Wavenumber-space wavefront sensorless adaptive-optics for optical coherence tomography

Abstract

Adaptive-optics optical coherence tomography (AO-OCT) allows the visualization of cellular-scale retinal structures; however, its adoption both at research and clinical levels has been restricted by hardware and software complexity. Based on the observation that aberrations other than defocus are depth-independent, we propose an approach for wavefront sensorless AO-OCT that utilizes the interferometric fringe modulation in wavenumber (k-) space to optimize the wavefront correction. This approach avoids the need for tomogram reconstruction at each optimization iteration and increases robustness against axial motion. The proposed routine combines k-space optimization with focal plane shifting (i.e., defocus optimization) and evaluates the objective function B-scan-wise, achieving 8 Zernike modes correction in ∼1.89 s. Experimental testing with a phantom model eye and computational complexity analysis show the proposed algorithm has a lower computational complexity and faster optimization time per mode while performing at least as well as depth-resolved optimization, using a LabVIEW implementation without the need for high-performance dedicated software or GPU acceleration. We demonstrate its performance in human retinal imaging in vivo.

1. Introduction

The integration of adaptive optics (AO) in optical coherence tomography (OCT) unlocked the ability to visualize the retina at the cellular scale [1], leading to profound insights into the characteristics of axonal bundles, capillaries, and retinal cells [2,3]. High-resolution imaging could facilitate investigations to understand the onset and progression of pathologies that originate from cellular changes, before any vision impairment or detectable macroscopic changes arise, such as age-related macular degeneration [4] and inner retina neurodegeneration [5,6]. Adaptive-optics systems incorporate an active component, such as a deformable mirror (DM), into the sample arm of the OCT interferometer to correct wavefront aberrations. These aberrations can arise from the optical system or from the eye itself (e.g., cornea and lens) during retinal imaging [7,8]. In a conventional adaptive optics approach, the optimal shape of the deformable mirror’s surface is determined by the wavefront measured with a pupil-conjugated wavefront sensor, integrated into the sample arm [1,9–11], or using a wavefront sensorless approach where an image quality metric, computed on the captured OCT signal, is optimized [12–15].

The sensorless AO (SLAO) approach can be used in low-order aberration regimes [7,8] (up to 3rd radial order in Zernike basis) corresponding to moderate entrance beam diameters $(\lesssim 4\text{mm})$, and when slow refresh rates <0.75 Hz are sufficient for static aberration correction in rodents and humans [13,15]. In contrast, the wavefront sensor-based AO (WSAO) approach is more suitable for fast correction of dynamic ocular aberrations and high-order aberration correction (up to 5th radial order in Zernike basis) when large entrance beam diameters are used (>5 mm) [11,16].

The adoption of AO-OCT at research and clinical levels has been hindered by its complexity. WSAO demands the use of cumbersome reflective optics in the sample arm to reduce reflections that interfere with the wavefront sensing of the weak backscattered light by the retina, in addition to an extra optical relay for placing the sensor at a conjugate plane [11,17]. The resulting optical system typically has meters-long, free-space optical paths. SLAO relaxes the sensitivity to reflections, enabling the use of lenses rather than focusing mirrors, thus reducing hardware complexity and footprint [14]. However, the current SLAO paradigm requires complex, computationally expensive, signal processing algorithms: the wavefront to be applied by the control element is parameterized as a weighted sum of Zernike modes with the coefficients found iteratively either one by one [13] or simultaneously [18], by optimizing an objective function that is determined from fully reconstructed, motion-corrected OCT volumes at each step of the optimization. The objective function is generally defined as the mean intensity within an en-face slab centered at the layer of interest, which requires dedicated algorithms with highly optimized GPU implementations for real-time volumetric reconstruction and digital axial tracking [14,15]. Polans et al. used the spectral interferograms of unprocessed B-scan data for computation of the objective function, avoiding the need for tomogram reconstruction, in a wavelength-swept source OCT system with balanced detection [19]. However, the objective function in wavenumber-space (k-space) eliminated the ability to select the focal plane depth since there is no depth selectivity in the optimization.

