorded hologram, one of the sideband terms is windowed and cropped. As shown in Fig. 1(b), Rc is the size of the cropped region, and the green area under the +f_{o} term is the noise induced by the DC spectrum. To reduce this unwanted phase in image processing, we have proposed PDA. Our proposed method crops the 2D Fourier spectrum from narrow to wide regions in regular intervals to reconstruct the correct phase information by reducing the effect of the DC spectrum and high-frequency noise.
Fig. 2 shows a schematic of the PDA. Fig. 2(a) shows the Fourier spectrum of the recorded hologram, and Fig. 2(b) shows each DC shifted +f_{o} term after cropping the sideband around +f_{o} with different-sized windows. Zero padding was applied, as shown in Fig. 2(b), to obtain images of the same resolution via inverse Fourier transform on the sidebands windowed using different sizes. Fig. 2(c) shows the phase calculated using the cropped Fourier spectrum. The average of these obtained phase profiles was then computed to yield the high-frequency thickness information of the specimen while reducing the effect of the DC spectrum. The PDA process is very simple as it selects the center of the sideband in the Fourier domain and crops it from the smallest to the largest window size in regular intervals, regardless of the trade-off between the phase error and high-frequency information of the specimen.
A detailed explanation of PDA implementation is as follows. Two holograms are recorded, one with the object in the field of view (object hologram) and one with the medium surrounding the object in the field of view (reference hologram), to obtain the change in phase of the wavefront caused by the spatially-varying optical path length of the object. In addition, this method cancels out the noise caused by the surface of the mirror, beam splitter, object lens, and image sensor. They are analyzed separately by applying different sized windows (from smallest to largest), and their phase difference is computed. The computed phase difference corresponding to any one of the windows can be written as [20]
where n_{s} and n_{s} are the constant refractive indices of the specimen and surrounding medium, respectively; h is the thickness information of the specimen; λ is the vacuum wavelength of the source; and Δn is the difference in the refractive index. Here, h is one thickness value obtained with a particular masking window (Eq. (2)). Thus, the thickness information with the smallest masking window size is h_{1}, and numbers are assigned in order of increasing window size. The average of all the thickness information with different window sizes is [20]
In this study, we wished to crop the phase region as wide as possible, so the fringe pattern was as narrow as possible. For this reason, we used modified Mach-Zehnder interferometry using two spherical waves.
Fig. 3 shows the experimental setup used for the demonstration of PDA. The setup used a 532 nm laser diode module (3 mW output power). To avoid overlapping of red blood cells (RBCs), we prepared thin blood smears as specimens by dripping blood on a glass slide and spreading it thinly. Both the object and reference beams were magnified by 40 × (0.65 NA) objective lenses. In this experiment, we used constant reflective indices for the RBCs and the surrounding medium of 1.42 and 1.34 (reflective index of the blood plasma), respectively [13]. Holograms were recorded using a CMOS sensor (Basler, acA2500-14uc) with 2590(H) × 1942(V) pixel resolution, and a pixel size of 2.2 μm (H) × 2.2 μm (V). The CMOS sensor was placed at the image plane. The beam splitter in front of the sensor was tilted to make the fringe pattern as dense as allowed by the sampling criteria.
Fig. 4 shows the unfiltered 3D profile of the object (red blood cell) obtained using the smallest phase region (90(H) × 90(V) pixels) and the largest phase region (660(H) × 660(V) pixels). In this study, we used a Goldstein phase-unwrapping algorithm [21] to obtain continuous phase information from the wrapped phase. As mentioned earlier, when the phase region is small, as shown in Fig. 4(a), the approximate shape information of the object and the noise decrease; when the region is wide, as shown in Fig. 4(b), the shape information of the object includes details and the noise increases. Therefore, we applied filtering using the proposed method. In this study, the window size of the data used for reconstruction of the proposed method was increased at intervals of 30 pixels horizontally and vertically from 90(H) × 90(V) pixels to 660(H) × 660(V) pixels. As a result, we obtained a total of 20 thickness datapoints with the same image resolution.
