- Email: [email protected]
Reconstruction method for samples with refractive index discontinuities in optical diffraction tomography
Optics and Lasers in Engineering 94 (2017) 58–62
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Reconstruction method for samples with refractive index discontinuities in optical diffraction tomography Xichao Ma, Wen Xiao, Feng Pan
MARK
⁎
Key Laboratory of Precision Opto-mechatronics Technology, School of Instrumentation Science & Optoelectronics Engineering, Beihang University, Beijing 100191, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Optical diffraction tomography Tomographic image processing Image reconstruction techniques
We present a reconstruction method for samples containing localized refractive index (RI) discontinuities in optical diffraction tomography. Abrupt RI changes induce regional phase perturbations and random spikes, which will be expanded and strengthened by existing tomographic algorithms, resulting in contaminated reconstructions. This method avoids the disturbance by recognition and separation of the discontinuous regions, and recombination of individually reconstructed data. Three-dimensional RI distributions of two fusion spliced optical fibers with different typical discontinuities are demonstrated, showing distinctly detailed structures of the samples as well as the positions and estimated shapes of the discontinuities.
1. Introduction Optical diffraction tomography (ODT) is a powerful technique for measurements of internal structures of micro-samples, providing access to quantitative three-dimensional (3D) distributions of refractive index (RI). It is one of the most important approaches for non-invasive observation of biomedical samples [1–5] as well as structure inspection of micro-optical devices [6–9]. In the process of tomographic measurement, it is required to record a series of complex wavefronts transmitted through the sample under test from different illumination angles using laser interferometry including digital holography [10–12] and phase shifting interferometry [13–15]. The restored amplitude and phase data are then numerically processed to reconstruct the 3D RI distribution of the sample using one of the tomographic algorithms such as filtered backprojection (FBPJ) [16], filtered backpropagation (FBPP) [17] or algebraic reconstruction technique (ART) [18]. Considerable effort has been put into improving image quality of ODT, including correction of radial run-out [19,20], extension of accurate reconstruction volumes [12,21–23], noise suppression [24] and approaches for complex scattering [25,26]. However, in certain but not rare conditions, samples could contain RI discontinuities which induce serious disturbance to reconstructed results. Ordinarily, ODT samples, such as cells or optical fibers, possess moderate fluctuations and continuous distributions of RI so that existing ODT algorithms are applicable. If drastic RI fluctuations exist on large scales, reconstruction usually depends on higher-order nonlinear algorithms, which are complicated and require extensive computations [27]. In practice, ⁎
Corresponding author. E-mail address: [email protected] (F. Pan).
http://dx.doi.org/10.1016/j.optlaseng.2017.03.003 Received 4 January 2017; Received in revised form 3 March 2017; Accepted 7 March 2017 0143-8166/ © 2017 Published by Elsevier Ltd.
abrupt RI changes are likely to exist within small, limited regions in typical ODT samples. These include tiny air bubbles enclosed in the sample, dust particles mixed in the medium, intrinsic structural jumps of some optical components, to name but a few. In this condition, scattering effects are localized, thus not impairing the entire phase image so that cumbersome nonlinear methods are not essential. However, the phase perturbations induced by regional RI discontinuities still contaminate the reconstructions seriously when using existing tomographic algorithms. Due to extra scattering caused by abrupt changes of RI, the phase values in the vicinity are impacted to be cluttered. When phase unwrapping is necessary, as is the usual case, the disarray is aggravated. Large phase jumps and random spikes abound in these regions, forming randomly fluctuating stripes in the sinograms, which obscure the useful data. More importantly, the localized phase disorder will be expanded and strengthened during tomographic reconstruction. When phase images are extended from 2D to 3D by back projecting or propagating, the disordered specks spread as well. After superposition of the extended images of all angles, the cluttered phase values within small regions expand to the entire range, making some important structures indistinct. Therefore, in this paper, we present a reconstruction method to avoid the spreading disturbance of regional RI discontinuities by recognition and separation of discontinuous regions, and recombination of individually reconstructed data. 3D reconstructions of RI distribution are available with detailed internal structures as well as the positions and estimated shapes of the discontinuous regions.
Optics and Lasers in Engineering 94 (2017) 58–62
X. Ma et al.
phase image is free from large phase jumps and random spikes. Therefore, each 2D phase image is divided into two images: one contains only the shapes of the discontinuous regions (termed as discontinuity images) and the other keeps the phase data without strong disturbance (termed as substituted phase images). Subsequently, both sets of images are processed individually. The discontinuity images are processed with FBPJ for positions and shape estimations of the discontinuities, the points of which are extracted with a threshold. FBPP is applied to reconstruct the RI distribution from the substituted phase images. In practice, the 2D amplitude images are not affected intensely by the discontinuities. Therefore, the amplitude images can be used for FBPP without the abovementioned modifications. Finally, a complete 3D distribution is acquired by recombination of the two individual maps. The points of discontinuities are inserted into the sample distribution to replace corresponding points.
