Anisotropic aberration correction using region of interest based digital adaptive optics in Fourier domain OCT

Abhishek Kumar; Tschackad Kamali; René Platzer; Angelika Unterhuber; Wolfgang Drexler; Rainer A. Leitgeb

doi:10.1364/BOE.6.001124

1. Introduction

Recently, the numerical defocus and higher order optical aberration correction methods have generated strong interest among researchers working in the field of digital holography and interferometric optical imaging techniques, such as OCT, where the phase of the signal can be accessed [1–12]. This is mainly because the numerical techniques obviate the need for any additional expensive adaptive optics hardware, spatial light modulator (SLM) or any additional wavefront sensing cameras, thereby reducing the system complexity and cost [13–18]. The numerical aberration correction is applied after the data acquisition, as a post processing step, on the computer. This also allows high speed image acquisition without the need of adjusting the optical system or the object being imaged. Especially, in case of optical coherence microscopy (OCM), where a high NA (close to unity) microscope objective (MO) is used, and the DOF is very shallow. In this case the effective DOF can be extended by numerical defocus and higher order aberration correction in the out of focus regions of the volume image. Thus the need of focus scanning through the imaging depth is avoided, enabling high speed volumetric imaging [10].

Numerical aberration correction requires an estimation of the phase correction factor that needs to be applied in the spatial Fourier domain, which can be done using either the optimization based methods [6–11] or by using the subaperture correlation based DAO demonstrated by Kumar et al. [12]. Optimization based techniques use an image metric to find the best estimate of the coefficients of the polynomial phase correction function in an iterative manner [7–9]. Whereas, the subaperture correlation based DAO technique gives the local slope information of the wavefront error, from which the phase correction polynomial function can be determined analytically in a single step [12]. The above methods work well in case of low NA imaging systems where the entire FOV is within the iso-planatic patch- the region across which the aberrations, and hence the point spread function (PSF), do not vary. This is due to the fact that the same phase correction is applied to the Fourier transform of the entire FOV. However, in case of a high resolution imaging system, where NA approaches unity and a large acceptance angle is involved, the point spread function is not the same across the lateral FOV. This situation is illustrated in Fig. 1. The anisotropic aberration across the FOV may be caused due to imperfect optics, an inhomogeneous sample, or misalignment.

Fig. 1 A situation where point objects A and B, lying within the same FOV, are imaged differently through an imperfect optics. Both points A and B see different wavefront aberration resulting in different PSFs.

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In this paper, a method is presented to deal with the problem of anisotropic aberration in a high resolution OCT system to achieve a uniform diffraction limited lateral resolution throughout the whole volume. The recorded enface image field at different depths in the tomogram is selected, digitally divided into smaller ROIs and processed individually using subaperture based DAO. The aberration corrected ROIs are finally stitched together to yield a final image with a uniform diffraction limited resolution across the entire FOV. In section 2, we describe in detail the proposed algorithm and the experimental setup for the fiber based point scanning SD OCT system. In section 3 we present our experimental results, showing the proof of principle using an iron(III) oxide nano-particle phantom sample imaged using a SD OCT setup at an effective NA of 0.6. The application of the method is also shown in ex vivo mouse adipose tissue. Finally we present our conclusions in section 4.

2. Methods

2.1 ROI based DAO for anisotropic aberration correction

After the volume data acquisition and standard Fourier domain OCT signal processing; i.e. zero-padding, $λ \to k$ mapping, dispersion compensation and $k \to z$ FFT, the complex valued enface image data at different depths is selected for the DAO based aberration correction. The schematic of the algorithm used for anisotropic aberration correction is shown in Fig. 2. In the first step, the entire FOV of $M \times M$ pixels is divided into smaller $κ \times κ$ sub-regions of size ~ $M / κ \times M / κ$ pixels. The value of $κ$ depends on how much the aberrations vary across the FOV. The ROI is selected by multiplying the image with mask of unity value of size $M / κ \times M / κ$ pixels. The size of the mask is empirically chosen such that the aberration within the ROI is about the same. In our experiments, value of $κ = 5$ was sufficient to achieve a uniform lateral resolution after the DAO processing.