As shown in Fig. 4(c), the noise in the 3D profile is reduced. In addition, we compared the proposed method with different conventional filtering methods. Figs. 4(d)-(h) show the results obtained using the conventional filters: a Gaussian filter (σ = 2), a Wiener filter (9(H) × 9(V) pixel filter size), an average filter and a median filter (5(H) × 5(V) pixels filter size), and a bilateral filter (σ = 2), respectively, applied to Fig. 4(b). The size of each conventional filter was chosen to retain as much high-frequency information as possible. As a result of comparing each 3D profile, no significant difference was found between the result of a Gaussian filter and the PDA. For numerical comparison, we calculated the signal-to-noise ratio (SNR) and mean squared error (MSE) of each 3D profile [22]. To compare only the noise coming from the DC spectrum, the comparison target of SNR and MSE uses the 3D profile of the smallest cropped phase region.
Table 1 shows the results of the numerical comparison between the PDA and various conventional filters. In Table 1, the MSE and SNR of the Gaussian filter show the best results among the conventional filtering methods. However, the Gaussian filtering results also show a higher MSE and lower SNR than PDA. Among the conventional filters compared in this section, the Gaussian filter, which shows the most effective value, is compared in more detail in the next subsection.
Units for magnetic properties
MSE | SNR (dB) | |
---|---|---|
Unfiltered | 0.4827 | 19.5095 |
PDA | 0.1882 | 23.5991 |
Gaussian | 0.2340 | 22.6533 |
Wiener | 0.2838 | 21.8156 |
Average | 0.2781 | 21.9044 |
Median | 0.3162 | 21.3470 |
Bilateral | 0.4795 | 19.5388 |
We used statistical analysis to determine the advantages of the proposed method. For statistical analysis, we randomly selected 20 different RBCs and obtained thickness profile data using small to large masking windows. The results obtained with PDA were compared with those obtained with unfiltered, Gaussian, and median filtering (Fig. 5).
The horizontal line in Fig. 5 indicates the result obtained using PDA. For the unfiltered method, it can be seen that the larger the size of the masking window, the lower the SNR, as shown in Fig. 5(a). However, the variation increases with a larger masking window. For single-window filtering, the SNR is highest for small window sizes, but it also removes high-frequency object information. It can be seen that the SNR of PDA is the same as that of a single filter window of 270(H) × 270(V) pixels. Above this window size, the single-filteringwindow SNR decreases, while the PDA SNR remains high. As the fringe density changes, the filtering window size should also change. This is automatically done when using PDA. In addition, from Fig. 5(b) and 5(c), it can be seen that a larger sigma value of the Gaussian filter and larger filter size of the median filter results in better SNR values. However, these filtering operations smooth the obtained depth profile from phase differences, resulting in the loss of high-frequency information. PDA also considers this aspect. The variances of the Gaussian and median filtering are much larger than the variance obtained using PDA. The variance value is related to the stability of filtering, which means that there is a possibility that random noise may remain. To verify this, the correlation between SNR and MSE was used in this study. Fig. 6 shows the correlation between SNR and MSE.
Fig. 6 shows the correlation between SNR and MSE for different RBCs generated using the same method. The vertical axis represents the SNR value and the horizontal axis represents the MSE value. As shown in Fig. 6, It can be seen that PDA provides a higher SNR with lower mean square error, while retaining high spatial frequency information of the object.
As shown in Fig. 7, when the sigma value of the Gaussian filter increases, a lower error and better SNR result. However, Gaussian filtering also suppresses higher spatial frequency information of the object. Among the randomly selected RBCs, 10% of RBCs have a serious phase error, and even if the sigma value of the filter increases, these errors lead to a lower SNR (Fig. 7).
Fig. 8 illustrates the advantages of our proposed method. PDA can provide high-resolution, high-SNR thickness profiles compared to a single masking window. This is the strength of the proposed method. The RBCs in Fig. 8 have a flat surface, unlike the donut-shaped RBCs in Fig. 4. This is because the pressing force was strong in the process of making the specimen. However, this did not affect the experimental results.
