Computational refocusing in phase-unstable polarization-sensitive optical coherence tomography

Abstract

We present computational refocusing in polarization-sensitive optical coherence tomography (PS-OCT) to improve spatial resolution in the calculated polarimetric parameters and extend the depth-of-field in phase-unstable, fiber-based PS-OCT systems. To achieve this, we successfully adapted SHort A-line Range Phase-stability adaptive optics (SHARP), a computational aberration correction technique compatible with phase-unstable systems, into a Stokes-based PS-OCT system with inter–A-line polarization modulation. Together with the spectral binning technique to mitigate system-induced chromatic polarization effects, we show that computational refocusing improves image quality in tissue polarimetry of swine eye anterior segment ex vivo with PS-OCT. The benefits, drawbacks, and potential applications of computational refocusing in anterior segment imaging are discussed.

Optical coherence tomography (OCT) can provide diffraction-limited imaging only within the depth of field determined by the system’s numerical aperture (NA). The compromise between depth of field and NA limits the applicability of medium- to high-NA imaging unless shallow penetration depth is sufficient, thin samples are imaged or axial stitching is used. In structural contrast and functional extensions of OCT, any blurring in the OCT signal is directly transferred through post-processing methods, such as in polarization-sensitive (PS-) OCT, which measures polarimetric properties of tissue [1]. PS-OCT has attracted clinical interest due to its sensitivity to the presence and organization of fibrillar tissue in intravascular [2], retinal [3] and anterior segment [4] imaging, among others. Few works have focused on achieving high-resolution PS-OCT imaging; notably, ocular anterior segment imaging [4] at a high resolution and with an extended depth of field using a Bessel beam, requiring custom hardware. Hardware-based adaptive optics (AO) was integrated into a PS-OCT system for aberration-free retinal imaging with improved effective lateral resolution [3]; however, the depth-independent correction applied in AO cannot extend the depth of field. Computational adaptive optics (CAO) is a numerical alternative to AO [5] that can correct aberrations and, additionally, extend the depth of field when the depth-dependent defocus is compensated by performing interferometric synthetic aperture microscopy (ISAM) [6] or computational refocusing (CR) [7] for low and moderate numerical aperture regimes.

Phase stability is crucial for the operation of CAO, CR, and ISAM, thus their adoption has been hindered in systems that suffer from system- and sample-induced phase noise, such as wavelength-swept source (SS-) OCT systems which are most commonly used in PS-OCT. Spectral-domain OCT has been preferred for performing CR since it provides greater phase stability compared with SS-OCT [6, 8]. Furthermore, those works employed a single incident polarization state, which is not robust enough to probe layered birefringent media in all scenarios. Recently, Zhu et al. demonstrated computational refocusing of the Jones matrix in a $k$-clocked SS-OCT system, thus avoiding phase jitter and requiring only to correct for phase offsets [9].

Previously, we introduced a fully computational technique to perform CAO in phase-unstable OCT systems such as raster-scanning, non-$k$–clocked SS-OCT systems, which we termed SHort A-line Range Phase-stability adaptive optics (SHARP) [10]. Here, we present its extension to Stokes-based PS-OCT, which required careful consideration of the signal acquisition and processing inherent to PS-OCT to improve the quality of tissue polarimetric parameters retrieved with an SS-PS-OCT system. We present an integrated processing strategy to computationally refocus and apply Stokes spectral binning processing [2] to tomograms acquired with a phase-unstable system using inter–A-line input polarization state modulation. We demonstrate image quality improvement in birefringence, optic axis, and degree of polarization (DOP) imaging of the anterior segment of an excised swine eye acquired with an SS-PS-OCT system having strong phase noise, showing that SHARP is suitable for inter–A-line modulated PS-OCT systems due to its ability to refocus in the presence of system-induced phase noise.

SHARP can correct two-dimensional (2D) $x$-$y$–separable aberrations in phase unstable systems by means of two one-dimensional (1D) steps consisting of phase stabilization and CAO performed sequentially along each of the two lateral scan axes $x$ and $y$, with an intermediate phase roll-back step [10]. The phase stabilization step consists of correcting for phase noise described as phase offsets and phase jitter that are frequently present in SS-OCT. For CAO, SHARP uses an optimization-based approach that describes the phase of the aberration correction filter as a weighted sum of a 1D polynomial basis with weights determined via optimization of an image quality metric [5].

