Combined hardware and computational optical wavefront correction

In many optical imaging applications, it is necessary to overcome aberrations to obtain high-resolution images. Aberration correction can be performed by either physically modifying the optical wavefront using hardware components, or by modifying the wavefront during image reconstruction using computational imaging. Here we address a longstanding issue in computational imaging: photons that are not collected cannot be corrected. This severely restricts the applications of computational wavefront correction. Additionally, performance limitations of hardware wavefront correction leave many aberrations uncorrected. We combine hardware and computational correction to address the shortcomings of each method. Coherent optical backscattering data is collected using high-speed optical coherence tomography, with aberrations corrected at the time of acquisition using a wavefront sensor and deformable mirror to maximize photon collection. Remaining aberrations are corrected by digitally modifying the coherently-measured wavefront during imaging reconstruction. This strategy obtains high-resolution images with improved signal-to-noise ratio of in vivo human photoreceptor cells with more complete correction of ocular aberrations, and increased flexibility to image at multiple retinal depths, field locations, and time points. While our approach is not restricted to retinal imaging, this application is one of the most challenging for computational imaging due to the large aberrations of the dilated pupil, time-varying aberrations, and unavoidable eye motion. In contrast with previous computational imaging work, we have imaged single photoreceptors and their waveguide modes in fully dilated eyes with a single acquisition. 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Introduction
Optical imaging is often limited by wavefront aberrations, which may be inherent to the imaging system itself or introduced by the imaged sample.By compensating for the wavefront error using adaptive optics, it is possible to acquire high resolution images even in the presence of aberrations [1][2][3][4].Conventionally, this has been achieved using a wavefront sensor and deformable mirror to measure and correct the aberrated optical wavefront.We term this method hardware adaptive optics (HAO).
HAO physically modifies the optical wavefront to allow tight focus of the imaging beam.This allows for high signal even in the presence of large aberrations.However, the HAO correction is optimized for a single depth within the sample, and thus, image quality is not optimal for other depth locations.The wavefront correction also suffers from limited sampling and inherent measurement and fitting errors, and is sensitive to system misalignment.Misalignment is particularly prevalent when acquiring data on living subjects due to involuntary head and eye motion and eye blinks.These effects can be reduced, but not eliminated.
Previous attempts to improve HAO images have relied upon amplitude-based deconvolution [5,6].These methods neglect the phase information and operate upon the amplitude or intensity image only.Small values in the optical transfer function lead to either signal loss or noise amplification, making these methods poorly suited for imaging in scattering samples.
Optical coherence tomography (OCT) is a broadband interferometric imaging method which measures coherently backscattered light, and can be combined with HAO for imaging samples that aberrate the wavefront [7][8][9][10].By acquiring interferometric data using OCT, the complex wavefront is measured, and the pupil phase can be digitally adjusted to compensate for aberrations [11].This is done by applying a phase-only filter in the spatial frequency domain and is analogous to the physical operation of a deformable mirror in adaptive optics.Therefore, we term this method computational adaptive optics (CAO).Using CAO, computational wavefront correction has been demonstrated for a variety of biological imaging applications [11][12][13][14][15].
With CAO, the image formation process can continue after data acquisition, and therefore the data does not need to be aberration free when acquired.This reduces the burden on HAO to provide optimal correction at the time of imaging and the need to maintain optimal sample alignment.Additionally, the aberration correction can be fine-tuned to each depth layer, field position, and time point of acquisition.However, because CAO does not physically modify the wavefront, any photons lost due to the presence of aberrations are not recovered using CAO.When the imaging beam is aberrated, the input signal strength is distributed away from the nominal focus.The back scattered photons are then rejected by the confocal detection of point-scanning OCT.This causes a drop in the detected OCT signal strength, leading to a loss in signal-to-noise ratio (SNR) that cannot be completely recovered using computational methods alone.
One alternative to scanned OCT is full-field OCT which removes the confocal gate and allows collection of aberrated photons.Using CAO, full-field OCT has acquired highresolution images of the retina through a dilated pupil without hardware wavefront correction [16].However, full-field OCT has stricter imaging speed requirements to achieve sufficient phase stability, lower diffraction-limited resolution, and reduced contrast due to the noise arising from scattered photons when compared to scanned OCT.
We combine HAO and CAO together to address the shortcomings of each method and to demonstrate how their strengths can be integrated to provide more complete correction of wavefront aberrations.A high-speed OCT system equipped with HAO is used to acquire 3D, phase-stable interferometric data with high SNR.This data is then reconstructed using CAO to remove aberrations left uncorrected by HAO.Together, HAO + CAO achieves improved resolution when compared to HAO, and improved SNR when compared to CAO.
This work is similar to, but distinct from point-spread function engineering [17], where hardware is used to introduce a known intensity profile onto the optical beam which can then be digitally corrected.For example, wavefront coding or airy beam imaging can create a distorted point-spread function which increases the depth-of-field [18][19][20].Likewise, artificially introduced astigmatism can be used for depth localization of sparse signals [21,22].In point-spread function engineering, the desired wavefront modification is known.In the case of combined HAO + CAO imaging, the wavefront aberrations are sample-induced, dynamic, and unknown.Rather than modify an ideal wavefront to suit some other purpose, the goal is to thoroughly correct the distorted wavefront for improved imaging capability, demonstrated here by in vivo retinal imaging.
Imaging the living human retina requires imaging through the optics of the eye itself, which are severely aberrated when the pupil of the eye is large [23].To obtain a high numerical aperture and therefore high resolution, the pupil must be dilated, resulting in strong ocular aberrations and photon loss.Because of this, previous demonstrations of in vivo CAOonly imaging using scanned OCT were performed on undilated subjects [13,24,25].Therefore, retinal imaging is an application well-suited for a combined hardware and computational approach.

