Polarization-sensitive (PS) optical coherence tomography (OCT) enables depth-resolved imaging of the polarization properties of biological tissues. Phase retardation measurement has been applied in several applications, such as birefringent tissue segmentation and quantification. OCT-based polarization uniformity measurement [1–3] has enabled selective visualization of pigmented tissues in the retina, and it can also be potentially useful for quantification of tissue polarization uniformity.

Currently utilized metrics of polarization uniformity are defined by the variation of polarization states of backscattered light from the local space in a tissue. The variation of polarization states does not only depend on the tissue property, but is also significantly affected by nontissue factors, such as the signal-to-noise ratio (SNR) [3], or spatial sampling density. Although polarization uniformity is useful as an endogenous contrast agent, for subjective visualization of pigmented tissues, quantitative and fully objective tissue evaluation is hard to achieve. The nontissue factors make it impossible to quantitatively compare the polarization uniformity among systems, subjects, protocols, and even among measurement sessions of the same subject.

In this Letter, we propose an improved metric to represent polarization uniformity. The performance of this metric is evaluated by numerical simulation and in vivo experiments measuring a human retina, and is compared with a conventional metric, i.e., degree-of-polarization uniformity (DOPU).

To quantify the multiple scattering of light, which is mainly caused by highly concentrated pigments in a tissue, variation in the polarization state of backscattered light can be used. The most commonly used metric to represent variation in the polarization state is DOPU, introduced by [1]. In the conventional calculation of DOPU, the detected signals of two orthogonal polarization detection channels of PS–OCT are directly considered as polarization components of backscattered light. However, this is valid only if noise does not exist.

If additive noise exists, the OCT signals detected at two polarization channels become ${[g_{\mathrm{H}},g_{\mathrm{V}}]}^{\mathrm{T}}={[E_{\mathrm{H}}+n_{\mathrm{H}},E_{\mathrm{V}}+n_{\mathrm{V}}]}^{\mathrm{T}}$, where $E_p$ and $n_p$ [$p=\mathrm{H}$ (horizontal) or V (vertical)] are the fields of backscattered light and additive noise at the $p$-polarization channel, respectively. In this Letter, we assume that $n_p$ obeys a complex zero-mean Gaussian distribution.

Conventional DOPU is calculated based on Stokes parameters derived by PS–OCT signals $\mathbf{s}={[s_0,s_1,s_2,s_3]}^{\mathrm{T}}=[{|g_{\mathrm{H}}|}^2+{|g_{\mathrm{V}}|}^2$, ${|g_{\mathrm{H}}|}^2-{|g_{\mathrm{V}}|}^2$, $2\mathrm{Re}[g_{\mathrm{H}}{g}_{\mathrm{V}}^{*}]$, and $2\mathrm{Im}{[g_{\mathrm{H}}{g}_{\mathrm{V}}^{*}]]}^{\mathrm{T}}$. Because $g_p\ne E_p$, the $\mathbf{s}$ term does not represent a correct Stokes parameter of the backscattered light ${[E_{\mathrm{H}},E_{\mathrm{V}}]}^{\mathrm{T}}$. Apparently, the discrepancy increases as the SNR decreases.

In our method, the noise-induced error is corrected by means of the statistical property of PS–OCT signals. The expectations of the Hermitian products of PS–OCT signals is expressed with energies of back-scattered light at each polarization detection channel (${|E_{\mathrm{H}}|}^2$, ${|E_{\mathrm{V}}|}^2$) and additive noise power (${\sigma }_{\mathrm{H}}^2$, ${\sigma }_{\mathrm{V}}^2$) as $〈{|g_{\mathrm{H}}|}^2〉=〈{|E_{\mathrm{H}}|}^2〉+{\sigma }_{\mathrm{H}}^2$, $〈{|g_{\mathrm{V}}|}^2〉=〈{|E_{\mathrm{V}}|}^2〉+{\sigma }_{\mathrm{V}}^2$, and $〈g_{\mathrm{H}}{g}_{\mathrm{V}}^{*}〉=〈E_{\mathrm{H}}{E}_{\mathrm{V}}^{*}〉$. Hence, the noise-error-corrected Stokes parameters are defined as

Some types of PS–OCT using a Jones-matrix [4] or Stokes-vector based analysis detect multiple pairs of PS–OCT data with different incident polarization states. For these systems, further noise suppression can be achieved by DOPUs obtained with multiple incident polarization states. The combined DOPU with $M$ input polarization states is obtained as

In addition to cross sectional polarization uniformity imaging, en-face polarization uniformity imaging can be obtained by changing the local spatial averaging of the Stokes parameters, $\overline{s_v^{(m)\prime }}$, to axial averaging along depth, $1/L\sum _{z=1}^N{s}_v^{(m)\prime }$, where $L$ is the number of pixels along depth. An en-face DOPU map would be resemble to de-polarization imaging with a polarimetry [5] and might provide a comprehensive understanding of the distribution of pigmented tissues.

