Accurate flow speed measurement through correlation ratios using a mode-selective photonic lantern in optical coherence tomography

Abstract

A novel method for measuring non-axial flow speed using optical techniques such as optical coherence tomography is introduced. The approach was based on the use of a modally-specific photonic lantern, which permits simultaneous probing of the sample with three distinct coherent spread functions. Transverse flow speed is measured from the ratio between the cross-correlation and autocorrelation of the signals. It achieved a 3 to 5 times higher accuracy than common autocorrelation approaches and measured flows as slow as 0.5 mm/s for an integration time of 1 second. Additionally, the method gives information on the flow’s three-dimensional orientation, does not require information about the diffusion coefficient, and is more robust to bias errors such as a gradient in the axial flow velocity.

1. Introduction

Optical coherence tomography (OCT) is an imaging technique that has revolutionized the field of ophthalmology by providing non-invasive, high-resolution (∼ 10 µm), three-dimensional visualization of the retina and its vasculature without the need for contrast agents [1,2]. Monitoring of blood flow in the retina is key for diagnosing several ophthalmologic pathologies [3,4], avoiding damage to important blood vessels during surgery or ensuring vascular integration during transplant [5,6]. As such, many techniques have been developed to image blood vessels and quantify their flow [7–20]. Although qualitative flow measurement is readily achieved, accurate quantitative measurements remain challenging and are needed for accurate monitoring and comparative studies.

Doppler OCT can precisely and accurately measure axial flow velocity even at subdiffusion speed [11]. However, for the retina, most blood vessels are located in-plane and orthogonal to the imaging axis. Some systems image the retina at three angles to overcome this limitation [14]. Such systems are technically challenging to implement, particularly for imaging small eyeballs such as found in mouse pup models of pediatric diseases [21]. Furthermore, studies have shown how Doppler OCT can produce inaccurate measurements [22] and its particular limitation to quantify high flow speeds [15].

Neural networks, which are increasingly popular, have also been applied to the challenge [23], with limited accuracy. Alternatively, calculating variance is used to highlight blood flow. Accurate estimation of the blood flow remains challenging [8,24]. To remedy this, it is often used in conjecture with other techniques such as speckle decorrelation for more quantitative results [24,25].

Many techniques rely on measuring blood flow-induced speckle pattern decorrelation [15,26,27]. The flow velocity is calculated by measuring the decorrelation at a specific delay or for multiple delays, in which case a fit of the correlation curve provides the particle speed. When measuring flow with a single delay, diffusion, signal-to-noise ratio, and bulk motion from the sample need to be accurately known or will otherwise induce a bias in the measurement. While fitting on a correlation curve is more robust to those errors, implementing the fitting approach is more challenging since it requires imaging the same point in very short intervals. Even when all those factors are adequately considered, inaccuracies occur due to axial gradient velocity (AGV) and shadow artifacts [22,28]. AGV refers to the axial component of the flow in a single pixel that is not uniform across the pixel. The sub-resolution variation of the axial flow causes an overestimation of the flow speed. Estimating sub-diffuse blood speed also remains problematic [29].

This work introduces a technique to accurately quantify flow using a 3-mode, mode-selective photonic lantern (MSPL) in OCT [30–33]. MSPLs are fiber-based n-by-1 couplers acting as a demultiplexer for higher-order fiber spatial modes such as LP01 and the two LP11 modes. Once demultiplexed into distinct single-mode fibers, the modes are imaged by independent OCT systems, creating a few-mode OCT (FM-OCT) [31,34–36].

This technique is based on the calculation of correlation ratios from 3 distinct complex spread functions (CSF) [37]. This approach has shown promising results using two modes of photonic lanterns [38] and other CSF engineering methods, giving improved measurements than just using correlation [39]. Here, the technique shows that using 3 modes provides more accurate measurements of flow velocity compared to other correlation-based approaches. It also determines the flow direction in three dimensions. Furthermore, such a CSF-engineered approach is not limited to OCT and could be applied to other imaging devices based on coherent light detection such as laser speckle imaging [40–43].

2. Theory

In this section, we derive an expression for the ratio of correlations using an OCT system with three distinct coherent spread functions.