Adaptive optics imaging system
A high-speed adaptive optics OCT system was used to physically correct aberrations and acquire retinal OCT data [26].Like most HAO systems, the sample arm used mirrors instead of lenses to eliminate back reflections that may interfere with the wavefront measurement.Unfortunately, the use of spherical mirrors introduces strong astigmatism that must be compensated by placing some mirrors out-of-plane.The system was a unique design which used toroidal mirrors to allow for in-the-plane alignment without strong system aberrations, the details of which were previously published in Ref [27].An updated version of the system published in Ref [26] was used for the experiments presented here.Key differences from the original design include the use of a single deformable mirror and four interleaved spectrometers.
The HAO system consisted of a deformable mirror (DM 97, ALPAO) and a custom Shack-Hartmann wavefront sensor, constructed with a 20 x 20 lenslet array in front of a sCMOS camera (Neo, Andor).A superluminescent diode centered at 790 nm with 47 nm bandwidth was used for both imaging and wavefront sensing, giving an axial resolution in tissue ( 1.38 n = ) of 4.7 µm.The pupil size was 6.67 mm at the eye, resulting in a theoretical diffraction limited transverse resolution of 2.4 µm.The spectral domain OCT system used four interleaved spectrometers operating at line rates of 250 kHz each, for an effective line rate of 1 MHz.All images were acquired at the 1 MHz rate.

Human subject imaging
Data was acquired from the right eyes of two healthy male subjects, ages 27 years and 26 years.These are referred to as Subject 1 and Subject 2, respectively.The left eye was covered by a patch, and the right eye was dilated using 0.5% tropicamide.Imaging was performed at 0.5°, 1°, 3.5°, 7.5°, and 12.5° temporal (T) to the foveal center.The OCT data was acquired with the HAO system running in closed loop feedback, and a real-time display was used to place the focus near the cone photoreceptors prior to data acquisition [28].Due to the relatively high numerical aperture of the dilated pupil, only posterior retinal layers had sufficient SNR for observation.
OCT data was acquired in bursts of 30 sequential volumes at 10 volumes per second, for a total acquisition time of 3 seconds.This acquisition scheme was standard for HAO to ensure that sufficient motion-free volumes were acquired for imaging of small features such as rods, and to avoid imaging during large motion artifacts such as saccades.This method was not altered for CAO and was sufficient to acquire many phase stable OCT volumes within a single burst.All procedures on the subjects adhered to the tenets of the Declaration of Helsinki and were approved by the Institutional Review Board of Indiana University.