To evaluate the performance of noise-corrected DOPU, a numerical simulation is performed, as follows. Random Stokes vectors uniformly distributed on a Poincaré sphere with unit norm, i.e., $\sqrt{s_1^2+s_2^2+s_3^2}=1$, are generated [6]. To obtain a data set with arbitrary randomness, a partial set of the Stokes vectors is extracted in the following manner. Consider a cone whose apex is located at the center of the Poincaré sphere and whose half opening angle is $\psi$. The Stokes vectors in the cone is extracted, and the polarization uniformity of this set of Stokes vectors is defined as the norm of the averaged vector of these Stokes vectors. A set of Stokes vectors with an arbitrary uniformity can be obtained by properly selecting the opening angle of the cone. These Stokes vectors are converted into Jones vectors as ${[E_{\mathrm{H}},E_{\mathrm{V}}]}^{\mathrm{T}}={[\sqrt{(s_0+s_1)/2}\exp [i{\tan }^{-1}(-s_3/s_2)],\sqrt{(s_0-s_1)/2}]}^{\mathrm{T}}$. Finally, numerically generated random numbers which simulate $n_p$ are magnified according to the set SNR and added to the Jones vector. In this Letter, the total signal intensity ${|E_{\mathrm{H}}|}^2+{|E_{\mathrm{V}}|}^2$ is a constant, and only additive noise is considered.

This data set of 30 realizations is used to numerically evaluate the performance of the propose Makita DOPU (M-DOPU) [Eq. (2)] and conventional Götzinger DOPU (G-DOPU) [1,7]. The uniformity without noise ($\widehat{g}$) is used as a control value. The G-DOPU is defined by following Ref. [7]. Namely, only the OCT signals that have higher intensity than the mean noise level $+3.5\times$ the standard deviation are used, and Stokes vectors are normalized prior to spatial averaging. A set of simulations is performed with several configurations of set uniformity $g$ and set SNR. For each configuration, 1000 trials are performed, and the mean of M-DOPU and G-DOPU are obtained.

The simulation results are shown in Fig. 1. The M-DOPU (blue curves) is in good agreement with the control (polarization uniformity without noise, green curve), except with low SNR and low uniformity. However, G-DOPU (orange curves) is biased from the control; the amount of the bias strongly depends on the set uniformity and the SNR. Especially at low SNR (5 dB, orange solid curve), the G-DOPU is not sensitive to the uniformity. This is because the number of Stokes vectors used for calculation is decreased by intensity thresholding. Although M-DOPU is moderately biased when the SNR and set uniformity are simultaneously low, it seems that the extremely low uniformity (lower than $\sim 0.4$) did not occur in the experiments and other studies [1,4,7]. M-DOPU may show good performance in practical cases, even with low SNR.

 

Fig. 1. Numerical simulation results of polarization uniformity measurements. Measured results of M-DOPU (blue) and G-DOPU (orange), with set SNR of 5 (solid), 10 (dash), and 15 (dashdot) dB, are shown. Green curve shows the uniformity measurement without noise, $\widehat{g}$.

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As shown in Fig. 1, all of the measurements, i.e., M-DOPU, G-DOPU, and even the control, do not become zero with a set uniformity of zero. In the case of a uniform distribution, the control value with zero uniformity followed $1/\sqrt{N}$, where $N$ is the number of Stokes vectors. More accurate estimation by correcting this bias will be the subject of future work.

The comparison of M-DOPU and G-DOPU was performed also by imaging an in vivo human retina. An eye of polypoidal choroidal vasculopathy (PCV) was scanned with a retinal Jones matrix OCT [4] in a $6\times 6\mathrm{mm}$ region filled with 500 axial lines (horizontal) times 255 sets (vertical) of 4 frames at a single vertical position. These four frames were averaged by an adaptive Jones averaging method [4,8]. A wavelength swept laser with a central wavelength of 1060 and 100 nm bandwidth was used. Noise levels were estimated at the vitreous region since the noise floor is almost flat over the imaging depth range. For calculation of the M-DOPU and G-DOPU, 3 pixels (36 μm, lateral) $\times 3$ pixels (12 μm, axial) surrounding the target pixel were used, where the lateral and axial resolutions will be 21.0 and 6.2 μm, respectively. For G-DOPU and M-DOPU [Eq. (2)], a pair of OCT signals among four Jones entries corresponding to higher sensitivity was used. For ${\mathrm{M}\text{-}\mathrm{DOPU}}^{\prime }$, all entries were used. The G-DOPU and M-DOPU images are shown in Fig. 2. Note that the G-DOPU cannot be correctly determined if the OCT signal is low. Hence, a large portion of the image was masked out as gray. This intensity mask is part of the G-DOPU imaging algorithm [7]. In the M-DOPU image, the pixels with ${\mathrm{M}\text{-}\mathrm{DOPU}}^{-1}<1$ are expressed with the same color as the M-DOPU of 1 (red). This anomaly can happen because the noise-correction coefficient in Eq. (1) uses measured noise powers that only asymptotically equal the expectation of noise powers.