In the present work, we employed a system that utilized two incident polarization states $p=\left\{p_1,p_2\right\}$ orthogonal on the Poincaré sphere and wherein the backscattered signal was measured by a receiver with two orthogonal polarization channels $c=\left\{c_1,c_2\right\}$, for a total of four interferometric spectral fringes $A_{p,c}$ forming two Jones vectors $A_{p_1}=\left[A_{p_1,c_1}{A}_{p_1,c_2}\right]$ and $A_{p_2}=\left[A_{p_2,c_1}{A}_{p_2,c_2}\right]$. The spectral binning processing utilized short-time Fourier transform of $A_{p,c}$ to reconstruct the complex-valued tomograms $T_{p,c,w}^K$ using $K$ spectral windows $w=\left\{w_1,w_2\dots w_K\right\}$ which, after transformation to Stokes vectors are used to calculate retardation vectors that are then aligned across $w$ to obtain an estimation less sensitive to wavelength-dependent contributions of the system on the polarization state, often referred to as polarization mode dispersion (PMD) [2]. Aberration correction with SHARP should be applied to each $T_{p,c,w}^K$; however, the determination of the phase noise corrections ${\widehat{\text{Δ}}}_{p,c}$ and the optimal phase filters for aberration correction $H_{p,c}$ is not robust when performed on the individual spectral bins due to the different amount of signal-to-noise ratio (SNR) across windows and the reduced axial resolution compared to the full-resolution tomograms $T_{p,c}$, i.e. the tomograms reconstructed without spectral binning. Therefore, our adaptation of SHARP for PS-OCT consists of two steps: to determine ${\widehat{\text{Δ}}}_{p,c}$ and $H_{p,c}$ from the full-resolution tomograms $T_{p,c}$ and then to apply the phase-noise and aberration corrections to each complex-valued spectral bin $T_{p,c,w}^K$ using the predetermined ${\widehat{\text{Δ}}}_{p,c}$ and $H_{p,c}$. This is valid since the phase filter applied in CAO is $k$-independent and the phase noise is identical across spectral bins.

Assuming $H_{p,c}=H$ (i.e., the phase filter is independent of the incident and detection polarization states), the image quality metric—entropy $S$ in our case—in the CAO optimization process can be evaluated from a single $p$ (either $p_1$ or $p_2$), and then the same $H$ is applied to all $T_{p,c,w}^K$. In contrast, phase noise corrections are different between incident polarization states (${\widehat{\text{Δ}}}_{p1,c}\ne {\widehat{\text{Δ}}}_{p2,c}$) since the signals are not acquired simultaneously when using inter-A-line modulation; however, they are equal for both detection channels. See Supplement 1 for further details about the procedure.

Aberration-corrected complex-valued Jones tomograms ${\tilde{T}}_{p_1,w}^K$ and ${\tilde{T}}_{p_2,w}^K$ were then transformed into two Stokes vectors $S_{p_1,w}^K$ and $S_{p_2,w}^K$ composed of the four Stokes parameters $I$, $Q$, $U$ and $V$. The positive effect of CAO applied to the complex-valued tomograms was translated into the Stokes parameters and subsequently into the polarimetric parameters. Each Stokes parameter was spatially averaged to reduce speckle using a kernel $h$. Polarimetric properties were finally computed from the aberration-corrected, averaged Stokes parameters, using the spectral binning approach [2], namely local phase retardation and cumulative optic axis in the QU plane.

To demonstrate our proposed workflow, we acquired PS-OCT datasets of the anterior segment of an excised porcine eye. The anterior segment possesses tissues with potential clinical interest but its relatively long axial extension and surface topography preclude the use of medium-NA, thus imaging is typically limited to a relatively coarse lateral resolution of around 20 μm [11–13]. Therefore, anterior segment imaging is an attractive application that could benefit from CR to achieve high-resolution polarimetric parameters of tissue micro-structures such as the arrangement of collagen fibrils in the stroma [4] and the palisades of Vogt [14]. We imaged the limbus, targeting the trabecular meshwork, and acquired an out-of-focus (OOF) tomogram with the focal plane located deep into the tissue. We then shifted up the focal plane to acquire a second tomogram to serve as an in-focus reference for comparison with the numerical result.

The datasets were acquired with an SS-OCT system having a light source that was affected by strong phase noise [15] (see Supplement 1). The system had a 1310 nm central wavelength, a 130 nm bandwidth, an A-line acquisition rate of 54 kHz, and a collimated input beam of 3.6 mm that, combined with an 18 mm effective focal length objective lens, produced an $e^{-2}$ diameter lateral resolution of 8.5 μm. The focus at the sample had a confocal parameter 180 μm (two times the Rayleigh range, $z_R$). The two input polarization states onto the sample were modulated with an electro-optic modulator in the sample arm between alternating A-lines. The acquired spectra were reconstructed using $K=5$ spectral windows with 66% overlap for spectral binning processing performed with a step size of 42 μm (7 px). The tomograms contained 1024 B-scans and 384 A-lines with 1024 depth samples, with an axial pixel sampling of 6 μm (assuming unity index of refraction) and 3.3 μm laterally to satisfy the Nyquist sampling requirement of CAO, from which we selected different regions of interest. We applied SHARP to the OOF tomogram, using only defocus correction since it was the dominant aberration in the experiment. We then processed the Stokes vector components of the original OOF tomogram before and after SHARP, and the reference tomogram, by filtering with $h$, a Gaussian kernel with $e^{-2}$ diameter 13×13 μm² (4×4 px²) along the lateral axes.