In this study, we proposed a new filtering method for reducing the phase error from the DC spectrum in DHM and compared our proposed method with general filtering methods. The proposed method showed strength in terms of stability compared to other filtering methods, and it had the advantage of obtaining 3D data with a high spatial frequency and SNR. Filtering methods using a single masking window either provided a low spatial frequency (small window) or high noise (large window) 3D profile of the object. However, our proposed method did not require the size of the filter to be determined, and it had the advantage of being able to filter effectively, even in severe phase error conditions. The proposed method retained the detailed (high-frequency) information of the object while removing unwanted noise. Furthermore, because the proposed method averaged the thickness information after phase unwrapping of windowed sidebands of 20 different sizes, it also eliminated the phase error that occurs after phase unwrapping. Off-axis DHM combined the object and reference beams at an angle, leading to the separation of the DC spectrum and sidebands. However, in many cases, especially while using low temporally coherent sources such as LEDs, to have a larger usable field of view (where interference fringes exist), the angle between the object and reference beams must be very small, leading to low-density fringes. In such cases, the sideband information overlapped with the DC spectrum, resulting in low lateral resolution owing to the smaller size of the filtering window used. The proposed method is very useful under such conditions. In this study, the width interval of the windowed sideband is an arbitrary value, and the height data also used an arbitrary number of data points for the average. As the width interval became narrower and the number of height data points used for the average increased, the filtering effect was expected to be excellent. However, because there is a trade-off relationship between processing time and filtering quality, it is important for the user to select an appropriate value.
This work was supported by the Japan-Korea Basic Scientific Cooperation Program between JSPS and NRF, Grant number (JPJSBP 120228811), and this work was supported under the framework of the international cooperation program managed by the National Research Foundation of Korea(NRF-2022K2A9A2A08000152, FY2022).
received his B.S. from Jeju National University, Jeju, Korea, in 2011 and M.S from Korea University, Seoul, Korea in 2015. He is currently a doctoral student at Kyushu Institute of Technology in Japan. He worked as a student researcher and a researcher at Korea Institute of Science and Technology, Korea, from 2011 to 2017. His research interests include autostereoscopic 3D display, integral imaging, 3D computational reconstruction, digital holography, and digital holographic microscopy.
received his B.S. degree from Hankyong National University, Anseong, South Korea, in 2018, and his M.S. degree from Kyushu Institute of Technology, Fukuoka, Japan, in 2020, where he is currently pursuing a Ph.D. His research interests include integral imaging, three-dimensional (3D) computational reconstruction, night vision, photon counting, digital holographic microscopy, information security and 3D visualization.
received his Ph.D. in applied physics from the M. S. University of Baroda, India, in 2003. From 2002 to 2008, he worked as a Scientist at the Institute for Plasma Research, Gandhinagar, India. Presently he is a Professor in the Department of Physics, Sardar Patel University, India. He has more than 100 publications including peer-reviewed journal articles, conference proceedings and invited conference papers. His research interests include 3-D microscopes for cell imaging and identification, biomedical optical systems, digital holography and optical instrumentation.
received his B.S. and M.S. in Telecommunication Engineering from Pukyong National University, Pusan, Korea, in 2003 and 2005, respectively, and M.S. and Ph.D. in electrical and computer engineering from the University of Connecticut, Storrs, CT, USA, in 2010 and 2011, respectively. Currently, he is a full professor at Hankyong National University, Korea. He worked as a researcher at Samsung Electronics in Korea from 2005 to 2007. His research interests include 3D display, 3D signal processing, 3D biomedical imaging, 3D photon counting imaging, 3D information security, 3D object tracking, 3D underwater imaging, digital holographic microscopy, photon counting, and 3D visualization of objects under inclement weather conditions.
received his B.S. in Telecommunication Engineering from Pukyong National University, Busan, Korea, in 1996, and his M.S. and Ph.D. from Kyushu Institute of Technology, Fukuoka, Japan, in 2000 and 2003, respectively. He is an associate professor at Kyushu Institute of Technology, Japan. His research interests include medical imaging, blood flow analysis, 3D display, 3D integral imaging, digital holographic microscopy, and 3D biomedical imaging.
Journal of information and communication convergence engineering 2023; 21(1): 90-97
Published online March 31, 2023 https://doi.org/10.56977/jicce.2023.21.1.90
Copyright © Korea Institute of Information and Communication Engineering.
