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Abstract
Optical coherence microscopy (OCM) is an interferometric technique providing 3D images of biological samples with micrometric resolution and penetration depth of several hundreds of micrometers. OCM differs from optical coherence tomography (OCT) in that it uses a high numerical aperture (NA) objective to achieve high lateral resolution. However, the high NA also reduces the depth-of-field (DOF), scaling with 1/NA2. Interferometric synthetic aperture microscopy (ISAM) is a computed imaging technique providing a solution to this trade-off between resolution and DOF. An alternative hardware method to achieve an extended DOF is to use a non-Gaussian illumination. Extended focus OCM (xfOCM) uses a Bessel beam to obtain a narrow and extended illumination volume. xfOCM detects back-scattered light using a Gaussian mode in order to maintain good sensitivity. However, the Gaussian detection mode limits the DOF. In this work, we present extended ISAM (xISAM), a method combining the benefits of both ISAM and xfOCM. xISAM uses the 3D coherent transfer function (CTF) to generalize the ISAM algorithm to different system configurations. We demonstrate xISAM both on simulated and experimental data, showing that xISAM attains a combination of high transverse resolution and extended DOF which has so far been unobtainable through conventional ISAM or xfOCM individually.
Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
Optical coherence tomography (OCT) has enabled rapid, highly sensitive, in vivo, volumetric imaging of tissues and cells and has been widely applied both in biomedical research and clinical settings [1–4]. OCM is based on the same principles as OCT but achieves higher transverse resolution by using higher-NA optics. This increase of resolution comes however at the expense of a reduced DOF, which is proportional to 1/NA2.
Various solutions have been proposed to overcome this inherent trade-off between resolution and DOF. Computational techniques have been developed to obtain a larger DOF [5–8] (for a review of these methods, see [9,10]). ISAM was presented as a solution to the inverse scattering problem, yielding spatially invariant transverse resolution throughout the entire volume [6,11,12]. Developments of ISAM such as real-time [13,14] or automated ISAM [15] have further extended the capabilities of this method.
Alternative hardware implementations of an extended DOF include dynamic focus control [16,17], mechanical depth scanning [18], and multi-channel imaging [19]. Another approach is to use non-Gaussian beam illumination via wavefront engineering using axicon lenses, phase plates, digital micro-mirror devices or spatial light modulators [16,20–27]. xfOCM uses an axicon to generate a Bessel-beam illumination and a decoupled Gaussian detection to guarantee an efficient collection of the scattered light. However, the use of a Gaussian detection comes at the cost of a reduced DOF. In this way, xfOCM achieves a better trade-off between lateral resolution and DOF than conventional OCM systems with equal Gaussian illumination and detection modes. xfOCM has been proven to be a powerful tool in many biomedical applications [21,28–32]. Nevetherless, xfOCM could still benefit from a further improvement of DOF.
In this paper, we propose to combine ISAM and xfOCM to simultaneously optimize transverse resolution, DOF and signal-to-noise ratio (SNR) to achieve unprecedented OCM imaging performance. Current ISAM implementations assume equal Gaussian illumination and detection apertures and are therefore not suitable for xfOCM systems. In 2012, Sheppard et al. showed that the ISAM reconstruction was not an exact but an approximate solution to the inverse scattering problem, as it assumes exact backscattering (i.e., incident light is backscattered in the same direction) [33]. A more general theory based on the 3D coherent transfer function (CTF) was proposed, enabling different system configurations to be explored.
In the present work, we exploit the 3D CTF formalism suggested in [33] in order to generalize the ISAM principle to decoupled apertures such as in xfOCM. We term our method extended ISAM (xISAM) from the combination of xfOCM and ISAM. While the ISAM reconstruction achieves focal-plane resolution throughout the volume, the xfOCM scheme allows illuminating the sample along an extended axial range. This results in a higher transverse resolution and SNR away from the focus as compared to conventional OCM systems. ISAM reconstruction critically relies on phase stability, which depends on the SNR [34–36]. Hence, xfOCM allows ISAM reconstruction at greater depths. We describe xISAM and validate it both on simulated and experimental data, showing a clear sensitivity advantage over conventional ISAM away from the focal plane.