Fig. 2 Schematic of the algorithm using ROI based DAO for anisotropic aberration correction for a given enface plane. The processing steps 2-7 belong to the sub-aperture based DAO method described in detail in Ref [12]. The above processing steps are repeated for each enface image in depth to achieve depth invariant lateral resolution.

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The sub-aperture based DAO is described in detail in Ref [12]. In brief, the 2-D FFT is calculated for the masked ROI to get to the spatial Fourier plane. The aperture in the Fourier plane is split into $K \times K$ subapertures and the 2-D IFFT of each sub-aperture is calculated to get the image of each sub-aperture. The image from the central subaperture is taken as the reference and cross-correlated with images of other sub-apertures to calculate the relative shifts in pixels between different images. The local slope of the wavefront error in the $m^{th}$ subaperture is calculated using the relative shift $Δ x_{m}$ and $Δ y_{m}$ as

s_{x, m} = \frac{π Δ x_{m}}{2} and s_{y, m} = \frac{π Δ y_{m}}{2} .

Note that the factor of

M

(size of the image in pixels) is missing in the above equation as compared to equation for the slope given in Ref [12]. This is because for the experiments presented this manuscript, the phase error is represented in terms of orthogonalized Zernike polynomials [18], instead of Taylor polynomials [12], as

ϕ_{e} (x, y) = \sum_{i = 1}^{P} a_{i} Z_{i} (x, y)

, where the co-ordinate (x, y) in the Fourier space is normalized such that the aperture lies within a circle of unit radius. By comparing the gradient of the phase error function with the calculated slope data, we have a solution in the matrix form as

S = G A

where

S = [S_{x}; S_{y}]

with

S_{x} = {[s_{x, 1}, ..., s_{x, N_{s}}]}^{T}

and

S_{y} = {[s_{y, 1}, ..., s_{y, N_{s}}]}^{T}

as the

N_{s} = K^{2}

element column vector containing x and y slope components,

G = [\partial Z / \partial x; \partial Z / \partial y]

is the gradient matrix with

\partial Z / \partial x

and

\partial Z / \partial y

as

(N_{s} \times P)

matrices of partial derivate of Zernike polynomials

Z_{i} (x, y)

with respect to x and y, and

A = {[a_{1}, ..., a_{P}]}^{T}

is vector containing the Zernike coefficients. The least square solution to estimate the coefficients in Eq. (2) is given by

\hat{A} = (G^{T} G^{- 1}) G^{T} S .

Once the coefficients are determined, the phase error correction is calculated. In our experiments, the pupil was divided into

K \times K = 5 \times 5

subapertures, and the first 25 Noll Zernike terms (till 6th order) were fitted to estimate the phase error [18]. The phase correction factor

\exp (- i ϕ_{e})

is multiplied with the whole aperture data in the Fourier plane and the IFFT is calculated to get an aberration free ROI. The procedure described above can then be repeated for each ROI. Finally, all ROIs can be stitched together to get an aberration free image with a uniform lateral resolution across the full FOV.

2.2 Experimental setup: fiber based point scanning SD OCT system

The schematic of the SD OCT system is shown in Fig. 3. It consists of a fiber based Michelson interferometer with a 90/10 coupler. A home built Ti-sapphire laser with a center wavelength of 800 nm and full broad band width of 290 nm is used as the light source [19]. The measured axial resolution is in air is 1.6 $μ m$ . The sample arm is equipped with a 40x NIR MO (Nikon CFI Apo 40xW, NA = 0.8). A spectrometer using a grating (1200 l/mm, Wasatch Photonics Inc.) with a 2048 pixel line scan camera (Amtel AviiVA EM4 2014) is used in the detection arm. The data is acquired at an A-scan rate of 37 kHz using a frame grabber (National Instruments, NI-PCIe 1427). The phase stability was maintained at this recording speed, and hence no phase noise correction was required [2]. A X-Y galvanometer scanner (Cambridge Technology, 6220H) in the sample arm scans a lateral FOV of ~250x250 $μ m^{2}$ . The mid-point between the galvo mirrors was imaged at the back aperture of the MO by a telescope ( $f_{1} = f_{2} = 75 mm$ ). The coherence gate curvature, which is difficult to avoid in case of high NA MO when the scanning is not perfectly telecentric, was corrected numerically in the post processing step [20]. The measured sensitivity of the system is 92 dB with 1.5 mW power incident on the sample.