Fig. 1. Schematic diagram of the experimental setup: M, mirror; PBS, polarizing beam splitter; λ/2, half-wave plate; SF, spatial filter with collimating lens; MPS, micropositioning stage with the measured sample; MO, microscope objective; TL, tube lens; BS, beam splitter; OB, object beam; RB, reference beam.
2. Reconstruction method 3. Experimental setup
The main idea of our method is separating the disturbing regions from the phase images and reconstructing individually. To recognize the discontinuities, the gradient magnitude of a phase image is calculated with a Sobel operator. The strong fluctuations in these regions make the gradient magnitude higher than in other areas. Therefore, a proper threshold is sufficient to detect these regions, generating a binary mask. The phase jumps and random spikes make the phase values in these regions highly disordered and thus unreliable, making it impossible to reconstruct correct RI from them. Consequently, the phase values in them are discarded and only the shapes of the specks are retained. In order to simulate the accumulation effect of phase distribution, the Euclidean distance transform [28] of the binary mask is computed, making bulges in the recognized regions. On the other hand, the recognized regions in the phase image are substituted with data from surface fitting of the surrounding area. The purpose of the substitution is to remove the drastic fluctuations. Therefore, it is sufficient to perform polynomial fitting on a relatively smooth area surrounding these regions. After data substitution, the
The experimental setup depicted in Fig. 1 is based on a MachZehnder off-axis holographic interferometer in the sample-rotating configuration [11,25]. The laser beam (λ=532 nm) is divided into an object beam and a reference beam, both of which are filtered and expanded by spatial filters and collimated to produce plane waves. The sample, an optical fiber here, is held vertically by a fiber holder on a micro-positioning stage. The object beam traverses the sample and is captured by an afocal imaging system composed of the MO (Sigma Koki, 20×, NA =0.4, infinity corrected) and the tube lens (f =200 mm). The interference pattern is recorded by the camera (Lu125M, Lumenera Co., Ontario, Canada) as a digital hologram. The sample is immersed in refractive-index-matching liquid (n =1.45) to reduce diffraction. 180 holograms are recorded with a 2° step during a full rotation of 360°. An extra image is recorded without the sample as a reference hologram for phase curvature compensation.
Fig. 2. Influences of discontinuities on phase images: (a) An unwrapped phase image of the spliced single-mode fiber with air bubbles marked with arrows; (b) Profile along the dashed line in (a) with the bubble sections marked with boxes; (c-d) Sinograms of Sections A and B, respectively, with θ indicating the illumination angle; (e) Profiles along the dashed lines in (c) and (d).
59
Optics and Lasers in Engineering 94 (2017) 58–62
X. Ma et al.
Fig. 3. Influences of discontinuities on 3D reconstructions: (a-b) Reconstructed 3D maps from raw phase data using FBPJ and 3D FBPP, respectively; (c-d) Cross-section images of Sections A and B in (b), respectively; (e) Profiles along the dashed lines in (c) and (d); (f-g) Amplitude distributions of the Fourier transform of (c) and (d), respectively.
Fig. 4. Procedures of the separation: (a) Raw phase image; (b) Gradient distribution of (a); (c) Recognized and separated regions; (d) Fitted surface of the area marked with the box in (a); (e) Substituted phase image.
4. Experimental results
avoid possible errors. Unwanted phase jumps and random spikes abound in the vicinity of the discontinuities, as illustrated in Fig. 2(b). Unlike the sinogram of a bubble-free slice [Fig. 2(c)], the sinogram of an influenced slice contains a randomly fluctuating stripe [Fig. 2(d)], which disturbs the data significantly, as shown in Fig. 2(e). The reconstructed results from raw phase data are presented in Fig. 3. It is clearly observed that the slices containing discontinuities are seriously contaminated in the 3D maps reconstructed with FBPJ (Fig. 3(a)) and the 3D version of FBPP (Fig. 3(b)), making it impossible to distinguish the cores and the cladding according to their different RI. Note that the disturbed part, in this case the fusion interface, is precisely the most anticipated area. Compared with a bubble-free slice (Fig. 3(c)), the influenced slices (Fig. 3(d)) does not reveal structural information of the sample, as illustrated in Fig. 3(e). The influence can also be expounded in the frequency domain. Each 2D scattered field
In order to verify the method, we measured two samples containing different typical discontinuities. The first sample is a fusion spliced optical fiber between two single-mode fibers, which contains core misalignment and tiny air bubbles in the fusion area. Air bubbles are very likely to appear in the fusion interface due to fiber-end contamination such as dust, fiber-glass powder, moisture, fiber-end roughness, as well as an improper fusion arc current. Therefore, this sample is of considerable practical significance. Due to the large difference between the RI inside the bubbles and that of the fiber, discontinuities occur in the interfaces. Fig. 2(a) demonstrates an unwrapped phase image of the sample with three air bubbles marked with arrows. We applied discrete cosine transform based unweighted least squares algorithm [29] for phase unwrapping to 60
Optics and Lasers in Engineering 94 (2017) 58–62
X. Ma et al.
Fig. 5. Reconstruction results of the first sample: (a) Sinogram from the separated discontinuity images with θ indicating the illumination angle; (b) Reconstruction from (a); (c) Sinogram from the substituted phase images; (d) Reconstruction from (c); (e) 3D map of the recombined reconstruction with discontinuities in red, cores in blue and the cladding in gray.