2. Materials & methods
2.1. Optical setup
The layout of the xfOCM instrument used in this work has been described elsewhere [28–30]. It consists of a Mach-Zehnder interferometer with separate illumination and detection paths, as displayed in Fig. 1. A broadband light source (Ti:Sapphire laser, Femtolasers) with a spectral bandwidth of 135 nm centered at 800 nm is coupled into a long single mode, polarization maintaining fiber with a mode field diameter of 4.2 μm (HB 750, Fibercore Ltd., Southampton). Propagation through the fiber broadens the pulses from the femtosecond laser source to reduce peak power. Light from the fiber is collimated by the lens LC1 (f = 8 mm) and split into reference (blue) and sample (green) arms by a first beam-splitter (BS1). In the sample arm, the beam passes either a normal lens (f = 75 mm) for the Gaussian illumination mode, or an axicon lens (Del Mar Photonics, 175° apex angle) for the Bessel illumination mode. The beam is then relayed through a scanning system to the tube lens (LT, f = 164 mm) and the objective (LS, 10×, 0.3 NA, Zeiss Neofluoar). The back-scattered light (red) is collected by the same objective and recombined to the reference beam by means of a second beam-splitter (BS2). The detection path is composed of a first lens LD (f = 200 mm) and a second lens LC2 (f = 20 mm) coupling the signal into a detection fiber. Finally, a custom spectrometer consisting of a transmission grating (1200 lines/mm), an objective lens (f = 135 mm) and a line detector (2048 pixels, Atmel AViiVA) records the interference pattern at a rate of 20 kHz. The illumination power, measured at the intermediate image plane between lens LT and the scan unit (see Fig. 1), was ∼ 10 mW.
Fig. 1 Schematic illustration of the xfOCM setup, as presented in [28–30]. Light from a Ti:Sapphire laser with a broad spectrum (Δλ = 135 nm) centered at 800 nm is split by beam-splitter BS1 into a sample (green) and reference (blue) arm. The sample arm contains either an axicon lens to generate a Bessel beam illumination or a normal lens for a Gaussian beam illumination. The light is then guided to the tube lens (LT) and objective (LS) via the X–Y scanner unit. The back-scattered light (red) from the sample is superimposed to the reference arm by beam-splitter BS2 and focused by LC2 into the detection fiber. The interference pattern is recorded by a custom-made spectrometer.
2.2. Coherent transfer function calculation
Following Sheppard et al. [33], we use the framework of the 3D CTF for a more accurate and generalized ISAM reconstruction which can be applied to different system configurations including decoupled illumination and detection apertures. The theory underlying 3D image formation and the CTF has been well studied in [37–39] and is based on the concept of generalized aperture introduced by McCutchen in 1964 [40]. Villiger and Lasser proposed a model for image formation in OCM setups using the generalized aperture and the CTF [37]. For each wavenumber channel of the system, the CTF is given by the convolution of the generalized illumination and detection apertures.
2.3. xISAM algorithm
For equal Gaussian illumination and detection apertures, the CTF has a maximum value on a sphere with a radius of twice the wavenumber k, corresponding to the maximum length of the scattering vector for a reflection geometry. Hence,
where x and y are the lateral spatial coordinates and where we use the lateral spatial frequency qr due to the cylindrical symmetry. As explained in [37], the curvature along qr is related to defocusing. The principle of ISAM is to perform a resampling in the frequency domain to flatten this spherical curvature and thus compensate for the defocus [6,11,12]. ISAM uses Eq. 1 for the k → qz resampling and assumes that the scattering comes from exact backscattering, corresponding to the maximum of the CTF.Sheppard et al. proposed a more accurate reconstruction taking the mean position of the CTF [33]. Taking this approach as a key ingredient, we use the center of mass of the 3D CTF at each lateral frequency as resampling curve for our xISAM reconstruction. Figure 2 illustrates this principle for two different systems, one with identical Gaussian illumination and detection modes (OCM) and one with decoupled Bessel illumination and Gaussian detection modes (xfOCM). Obviously, xfOCM data are incompletely represented in classical ISAM, which uses a spherical resampling surface. In contrast, our method allows refocusing data acquired with potentially any system configuration.