Fig. 3 Schematic of the fiber based point scanning SD OCT system.

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The lateral scanning step size, determined using a high-frequency resolution test target (Thorlabs, NBS 1963A), was found to be 0.54 $μ m$ . Figure 4(a) shows the enface OCT image of scanned horizontal and vertical line sets with frequency of 203 cycles/mm. The peak-to- peak distance of a cycle, both along vertical and horizontal direction, is 4.9 $μ m$ and corresponds to 9 pixels. The lateral resolution was measured using an iron (III) oxide nano-particles (mean diameter < 400 nm, embedded in polyurethane matrix) phantom sample. Figure 4(b) shows the lateral point spread functions (PSF) corresponding to the nano-particles in the focal plane, and a cut through one of the PSF marked by white arrows is shown in Fig. 4(c). The measured full width half maximum (FWHM) of the cut through the intensity profile, which corresponds to the system lateral resolution, is 0.81 $μ m$ . We use in the present work the FWHM of the lateral linearly scaled PSF as quantitative figure of the lateral resolution. The effective NA of the imaging is only 0.6, even though the MO has NA of 0.8. This is due to the fact that the scanning beam under-fills the back aperture of the MO. The measured DOF is ~7 $μ m$ .

Fig. 4 (a) Enface OCT image of NBS 1963A resolution test target showing line set with frequency of 203 cycles/mm. The measured lateral scanning step size is 0.54 $μ m$ . (b) Enface image showing lateral PSFs corresponding to nano-particles in the focal plane, (c) the cut through the intensity profile of the PSF marked by white arrows in (b). The measured FWHM of the intensity profile, and hence the lateral resolution of the system, is 0.81 $μ m$ . The scale bar in (b) corresponds to 20 $μ m$ .

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3. Experimental results

3.1 Nano-particle phantom imaging

Figure 5(a) shows the enface image of iron (III) oxide nano particle phantom at a distance of 46 $μ m$ from the focal plane. Here, we can clearly see the effect of anisotropic aberration as the PSFs across the field of view are different. For example, in the region near the green dotted box, the spots appear to be suffering from spherical aberration, whereas in the region near red dotted box, they appear to suffer from coma. This observation is confirmed by the wavefront error maps shown in Figs. 5(l) and 5(m). The wavefront error map shown in Fig. 5(l) for the ROI at the location of green dotted box has a dominant primary spherical aberration and has a root mean square (RMS) wavefront error value of 2.96 radians. Primary horizontal coma aberration dominates the phase error map shown in Fig. 5(m) for the ROI at the location of red dotted box and has a RMS wavefront error value of 5.11 radians. As shown in Fig. 5(b), the overall aberration has reduced after global DAO correction where phase correction is applied to Fourier data of the whole image. However, we can still see smearing around the PSFs. After ROI based corrections, in the final stitched image in Fig. 5(c), we can see that the spots appear sharper and smearing has been reduced. This is clearly evident in the zoomed-in views in Figs. 5(d)-5(f) and Figs. 5(h)-5(j). Profile plots across the spots at the locations marked by white arrows in zoomed-in images, shown in Figs. 5(g) and 5(k), show the quantitative improvement in resolution and SNR. We observe that the intensity of spots is almost doubled and the FWHM spot size is almost halved after the ROI based DAO correction as compared to after the global DAO correction. The FWHM of the cuts through the spots in Figs. 5(g) and 5(k) after ROI based DAO correction is ~0.95 $μ m$ , which is close to the measured diffraction limited resolution of 0.81 $μ m$ .

Fig. 5 (a) Original enface image of iron oxide nano particle phantom at a distance of 46 $μ m$ above the focal plane, (b) image obtained after applying global phase correction, (c) final image after stitching phase corrected ROIs together, (d)-(f) zoomed in images at the location of green dotted box in (a)-(c), (g) cut through plots at the location of arrow in (f) for each case, (h)-(j) zoomed in images at the location of red dotted box in (a)-(c), (k) cut through plots at the location of arrow in (j) for each case, (l) and (m) show wavefront error maps in radians for ROIs at the location of green and red dotted box respectively. O: original, G: global in the plots (g) and (k). Scale bars in (a) represent 20 $μ m$ and applies to (b) and (c). Same gamma correction is applied to all the images for the visual comparison of SNR levels.