Fig. 6. Reconstruction results of the second sample: (a) An unwrapped phase image of the spliced fiber between a single-mode fiber and a multi-mode fiber with discontinuities marked with boxes; (b) Sinogram of Section A with θ indicating the illumination angle; (c) 3D map reconstructed from the raw phase data with FBPP; (d) Recognized and separated discontinuous regions; (e) Substituted phase image; (f) Recombined 3D reconstruction with discontinuities in red, cores in blue and the cladding in gray.
image (Fig. 4(e)) is smoother in contrast to the raw image (Fig. 4(a)). The discontinuity images of all illumination angles form a sinogram for each slice of the sample, as shown in Fig. 5(a). The estimated shapes of the discontinuities reconstructed from them with FBPJ are displayed in Fig. 5(b). Reconstructed slices are then stacked up to form a 3D map. The sinograms from the substituted phase images (Fig. 5(c)) are free from strong disturbance, as compared with Fig. 2(d). FBPP is preferred to reconstruct the RI distribution for better accuracy (Fig. 5(d)). Here the 3D version of FBPP is applied in consideration of the structural variation along the vertical axis and the amplitude images restored from the holograms are used without the abovementioned procedures. Finally, the two maps are recombined for a complete reconstruction. In Fig. 5(e), the cores (blue parts) and the cladding (gray parts) are
corresponds to a hemispherical surface in the frequency domain, according to the Fourier diffraction theorem [30]. The random phase perturbations enhance high frequency components unevenly, making the restored frequency spectrum highly inhomogeneous (Fig. 3(g)), in contrast with a regular one (Fig. 3(f)). As a result, informative reconstructions are damaged. The procedures of separation of the phase images are shown in Fig. 4. The regions of discontinuities present higher gradient magnitude due to strong fluctuations, as in Fig. 4(b). These regions are separated with a proper threshold and the Euclidean distance transform is calculated, as displayed in Fig. 4(c). The recognized regions in the phase image are substituted with data from the cubic polynomial surface fitting of the marked area (Fig. 4(d)). The substituted phase 61
Optics and Lasers in Engineering 94 (2017) 58–62
X. Ma et al.
distinguished by their distinct RI values. Because there are no solid RI values in the air bubbles, the points in them (red parts) are set to a distinctive constant to highlight their outlines. The misalignment of cores is readily observable, along with the positions and approximate shapes of the air bubbles, which are indistinct before (Fig. 3(a) and (b)). Some additional fusion defects (blue parts other than the cores) are also visible, which are likely to be induced by disturbed internal stress. The other sample we studied is a fusion spliced optical fiber between a single-mode fiber and a multi-mode fiber. The diameter of the core of the multi-mode fiber (about 60 µm) is much larger than that of the single-mode fiber (about 8 µm). Therefore, RI discontinuities occur at the joint interface, which cause irregular boundaries with steep phase changes, as shown in Fig. 6(a). Unlike air bubbles, this kind of discontinuity is intrinsic in the structure. In the sinogram of the interface (Fig. 6(b)), abrupt changes are abundant. Consequently, a direct reconstruction from the raw phase data suffers from a clutter of fragments (Fig. 6(c)), so that the fusion area is indiscernible. The recognition of discontinuities is also based on the gradient magnitude of the phase images (Fig. 6(d)). The recognized regions are substituted with the cubic polynomial surface fitting data and a flat border, as illustrated in Fig. 6(e). Finally, the 3D RI distribution is obtained by recombination of the reconstructions from the discontinuity images by FBPJ and the substituted phase images by 3D FBPP, in which the amplitude images are unprocessed. The 3D map presented in Fig. 6(f) reveals the detailed structures of the fusion interface (red parts), including slight deformation of the cores. Another discontinuous part (outlying red part) is also exhibited, which appears to be a dust particle attached to the fiber.
[2]
[3]
[4] [5]
[6] [7] [8] [9] [10] [11]
[12]
[13] [14] [15] [16] [17]
5. Conclusion
[18]
In summary, this paper describes a method for tomographic reconstruction of samples containing localized RI discontinuities. These discontinuities cause regional perturbations and random spikes in 2D phase images, which will be expanded and strengthened by existing tomographic algorithms, resulting in contaminated reconstructions. The disturbance is avoided by recognition and separation of these regions. The eventual 3D RI distribution is acquired by recombination of individually reconstructed data from the separated discontinuity images and the substituted phase images filled with surface fitting data. Two samples with different typical RI discontinuities are demonstrated: one is a fusion spliced optical fiber with tiny air bubbles enclosed and the other also a spliced fiber but containing intrinsic structural RI jumps. The 3D maps obtained with this method are free from spreading disturbance and show obviously important structures of the samples as well as the positions and estimated shapes of the discontinuities. The proposed method is of practical significance and will find important applications in 3D imaging of micro-optical components, as well as biomedical samples in case of accidental foreign matters or air bubbles in the culture media.