Hyun-Woo Kim ^{1}, Jaehoon Lee ^{1}, Arun Anand ^{2}, Myungjin Cho ^{3*}, and Min-Chul Lee^{1* }, Member, KIICE
^{1}Department of Computer Science and Networks, Kyushu Institute of Technology, Iizuka, Fukuoka 820-8502, Japan
^{2}Department of Physics, Sardar Patel University, Vallabh Vidyanagar, Anand 388120, India
^{3}School of ICT, Robotics, and Mechanical Engineering, IITC, Hankyong National University, Kyonggi-do 17579, South Korea
Correspondence to:Myungjin Cho^{1} (E-mail: mjcho@hknu.ac.kr), Min-chul Lee^{2} (E-mail: lee@csn.kyutech.ac.jp, Tel: +81-948-29-7699)
^{1}School of ICT, Robotics, and Mechanical Engineering, IITC, Hankyong National University, Kyonggi-do 17579, Republic of Korea
^{2}Department of Computer Science and Networks, Kyushu Institute of Technology, Fukuoka 820-8502, Japan
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Digital holographic microscopy (DHM) is a three-dimensional (3D) imaging technique that uses the phase information of coherent light. In the reconstruction process of DHM, a narrow region around the positive or negative sideband from the Fourier domain is windowed to avoid noise due to the DC spectrum of the hologram spectrum. However, the limited size of the window also degrades the high-frequency information of the 3D object profile. Although a large window can have more detailed information of the 3D object shape, the noise is increased. To solve this trade-off, we propose phase difference averaging (PDA). The proposed method yields high-frequency information of the specimen while reducing the DC noise. In this paper, we explain the reconstruction algorithm for this method and compare it to various conventional filtering methods including Gaussian, Wiener, average, median, and bilateral filtering methods.
Keywords: Digital holographic microscopy, Noise reduction, Phase error, Three-dimensional imaging
Light sensors, including the human eye, and digital arrays cannot detect phase information directly, but can only detect the intensity of light. For this reason, the interference characteristics of light have been used for obtaining phase information. In particular, holography is a technology that uses interference to detect the phase of light. It utilizes the phase difference between a reference beam and an object beam to record three-dimensional (3D) information for objects. This technique has been studied extensively since it was first proposed in 1948 [1]. Digital holography (DH) uses the same basic principles as holography, but uses image sensors instead of films for recording. Unlike films that cannot be reused after recording, DH can record images without storage limitations. Recently, DH has been applied to quantitative phase imaging (QPI) [2,3], microscopy [2-10], 3D image encryption [11], and 3D object recognition [12]. Among these applications, as digital holographic microscopy (DHM) can obtain the thickness profile of an object with high resolution, it is used for disease diagnosis [13,14], 3D profiling of microstructures [15-17], and microbial research [18, 19]. In DHM, one of the sidebands (corresponding to object information) in the Fourier domain is windowed to obtain the depth information. In this process, when a wide window is used, high frequency information of the object is obtained. However, the information in the DC spectrum, as well as the noise, also increases, leading to phase error. In contrast, when a narrow phase region is masked, the phase error decreases, but high frequency information about the object is lost. Furthermore, in interferometers, it is difficult to adjust the interference fringe density and set the size of the window used to mask the sideband. In this paper, we propose a phase error filtering method for DHM, which can address this ambiguity. We refer to this filtering method as phase differences averaging (PDA).
Fig. 1(a) and (b) show the 2D and 1D Fourier spectra of the recorded hologram, respectively. To obtain depth information from the recorded hologram, one of the sideband terms is windowed and cropped. As shown in Fig. 1(b), Rc is the size of the cropped region, and the green area under the +f_{o} term is the noise induced by the DC spectrum. To reduce this unwanted phase in image processing, we have proposed PDA. Our proposed method crops the 2D Fourier spectrum from narrow to wide regions in regular intervals to reconstruct the correct phase information by reducing the effect of the DC spectrum and high-frequency noise.
Fig. 2 shows a schematic of the PDA. Fig. 2(a) shows the Fourier spectrum of the recorded hologram, and Fig. 2(b) shows each DC shifted +f_{o} term after cropping the sideband around +f_{o} with different-sized windows. Zero padding was applied, as shown in Fig. 2(b), to obtain images of the same resolution via inverse Fourier transform on the sidebands windowed using different sizes. Fig. 2(c) shows the phase calculated using the cropped Fourier spectrum. The average of these obtained phase profiles was then computed to yield the high-frequency thickness information of the specimen while reducing the effect of the DC spectrum. The PDA process is very simple as it selects the center of the sideband in the Fourier domain and crops it from the smallest to the largest window size in regular intervals, regardless of the trade-off between the phase error and high-frequency information of the specimen.