Here, we generalized the k-space objective function to account for unbalanced detection exhibited in systems lacking balanced-detection hardware such as most spectral-domain OCT configuration or from imperfections in the balanced detection. Furthermore, we combined k-space optimization for aberrations other than defocus with depth-space (z-space) optimization for enabling automatic focal plane shifting capability. During the optimization procedure, we evaluate our objective function B-scan-wise rather than volume-wise, thus reducing the acquisition and processing time of each step [19]. Experiments with a phantom eye and computational complexity analysis suggest the proposed approach performs at least as well as volume-wise depth-resolved optimization, while being more computationally efficient. We demonstrate the performance of k-space SLAO in human retina imaging of a healthy volunteer using a lens-based spectral-domain OCT system with a deformable mirror, achieving aberration correction up to 8 Zernike modes (up to 3rd order plus spherical aberration) in ∼1.89 s in a LabVIEW implementation with no need for complex dedicated software or GPU.

2. Materials and methods

The conventional depth-resolved SLAO optimization approach [13–15] was motivated by the observation made in the pioneering AO-OCT work [1] that shifting the focal plane is necessary to accommodate the layer of interest within the depth of field that is typically shorter than the retinal thickness in AO imaging. This observation may have led to the common interpretation that aberrations are depth-dependent, and thus they need to be optimized for a fixed depth [15]. However, aberrations other than defocus in the retina are, in principle, independent of depth. Defining an objective function in wavenumber-space is possible as demonstrated in a previous work [19], circumventing the need for OCT volume computation and digital axial tracking at each step of optimization. Intuitively, the surfaces that induce aberrations in the eye (i.e., cornea and lens) are relatively far from the retina and close to the conjugate plane. Thus, there is minimal change under propagation through these surfaces when correcting defocus at a given depth in the retina with a converging or diverging beam and there is minimal change in aberrations with depth. In fact, the wavefront sensor used in WSAO does not have depth selectivity: in WSAO, aberrations are corrected without explicit depth selectivity and then the focal plane can be shifted to the layer of interest.

2.1. Experimental setup

We built a spectral-domain AO-OCT system, as illustrated in Fig. 1, based on a 55 nm bandwidth superluminescent laser diode (SLD830S-A20W, Thorlabs) centered at 830 nm, two spectrometers (CS800-840/180, Wasatch Photonics) for polarization-diverse detection, interfaced with frame grabbers (Xtium2-CL MX4, Teledyne) for full CameraLink acquisition at a maximum 250 kHz A-line rate, a two-axis galvanometer scanner (Saturn 1B, ScannerMax) that was synchronized with the frame grabber’s acquisition via a DAQ board (PCIe-6351, National Instruments), and a deformable mirror (Mirao 52e, Imagine Optic) with 52 actuators, peak-to-valley stroke of ±50 µm and 2.4 ms rise time. This stroke was sufficient to correct for typical aberrations of the human eye [7], including defocus. Furthermore, in Supplement 1 ^{(1.9MB, pdf)} we show that the DM stroke for defocus correction is conserved at the image plane even after passing through the two 4F telescopes formed by the lenses L₃–L₆. We note that in the context of the present work, polarization-diverse detection is not required.

Fig. 1.

Schematic of the custom-built spectral-domain OCT system used for SLAO retinal imaging. C: Collimator; DC: Dispersion compensator; DM: Deformable mirror; GS: Galvanometer scanner; L: Lens; M: Mirror; NDF: Neutral density filter; PBS: Polarizing beam splitter; SLD: Superluminescent diode; SP: Spectrometer; TS: Translation stage.

The light from the source was split into the sample and reference arms. The sample arm included an achromatic fiber collimator C₂ (PAF2P-A10B, Thorlabs) and achromatic lenses L₁-L₆ (B-coated AC254 series, Thorlabs) to conjugate the DM plane with the fast axis mirror and the pupil plane. Despite the slow axis mirror not been exactly conjugated to the pupil plane, no pupil wobble effects were observed on image quality or signal-to-noise ratio (SNR) along the slow-scan axis in the recorded tomograms, possibly due to the relatively small field of view and magnification of the final telescope [20]. The reference arm was configured with air and fiber path lengths intended to matched those of the sample arm; a 3 cm Shot glass N-SF11 dispersion compensation block was used to match the glass in the sample arm, with residual dispersion that was corrected numerically. A neutral density filter was used in the reference path to control the light intensity at the detectors. An acquisition and control framework was developed in LabVIEW (National Instruments, USA) integrating live preview, data recording and SLAO routine.