[19] [20] [21] [22]
[23] [24] [25] [26] [27]
[28]
[29]
References
[30]
[1] Kim Y, Shim H, Kim K, Park H, Heo JH, Yoon J, et al. Common-path diffraction
62
optical tomography for investigation of three-dimensional structures and dynamics of biological cells. Opt Express 2014;22:10398–407. Su J-W, Hsu W-C, Chou C-Y, Chang C-H, Sung K-B. Digital holographic microtomography for high-resolution refractive index mapping of live cells. J Biophotonics 2013;6:416–24. Kim K, Yoon H, Diez-Silva M, Dao M, Dasari RR, Park Y. High-resolution threedimensional imaging of red blood cells parasitized by Plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography. J Biomed Opt 2013;19:011005. Charrière F, Marian A, Montfort F, Kuehn J, Colomb T, Cuche E, et al. Cell refractive index tomography by digital holographic microscopy. Opt Lett 2006;31:178–80. Charrière F, Pavillon N, Colomb T, Depeursinge C, Heger TJ, Mitchell EAD, et al. Living specimen tomography by digital holographic microscopy: morphometry of testate amoeba. Opt Express 2006;14:7005–13. Kim K, Yoon J, Park Y. Large-scale optical diffraction tomography for inspection of optical plastic lenses. Opt Lett 2016;41:934–7. Pan Z, Liang Z, Li S, Weng J, Zhong J. Microtomography of the polarizationmaintaining fiber by digital holography. Opt Fiber Technol 2015;22:46–51. Lin Y-C, Cheng C-J. Sectional imaging of spatially refractive index distribution using coaxial rotation digital holographic microtomography. J Opt 2014;16:065401. Gorski W, Osten W. Tomographic imaging of photonic crystal fibers. Opt Lett 2007;32:1977–9. Malek M, Khelfa H, Picart P, Mounier D, Poilâne C. Microtomography imaging of an isolated plant fiber: a digital holographic approach. Appl Opt 2016;55:A111–21. Belashov AV, Petrov NV, Semenova IV. Accuracy of image-plane holographic tomography with filtered backprojection: random and systematic errors. Appl Opt 2016;55:81–8. Kostencka J, Kozacki T, Kuś A, Kemper B, Kujawińska M. Holographic tomography with scanning of illumination: space-domain reconstruction for spatially invariant accuracy. Biomed Opt Express 2016;7:4086–101. Sung Y, Choi W, Fang-Yen C, Badizadegan K, Dasari RR, Feld MS. Optical diffraction tomography for high resolution live cell imaging. Opt Express 2009;17:266–77. Choi W, Fang-Yen C, Badizadegan K, Oh S, Lue N, Dasari RR, et al. Tomographic phase microscopy. Nat Methods 2007;4:717–9. Xiu P, Zhou X, Kuang C, Xu Y, Liu X. Controllable tomography phase microscopy. Opt Lasers Eng 2015;66:301–6. Kak AC, Slaney M. Principles of Computerized Tomographic Imaging. New York, US: Society for Industrial and Applied Mathematics; 2001. Devaney AJ. A filtered backpropagation algorithm for diffraction tomography. Ultrason Imaging 1982;4:336–50. Gordon R. A tutorial on art (algebraic reconstruction techniques). IEEE Trans Nucl Sci 1974;21:78–93. Gorski W. Tomographic microinterferometry of optical fibers. Opt Eng 2006;45:125002. Jeon Y, Hong CK. Rotation error correction by numerical focus adjustment in tomographic phase microscopy. Opt Eng 2009;48:105801. Ma X, Xiao W, Pan F. Accuracy improvement in digital holographic microtomography by multiple numerical reconstructions. Opt Lasers Eng 2016;86:338–44. Choi W, Fang-Yen C, Badizadegan K, Dasari RR, Feld MS. Extended depth of focus in tomographic phase microscopy using a propagation algorithm. Opt Lett 2008;33:171–3. Yablon AD. Multifocus tomographic algorithm for measuring optically thick specimens. Opt Lett 2013;38:4393–6. Kostencka J, Kozacki T, Dudek M, Kujawińska M. Noise suppressed optical diffraction tomography with autofocus correction. Opt Express 2014;22:5731–45. Kostencka J, Kozacki T, Kuś A, Kujawińska M. Accurate approach to capillarysupported optical diffraction tomography. Opt Express 2015;23:7908–23. Kamilov US, Papadopoulos IN, Shoreh MH, Goy A, Vonesch C, Unser M, et al. Learning approach to optical tomography. Optica 2015;2:517–22. Tsihrintzis GA, Devaney AJ. Higher-order (nonlinear) diffraction tomography: reconstruction algorithms and computer simulation. IEEE Trans Image Process 2000;9:1560–72. Maurer CR, Qi R, Raghavan V. A linear time algorithm for computing exact Euclidean distance transforms of binary images in arbitrary dimensions. IEEE Trans Pattern Anal Mach Intell 2003;25:265–70. Backoach O, Kariv S, Girshovitz P, Shaked NT. Fast phase processing in off-axis holography by CUDA including parallel phase unwrapping. Opt Express 2016;24:3177–88. Wolf E. Three-dimensional structure determination of semi-transparent objects from holographic data. Opt Commun 1969;1:153–6.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Reconstruction method for samples with refractive index discontinuities in optical diffraction tomography Xichao Ma, Wen Xiao, Feng Pan
MARK
⁎
Key Laboratory of Precision Opto-mechatronics Technology, School of Instrumentation Science & Optoelectronics Engineering, Beihang University, Beijing 100191, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Optical diffraction tomography Tomographic image processing Image reconstruction techniques
We present a reconstruction method for samples containing localized refractive index (RI) discontinuities in optical diffraction tomography. Abrupt RI changes induce regional phase perturbations and random spikes, which will be expanded and strengthened by existing tomographic algorithms, resulting in contaminated reconstructions. This method avoids the disturbance by recognition and separation of the discontinuous regions, and recombination of individually reconstructed data. Three-dimensional RI distributions of two fusion spliced optical fibers with different typical discontinuities are demonstrated, showing distinctly detailed structures of the samples as well as the positions and estimated shapes of the discontinuities.
