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This work is related to an Independent study of a senior ECEE student at the University of Memphis, Kamesh Balachandran, finished in May 2023. Results of this study were presented as a Poster in the 2023 Optica Imaging and Applied Optics Congress held in Boston (August 14-17).
This work is related to an Independent study of a senior ECEE student at the University of Memphis, Kamesh Balachandran, finished in May 2023. Results of this study were presented as a Poster in the 2023 Optica Imaging and Applied Optics Congress held in Boston (August 14-17).
A Digital Holographic Microscope (DHM) is an advanced optical interferometer that utilizes a microscopic imaging system to capture interference patterns, allowing for the reconstruction of amplitude and phase images of unstained samples. In-line DHM systems are characterized by completely aligned interfering beams, typically achieved by setting the interference angle to zero. Traditionally, reconstruction algorithms for in-line DHM systems rely on phase-shifting methods, which involve recording multiple interference patterns with laterally shifted fringes. These conventional methods require accurate knowledge of the phase shifts between the recorded holograms, which can be challenging to obtain experimentally and may lead to distorted and unreliable phase maps when inaccurate values are used. As an alternative, blind phase-shifting algorithms, which do not require prior knowledge of the phase shifts, have been proposed. In this undergraduate research project, we aim to explore the performance of an alternative approach based on Principal Component Analysis (PCA) for phase reconstruction in an in-line DHM system. In 2011, Vargas et al. proposed an approach based on PCA for reconstructing both amplitude and phase distributions in DHM [1]. In this undergraduate research project, we aim to investigate the performance of the PCA-based method for phase reconstruction in an in-line DHM system. Specifically, we will evaluate the accuracy and robustness of PCA-based phase reconstruction algorithms based on the number of phase-shifted images and the value of the phase step. The PCA algorithm will be tested under both noiseless and noisy conditions to assess its reliability in practical imaging scenarios. This research project seeks to contribute to the understanding of the potential of PCA as a viable method for phase reconstruction in DHM and its applicability in real-world imaging applications.
A Digital Holographic Microscope (DHM) is an advanced optical interferometer that utilizes a microscopic imaging system to capture interference patterns, allowing for the reconstruction of amplitude and phase images of unstained samples. In-line DHM systems are characterized by completely aligned interfering beams, typically achieved by setting the interference angle to zero. Traditionally, reconstruction algorithms for in-line DHM systems rely on phase-shifting methods, which involve recording multiple interference patterns with laterally shifted fringes. These conventional methods require accurate knowledge of the phase shifts between the recorded holograms, which can be challenging to obtain experimentally and may lead to distorted and unreliable phase maps when inaccurate values are used. As an alternative, blind phase-shifting algorithms, which do not require prior knowledge of the phase shifts, have been proposed. In this undergraduate research project, we aim to explore the performance of an alternative approach based on Principal Component Analysis (PCA) for phase reconstruction in an in-line DHM system. In 2011, Vargas et al. proposed an approach based on PCA for reconstructing both amplitude and phase distributions in DHM [1]. In this undergraduate research project, we aim to investigate the performance of the PCA-based method for phase reconstruction in an in-line DHM system. Specifically, we will evaluate the accuracy and robustness of PCA-based phase reconstruction algorithms based on the number of phase-shifted images and the value of the phase step. The PCA algorithm will be tested under both noiseless and noisy conditions to assess its reliability in practical imaging scenarios. This research project seeks to contribute to the understanding of the potential of PCA as a viable method for phase reconstruction in DHM and its applicability in real-world imaging applications.