The theoretical axial resolution of the system was 5.5 µm in air and the $e^{-2}$ beam diameter at the pupil was ∼3.4 mm, equivalent to a focused Gaussian spot with $e^{-2}$ diameter 5.2 µm in a 16.7 mm focal-length eye, without considering the double-pass effect. This intermediate numerical aperture (NA) regime was chosen with the interest of achieving near-isotropic resolution while avoiding the need for mydriasis. We also designed and integrated a chin and head rest and a fixation target using a fiber-coupled LED (M660FP1, Thorlabs) and a dichroic mirror (DMLP650, Thorlabs) to improve gaze stability and reduce motion artifacts, which could affect illumination/detection coupling efficiency leading to SNR fluctuation artifacts.

2.2. k-space sensorless adaptive-optics

The goal in SLAO is to find the optimal DM shape that minimizes aberrations. The control command v applied to the DM is an $n\times 1$ vector with the value of all n individual actuators in the DM ( $n=52$ in our case). We can define $\mathit{v}=Z\mathit{w}$ as a weighted sum of p Zernike modes, vectorized into an $n\times p$ matrix Z, with a $p\times 1$ coefficients vector w. The goal of SLAO becomes to find the optimal set of coefficients ${\mathit{w}}^{\ast }$ that minimizes aberrations. Under the assumption that aberrations are minimal when signal intensity is maximal, the signal “energy” can be used as a proxy for the optimization

$${\mathit{w}}^{\ast }=\arg \underset{\mathit{w}}{max}\left\{\sum _{(x,y)\in R}\sum _{j\in J}{|S_{\mathit{w}}(j,x,y)|}^2\right\},$$ (1)

where $S_{\mathit{w}}(j,x,y)$ is the OCT signal as a function of either wavenumber $(j=k)$ or depth $(j=z)$ and transverse spatial coordinates $(x,y)$ . J and R represent the domain over which the metric is evaluated. In conventional SLAO, the optimization is computed in depth-space and volume-wise, therefore $S_{\mathit{w}}(j,x,y)=T_{\mathit{w}}(z,x,y)$ is the motion-free reconstructed tomogram, the domain J is a slab in depth and the transverse domain R is two-dimensional.

In our proposed SLAO, the optimization was computed in wavenumber-space and B-scan-wise [19] using the interferometric fringe modulation

$$S_{\mathit{w}}(j,x,y)=\frac{t_{\mathit{w}}(k,x)-{\overline{t}}_{\mathit{w}}(k)}{1+{\overline{t}}_{\mathit{w}}(k)}H(k),$$ (2)

when optimizing any Zernike mode other than defocus (i.e., $w\ne 4$ in the OSA/ANSI standard indices that are adopted here), thus the domain J was the entire measured wavenumber domain and the transverse domain R was one-dimensional. $t_{\mathit{w}}(k,x)$ are the raw detected interferometric fringes, ${\overline{t}}_{\mathit{w}}(k)=\sum _x{t}_{\mathit{w}}(k,x)/n_x$ estimates the reference signal by averaging across the $n_X$ A-lines, and $H(k)$ is a rectangular function with width equal to the full width at 10% of the maximum of ${\overline{t}}_{\mathit{w}}(k)$ , to ignore the noisy regions at the end of the spectrum. We added 1 in the denominator to prevent numerical diverging values at the edges of the spectrum. The use of the entire spectral fringe modulation for driving the optimization makes it more robust since the signal from all depths contribute to the metric, which we believe is what makes B-scan-wise optimization possible, as opposed to a few depth samples as in conventional SLAO. Using the fringe modulation, obtained through background subtraction and spectral homogenization in Eq. (2), rather than the raw fringes was necessary to mitigate the impact of light intensity fluctuations coming from the light source that affect the DC component of the fringes, which is much larger than the fringe modulation, and thus can mislead the optimization.