1. Introduction Optical diffraction tomography (ODT) is a powerful technique for measurements of internal structures of micro-samples, providing access to quantitative three-dimensional (3D) distributions of refractive index (RI). It is one of the most important approaches for non-invasive observation of biomedical samples [1–5] as well as structure inspection of micro-optical devices [6–9]. In the process of tomographic measurement, it is required to record a series of complex wavefronts transmitted through the sample under test from different illumination angles using laser interferometry including digital holography [10–12] and phase shifting interferometry [13–15]. The restored amplitude and phase data are then numerically processed to reconstruct the 3D RI distribution of the sample using one of the tomographic algorithms such as filtered backprojection (FBPJ) [16], filtered backpropagation (FBPP) [17] or algebraic reconstruction technique (ART) [18]. Considerable effort has been put into improving image quality of ODT, including correction of radial run-out [19,20], extension of accurate reconstruction volumes [12,21–23], noise suppression [24] and approaches for complex scattering [25,26]. However, in certain but not rare conditions, samples could contain RI discontinuities which induce serious disturbance to reconstructed results. Ordinarily, ODT samples, such as cells or optical fibers, possess moderate fluctuations and continuous distributions of RI so that existing ODT algorithms are applicable. If drastic RI fluctuations exist on large scales, reconstruction usually depends on higher-order nonlinear algorithms, which are complicated and require extensive computations [27]. In practice, ⁎
Corresponding author. E-mail address: [email protected] (F. Pan).
http://dx.doi.org/10.1016/j.optlaseng.2017.03.003 Received 4 January 2017; Received in revised form 3 March 2017; Accepted 7 March 2017 0143-8166/ © 2017 Published by Elsevier Ltd.
abrupt RI changes are likely to exist within small, limited regions in typical ODT samples. These include tiny air bubbles enclosed in the sample, dust particles mixed in the medium, intrinsic structural jumps of some optical components, to name but a few. In this condition, scattering effects are localized, thus not impairing the entire phase image so that cumbersome nonlinear methods are not essential. However, the phase perturbations induced by regional RI discontinuities still contaminate the reconstructions seriously when using existing tomographic algorithms. Due to extra scattering caused by abrupt changes of RI, the phase values in the vicinity are impacted to be cluttered. When phase unwrapping is necessary, as is the usual case, the disarray is aggravated. Large phase jumps and random spikes abound in these regions, forming randomly fluctuating stripes in the sinograms, which obscure the useful data. More importantly, the localized phase disorder will be expanded and strengthened during tomographic reconstruction. When phase images are extended from 2D to 3D by back projecting or propagating, the disordered specks spread as well. After superposition of the extended images of all angles, the cluttered phase values within small regions expand to the entire range, making some important structures indistinct. Therefore, in this paper, we present a reconstruction method to avoid the spreading disturbance of regional RI discontinuities by recognition and separation of discontinuous regions, and recombination of individually reconstructed data. 3D reconstructions of RI distribution are available with detailed internal structures as well as the positions and estimated shapes of the discontinuous regions.
Optics and Lasers in Engineering 94 (2017) 58–62
X. Ma et al.
phase image is free from large phase jumps and random spikes. Therefore, each 2D phase image is divided into two images: one contains only the shapes of the discontinuous regions (termed as discontinuity images) and the other keeps the phase data without strong disturbance (termed as substituted phase images). Subsequently, both sets of images are processed individually. The discontinuity images are processed with FBPJ for positions and shape estimations of the discontinuities, the points of which are extracted with a threshold. FBPP is applied to reconstruct the RI distribution from the substituted phase images. In practice, the 2D amplitude images are not affected intensely by the discontinuities. Therefore, the amplitude images can be used for FBPP without the abovementioned modifications. Finally, a complete 3D distribution is acquired by recombination of the two individual maps. The points of discontinuities are inserted into the sample distribution to replace corresponding points.
Fig. 1. Schematic diagram of the experimental setup: M, mirror; PBS, polarizing beam splitter; λ/2, half-wave plate; SF, spatial filter with collimating lens; MPS, micropositioning stage with the measured sample; MO, microscope objective; TL, tube lens; BS, beam splitter; OB, object beam; RB, reference beam.