We centered our analysis on the limbal region, depicted in the OOF B-scan of Figure 1(a), where the short vertical white line represents the length of the confocal parameter. This configuration is representative of medium-NA imaging of the limbus, where the axial extent of the tissue makes it impossible to have the whole ROI in focus. The optimal defocus weights obtained with SHARP suggest that the focus was located toward the bottom of the box. Figures 1(b–c) show en face intensity views before and after SHARP, respectively, at the depth $4.4z_R$ indicated by the dotted line in Fig. 1(a). The intensity after SHARP exhibits higher contrast and better resolution given the correction of defocus, approaching the quality observed in the corresponding in-focus reference en face in Fig. 1(d), as noted when comparing the bright structure within the rectangles. Notably, superficial structures project shadows that appear as dark horizontal bands in Fig. 1; these bands are sharp in Fig. 1(d) because the obstructing structures are close to the reference focal plane, while they are less prominent when imaging at the OOF plane because the obstructing structures are out-of-focus allowing light to reach deeper structures and reducing the shadowing effect, resulting in bands with less contrast in Fig. 1(c).

Fig. 1.

(a) B-scan of porcine eye anterior segment. The limited depth of field is very apparent. The focus is at the bottom of the solid green box. The dotted line indicates the location of en face views: (b) original OOF, (c) after SHARP, and (d) reference. Orange arrows indicate projected shadows that are not expected to be replicated in OOF image with CR.

Figure 2 presents results after PS-OCT processing of the original OOF, SHARP, and reference tomograms. Fig. 2(a) shows an intensity B-scan with green and cyan lines marking the location for the tilted cross-sectional views on Figs. 2(b–j), which from the optimal defocus weights are estimated to be $4.4z_R$ away from the focal plane (see Visualization 1 for en-face intensity comparisons). We show different tissue polarimetric parameters of the original, SHARP, and reference data side-by-side. The DOP is overlaid with the intensity in Figs. 2(b–d), using the DOP as the hue and the intensity as the brightness [2] (see Visualization 2); local phase retardation $\text{Δ}n$ is shown in Figs. 2(e–g) (see Visualization 3); and the optic axis orientation $\phi$ is overlaid as the hue with the local phase retardation as the brightness in Figs. 2(h–j) (see Visualization 4). The stroma is mostly composed of lamellar, homogeneous scattering tissue of randomly aligned collagen fibrils [4], which induces phase retardation with random optic axis orientations. Structures in the original OOF images appear to be highly affected by a combination of defocus, spatial averaging, and spectral binning processing. CR did not induce any corruption or artifacts in the polarimetric parameters; instead, successful refocusing is indicated by the clear improvement in spatial resolution in the polarimetric parameters, suggesting that the proposed implementation of SHARP for PS-OCT is compatible with spectral binning processing even in the presence of phase-jitter noise. Correcting for defocus reveals structures that appeared blurred in the original tomogram, and all polarimetric images exhibit better contrast and sharpness associated with the improvement of lateral resolution.

Fig. 2.

(a) Intensity B-scan. En face views of tissue polarimetric parameters, showing original OOF, after SHARP and reference side-to-side: (b)–(d) DOP (isoluminant colormap) overlaid with intensity (brightness); (e)–(g) local phase retardation; (h)–(j) optic axis orientation (cyclic isoluminant colormap) overlaid with local phase retardation (brightness). $z$–$y$ cross-section views at the location marked by the vertical line in (a) showing DOP overlaid with intensity: (k) original OOF and (l) SHARP. (m) $\text{ΔCCP}$ as a function of depth for intensity and polarimetric parameters. $\text{ΔCCP}$ DOP is shown with a different vertical axis (right) to match the others (left).

In Figs. 2(k–l) we present $z$–$y$ cross-sectional views of the DOP overlaid with intensity, where vertically oriented ridge-like structures are revealed in the SHARP image [some marked with black arrows in Fig. 2(l)] that are distinctive of the stromal striae as have been observed in depolarization images of sheep eye’s limbal region [4]. To validate the improvement provided by SHARP, we computed $\text{ΔCCP}={\text{CCP}}_S-{\text{CCP}}_O$, the difference in the peaks ${\text{CCP}}_O$ and ${\text{CCP}}_S$ of the cross-correlation between the reference images and the images without and with SHARP, respectively. Figure 2(m) shows that $\text{ΔCCP}$ is strictly positive for intensity and all polarimetric parameters and increases with distance from the focal plane, as expected, because the effect of refocusing is more significant far from the focal plane.