In order to freely select the focal plane position, we optimized the coefficient for defocus $w_4$ in z-space, using a depth slab $\mathrm{\Delta }z$ of 11 depth pixels. The peak axial position of the cross-correlation between two consecutive B-scans computed via fast Fourier transform was used to track the axial position of the user-defined target depth. Additionally, to evaluate the objective function we used a one-dimensional transverse domain R spanning the length of a B-scan corresponding to a B-scan-wise objective function. Therefore, Eq. (1) in our SLAO routine was

$${\mathit{w}}^{\ast }=\arg \underset{\mathit{w}}{max}\{\begin{array}{ll}\sum _x\sum _{z\in \mathrm{\Delta }z}{|T_{\mathit{w}}(z,x)|}^2& \text{if}\mathrm{ }w=4\\ \sum _x\sum _k{\left|\frac{t_{\mathit{w}}(k,x)-{\overline{t}}_{\mathit{w}}(k)}{1+{\overline{t}}_{\mathit{w}}(k)}\right|}^{2}H(k)& \text{if}\mathrm{ }w\ne 4.\end{array}$$ (3)

We started by optimizing for defocus in z-space, evaluating the metric in a depth-slab centered at the target depth, then updated the DM with the optimal coefficient $w_4^{\ast }$ and continued with the remaining modes by optimizing in k-space and updating the DM one mode at a time. The target depth for defocus correction was selected manually in the control software by locating a line overlaid in the B-scan preview. The idea of depth-independent optimization in conjunction with focal plane shifting is similar to WSAO, where there is no depth selectivity.

2.3. Optimization procedure

The optimization approach that was used is similar to the conventional one-mode-at-a-time approach with a heuristic to track the objective function and prevent the process from getting stuck in plateaus [13], in combination with the novel objective function in Eq. (3). The optimization of the i–th Zernike mode started with creating a linearly-spaced grid of q coefficient values covering a search range ${\mathrm{\Delta }}_w$ centered around the current value, then each value in the grid was projected onto the sample plane by the deformable mirror followed by a 5 ms delay to wait for the mirror to settle, after which a B-scan was acquired and used to evaluate the objective function. Robustness was observed to be reduced if the wait time was set close to the deformable mirror rise time of 2.4 ms; therefore, 5 ms was chosen as a good trade-off between speed and robustness. The exhaustive search was repeated if the index of the optimal coefficient was outside the central half segment of the grid, similarly to previous works [13], using the current optimal value as the center of the new grid. To prevent the process from getting stuck in plateaus, this heuristic procedure was evaluated once and in the second iteration the algorithm continued regardless of the location of the optimal coefficient. The number of points $q=21$ was found empirically with phantom samples and imaging in vivo and the search range ${\mathrm{\Delta }}_w$ is detailed in each experiment. The Zernike modes included in the matrix Z were those that have been observed to have the largest contribution in the ocular wavefront in the eye at our NA, namely second and third order Zernike modes, in addition to primary spherical aberration [7,8] for a total of 8 Zernike modes.

In summary, the SLAO routine was performed as follows: first, a reference B-scan was acquired and displayed on the screen. The operator set the target plane for defocus optimization by using a drag-and-drop line projected in the reference B-scan image. The SLAO optimization started by optimizing defocus based on the z-domain image metric and the user-defined target depth, adjusted accordingly to the instantaneous axial shift estimated from the cross-correlation between the past and current B-scans. Then, the following Zernike modes were optimized using the k-domain image metric, dramatically increasing the performance of the optimization procedure because of the reduced computation complexity in k-space compared to z-space (i.e., no need for axial tracking and tomogram reconstruction involving Fourier transforms). Data recording was performed right after the SLAO optimization.

2.4. Calibration of the deformable mirror

In practice, the actual wavefront $\mathit{\phi }$ at the sample plane differs from the command v applied to the DM because of crosstalk between the n actuators and propagation through the optical system. The wavefront $\mathit{\phi }$ at the pupil plane when the control command is v is characterized by the $n\times n$ interaction matrix A, the $n\times 1$ offset vector ${\mathit{v}}_{\mathbf{0}}$ , and the $n\times 1$ baseline wavefront ${\mathit{\phi }}_{\mathbf{0}}$ ,

$$\mathit{\phi }=A(\mathit{v}-{\mathit{v}}_{\mathbf{0}})+{\mathit{\phi }}_{\mathbf{0}},$$ (4)

where ${\mathit{v}}_{\mathbf{0}}$ is the command that makes the deformable mirror flat, and was provided by the manufacturer, therefore we defined $\mathit{v}=\mathit{\psi }+{\mathit{v}}_{\mathbf{0}}$ and parameterized $\mathit{\psi }$ as a weighted sum of p Zernike modes, vectorized into an $n\times p$ matrix Z, with a $p\times 1$ coefficients vector w, namely $\mathit{\psi }=Z\mathit{w}$ . Substituting in Eq. (4), we obtain

$$\mathit{\phi }=A(\mathit{\psi }+{\mathit{v}}_{\mathbf{0}}-{\mathit{v}}_{\mathbf{0}})+{\mathit{\phi }}_{\mathbf{0}}=AZ\mathit{w}+{\mathit{\phi }}_{\mathbf{0}}.$$ (5)