 

Fig. 2. In vivo human retinal image obtained by Jones matrix OCT. Cross sectional polarization uniformity images obtained with (a) G-DOPU, (b) M-DOPU [Eq. (2)], and (c) ${\mathrm{M}\text{-}\mathrm{DOPU}}^{\prime }$ [Eq. (3)]. (d)–(f) Set of examples with 5 dB SNR penalty for each polarization uniformity measurement.

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For quantitative evaluation, the SNR dependency has been investigated by adding numerically generated complex Gaussian white noise to a retinal OCT cross section. Before adding the numerical noise, the pixels in several types of DOPU images are grouped into 40 groups based on its numerical-noise free G-DOPU, M-DOPU, or ${\mathrm{M}\text{-}\mathrm{DOPU}}^{\prime }$ values. After adding the numerical noise, the three types of DOPU values were calculated, and the mean values were obtained for each group. The means of nine representative groups are plotted in Figs. 3(a)–3(c). The G-DOPU shows a strong dependency on SNR, especially at low-valued groups. The trend is opposite between high-valued and low-valued groups; i.e., the G-DOPUs of the low-valued groups increase as SNR decrease, while that of high-valued groups decrease. On the other hand, the M-DOPU exhibits negligible SNR dependency over the 7 dB SNR reduction range [Figs. 3(b) and 3(c)].

 

Fig. 3. SNR dependencies of polarization uniformity measurements.

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The polarization uniformity images with a 5 dB SNR penalty are shown in Figs. 2(d)–2(f). Because of reduced SNR, a vast region is masked out in the G-DOPU image [Fig. 2(d)]. However, without numerical noise, the appearances of low-uniformity regions, such as the retinal pigment epithelium (RPE), are almost consistent among all types of DOPU images [Figs. 2(a), 2(c)]. In the G-DOPU image, some part of the RPE under exudation (white circle) disappeared with lower SNR [Fig. 2(d)], while it existed in the image with higher SNR [Fig. 2(a)]. Conversely, M-DOPU [Figs. 2(b) and 2(e)] and ${\mathrm{M}\text{-}\mathrm{DOPU}}^{\prime }$ [Figs. 2(c) and 2(f)] images were not accompanied by this artifact. Therefore, the proposed M-DOPU will be suitable for quantitative evaluation of retinal pathologies. Furthermore, noise in the images is clearly reduced by using two input polarization states [Fig. 2(f)]. ${\mathrm{M}-\mathrm{DOPU}}^{\prime }$, which uses multiple input polarization states, is more robust to noise than M-DOPU.

En-face images of several modalities, and a cross sectional composite image, ${\mathrm{M}-\mathrm{DOPU}}^{\prime }$, as color and OCT logarithmic intensity as opacity, are shown in Fig. 4. Around the center of the imaging region (blue arrows), a high M-DOPU (black) region is visible in an en-face M-DOPU image [Fig. 4(d)] and the corresponding cross sectional composite image [Fig. 4(a)]. Because a high M-DOPU indicates less multiple scattering in the entire depth, it indicates low pigmentation in these regions. According to low fluorescence in a fundus auto-fluorescence image [FAF, Fig. 4(c)] and hyper-fluorescence in a late phase fluorescein angiography [FA, Fig. 4(e)], this region is a region with RPE corruption. The u-shaped hyperautofluorescence [Fig. 4(c)] might be coming from devitalized blood [9]. Because the influence of the RPE decreased due to the high attenuation of this region, the en-face ${\mathrm{M}\text{-}\mathrm{DOPU}}^{\prime }$ becomes high at the corresponding location [Fig. 4(d)]. Visualization with other contrast, e.g., Doppler blood flow imaging, will provide comprehensive understanding of some other eye diseases, such as multifunctional imaging [10]. En-face projection image of a squared Doppler phase shift [4], in which the color is defined by the integrated values of $(1\text{-}\mathrm{M}\text{-}\mathrm{DOPU})$ over the Doppler signals to represent relative depth [Fig. 4(f)], shows that Doppler blood flow signals from the choroid are permeated at the locations of low pigmentation. There are two spots (green arrows) exhibiting a high Doppler flow signal from deep tissue in OCT angiography [Fig. 4(f)]. Their locations are in good agreement with hyper-fluorescent spots in FA [Fig. 4(e)].

 

Fig. 4. (a) OCT-${\mathrm{M}\text{-}\mathrm{DOPU}}^{\prime }$ composite cross section showing the structure of the retina and the location of pigments. (b) OCT en-face projection, (c) fundus autofluorescence, (d) en-face M-DOPU, (e) fluorescence angiography, and (f) en-face OCT angiography employing M-DOPU.

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A new metric, M-DOPU, has been introduced to represent the spatial uniformity of the polarization state of backscattered light. The performance of this new metric has been evaluated numerically and experimentally. In vivo retinal imaging of a diseased eye with Jones matrix OCT visualizes fine structure of pigmented tissues. Because M-DOPU is highly immunized to noise, it will be suitable not only for qualitative imaging, but also for quantitative analysis of the polarization property of tissues.