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Figure 6(a) shows a B-scan slice form the original phantom volume image affected by anisotropic aberration. We can see the smearing in the lateral direction of the PSFs corresponding to the nano-particles located away from the focal plane in depth, indicated by the yellow arrow in Fig. 6(a), due to defocus and other higher order aberrations. After the global DAO aberration correction, shown in Fig. 6(b), the smearing has visibly reduced in the regions away from the focus. However, after applying ROI based DAO correction, we appreciate more uniform lateral resolution throughout the depth of 218 $μ m$ , which indicates an increase in DOF by a factor ~30x. The zoomed in view of the regions at the top and bottom layers of the tomograms marked by the green and the red dotted box respectively clearly show the improvement after the application of global and ROI based DAO aberration correction. The profile plots along lateral direction across one of the PSF corresponding to nano-particle located at a distance of $57 μ m$ above the focal plane, in the region marked by the green dotted box, show that the FWHM of the lateral spread improves from 3.5 $μ m$ in case of the original to 2.1 $μ m$ after the global aberration correction. It further improves to 0.8 $μ m$ after ROI based correction, which is the same as the measured diffraction limited resolution. This indicates that an improvement of lateral resolution by 4.4x is achieved after ROI based correction over the original at a distance of ~8 times the DOF from the focal plane. Similar improvement can be seen for the PSF located at a distance of 90 $μ m$ below the focal plane (~13x DOF) marked by the red dotted box. Figure 6(d) shows the variation of SNR with distance from the focal plane for the original volume image, after global aberration correction and after ROI based correction. The SNR values at several enface planes in depth are calculated using the formula: $10 \log_{10} [Peak intensity pixel value / variance(Noise)]$ . We notice that the SNR is improved after global and ROI based aberration correction, and the improvement is more pronounced as the distance from the focal plane increases. SNR improvement of ~8 dB and 10 dB is observed after ROI based correction over the global aberration corrected image and the original image respectively at distance of −60 $μ m$ (8.5x DOF) above the focal plane. Also, SNR improvement of ~6 dB and 10 dB is observed after ROI based correction over the global aberration corrected image and the original image respectively at distance of + 150 $μ m$ (21.4x DOF) below the focal plane. Figure 6(e) shows the 3-D rendering of the original, global aberration corrected and ROI based aberration volume image over the depth range of 218 $μ m$ , where we can clearly appreciate the improvement in resolution after global and ROI based aberration correction, and the fact that the resolution is more uniform after ROI based correction.

Fig. 6 (a) Original B-scan of iron oxide nano particle phantom showing a depth range of 218 $μ m$ , (b) after applying global aberration correction, and (c) after ROI based correction. (d) Variation of SNR with the distance from the focal plane for each case. O: original, G: global in the plot (d). (e) 3-D rendering of the images for each case. Horizontal scale bars in (a)-(c) represent 30 $μ m$ . Same gamma correction is applied to all the images.

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3.2 Ex vivo imaging of mouse adipose tissue

Ex vivo imaging of mouse white adipose tissue was also performed using the high resolution SD OCT setup described in section 2.2. White adipose tissue was excised from the inguinal and properitoneal fat pads, shortly immersed in Dulbecco's Modified Eagle Medium (DMEM, Gibco) and fixed with 4% Paraformaldehyde (PFA, Sigma) adjusted to pH 7.0 with 10 x Dulbecco’s Phosphate Buffered Saline (DPBS, Gibco). PFA fixed samples were washed twice with 1 x DPBS and stored in 1x DPBS supplemented with 20% Ethanol (Sigma) until use. Figure 7(b) shows the result of application of ROI based DAO correction to a layer of ex vivo mouse adipose tissue located at a distance of 54 $μ m$ above the focal plane. Here, we can see that the resolution has improved in comparison to the original image in Fig. 7(a) as cell boundaries appear sharper and thinner. Zoomed in images at the location of the asterisk in Fig. 7(a) and cuts through the cell boundary, shown in the insets, in Figs. 7(c)-7(e) clearly show that the cell walls of the adipocytes appear sharper and thinner in the ROI based DAO corrected image in comparison to the global DAO corrected image and the original image. The measured FWHM thickness of the cell wall in the original image is 8.7 $μ m$ . After ROI based correction it reduces to ~1.5 $μ m$ , which is within the known anatomical mouse adipose cell wall thickness range of 1-5 $μ m$ , indicating a lateral resolution improvement by almost 6x. The bright spots, marked by white arrows in Fig. 7(e), representing highly scattering nuclei appear more tightly focused after ROI based DAO correction. Figure 7(f) shows the 3-D rendering of the original, the global aberration corrected and the ROI based aberration corrected ex vivo mouse adipose tissue image. We can observe that in the ROI based aberration corrected image cell boundaries appear shaper than the global aberration corrected image, and the resolution has improved dramatically as compared to the original image.