A detailed explanation of PDA implementation is as follows. Two holograms are recorded, one with the object in the field of view (object hologram) and one with the medium surrounding the object in the field of view (reference hologram), to obtain the change in phase of the wavefront caused by the spatially-varying optical path length of the object. In addition, this method cancels out the noise caused by the surface of the mirror, beam splitter, object lens, and image sensor. They are analyzed separately by applying different sized windows (from smallest to largest), and their phase difference is computed. The computed phase difference corresponding to any one of the windows can be written as [20]
where n_{s} and n_{s} are the constant refractive indices of the specimen and surrounding medium, respectively; h is the thickness information of the specimen; λ is the vacuum wavelength of the source; and Δn is the difference in the refractive index. Here, h is one thickness value obtained with a particular masking window (Eq. (2)). Thus, the thickness information with the smallest masking window size is h_{1}, and numbers are assigned in order of increasing window size. The average of all the thickness information with different window sizes is [20]
In this study, we wished to crop the phase region as wide as possible, so the fringe pattern was as narrow as possible. For this reason, we used modified Mach-Zehnder interferometry using two spherical waves.
Fig. 3 shows the experimental setup used for the demonstration of PDA. The setup used a 532 nm laser diode module (3 mW output power). To avoid overlapping of red blood cells (RBCs), we prepared thin blood smears as specimens by dripping blood on a glass slide and spreading it thinly. Both the object and reference beams were magnified by 40 × (0.65 NA) objective lenses. In this experiment, we used constant reflective indices for the RBCs and the surrounding medium of 1.42 and 1.34 (reflective index of the blood plasma), respectively [13]. Holograms were recorded using a CMOS sensor (Basler, acA2500-14uc) with 2590(H) × 1942(V) pixel resolution, and a pixel size of 2.2 μm (H) × 2.2 μm (V). The CMOS sensor was placed at the image plane. The beam splitter in front of the sensor was tilted to make the fringe pattern as dense as allowed by the sampling criteria.
Fig. 4 shows the unfiltered 3D profile of the object (red blood cell) obtained using the smallest phase region (90(H) × 90(V) pixels) and the largest phase region (660(H) × 660(V) pixels). In this study, we used a Goldstein phase-unwrapping algorithm [21] to obtain continuous phase information from the wrapped phase. As mentioned earlier, when the phase region is small, as shown in Fig. 4(a), the approximate shape information of the object and the noise decrease; when the region is wide, as shown in Fig. 4(b), the shape information of the object includes details and the noise increases. Therefore, we applied filtering using the proposed method. In this study, the window size of the data used for reconstruction of the proposed method was increased at intervals of 30 pixels horizontally and vertically from 90(H) × 90(V) pixels to 660(H) × 660(V) pixels. As a result, we obtained a total of 20 thickness datapoints with the same image resolution.
As shown in Fig. 4(c), the noise in the 3D profile is reduced. In addition, we compared the proposed method with different conventional filtering methods. Figs. 4(d)-(h) show the results obtained using the conventional filters: a Gaussian filter (σ = 2), a Wiener filter (9(H) × 9(V) pixel filter size), an average filter and a median filter (5(H) × 5(V) pixels filter size), and a bilateral filter (σ = 2), respectively, applied to Fig. 4(b). The size of each conventional filter was chosen to retain as much high-frequency information as possible. As a result of comparing each 3D profile, no significant difference was found between the result of a Gaussian filter and the PDA. For numerical comparison, we calculated the signal-to-noise ratio (SNR) and mean squared error (MSE) of each 3D profile [22]. To compare only the noise coming from the DC spectrum, the comparison target of SNR and MSE uses the 3D profile of the smallest cropped phase region.
Table 1 shows the results of the numerical comparison between the PDA and various conventional filters. In Table 1, the MSE and SNR of the Gaussian filter show the best results among the conventional filtering methods. However, the Gaussian filtering results also show a higher MSE and lower SNR than PDA. Among the conventional filters compared in this section, the Gaussian filter, which shows the most effective value, is compared in more detail in the next subsection.
Units for magnetic properties.
MSE | SNR (dB) | |
---|---|---|
Unfiltered | 0.4827 | 19.5095 |
PDA | 0.1882 | 23.5991 |
Gaussian | 0.2340 | 22.6533 |
Wiener | 0.2838 | 21.8156 |
Average | 0.2781 | 21.9044 |
Median | 0.3162 | 21.3470 |
Bilateral | 0.4795 | 19.5388 |
We used statistical analysis to determine the advantages of the proposed method. For statistical analysis, we randomly selected 20 different RBCs and obtained thickness profile data using small to large masking windows. The results obtained with PDA were compared with those obtained with unfiltered, Gaussian, and median filtering (Fig. 5).