Phase stability
Computational wavefront correction requires a stable phase relationship during the measurement of each location within the imaging volume.In a dynamic sample such as the human eye, there is significant motion which is largely overcome by using high imaging speeds.Although the 1 MHz line rate was sufficient to overcome much of the eye motion, it was important to measure the system stability due to the use of galvanometer scanning mirrors at such high frame rates and the fluctuations of the deformable mirror.This was analyzed using a model eye with the HAO system operating with and without closed-loop feedback.Repeated frames were acquired at the same location by fixing the position of the slow-axis scanning mirror.Complex conjugate multiplication was performed between consecutively acquired frames (along the temporal axis), and the result was averaged along both depth and the fast-scanning axis.This canceled out any transverse motion and measured purely axial motion, to which computational OCT is most sensitive.Following volume phase

stabilization [ previously de
Phase mot method outlin length of the f Each depth im Fourier spectr was done by was to filter dimension by The result wa

CAO pro
Residual aber CAO was use a phase-only en face OCT transform.Th wavefront cor Depth layers HAO and HA  Cone mosaic images were generated by a maximum projection of the inner segment/outer segment junction (IS/OS) and cone outer segment tip (COST) depth layers [10].The rod mosaic image was taken directly from the rod outer segment tip (ROST) depth layer.Prior to extracting individual cell layers, the HAO and HAO + CAO data were co-registered to remove any translation introduced by CAO processing, and flattened to remove tip, tilt, and slowly-varying axial eye motion [31,32].
The optimal aberration correction filter was determined via stochastic optimization of the image sharpness over the first five Zernike orders, using the resilient backpropagation procedure outlined in [33].The CAO phase filter extended to the maximum theoretical cutoff frequency of the confocal system, defined as two times the spatial frequency coverage of the 6.67 mm pupil at the eye [34].A single correction was used for the entire field-of-view, which is roughly half the size of the expected isoplanatic patch on the retina [35].The image sharpness metric was calculated from the complex OCT signal ( , )  S x y as the sum of the squared intensity, 2 * , ( , ) ( , ) .
The CAO procedure was tested on the COST layer at 12.5°T in Subject 2 to determine the run time and image sharpness improvement for an increasing number of Zernike modes.The maximum Zernike mode was increased from 2 nd to 10 th order (excluding piston, tip, and tilt), and the optimization was run 10 times at each step.The optimization was performed on the 300 x 300 pixel image using MATLAB 2015b on an Intel Core i7-6950X processor.Optimization up to 5 th order (20 th Zernike term) was determined to be a good balance between optimization time and image improvement, with an average runtime and sharpness improvement of 12 seconds and 42%, respectively.This was used as the default setting for processing other retinal data sets.