We can then decompose the obtained wavefront $\mathit{\phi }$ into the same Zernike basis with coefficients $\mathit{\alpha }$ , resulting in

$$Z\mathit{\alpha }=AZ\mathit{w}+Z{\mathit{\alpha }}_{\mathbf{0}}$$ (6)

$$\mathit{\alpha }=A_z\mathit{w}+{\mathit{\alpha }}_{\mathbf{0}},$$ (7)

where ${\mathit{\phi }}_{\mathbf{0}}=Z{\mathit{\alpha }}_{\mathbf{0}}$ and $A_z=Z^{-1}AZ$ . Equation (7) is a modal representation of the original element-basis expression in Eq. (4).

We performed modal calibration by using Eq. (7) and a wavefront sensor (WFS20-14AR, Thorlabs) placed at the sample plane that directly outputs the Zernike coefficients $\overline{\mathit{\alpha }}$ of the measured wavefront. For p Zernike modes, a total of $2p+1$ measurements were performed, each one with the coefficients vector w composed of a single Zernike mode, i.e., w is a one-hot vector, which allowed to retrieved the matrix $A_z$ one column at a time. Concretely, for the i-th column we projected the coefficients ${\mathit{w}}^{(+/-)}=(0,0,\dots \pm \delta ,\dots ,0,0)$ having a value of $\pm \delta$ in the i–th entry and zero elsewhere, and recorded their associated measurements ${\overline{\mathit{\alpha }}}^{(+)}$ and ${\overline{\mathit{\alpha }}}^{(-)}$ . The i–th column of $A_z$ is then populated with the value $({\overline{\mathit{\alpha }}}^{(+)}-{\overline{\mathit{\alpha }}}^{(-)})/(2\delta )$ . This approach eliminates the baseline coefficient ${\alpha }_0$ since it is averaged out within the two measurements. An extra measurement ${\overline{\mathit{\alpha }}}_0$ , which adds up to $2p+1$ , was used to retrieved the baseline coefficients by projecting a command with coefficients set to zero, $\mathit{w}=(0,\dots ,0)$ . The calibrated interaction matrix ${\overline{A}}_z$ was then inverted using the pseudo-inverse to get ${\overline{A}}_z^{(-1)}$ . Once calibrated, the command applied to the deformable mirror to obtain a waveform composed of a given set of Zernike coefficients ${\mathit{\alpha }}^{\ast }$ was found as

$$\mathit{v}=Z{\overline{A}}_z^{(-1)}({\mathit{\alpha }}^{\ast }-{\overline{\mathit{\alpha }}}_0)+{\mathit{v}}_0.$$ (8)

2.5. Experiments and data acquisition

We imaged the retina of a healthy subject with a field of view (FOV) of 2.5 × 2.5 deg² at an approximate eccentricity of 4 deg nasal to the fovea, with 512 A-lines per B-scan and 512 B-scans. SLAO optimizations covered the same FOV along the fast-scan axis of 2.5 deg over 512 A-lines, with a Zernike weights search range of ${\mathrm{\Delta }}_w=\pm 0.2\text{µm}$. Standard OCT processing consisting in background subtraction, spectrum flattening followed by windowing, k-mapping and dispersion correction were applied to all datasets in post-processing. The relatively short confocal parameter (∼60 µm) prevents achieving in-focus images across the entire retinal thickness simultaneously, therefore, we selected an appropriate target depth for optimizing defocus $Z_4$ at either the retinal nerve fiber layer (RNFL) or the inner segment/outer segment (IS/OS) junction.