PCA-based framework
The PCA-based phase reconstruction method employed in this research project is designed for in-line DHM systems, which utilize asynchronous phase-shifted holograms. In an in-line DHM system, the irradiance distribution of a hologram (h) can be expressed as hn (x, y) = a(x, y) + b(x, y) cos[ϕ(x, y) + δn] , where (x, y) represent the lateral spatial coordinates, a(x, y) represents the background illumination, b(x, y) represents the distribution of the fringes' contrast, ϕ(x, y) represents the phase distribution, and δn represents the phase shift (or phase step) introduced by the reference wave. By expanding the cosine of the sum of two angles, the hologram distribution (h) can be rewritten as hn (x, y) = a(x, y) + b(x, y) cos[ϕ(x, y)] cos[δn] – b(x, y) sin[ϕ(x, y)] sin[δn]. Consequently, the normalized hologram distribution is decomposed into two uncorrelated quadrature signals: hnorm,n (x, y) = hn (x, y) – a (x, y) = bcn(x, y) cos[δn] + bsn(x, y) sin[δn] being bc,n(x, y) = b cos[ϕ] and bs,n(x, y) = b sin[ϕ]. The PCA method estimates the bc,n(x, y) and bs,n(x, y) distributions without any prior knowledge of the phase shifts, allowing the reconstruction of the phase images as ϕ(x, y) = tan-1[bs,n / bc,n].
The PCA-based phase reconstruction method employed in this research project is designed for in-line DHM systems, which utilize asynchronous phase-shifted holograms. In an in-line DHM system, the irradiance distribution of a hologram (h) can be expressed as hn (x, y) = a(x, y) + b(x, y) cos[ϕ(x, y) + δn] , where (x, y) represent the lateral spatial coordinates, a(x, y) represents the background illumination, b(x, y) represents the distribution of the fringes' contrast, ϕ(x, y) represents the phase distribution, and δn represents the phase shift (or phase step) introduced by the reference wave. By expanding the cosine of the sum of two angles, the hologram distribution (h) can be rewritten as hn (x, y) = a(x, y) + b(x, y) cos[ϕ(x, y)] cos[δn] – b(x, y) sin[ϕ(x, y)] sin[δn]. Consequently, the normalized hologram distribution is decomposed into two uncorrelated quadrature signals: hnorm,n (x, y) = hn (x, y) – a (x, y) = bcn(x, y) cos[δn] + bsn(x, y) sin[δn] being bc,n(x, y) = b cos[ϕ] and bs,n(x, y) = b sin[ϕ]. The PCA method estimates the bc,n(x, y) and bs,n(x, y) distributions without any prior knowledge of the phase shifts, allowing the reconstruction of the phase images as ϕ(x, y) = tan-1[bs,n / bc,n].
Framework of the study
This study focuses on evaluating the performance of the PCA-based algorithm using simulated phase-shifted holograms. The phantom phase image was utilized as the reference to generate a series of phase-shifted holograms. These phase-shifted holograms were then used as input images for the PCA-based algorithm to reconstruct the phantom phase distribution. To assess the accuracy of the reconstructed phase map, we have employed two quantitative metrics: mean-square error (MSE) and correlation values, which are calculated by comparing the true phase map with the estimated phase map obtained from the PCA-based algorithm. These evaluation metrics provide quantitative insights into the estimated phase distribution.
This study focuses on evaluating the performance of the PCA-based algorithm using simulated phase-shifted holograms. The phantom phase image was utilized as the reference to generate a series of phase-shifted holograms. These phase-shifted holograms were then used as input images for the PCA-based algorithm to reconstruct the phantom phase distribution. To assess the accuracy of the reconstructed phase map, we have employed two quantitative metrics: mean-square error (MSE) and correlation values, which are calculated by comparing the true phase map with the estimated phase map obtained from the PCA-based algorithm. These evaluation metrics provide quantitative insights into the estimated phase distribution.