Fig. 7 (a) Original enface image showing a layer of ex vivo mouse adipocytes at a distance of 54 $μ m$ above the focal plane, (b) final image after ROI based DAO correction. Zoomed in images at the location of asterisk sign in (a): (c) original, (d) obtained from the image after global phase correction (figure not shown here), (e) after ROI based phase correction. Cut- through the cell boundary at the location marked by the green arrows in (c)-(e) is shown in the insets. Scale bars in (a) and (b) represents 50 $μ m$ . Scale bar in (c) denotes 20 $μ m$ and applies to (d) and (e). (f) 3-D rendering of original, global aberration corrected and ROI based aberration corrected image.

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4. Conclusions/discussion

We have demonstrated that by using ROI based DAO we can correct anisotropic aberration across the lateral FOV that affect high resolution OCT setups. The proof of principle is demonstrated using an iron (III) oxide nano- particle phantom imaged with a SD OCT system at an effective NA of 0.6. After the ROI based correction, the overall SNR is improved and sub-micron lateral resolution is achieved throughout the whole volume, with an increase in DOF by a factor of ~30x. The application of this method is also shown in ex vivo mouse adipose tissue. The method uses sub-aperture based DAO, which requires splitting of aperture into smaller sub-apertures and cross-correlation of the image formed by each sub-aperture with reference image of the central sub-aperture in order to estimate local wavefront error slopes [12]. However, these steps are independent of each other and can be parallelized in GPU. Parallelization using GPU in optimization based techniques that require long iterations especially for higher order phase correction is a challenge, as each iterative step is dependent on the preceding iterative steps. Thus, we believe that with its implementation in GPU, the ROI based DAO using subaperture method has the potential to outperform optimization based techniques in terms of computational speed. However, a more detailed study is required in the future to compare sub-aperture method with optimization based methods, which is out of the scope of the current paper. Also, in the iso-planatic volume, the knowledge about aberration in a given depth plane may be used to correct aberration in the other depth planes which can further speed up the overall processing time for the subaperture method. Furthermore, with the numerical motion correction techniques, it may be possible to apply the DAO technique to in vivo biological samples [21]. The ROI based DAO can also be used in conjunction with AO (hardware based) aided OCT or holography systems to numerically correct for aberrations in the image regions outside the iso-planatic patch, which are left uncorrected by the AO. Like other digital phase correction methods, the sub-aperture based DAO works in the single-scattering regime where the OCT signal is strong. With the recent development of techniques for increasing the penetration depth and suppressing the multiple scattering of light in the tissue [22–24], it is promising to state that numerical/digital phase correction methods can be extended to the multiple-scattering regime in the future.

Acknowledgments

The financial support by the Austrian Federal Ministry of Economy, Family and Youth and the Austrian National Foundation of Research, Technology and Development is gratefully acknowledged. We further acknowledge financial support by Carl Zeiss Meditec, Inc, Dublin, USA.

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Anisotropic aberration correction using region of interest based digital adaptive optics in Fourier domain OCT

Abstract

1. Introduction

2. Methods

2.1 ROI based DAO for anisotropic aberration correction

2.2 Experimental setup: fiber based point scanning SD OCT system

3. Experimental results

3.1 Nano-particle phantom imaging

3.2 Ex vivo imaging of mouse adipose tissue

4. Conclusions/discussion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (3)

Biomedical Optics Express