For image visualization, bulk motion correction, despeckling and tissue flattening with respect to the RNFL and the IS/OS were applied after data acquisition. For motion correction, sub-pixel bulk shifts between adjacent B-scans were estimated via intensity image cross-correlation and corrected on the complex-valued tomograms [21] after phase-stabilization [22]. For despeckling, we applied the technique TNode [23] using three-dimensional search and similarity windows with isotropic sizes 21 px and 7 px, respectively, and filtering parameter $h_0=0.11$ .

We also performed two experiments to validate the proposed approach and assess its repeatability. First, for validation, we imaged the same static sample (an aberrated model eye) using four different optimization configurations: i) volume-wise in z-space (conventional technique), ii) B-scan-wise in z-space, iii) B-scan-wise in k-space (proposed technique) and iv) volume-wise in k-space. B-scan-wise configurations used a FOV of 2.5 deg over 484 A-lines, while the volume-wise configurations used a FOV of 2.5 × 2.5 deg² over 22 A-lines per B-scan and 22 B-scans, thus keeping the same total number of A-lines. For all configurations, we recorded data twelve times; each time we ran the SLAO routine with ${\mathrm{\Delta }}_w=\pm 0.1\text{µm}$ and computed the increase in signal-to-noise ratio (SNR) with respect to a reference tomogram recorded with the DM flat. The SNR was computed as $R=⟨I⟩/⟨N⟩$ , where $⟨I⟩$ is the average intensity of a 40 µm (21 px) depth slab within the sample sub-surface (directly underneath the surface, to avoid saturation from the surface) and $⟨N⟩$ is the average noise intensity computed in a region without signal above to the sample. The aberrated model eye consisted of an aspheric lens (C280TMD-B, Thorlabs) placed backwards and misaligned right after a microscope coverslip to purposefully induce aberrations, and the sample was a scattering paper card placed near the focal plane. For this experiment, because the sample was static, real-time axial tracking was disabled.

Second, to assess the repeatability in vivo, we imaged the same subject ten times, performing an independent optimization for SLAO in each measurement, with the target focal position set at the RNFL, and having a short rest time in between recordings where the subject moved away from the chin rest and then positioned back for the next measurement. After data acquisition, the recorded tomograms were corrected for motion as explain before and for system sensitivity roll-off to eliminate differences in signal intensity between measurements due to the system. Finally, the SNR increase with respect to a reference tomogram recorded with the DM flat was computed for each tomogram as described before, using a 60 µm (31 px) depth slab in the RNFL.

3. Experimental results

3.1. Validation and repeatability experiments

Box plots of the validation experiments with the sample ex vivo and the repeatability experiment in vivo are shown in Fig. 2. t–test was used to compare the average increase in signal-to-noise ratio (SNR) with each configuration, with 20 degrees of freedom $\text{df}=20$ after removing outliers, defined as points beyond 1.5 times the interquartile range from the first and third quartiles. In all cases, SLAO provided an increase in SNR with respect to the reference. k-space optimization provided, on average, a significantly higher increase in SNR than z-space optimization $(p<0.001)$ . With z-space optimization, the average increase in SNR of 3.89 ±0.24 dB with B-scan-wise metric (referred to as 2D) was significantly lower $(p<0.001)$ than the increase of 4.35 ±0.20 dB obtained with en-face-wise metric (referred to as 3D). This is in agreement with prior (qualitative) observations, that B-scan-wise optimization is not sufficiently robust in depth-resolved SLAO [15]. In contrast, there is not significant difference between B-scan and volume optimizations in k-space $(p=0.611)$ , 4.76 ±0.06 dB and 4.78 ±0.09 dB respectively. This also suggests that the large under-sampling used in the volume-wise FOV is not affecting the spectral fringes in a way that can mislead the optimization, for instance with fringe washout coming from the fast motion of the galvanometer mirrors. Importantly, there is a significant difference between the conventional approach based on z-space optimization and our proposed approach based on k-space optimization $(p<0.001)$ , in addition to a reduction in variability. We attribute this improvement to the fact that more data points per A-line are included in the metric, i.e. the entire spectrum, compared to z-space optimization which pools information only from a few depth samples.

Fig. 2.

Box plot of increase in SNR for (a) the validation experiment comparing different SLAO routines and (b) in vivo repeatability experiment. (c) Optimal weight values and corresponding average DM commands obtained with conventional depth-resolved SLAO and the proposed routine. The weight values are plotted as rectangles at the mean ± two standard deviations of all the repetitions. The DM command images display the fraction of voltage applied to each element with respect to the maximum voltage.