The horizontal line in Fig. 5 indicates the result obtained using PDA. For the unfiltered method, it can be seen that the larger the size of the masking window, the lower the SNR, as shown in Fig. 5(a). However, the variation increases with a larger masking window. For single-window filtering, the SNR is highest for small window sizes, but it also removes high-frequency object information. It can be seen that the SNR of PDA is the same as that of a single filter window of 270(H) × 270(V) pixels. Above this window size, the single-filteringwindow SNR decreases, while the PDA SNR remains high. As the fringe density changes, the filtering window size should also change. This is automatically done when using PDA. In addition, from Fig. 5(b) and 5(c), it can be seen that a larger sigma value of the Gaussian filter and larger filter size of the median filter results in better SNR values. However, these filtering operations smooth the obtained depth profile from phase differences, resulting in the loss of high-frequency information. PDA also considers this aspect. The variances of the Gaussian and median filtering are much larger than the variance obtained using PDA. The variance value is related to the stability of filtering, which means that there is a possibility that random noise may remain. To verify this, the correlation between SNR and MSE was used in this study. Fig. 6 shows the correlation between SNR and MSE.
Fig. 6 shows the correlation between SNR and MSE for different RBCs generated using the same method. The vertical axis represents the SNR value and the horizontal axis represents the MSE value. As shown in Fig. 6, It can be seen that PDA provides a higher SNR with lower mean square error, while retaining high spatial frequency information of the object.
As shown in Fig. 7, when the sigma value of the Gaussian filter increases, a lower error and better SNR result. However, Gaussian filtering also suppresses higher spatial frequency information of the object. Among the randomly selected RBCs, 10% of RBCs have a serious phase error, and even if the sigma value of the filter increases, these errors lead to a lower SNR (Fig. 7).
Fig. 8 illustrates the advantages of our proposed method. PDA can provide high-resolution, high-SNR thickness profiles compared to a single masking window. This is the strength of the proposed method. The RBCs in Fig. 8 have a flat surface, unlike the donut-shaped RBCs in Fig. 4. This is because the pressing force was strong in the process of making the specimen. However, this did not affect the experimental results.
In this study, we proposed a new filtering method for reducing the phase error from the DC spectrum in DHM and compared our proposed method with general filtering methods. The proposed method showed strength in terms of stability compared to other filtering methods, and it had the advantage of obtaining 3D data with a high spatial frequency and SNR. Filtering methods using a single masking window either provided a low spatial frequency (small window) or high noise (large window) 3D profile of the object. However, our proposed method did not require the size of the filter to be determined, and it had the advantage of being able to filter effectively, even in severe phase error conditions. The proposed method retained the detailed (high-frequency) information of the object while removing unwanted noise. Furthermore, because the proposed method averaged the thickness information after phase unwrapping of windowed sidebands of 20 different sizes, it also eliminated the phase error that occurs after phase unwrapping. Off-axis DHM combined the object and reference beams at an angle, leading to the separation of the DC spectrum and sidebands. However, in many cases, especially while using low temporally coherent sources such as LEDs, to have a larger usable field of view (where interference fringes exist), the angle between the object and reference beams must be very small, leading to low-density fringes. In such cases, the sideband information overlapped with the DC spectrum, resulting in low lateral resolution owing to the smaller size of the filtering window used. The proposed method is very useful under such conditions. In this study, the width interval of the windowed sideband is an arbitrary value, and the height data also used an arbitrary number of data points for the average. As the width interval became narrower and the number of height data points used for the average increased, the filtering effect was expected to be excellent. However, because there is a trade-off relationship between processing time and filtering quality, it is important for the user to select an appropriate value.
This work was supported by the Japan-Korea Basic Scientific Cooperation Program between JSPS and NRF, Grant number (JPJSBP 120228811), and this work was supported under the framework of the international cooperation program managed by the National Research Foundation of Korea(NRF-2022K2A9A2A08000152, FY2022).
Units for magnetic properties.
MSE | SNR (dB) | |
---|---|---|
Unfiltered | 0.4827 | 19.5095 |
PDA | 0.1882 | 23.5991 |
Gaussian | 0.2340 | 22.6533 |
Wiener | 0.2838 | 21.8156 |
Average | 0.2781 | 21.9044 |
Median | 0.3162 | 21.3470 |
Bilateral | 0.4795 | 19.5388 |
Byung-Gil Lee, Dong-hee Lim, Jin-soo Kim
Journal of information and communication convergence engineering 2017; 15(3): 175-181 https://doi.org/10.6109/jicce.2017.15.3.175