Fig. 2 Resampling curves derived from the CTFs for Gaussian and Bessel illuminations. CTF for (left) Gaussian illumination and detection modes, and (right) Bessel illumination and Gaussian detection modes. A xISAM resampling curve (yellow) extracted from the center of mass of the CTF for each lateral frequency is shown.
Figure 3 illustrates the processing steps for xISAM reconstruction. We start from the tomogram SOCM(x, y, z) recovered from the recorded interference signal using the conventional λ → k mapping and 1D Fourier transform. A phase correction is first performed to ensure phase stability for the reconstruction [36, 41]. In this work, we used the phase correction presented in [41] using a glass coverslip placed on top of the sample as reference. More elaborate algorithms not requiring a reference object could be used to correct for phase instabilities, as presented in [42–44] for example. The next step is a coordinate change from z to z′ consisting of a circular shift of the data so that the focal plane appears at z′ = 0. As shown in Fig. 3, the shifted data is also zero-padded with 2nzo lines to prevent aliasing, where nzo is the number of pixels from the focus to the center of S(x, y, z). The spatial frequency domain OCM signal is then obtained via 3D Fourier transformation. The xISAM k → qz interpolation is then performed using a resampling grid derived from the CTF as explained in the previous section. Figure 3 shows how the xISAM resampling compensate for the defocus by flattening the phase. Finally, a 3D inverse Fourier transform and an inverse coordinate change allow retrieving the defocus-corrected tomogram SxISAM(x, y, z).
Fig. 3 xISAM processing steps. The processing is the same as in ISAM, but with the resampling grid obtained from the CTF as shown in Fig. 2. It consists of a phase correction procedure to ensure phase stability, followed by a coordinate change from z to z′ to place the focal plane at z′ = 0. The 3D signal is then Fourier transformed to obtain the Fourier domain OCM signal. The amplitude of the whole signal and the phase of a single out-of-focus scatterer is shown. The crucial step is the xISAM resampling using the grid extracted from the CTF. Finally, a 3D inverse Fourier transform and coordinate change retrieve the defocus-corrected tomogram.
3. Results
3.1. Simulations
As a proof-of-concept, we simulated data composed of point scatterers aligned along the optical axis. OCM and xfOCM signals S(x, y, k) were calculated as the convolution of the scattering potential η(x, y, z) and the system (complex) point-spread-function (PSF) h(x, y, z, k):
The PSF at each k was obtained from the CTF by 3D Fourier transformation.Figures 4 and 5 show the results of the simulations for the OCM and xfOCM modalities, respectively. In both cases, the unprocessed tomogram and the xISAM reconstructed image are shown, as well as the phase associated with an out-of-focus scatterer (red arrow) before and after xISAM processing. Line profiles through another scatterer are also plotted alongside a profile through the in-focus scatterer. For better comparison, the profiles were normalized relative to their energy.
Fig. 4 xISAM simulations on OCM data. (a) Simulated OCM tomogram from a Gaussian-Gaussian configuration, with point scatterers aligned along the optical axis. (b) Phase profile associated with the out-of-focus scatterer indicated by the red arrow in (a). (c) xISAM reconstructed tomogram. (d) Phase profile as in (b). (e) Energy-normalized line profiles through the scatterers indicated by the colored lines in (a) and (b), showing an almost perfect recovery of the focal-plane resolution. The data is displayed in linear scale.