The optimal Zernike weights obtained with conventional depth-resolved SLAO and our proposed approach shown in Fig. 2(c) follow a similar trend, however, there are small differences, such as the weight of spherical aberration $\left(Z_{12}\right)$ , which dominates the wavefront correction due to the aspheric lens being backwards. The optimal DM commands averaged across all the repetitions, shown also in Fig. 2(c), are qualitatively similar to each other and have a root mean squared error of 0.01. Despite yielding similar results, the subtle differences between the optimal weights and resulting wavefronts between the two methods may explain the differences in SNR increase observed in Fig. 2(a).

In the repeatability experiment in vivo, the proposed k-space SLAO routine provided consistent gain in SNR (14.21 dB, with a standard deviation of 1.35 dB) after optimization. Figure 2(b) shows a box plot of the increase in SNR between each tomogram with SLAO versus no AO.

3.2. In vivo retinal imaging

Structural images of the healthy retina are shown in Fig. 3. We present a reference B-scan image from the tomogram without AO in Fig. 3(a), and B-scan and en-face images from the tomograms with k-space SLAO having the target focal position located at the RNFL in Figs. 3(b)-(c), and at the IS/OS junction in Figs. 3(e)-(f). The en-face images are average intensity projections within 14 µm (7 px) depth slabs indicated by the color lines in the corresponding B-scan images in Figs. 3(b) and (e), and the peak SNR is reported in each image. Structural images were despeckled with TNode, except for the IS/OS en-face images since the point-like structure of photoreceptors violate the fully-developed speckle assumption of TNode. Images of the RNFL without TNode can be found in Supplement 1 ^{(1.9MB, pdf)} . A full en-face fly-through is provided in Visualization 1 ^{(91.3MB, avi)} .

Fig. 3.

Retinal imaging with k-space SLAO-OCT reveals nerve fiber bundles and photoreceptors. (a) Cross-sectional image without AO exhibiting low SNR and resolution. Cross-sectional and en-face images with SLAO setting the target focal plane depth at (b)-(c) the RNFL and (e)-(f) the IS/OS junction. En-face images with only focal plane shifting set to (d) the RNFL and (g) the IS/OS junction. Dashed lines in the cross-sectional and en-face images indicate the locations of each other’s viewing planes. (h) Intensity histograms at the RNFL showing the SNR improvement. (i) Radial power spectrum of the insets in (f) and (g) showing a peak at the photoreceptors spacing when using full AO. S: superior, I: inferior, N: nasal, T: temporal.

The tomogram without AO has a weak signal, hindering the visualization of the retinal structures and providing poor contrast between layers, specially in the inner retina. Furthermore, aberrations, including defocus, hinder the visualization of structural information in the en-face images such as axonal bundles in the RNFL and photoreceptors in the IS/OS junction. The improvement in SNR and image sharpness provided by SLAO is clearly observed in both layers of interest, and demonstrated by the increase in average signal intensity. Figure 3(h) shows histograms of the en-face intensity images within the RNFL, and demonstrates the improvement in SNR brought by defocus and further correction of higher order aberrations. The bi-modal shape of the histograms reflects the difference in intensity between axonal and glial tissue.

In the RNFL, the axonal bundles are better resolved and distinctly run along the nasal-temporal orientation, the characteristic distribution for this eccentricity. In the IS/OS layer, averaging over the z-slab reduced speckle and k-space SLAO enabled better visualization of well-defined, point-like photoreceptor cells, as observed in the zoomed insets. In both cases, while the structures are visible with defocus correction alone as shown in Figs. 3(d) and (g), they are better defined with full aberration correction, as shown in the zoomed insets. In particular, photoreceptors are not resolved when only defocus is corrected, and the characteristic bright circular spots are better resolved when the full AO correction is applied. Figure 3(i) shows the radial power spectrum of the insets in Figs. 3(f) and (g), computed by taking the Fourier transform of the log-scale intensities and averaging along the azimuthal direction [24]. A clear peak beyond the background components is observed only with full AO correction, at a spatial frequency of around 0.12 µm⁻¹ that reflects the photoreceptors spacing at this parafoveal eccentricity [24].