Results
Results
This undergraduate research study comprehensively evaluates the quality of reconstructed phase images using the PCA-based algorithm. Firstly, we analyze the reconstructed phase images for different numbers of phase-shifted in-line holograms, ranging from 2 to 20. In particular, we analyze 2, 3, 5, 7, 9, 11, 15, and 20 holograms. For each number of phase-shifted holograms, we analyze 100,000 combinations of random phase steps of the phase-shifted holograms from the range of 0 to 2π. Our findings reveal two important results. Firstly, we determine that the minimum number of phase-shifted holograms required for the PCA-based algorithm to perform accurately is 3. This implies that at least 3 phase-shifted inline holograms are required to reconstruct a phase image. Secondly, we observe that the quality of the reconstructed phase depends on the phase steps. As the below figure shows, incorrect values of the phase steps lead to imprecise phase measurements (e.g., high MSE value and low correlation). In fact, accurate phase images, with an MSE value lower than 0.001, were only obtained for specific combinations of phase steps. For instance, out of the 100,000 possibilities, only 2,260 phase-steps combinations provided an accurate estimated phase map, indicating a success rate of 2.26% using 3 holograms. The highest success rate is 2.97% using 9 holograms.
This undergraduate research study comprehensively evaluates the quality of reconstructed phase images using the PCA-based algorithm. Firstly, we analyze the reconstructed phase images for different numbers of phase-shifted in-line holograms, ranging from 2 to 20. In particular, we analyze 2, 3, 5, 7, 9, 11, 15, and 20 holograms. For each number of phase-shifted holograms, we analyze 100,000 combinations of random phase steps of the phase-shifted holograms from the range of 0 to 2π. Our findings reveal two important results. Firstly, we determine that the minimum number of phase-shifted holograms required for the PCA-based algorithm to perform accurately is 3. This implies that at least 3 phase-shifted inline holograms are required to reconstruct a phase image. Secondly, we observe that the quality of the reconstructed phase depends on the phase steps. As the below figure shows, incorrect values of the phase steps lead to imprecise phase measurements (e.g., high MSE value and low correlation). In fact, accurate phase images, with an MSE value lower than 0.001, were only obtained for specific combinations of phase steps. For instance, out of the 100,000 possibilities, only 2,260 phase-steps combinations provided an accurate estimated phase map, indicating a success rate of 2.26% using 3 holograms. The highest success rate is 2.97% using 9 holograms.
The below reports these probability rates. It is important to highlight that traditional phase-shifting methods are quite accurate, providing a MSE of the order of 10-32 and a correlation of 1 for three phase-shifted inline holograms if one knows the exact phase shifts. Despite our best efforts, we were unable to identify any discernible relationship between the phase steps in the successful reconstructions. This suggests that accurate reconstructed phase images using the PCA-based algorithm are not solely dependent on the phase steps, but rather on complex interactions among various factors. Further research may be needed to understand why selected phase steps only reconstruct accurate phase images.
The below reports these probability rates. It is important to highlight that traditional phase-shifting methods are quite accurate, providing a MSE of the order of 10-32 and a correlation of 1 for three phase-shifted inline holograms if one knows the exact phase shifts. Despite our best efforts, we were unable to identify any discernible relationship between the phase steps in the successful reconstructions. This suggests that accurate reconstructed phase images using the PCA-based algorithm are not solely dependent on the phase steps, but rather on complex interactions among various factors. Further research may be needed to understand why selected phase steps only reconstruct accurate phase images.
Finally, we analyze the sensitivity of the PCA-based algorithm to noisy conditions. To generate noisy phase distributions, we added a white Gaussian distribution to the true phase map with varying signal-to-noise ratios (SNR), ranging from 5 dB to 45 dB. For a each SNR value, 50 noisy phase images were generated by the MATLAB built-in function awgn.
Finally, we analyze the sensitivity of the PCA-based algorithm to noisy conditions. To generate noisy phase distributions, we added a white Gaussian distribution to the true phase map with varying signal-to-noise ratios (SNR), ranging from 5 dB to 45 dB. For a each SNR value, 50 noisy phase images were generated by the MATLAB built-in function awgn.