Fig. 5 xISAM simulations on xfOCM data. xISAM proof-of-concept with a simulated xfOCM tomogram of point scatterers, similarly to Fig. 4. (a) Unprocessed and (b) xISAM reconstructed data, respectively. (c) and (d) Associated phase profile through an out-of-focus scatterer. (e) Energy-normalized line profiles through the indicated scatterers. The xISAM reconstruction leads to a clear reduction of the energy in the side lobes and an increase of the central peak.
As seen in the line profiles and as known from [21], xfOCM increases the focal range with respect to OCM (blue curves in Figs. 4 and 5). The xISAM reconstruction allows recovering the focal-plane transverse resolution almost perfectly in the OCM case (orange, Fig. 4). On the other hand, the reconstruction of the xfOCM signal (Fig. 5) is not as ideal but still shows a significant improvement with a reduction of the side lobes to the benefit of the central peak. The reason for this difference is presumably that the reconstruction of xfOCM data is more delicate due to the more complicated shape of the resampling curve. In other words, it is more difficult to perfectly flatten the CTF and thus completely compensate for the defocus. This imprecision is visible in Fig. 5 in the phase profile after reconstruction, which is not perfectly flat. Finally, we note for both cases that despite the refocusing, the reconstructions still drop in intensity with increasing distance from the focal plane due to reduced signal collection.
3.2. Microbeads sample
To validate the xISAM procedure experimentally, we first imaged 1 μm-sized polystyrene microspheres embedded in silicone. This technical phantom was prepared by drying the water suspended beads, re-suspending them in the silicone and adding a curing agent. Prior to baking, air bubbles were removed from the sample using a vacuum bell.
Figure 6 shows the acquired OCM and xfOCM data and their respective xISAM reconstruction. En face views at three different depths and a maximum projection along the 256 B-scans in the x-direction are shown for all four cases. Beads at ±45 μm depth in the OCM tomogram are almost not discernible in the unprocessed data but are well refocused after xISAM resampling. Compared with classical OCM, the xfOCM scheme exhibits a larger focal range allowing the observation of scatterers positioned at ± 50 μm with improved contrast. In accordance with the simulated data, the reconstruction of the xfOCM tomograms redistributes the energy from the side lobes towards the central peak. This energy redistribution leads to a general improvement of the image quality. We note that a jitter caused by the scanning mirrors is apparent in the en face views at the focal plane, but did not prevent a successful reconstruction. The capability of xISAM is well observable in the maximum projections. Even though the reconstruction of OCM data yields focal-plane resolution at all depths, the xfOCM and even more particularly its xISAM reconstruction display a clear SNR advantage in out-of-focus planes.
Fig. 6 OCM and xfOCM tomograms of polystyrene beads, and their corresponding reconstruction. The three first rows show en face images at various depths, while the last row shows maximum projections along the 256 B-scans. The different depths are indicated by the yellow lines in the maximum projections. All scale bars indicate 20 μm. The data is displayed in logarithmic scale with a range covering 20 dB for the en face views and 35 dB for the maximum projections.
To better characterize the performance of OCM, xfOCM and xISAM, we plotted in Fig. 7 several lines profiles along different beads. The results are in excellent agreement with the simulations presented in Figs. 4–5, showing a clear refocusing for the OCM configuration and an energy increase in the central peak with a diminution of the side lobes for the xfOCM configuration. To further compare the four cases, we defined an image quality metrics as the sum of the 20 highest pixels at each depth. xISAM achieves a clear improvement for both system configurations. Undeniably, the extended focus configuration brought by the Bessel beam illumination procures a substantial advantage in terms of signal collection. This demonstrates the usefulness of combining the ISAM concept with a hardware implementation to obtain a greatly extended focal range.