It is also important to note that even when the focus is at the RNFL, all layers including the bright outer retina still contribute to the optimization since k-space SLAO benefits from signal contributions across the entire depth. This benefit is expected to be even larger when the focus is at more weakly scattering layers, such as the inner/outer nuclear layers.

3.3. Computational complexity and optimization time evaluation

We compared the computational performance of the k-space and z-space optimization approaches in terms of computational complexity and measured optimization time. Computational complexity estimates the number of operations required to evaluate the objective function for a single A-line with N wavenumber samples. Tomogram reconstruction in z-space optimization involves fringe resampling, DC subtraction, dispersion correction, and a fast Fourier transform (FFT) [25]. Using FFT-based interpolation, the forward FFT has a complexity of $\mathcal{O}(N{\log }_{2}N)$ , the inverse FFT with interpolation factor m has $\mathcal{O}(mN[{\log }_{2}N+{\log }_{2}m])\approx \mathcal{O}(mN{\log }_{2}N)$ , and linear resampling adds $\mathcal{O}(N)$ , yileding a total complexity of $\mathcal{O}(N[m+1]{\log }_{2}N+N)$ . DC subtraction has $\mathcal{O}(1)$ complexity since no multiplications or divisions are required, dispersion correction adds $\mathcal{O}(N)$ for point-wise products, and the final FFT contributes another $\mathcal{O}(N{\log }_{2}N)$ . Thus, the overall complexity per A-line in z-space optimization is $\mathcal{O}(N[m+2]{\log }_{2}N+2N)$ . For a typical configuration $(N=2048,m=2)$ , this corresponds to approximately $\mathcal{O}(46N)$ operations. In contrast, the k-space approach requires only DC subtraction, $\mathcal{O}(1)$ , and spectral regularization and normalization, $\mathcal{O}(2N)$ . Therefore, its per-A-line complexity is roughly 23 times lower than that of z-space optimization.

In practice, GPU-based implementations can achieve real-time reconstruction despite high computational loads [25]. Hence, measured optimization time and number of voxels processed per mode provide a more meaningful benchmark. For our implementation, the average optimization time was 0.42 s for defocus and 0.21 s per mode for higher-order modes, using 21 weight values and 10.75 kA-lines per mode. The longer time to optimize defocus arises from the additional axial tracking and reconstruction steps. For comparison, z-space SLAO achieved 0.27 s per mode [15], using a small $150\times 50$ en-face region and 9 weights (67.5 kA-lines per mode). Our routine also included a 5 ms delay per iteration to allow the deformable mirror (DM) to settle —a hardware constraint rather than a method limitation. Previous works avoided this by using a faster although more costly DM with response time 0.45 ms. Neglecting this dead time and focusing only on the objective function evaluation, our k-space optimization would reach 0.105 s per mode, making it 2.5× faster than the reported z-space implementation, while requiring six times fewer A-lines per mode optimization. Despite using fewer A-lines, note that the k-space objective function [Eq. (3)] averages over all k, effectively increasing the number of observations guiding the optimization, whereas z-space optimization focuses on one or a few depth samples.

In summary, k-space optimization offers significantly lower computational complexity, faster optimization, and fewer A-lines per mode compared to z-space methods. Further improvements are possible by using faster DMs or incorporating a defocus look-up table (LUT) that pre-calibrates the DM stroke versus focal plane displacement, eliminating the need to optimize the defocus weight directly.

4. Conclusions

We demonstrated the possibility to combine ocular aberration correction using k-space SLAO with focal plane shifting capability using z-space optimization for high-resolution OCT imaging. The proposed SLAO routine does not rely on real-time, motion-corrected tomogram reconstruction, which requires sophisticated implementations and GPU acceleration, and is more robust to axial motion than z-space optimization. Furthermore, validation experiments showed that the k-space metric is more robust than the z–space metric, and repeatability testing in vivo suggested it offers consistent SNR improvement. The proposed SLAO routine achieved faster optimization time per mode to previous works, without the need of GPU-acceleration and using a less expensive DM with longer settle time. Reducing the computational complexity and decreasing the optimization time and A-line acquisitions per mode offers a simple and efficient alternative that lowers the complexity of implementation of sensorless AO. This was possible by combining k-space optimization with B-scan-wise objective function evaluation.

Funding

National Institutes of Health 10.13039/100000002 ( P41 EB015903).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 ^{(1.9MB, pdf)} for supporting content.