For evaluating the algorithm's performance in the presence of noise, we selected one successful case each for 3 and 15 phase-shifted holograms. Tables 2 and 3 report the mean and standard deviation (std) of the MSE and correlation values between the ground truth phase image and the reconstructed one. These results reveal that the PCA algorithm is not suitable for holograms with low SNR values below 20 dB, as the MSE values between the reconstructed phase images and the ground truth one are higher than 0.001. However, if the noise is minimal (SNR > 20 dB), the PCA algorithm is robust regardless of the number of phase-shifted images used. This suggests that the PCA-based can reliably reconstruct accurate phase images even in the presence of moderate levels of noise when SNR is higher than 20 dB. It is important to mention that traditional PS-DHM methods are quite robust to noise conditions, providing a MSE of the order of 10-32 for the range from 5 dB to 45 dB
For evaluating the algorithm's performance in the presence of noise, we selected one successful case each for 3 and 15 phase-shifted holograms. Tables 2 and 3 report the mean and standard deviation (std) of the MSE and correlation values between the ground truth phase image and the reconstructed one. These results reveal that the PCA algorithm is not suitable for holograms with low SNR values below 20 dB, as the MSE values between the reconstructed phase images and the ground truth one are higher than 0.001. However, if the noise is minimal (SNR > 20 dB), the PCA algorithm is robust regardless of the number of phase-shifted images used. This suggests that the PCA-based can reliably reconstruct accurate phase images even in the presence of moderate levels of noise when SNR is higher than 20 dB. It is important to mention that traditional PS-DHM methods are quite robust to noise conditions, providing a MSE of the order of 10-32 for the range from 5 dB to 45 dB
Outcomes and Learning Objectives
For this research undergraduate project, we have modified the original MATLAB code for the PCA-based algorithm [2], which was designed for macroscopic interferometry, to be used in in-line digital holographic microscopy system. Our results reveal that the minimum number of phase-shifted interferograms required for the PCA-based algorithm is 3. Furthermore, accurate reconstructed phase maps (i.e., MSE value < 0.001) can only be achieved for specific combinations of phase steps. Out of the 100,000 possibilities tested, only 2,260 phase-step combinations resulted in accurate estimated phase maps, indicating a success rate of 2.26% when using 3 interferograms. The highest success rate achieved was 2.97% when using 9 interferograms. These results suggest that the PCA-based method is not deterministic, and the quality of the reconstructed phase image is highly dependent on the chosen phase steps. In addition, the PCA algorithm is not suitable for noisy interferograms with a SNR value lower than 20 dB. However, if the SNR value is equal to or higher than 20 dB, the algorithm exhibits robustness, providing accurate phase maps independently of the number of phase-shifted images. This indicates that the PCA-based algorithm is resilient to moderate noise, but its performance is highly dependent on the chosen phase steps.
The learning objectives of this research undergraduate project were:
1. Understand the generation of inline holograms.
2. Understand the PCA algorithm created by Vargas et al. and modify it for inline DHM holograms.
3. Analyze the performance of the PCA algorithm for a different number of phase-shifted holograms.
4. Generate noisy inline holograms by using MATLAB built-in functions.
5. Analyze the performance of the PCA algorithm for different noisy conditions.
6. Assess the accuracy and robustness of the PCA algorithm.
Credits
Credits
[1] J. Vargas, J. Antonio Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis," Opt. Letters 36, 1326–1328 (2011).
[2] Original codes written by Vargas et al., https://drive.google.com/file/d/0B_agtNXaoRd6OTY5YTdhZDQtOWJlZi00NTQ2LWFhNzktZDM0OGU3NDg5OTAx/view?sort=name&layout=list&num=50&pli=1&resourcekey=0-0oGv8utqCqwelaUBDjdpcw
[3] Codes were developed in Matlab R2023a. Natick, Massachusetts: The MathWorks Inc.
Citation
Citation
If using our modified codes for publication, please kindly cite the following:
If using our modified codes for publication, please kindly cite the following:
- K. Balachandran, R. Castaneda, A. Doblas, “Evaluation of the robustness and accuracy of PCA-based algorithms for in-line Digital Holographic Microscopy,” Optica Imaging and Applied Optics Congress, paper JTu4A.35 (2023).
Poster_OSA2023-PCA_FINAL.pdfSupport or Contact The Principal Investigator is Dr. Ana Doblas (adoblas@umassd.edu)
Support or Contact The Principal Investigator is Dr. Ana Doblas (adoblas@umassd.edu)
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