Fig. 7 Profiles through the tomograms of Fig. 6. (a), (b), (d) and (e) Line profiles across different beads at various depths, in excellent agreement with the simulated data (Figs. 4–5). Insets: en face views of the beads before and after reconstruction. (c) and (f) Metrics to evaluate image quality over depth in the four cases. The metrics consists in the sum of the highest 20 pixels at each depth z and shows the benefit of xfOCM and its xISAM reconstruction compared to both OCM unprocessed and processed data.
3.3. Mandarin sample
We further demonstrate our method taking a mandarin as a sample for the xISAM reconstruction. A slice of mandarin was cut with a razor blade. The cut surface with fresh pulp was then simply covered by a coverslide. As for the microspheres measurement, this sample was imaged with the conventional OCM configuration and the xfOCM system. Both tomograms were then resampled using the xISAM procedure. The results are shown in Fig. 8. The en face views exhibit sharper features in both reconstructions with respect to the unprocessed data. Moreover, the maximum projections show a clear improvement in both resolution and contrast (see zoomed-in areas). Similarly to the microspheres measurement, the benefit of the xfOCM scheme in terms of signal collection away from the focus is evident in these projections. The xISAM processing brings an additional gain of transverse resolution even far from the focal plane, as can be seen in the zoomed-in areas.
Fig. 8 Slice of mandarin imaged with OCM and xfOCM and processed with xISAM. En face planes and maximum projections in the x-direction of the OCM and xfOCM tomograms are displayed with their respective xISAM reconstruction. Zoomed views of the areas indicated by a white rectangle are also shown. The signal range spans 25 dB for the en face views and 35 dB for the maximum projections. Scale bars in all directions correspond to 50 μm.
4. Discussion & conclusion
We introduced xISAM, a new computational method for high-resolution volumetric imaging combining the approaches of xfOCM and ISAM. Using the framework of the CTF as suggested in [33], we generalized the ISAM technique to different system configurations. This was achieved by taking the center of mass of the CTF as resampling curve for the k → qz interpolation. Our method was applied to a setup with NA = 0.3 that allowed both OCM and xfOCM modalities. Moreover, xISAM can be applied to any system configuration provided that the CTF can be calculated or measured.
For the experimental data, a phase correction procedure was applied based on a glass coverslip. Other algorithms not requiring the use of a reference object might need to be implemented for future applications, for instance based on existing techniques such as presented in [42–44]. Moreover, it was demonstrated that performing an aberration correction alongside with ISAM yields superior results [45, 46]. Aberrations are caused by imperfections of the setup or the sample itself, and can significantly degrade the resolution and image quality. These effects are increasingly important with increasing NA. To apply xISAM to other OCM configurations such as dark-field [47–49] or visible OCM [50] using a NA ≥ 0.8, an aberration correction step becomes necessary. Various methods have been developed to address this issue, two notable examples being computational adaptive optics (CAO) [15,45,51] and subaperture correlation based digital AO [52,53]. Future developments of our method aim at including such an aberration correction procedure.
In summary, we first demonstrated here the principle of xISAM on simulated data of point scatterers. Our method achieved a recovery of the focal-plane resolution for the OCM configuration and a diminution of the side lobes and increase of the central peak in the extended focus scheme, improving the image quality. We then performed validation on real experimental measurements of polystyrene beads and a mandarin slice. The experimental results were in good agreement with the simulations. We noted that xfOCM exhibits a significant signal collection advantage over OCM data, even after ISAM reconstruction. The increased signal collection away from the focus offers higher phase stability and thus enables reconstruction at greater depths. This better SNR over depth shows the benefit of implementing a hardware-based DOF in addition to performing a software reconstruction. Combining both methods further increases the focal range, enabling high-resolution imaging even at large depths.
Funding
This work was partially funded by the Swiss National Science Foundation (grants 205321L_135353 & 205320L_150191). A.B. and T.L. acknowledge support from the Seventh Framework Programme and the Horizon 2020 Framework Programme of the European Union via grants 602812 and 686271, respectively.
Acknowledgments
We would like to thank Martin Villiger for all the experimental data presented in this work.
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