2. Reconstruction method 3. Experimental setup
The main idea of our method is separating the disturbing regions from the phase images and reconstructing individually. To recognize the discontinuities, the gradient magnitude of a phase image is calculated with a Sobel operator. The strong fluctuations in these regions make the gradient magnitude higher than in other areas. Therefore, a proper threshold is sufficient to detect these regions, generating a binary mask. The phase jumps and random spikes make the phase values in these regions highly disordered and thus unreliable, making it impossible to reconstruct correct RI from them. Consequently, the phase values in them are discarded and only the shapes of the specks are retained. In order to simulate the accumulation effect of phase distribution, the Euclidean distance transform [28] of the binary mask is computed, making bulges in the recognized regions. On the other hand, the recognized regions in the phase image are substituted with data from surface fitting of the surrounding area. The purpose of the substitution is to remove the drastic fluctuations. Therefore, it is sufficient to perform polynomial fitting on a relatively smooth area surrounding these regions. After data substitution, the
The experimental setup depicted in Fig. 1 is based on a MachZehnder off-axis holographic interferometer in the sample-rotating configuration [11,25]. The laser beam (λ=532 nm) is divided into an object beam and a reference beam, both of which are filtered and expanded by spatial filters and collimated to produce plane waves. The sample, an optical fiber here, is held vertically by a fiber holder on a micro-positioning stage. The object beam traverses the sample and is captured by an afocal imaging system composed of the MO (Sigma Koki, 20×, NA =0.4, infinity corrected) and the tube lens (f =200 mm). The interference pattern is recorded by the camera (Lu125M, Lumenera Co., Ontario, Canada) as a digital hologram. The sample is immersed in refractive-index-matching liquid (n =1.45) to reduce diffraction. 180 holograms are recorded with a 2° step during a full rotation of 360°. An extra image is recorded without the sample as a reference hologram for phase curvature compensation.
Fig. 2. Influences of discontinuities on phase images: (a) An unwrapped phase image of the spliced single-mode fiber with air bubbles marked with arrows; (b) Profile along the dashed line in (a) with the bubble sections marked with boxes; (c-d) Sinograms of Sections A and B, respectively, with θ indicating the illumination angle; (e) Profiles along the dashed lines in (c) and (d).
59
Optics and Lasers in Engineering 94 (2017) 58–62
X. Ma et al.
Fig. 3. Influences of discontinuities on 3D reconstructions: (a-b) Reconstructed 3D maps from raw phase data using FBPJ and 3D FBPP, respectively; (c-d) Cross-section images of Sections A and B in (b), respectively; (e) Profiles along the dashed lines in (c) and (d); (f-g) Amplitude distributions of the Fourier transform of (c) and (d), respectively.
Fig. 4. Procedures of the separation: (a) Raw phase image; (b) Gradient distribution of (a); (c) Recognized and separated regions; (d) Fitted surface of the area marked with the box in (a); (e) Substituted phase image.
4. Experimental results
avoid possible errors. Unwanted phase jumps and random spikes abound in the vicinity of the discontinuities, as illustrated in Fig. 2(b). Unlike the sinogram of a bubble-free slice [Fig. 2(c)], the sinogram of an influenced slice contains a randomly fluctuating stripe [Fig. 2(d)], which disturbs the data significantly, as shown in Fig. 2(e). The reconstructed results from raw phase data are presented in Fig. 3. It is clearly observed that the slices containing discontinuities are seriously contaminated in the 3D maps reconstructed with FBPJ (Fig. 3(a)) and the 3D version of FBPP (Fig. 3(b)), making it impossible to distinguish the cores and the cladding according to their different RI. Note that the disturbed part, in this case the fusion interface, is precisely the most anticipated area. Compared with a bubble-free slice (Fig. 3(c)), the influenced slices (Fig. 3(d)) does not reveal structural information of the sample, as illustrated in Fig. 3(e). The influence can also be expounded in the frequency domain. Each 2D scattered field
In order to verify the method, we measured two samples containing different typical discontinuities. The first sample is a fusion spliced optical fiber between two single-mode fibers, which contains core misalignment and tiny air bubbles in the fusion area. Air bubbles are very likely to appear in the fusion interface due to fiber-end contamination such as dust, fiber-glass powder, moisture, fiber-end roughness, as well as an improper fusion arc current. Therefore, this sample is of considerable practical significance. Due to the large difference between the RI inside the bubbles and that of the fiber, discontinuities occur in the interfaces. Fig. 2(a) demonstrates an unwrapped phase image of the sample with three air bubbles marked with arrows. We applied discrete cosine transform based unweighted least squares algorithm [29] for phase unwrapping to 60
Optics and Lasers in Engineering 94 (2017) 58–62
X. Ma et al.
Fig. 5. Reconstruction results of the first sample: (a) Sinogram from the separated discontinuity images with θ indicating the illumination angle; (b) Reconstruction from (a); (c) Sinogram from the substituted phase images; (d) Reconstruction from (c); (e) 3D map of the recombined reconstruction with discontinuities in red, cores in blue and the cladding in gray.