Blurring due to defocus not only causes a reduction in the spatial resolution but also changes the quantitative estimation of the polarimetric parameters since a larger beam mixes backscattered light from a larger portion of the tissue. This results, for example, in an underestimation of the DOP. Therefore, the benefit of CR is observed as an improvement of both the spatial resolution and the quantitative estimation of the polarimetric parameters. For instance, the DOP after SHARP shown in Figs. 2(c,l) is, on average, greater than the out-of-focus counterparts. Similarly, the local phase retardation and optic axis orientation images obtained with SHARP present a quantitative estimation closer to the reference images. For instance, in the inset of Fig. 2(f), it can be seen that the overall magnitude of local phase retardation is improved with SHARP while preserving structures observed in the corresponding reference frames, and the inset of Fig. 2(i) exhibits better-defined optic axis orientation as observed in the color of structures and their boundaries (e.g., structure pointed by arrows). From our results, it is clear that the geometry of the cornea is a challenge for imaging settings in which the DOF is smaller than the apex-to-limbus distance (roughly 3 mm). Not only it is not possible to have the entire cornea in focus, but the focal plane becomes a curved surface with focal depth highly dependent on the lateral coordinates.

In OCT, there is a drop in signal-to-noise ratio (SNR) that increases with distance from the focal plane due to confocal gating. Therefore, locations far from the focal plane—in addition to increased contribution of multiply-scattered photons not considered in standard deconvolutional models—exhibit lower SNR that results in a less robust estimation of polarimetric parameters in computationally refocused data when compared with the experimentally in-focus counterparts. This difference is especially obvious in the DOP as seen in Figs. 2(b–d) where the reference image exhibits a greater average DOP.

There are two main limitations to the use of CR in PS-OCT. First, the axial resolution of local phase retardation and optic axis orientation depends on the step size for local PS processing and the magnitude of spectral binning that is determined by the number of windows and the overlap used [2]. Thus, cross-sectional views show a less remarkable improvement than en face views, where the improvement is 2D and less affected by the axial resolution loss. Second, the improvement in the lateral resolution after CR is partially neutralized by the spatial averaging of Stokes vectors required in PS processing. Here, we used the smallest possible kernel $h$ that proved effective at speckle mitigation while avoiding a significant lateral resolution loss.

The ability to extend the depth-of-field in medium-NA systems using CR represents an opportunity to fill the gap between low-NA systems with insufficient resolution and high-NA systems with short depth-of-field. For this reason, we believe CR has the potential to assist in research and clinical applications of PS-OCT in anterior segment imaging. For instance, the trabecular meshwork (TM) is known to regulate intraocular pressure (IOP) and its precise identification can be useful in studies related to the understanding, evaluation, and treatment of glaucoma [1, 13]. High-resolution polarimetric contrast could complement structural contrast in which the TM is not always clearly identified [11]. Furthermore, TM stiffness is known to be related to the IOP, and its quantification based on structural OCT imaging was used for the identification of eyes with glaucoma [13]. PS-OCT could be an alternative or a complement to IOP measurement approaches given its sensitivity to the organization of fibrous tissue, such as the collagen in the TM [11]. Indeed, a relationship between birefringence and IOP has been suggested in the sclera [12]. PS-OCT in combination with CR could also benefit future explorations of polarimetric properties of the palisades of Vogt, which requires a medium-NA system to resolve its ridge-like structure with ∼50 μm periodicity [14]. These organized fibrovascular structures contain stem cells responsible for the maintenance and regeneration of corneal tissue. The potential use of CR in OCT imaging of the palisades of Vogt has been suggested [14]. CR could also be used in tissue polarimetry for understanding scarring processes in the limbus, such as blebs scarring in glaucoma patients treated with trabeculectomy [1].

We have demonstrated the possibility of performing computational refocusing in PS-OCT for improving resolution and contrast in polarimetric parameters of tissue obtained with phase-unstable systems using SHARP in combination with spectral binning processing. Experimental results show that CR can be a powerful tool to provide high-resolution tissue polarimetry in an extended depth of field. This PS implementation of CR can be a straightforward approach to increase lateral resolution in standard phase unstable, Stokes-based SS-PS-OCT systems, with no or few system modifications. We showed an improvement in image quality of polarimetric parameters of the anterior segment of a swine eye, revealing structures not clearly visualized in the out-of-focus images, approaching the visualization of the reference in-focus images despite the large defocus offset. A MATLAB implementation of SHARP with exemplary scripts is available in Code 1, Ref. [16].