Results and discussion
Retinal data was acquired from living human subjects with fully dilated pupils using a highspeed adaptive optics OCT system.Representative cone photoreceptor mosaics for each possible combination of HAO and CAO are given in Fig. 2, along with the peak SNR in each case.Each image shows the OCT amplitude presented on a common grayscale normalized to the HAO + CAO image.The peak SNR was calculated as Images acquired without HAO (Fig. 2(a)) correspond to a fixed defocus applied to the deformable mirror based upon the subject's eyeglass prescription (−2 diopter).In the no-AO case, the strong ocular aberrations lead to poor SNR and poor resolution.Note that in many subjects, the SNR without adaptive optics may be so low that no photoreceptors are visible.CAO recovers the diffraction-limited resolution, and the peak SNR increases due to higher peak signal of the corrected point-spread function (PSF).However, the total signal collected remains constant before and after computational correction.The greatest improvement in SNR comes from the addition of HAO (Fig. 2(b)), which increased the peak signal by nearly an order of magnitude over the no-AO case.Still, the point-spread function remains somewhat aberrated.The HAO + CAO image shows both improved resolution and the greatest increase in SNR when compared to the no-AO image, a 12.7 dB increase.This demonstrates the synergy between the hardware and computational wavefront corrections.A compar mosaics at in compiled from (COST) depth visualization residual aberr fovea.Signal improved reso between the t 0.5°T, 1°T, a profiles.At g modes [36], a extend these f 7.5°T and 12 delineated fol  The root m the correspon peak IS/OS a outer segment orrection, mult e rods themsel adows of the c yer onto the R e color coded respond to the at imaging of t he rod photor ffraction-limite n limit using c within the field eparated are res n to being opt imized for eac shown in Fig. 5 The optimized by HAO, but c a general pro (Z 5 ), and posi weights vary gth.The tempo dynamics of oc mirror, resulting  CAO and ing to the d over the spatial frequencies corresponding to the 6.67 mm physical pupil, and is given as a fraction of the central wavelength, λ.Note that there is a non-trivial relationship between residual aberration RMS and sharpness improvement, as each aberration mode has a unique influence upon the value of the metric [46].
The impact of ocular aberrations typically increases with retinal eccentricity, placing a greater burden on the HAO system, which could be shared by CAO.There is also increased difficultly of obtaining optimal alignment of the subject pupil with the HAO system at larger retinal eccentricities.The results in Table 1 follow that general trend, showing more improvement in the image sharpness metric with increasing eccentricity.The image sharpness improvement was calculated as the increase in the sharpness metric defined in Eq. ( 2), with the modification that the metric was normalized by the sum of the pixel intensities to account for any small variations in image power resulting from the computational correction.These results were achieved by computationally correcting up to the 20 th Zernike mode, while the HAO system corrected up to 70 singular-value modes.Therefore, the residual aberrations do not result from a limited number of modes corrected by HAO, but from the accuracy with which the modes are measured and corrected.Calibration error, fitting error, measurement error, and bandwidth error all contribute to the presence of residual aberrations [47].The computed pupil also has many more adjustable elements than the number of actuators on the deformable mirror used in this study, which may partially explain the improvement gained from CAO.
The term computed pupil refers to the spatial frequency coverage of the OCT system accessed by taking the 2D transverse Fourier transform of the OCT signal, as illustrated in Fig. 1(a).The computed pupil is circular and extends to the cutoff frequency of the imaging system.Within the computed pupil, the phase of each pixel is digitally modified using double-precision floating-point numbers, making the pixels equivalent to piston-only actuators with nearly infinite stroke.Pixels outside the computed pupil are left unmodified.The number of pixels across the computed pupil is termed the number of computational actuators.
For the imaging protocol used here, the en face image size was 300 x 300 pixels originally acquired with 0.4 x 0.5 µm spacing and a 6.67 mm physical pupil.This resulted in 14,111 piston-only computational actuators within the computed pupil.For comparison, the Alpao DM 97 used in this study had 97 discrete actuators with approximately Gaussian influence functions.For a piston-only wavefront corrector to achieve equivalent performance to a corrector with a Gaussian influence function, the required number of actuators is predicted to increase by a factor of 10 to 40 depending upon pupil size, among other factors [23].In this study, the piston-only computed pupil had approximately 150 times more actuators than the

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The result method on the as a gold stan the cones are Photoreceptor correction, wi images showi cones which a the increase in  stability.Additionally, full integration of HAO and CAO for clinical use would require a near real-time implementation of the automated CAO optimization procedure.Manual CAO correction has previously been implemented for real-time display using a graphics processing unit [50].It is anticipated that parallelization will enable optimization of multiple depth layers simultaneously to produce HAO + CAO images at an acceptable speed for clinical use.This combined hardware and computational approach to wavefront correction is expected to be useful in other applications outside of retinal imaging.HAO for optical microscopy with direct wavefront sensing is often inaccurate due to poor performance of the wavefront sensor in thick samples [4,51].The aberrated image acquired using direct wavefront sensing could be used as a high-SNR starting point for further computational correction.Sensorless adaptive optics, in which a deformable mirror is adjusted to maximize an appropriate image sharpness metric, must optimize the mirror shape for each scan point at the time of acquisition [10,46,52].Therefore, the image acquisition time is dramatically increased when obtaining an optimal wavefront correction.Using CAO, the sensorless AO correction does not need to be optimal, since uncorrected aberrations can be removed post-acquisition.This could dramatically reduce the acquisition time for sensorless AO, enabling imaging of more dynamic samples.
Fig. 1 the re domai HAO are sh and ro tips.R Fig. 2 witho recove is imp displa is give Fig. 4 Zoom indivi the ro presen indica Because r with retinal ec Fig. 5 corres (numb CAO encirc tempo Fig. 6 autom fovea CAO.