Fig. 6. Reconstruction results of the second sample: (a) An unwrapped phase image of the spliced fiber between a single-mode fiber and a multi-mode fiber with discontinuities marked with boxes; (b) Sinogram of Section A with θ indicating the illumination angle; (c) 3D map reconstructed from the raw phase data with FBPP; (d) Recognized and separated discontinuous regions; (e) Substituted phase image; (f) Recombined 3D reconstruction with discontinuities in red, cores in blue and the cladding in gray.
image (Fig. 4(e)) is smoother in contrast to the raw image (Fig. 4(a)). The discontinuity images of all illumination angles form a sinogram for each slice of the sample, as shown in Fig. 5(a). The estimated shapes of the discontinuities reconstructed from them with FBPJ are displayed in Fig. 5(b). Reconstructed slices are then stacked up to form a 3D map. The sinograms from the substituted phase images (Fig. 5(c)) are free from strong disturbance, as compared with Fig. 2(d). FBPP is preferred to reconstruct the RI distribution for better accuracy (Fig. 5(d)). Here the 3D version of FBPP is applied in consideration of the structural variation along the vertical axis and the amplitude images restored from the holograms are used without the abovementioned procedures. Finally, the two maps are recombined for a complete reconstruction. In Fig. 5(e), the cores (blue parts) and the cladding (gray parts) are
corresponds to a hemispherical surface in the frequency domain, according to the Fourier diffraction theorem [30]. The random phase perturbations enhance high frequency components unevenly, making the restored frequency spectrum highly inhomogeneous (Fig. 3(g)), in contrast with a regular one (Fig. 3(f)). As a result, informative reconstructions are damaged. The procedures of separation of the phase images are shown in Fig. 4. The regions of discontinuities present higher gradient magnitude due to strong fluctuations, as in Fig. 4(b). These regions are separated with a proper threshold and the Euclidean distance transform is calculated, as displayed in Fig. 4(c). The recognized regions in the phase image are substituted with data from the cubic polynomial surface fitting of the marked area (Fig. 4(d)). The substituted phase 61
Optics and Lasers in Engineering 94 (2017) 58–62
X. Ma et al.
distinguished by their distinct RI values. Because there are no solid RI values in the air bubbles, the points in them (red parts) are set to a distinctive constant to highlight their outlines. The misalignment of cores is readily observable, along with the positions and approximate shapes of the air bubbles, which are indistinct before (Fig. 3(a) and (b)). Some additional fusion defects (blue parts other than the cores) are also visible, which are likely to be induced by disturbed internal stress. The other sample we studied is a fusion spliced optical fiber between a single-mode fiber and a multi-mode fiber. The diameter of the core of the multi-mode fiber (about 60 µm) is much larger than that of the single-mode fiber (about 8 µm). Therefore, RI discontinuities occur at the joint interface, which cause irregular boundaries with steep phase changes, as shown in Fig. 6(a). Unlike air bubbles, this kind of discontinuity is intrinsic in the structure. In the sinogram of the interface (Fig. 6(b)), abrupt changes are abundant. Consequently, a direct reconstruction from the raw phase data suffers from a clutter of fragments (Fig. 6(c)), so that the fusion area is indiscernible. The recognition of discontinuities is also based on the gradient magnitude of the phase images (Fig. 6(d)). The recognized regions are substituted with the cubic polynomial surface fitting data and a flat border, as illustrated in Fig. 6(e). Finally, the 3D RI distribution is obtained by recombination of the reconstructions from the discontinuity images by FBPJ and the substituted phase images by 3D FBPP, in which the amplitude images are unprocessed. The 3D map presented in Fig. 6(f) reveals the detailed structures of the fusion interface (red parts), including slight deformation of the cores. Another discontinuous part (outlying red part) is also exhibited, which appears to be a dust particle attached to the fiber.
[2]
[3]
[4] [5]
[6] [7] [8] [9] [10] [11]
[12]
[13] [14] [15] [16] [17]
5. Conclusion
[18]
In summary, this paper describes a method for tomographic reconstruction of samples containing localized RI discontinuities. These discontinuities cause regional perturbations and random spikes in 2D phase images, which will be expanded and strengthened by existing tomographic algorithms, resulting in contaminated reconstructions. The disturbance is avoided by recognition and separation of these regions. The eventual 3D RI distribution is acquired by recombination of individually reconstructed data from the separated discontinuity images and the substituted phase images filled with surface fitting data. Two samples with different typical RI discontinuities are demonstrated: one is a fusion spliced optical fiber with tiny air bubbles enclosed and the other also a spliced fiber but containing intrinsic structural RI jumps. The 3D maps obtained with this method are free from spreading disturbance and show obviously important structures of the samples as well as the positions and estimated shapes of the discontinuities. The proposed method is of practical significance and will find important applications in 3D imaging of micro-optical components, as well as biomedical samples in case of accidental foreign matters or air bubbles in the culture media.
[19] [20] [21] [22]
[23] [24] [25] [26] [27]
[28]
[29]
References
[30]
[1] Kim Y, Shim H, Kim K, Park H, Heo JH, Yoon J, et al. Common-path diffraction
62
optical tomography for investigation of three-dimensional structures and dynamics of biological cells. Opt Express 2014;22:10398–407. Su J-W, Hsu W-C, Chou C-Y, Chang C-H, Sung K-B. Digital holographic microtomography for high-resolution refractive index mapping of live cells. J Biophotonics 2013;6:416–24. Kim K, Yoon H, Diez-Silva M, Dao M, Dasari RR, Park Y. High-resolution threedimensional imaging of red blood cells parasitized by Plasmodium falciparum and in situ hemozoin crystals using optical diffraction tomography. J Biomed Opt 2013;19:011005. Charrière F, Marian A, Montfort F, Kuehn J, Colomb T, Cuche E, et al. Cell refractive index tomography by digital holographic microscopy. Opt Lett 2006;31:178–80. Charrière F, Pavillon N, Colomb T, Depeursinge C, Heger TJ, Mitchell EAD, et al. Living specimen tomography by digital holographic microscopy: morphometry of testate amoeba. Opt Express 2006;14:7005–13. Kim K, Yoon J, Park Y. Large-scale optical diffraction tomography for inspection of optical plastic lenses. Opt Lett 2016;41:934–7. Pan Z, Liang Z, Li S, Weng J, Zhong J. Microtomography of the polarizationmaintaining fiber by digital holography. Opt Fiber Technol 2015;22:46–51. Lin Y-C, Cheng C-J. Sectional imaging of spatially refractive index distribution using coaxial rotation digital holographic microtomography. J Opt 2014;16:065401. Gorski W, Osten W. Tomographic imaging of photonic crystal fibers. Opt Lett 2007;32:1977–9. Malek M, Khelfa H, Picart P, Mounier D, Poilâne C. Microtomography imaging of an isolated plant fiber: a digital holographic approach. Appl Opt 2016;55:A111–21. Belashov AV, Petrov NV, Semenova IV. Accuracy of image-plane holographic tomography with filtered backprojection: random and systematic errors. Appl Opt 2016;55:81–8. Kostencka J, Kozacki T, Kuś A, Kemper B, Kujawińska M. Holographic tomography with scanning of illumination: space-domain reconstruction for spatially invariant accuracy. Biomed Opt Express 2016;7:4086–101. Sung Y, Choi W, Fang-Yen C, Badizadegan K, Dasari RR, Feld MS. Optical diffraction tomography for high resolution live cell imaging. Opt Express 2009;17:266–77. Choi W, Fang-Yen C, Badizadegan K, Oh S, Lue N, Dasari RR, et al. Tomographic phase microscopy. Nat Methods 2007;4:717–9. Xiu P, Zhou X, Kuang C, Xu Y, Liu X. Controllable tomography phase microscopy. Opt Lasers Eng 2015;66:301–6. Kak AC, Slaney M. Principles of Computerized Tomographic Imaging. New York, US: Society for Industrial and Applied Mathematics; 2001. Devaney AJ. A filtered backpropagation algorithm for diffraction tomography. Ultrason Imaging 1982;4:336–50. Gordon R. A tutorial on art (algebraic reconstruction techniques). IEEE Trans Nucl Sci 1974;21:78–93. Gorski W. Tomographic microinterferometry of optical fibers. Opt Eng 2006;45:125002. Jeon Y, Hong CK. Rotation error correction by numerical focus adjustment in tomographic phase microscopy. Opt Eng 2009;48:105801. Ma X, Xiao W, Pan F. Accuracy improvement in digital holographic microtomography by multiple numerical reconstructions. Opt Lasers Eng 2016;86:338–44. Choi W, Fang-Yen C, Badizadegan K, Dasari RR, Feld MS. Extended depth of focus in tomographic phase microscopy using a propagation algorithm. Opt Lett 2008;33:171–3. Yablon AD. Multifocus tomographic algorithm for measuring optically thick specimens. Opt Lett 2013;38:4393–6. Kostencka J, Kozacki T, Dudek M, Kujawińska M. Noise suppressed optical diffraction tomography with autofocus correction. Opt Express 2014;22:5731–45. Kostencka J, Kozacki T, Kuś A, Kujawińska M. Accurate approach to capillarysupported optical diffraction tomography. Opt Express 2015;23:7908–23. Kamilov US, Papadopoulos IN, Shoreh MH, Goy A, Vonesch C, Unser M, et al. Learning approach to optical tomography. Optica 2015;2:517–22. Tsihrintzis GA, Devaney AJ. Higher-order (nonlinear) diffraction tomography: reconstruction algorithms and computer simulation. IEEE Trans Image Process 2000;9:1560–72. Maurer CR, Qi R, Raghavan V. A linear time algorithm for computing exact Euclidean distance transforms of binary images in arbitrary dimensions. IEEE Trans Pattern Anal Mach Intell 2003;25:265–70. Backoach O, Kariv S, Girshovitz P, Shaked NT. Fast phase processing in off-axis holography by CUDA including parallel phase unwrapping. Opt Express 2016;24:3177–88. Wolf E. Three-dimensional structure determination of semi-transparent objects from holographic data. Opt Commun 1969;1:153–6.