2.1. Coherent spread function in optical coherence tomography

The OCT signal (E) of a single pixel can be approximated by the light reflectance from multiple particle scatterers randomly distributed in space such as [26,27]:

$$E=\underset{i=1}{\overset{N}{\sum }}{r}_i\cdot h_{xy}(x_i,y_i)\cdot h_z(z_i),$$ (1)

where N is the number of scatterers, $r_i$ is the reflectivity of each i^th scatterer, $x_i$ , $y_i$ , and $z_i$ describe the 3D position of a scatterer, and $h_{xy}$ and $h_z$ are the CSF in the lateral and axial directions, respectively. For simplicity and without impacting the accuracy, $r_i$ can be approximated to 1 [26]. At the beam focus location, the axial CSF is often approximated as a Gaussian curve by [27]:

$$h_z(z)=\sqrt{\frac{2}{\sqrt{\pi }{w}_z}}\cdot \exp \left[-iqz\right]\cdot \exp \left[-2\frac{z^2}{w_z^2}\right]$$ (2)

where $q=2k_{0}n$ , $k_0$ is the central wavenumber, n is the refractive index, z is the axial position in depth orientation, and $w_z$ is the beam waist. The first term is a normalization factor, the second is the phase response, while the last is the amplitude response. The lateral CSF is defined by a combination of the imaging lens, collimating lens, and the optical fiber-guided modes used for illumination and collection. The system is assumed to be free of aberrations. Since OCT typically uses single-mode fibers, the illumination and collection are performed by the LP01 mode which can be approximated by a Gaussian function [44] :

$$f_{\text{LP01}}(x,y)=\exp [-\frac{x^2+y^2}{w_{xy}^2}]$$ (3)

where x and y are the lateral positions and $w_{xy}$ is the beam waist size.

The system was based on a 3-mode MSPL replacing the single-mode fiber used for both illumination and collection of light [30–33]. These optical components have been used with OCT before to reduce speckle noise and explore a new type of contrast [36,45]. Illumination was performed through the LP01 mode while light was collected from all 3 modes including LP01 and the two orthogonal LP11 modes: horizontal $\left({\text{LP11}}_{\text{h}}\right)$ and vertical $\left({\text{LP11}}_{\text{v}}\right)$ . To simplify notation, the OCT signal collected by the LP01 mode is defined by Channel 1 (Ch1) while the ones collected by ${\text{LP11}}_{\text{h}}$ and ${\text{LP11}}_{\text{v}}$ are defined Channel 2 (Ch2) and Channel 3 (Ch3), respectively.

The lateral CSF is then defined by the product of the illumination and collection modes [46]:

$$h_{xy}(x,y)=f_{\text{LP01}}(x,y)\cdot f_{\text{LP01}}(x,y)=\exp \left[-2\frac{x^2+y^2}{w_{xy}^2}\right].$$ (4)

To facilitate further mathematical calculations, $h_{xy}$ can be normalized as follows:

$${h_{xy}(x,y)|}_{\text{Ch1}}=\frac{2}{\sqrt{\pi }{w}_{xy}}\exp \left[-2\frac{x^2+y^2}{w_{xy}^2}\right].$$ (5)

Similarly to the approximation of $f_{\text{LP01}}$ from Eq. (3), the horizontal and vertical fiber mode distributions of the higher order modes $f_{\text{LP11}}$ are expressed as:

$$f_{\text{LP11,h}}(x,y)=\frac{x}{\zeta \cdot w_{xy}}\exp [-\frac{x^2+y^2}{{(\gamma \cdot w_{xy})}^2}]\text{and}{f}_{\text{LP11,v}}(x,y)=\frac{y}{\zeta \cdot w_{xy}}\exp [-\frac{x^2+y^2}{{(\gamma \cdot w_{xy})}^2}]$$ (6)

where the fitting parameters $\zeta =0.7036$ and $\gamma =0.9279$ were obtained by fitting $f_{\text{LP11}}$ over the Bessel and modified Bessel functions used to define LP11. The resulting normalized lateral CSF for Ch2 $\left(h_{xy,2}\right)$ and Ch3 $\left(h_{xy,3}\right)$ were obtained by multiplying the $f_{\text{LP01}}$ (for illumination) and $f_{\text{LP11,h}}$ or $f_{\text{LP11,v}}$ (for collection), respectively. This is expressed as:

$${h_{xy}(x,y)|}_{\text{Ch2}}=\sqrt{\frac{2}{\pi }}\frac{\xi }{w_{xy}^2}\cdot x\cdot \exp [-\xi \frac{x^2+y^2}{{(w_{xy})}^2}]\text{and}{h_{xy}(x,y)|}_{\text{Ch3}}=\sqrt{\frac{2}{\pi }}\frac{\xi }{w_{xy}^2}\cdot y\cdot \exp [-\xi \frac{x^2+y^2}{{(w_{xy})}^2}]$$ (7)

where $\xi =(1+{\gamma }^2)/{\gamma }^2$ .

2.2. Signal correlation

The first-order correlation can be calculated as follows:

$$g_{AB}^{(1)}(\tau )=\frac{⟨E_A^{\ast }(0)\odot E_B(\tau )⟩}{\sqrt{⟨|E_A^{\ast }(0)|⟩\cdot ⟨|E_B(\tau )|⟩}},$$ (8)

where A and B are the signals being correlated and acquired by an unspecified channel A and B, $\tau$ is the correlation delay, E is the OCT tomogram complex signal, $E^{\ast }$ is the complex conjugate of E, $\odot$ is the Hadamard product, and $⟨\cdots ⟩$ is the ensemble average. The subscripts 1, 2, and 3 are used to refer to Ch1, Ch2, and Ch3, respectively, when the correlated channels A and B are specified. The theoretical expected value of Eq. (8) is affected by flow and can be determined as follows. The correlation values of a single moving scatterer are calculated over all possible initial positions $(x_0,y_0,z_0)$ and averaged using an integral with normalized CSFs [26,27]. By following this approach, we obtain this expression:

$$g_{AB}^{(1)}(\tau )={\iiint }_{\mathrm{\infty }}^{\mathrm{\infty }}{h_{xy}^{\ast }(x(0),y(0))|}_A\cdot {h_{xy}(x(\tau ),y(\tau ))|}_B\cdot h_z^{\ast }(z(0))\cdot h_z(z(\tau ))d{x}_{0}d{y}_{0}d{z}_0,$$ (9)

where x, y, and z stand for the scatter position at a particular time $\tau$ . Those positions vary in time as follows:

$$\begin{array}{l}x(\tau )=x_0+v_x\cdot \tau +x^{\prime }(\tau ),\\ y(\tau )=y_0+v_y\cdot \tau +y^{\prime }(\tau ),\\ z(\tau )=z_0+v_z\cdot \tau +z^{\prime }(\tau ),\end{array}$$ (10)

where $v_x,v_y$ , and $v_z$ are the flow velocity components along those axes and $x^{\prime },y^{\prime }$ , and $z^{\prime }$ are the displacements caused by diffusion. Thus, $x_0,y_0$ , and $z_0$ are possible positions of the scatterer at time $\tau =0$ .

Separating the decorrelation signal generated by the flow $(g_{AB}^{(1)}(\tau ){|}_f)$ and diffusion $(g_{AB}^{(1)}(\tau ){|}_D)$ into two terms was previously proposed since they are uncorrelated phenomena [27]:

$$g_{AB}^{(1)}(\tau )={g_{AB}^{(1)}(\tau )|}_f\cdot {g_{AB}^{(1)}(\tau )|}_D,$$ (11)

where $g_{AB}^{(1)}(\tau ){|}_f$ can be calculated from Eq. (9), but without the diffusion motion in Eq. (10).

The decorrelation due to diffusion is primarily related to the random phase variations of the signal of the scatterers caused by their axial motions instead of signal amplitude variations caused by 3D motions [27]. Thus, the diffusion decorrelation is invariant between channels and can be expressed by:

$${g_{AB}^{(1)}(\tau )|}_D=\exp [-q^{2}D\mid \tau \mid ],$$ (12)

where D is the diffusion coefficient. Taking the absolute value of $\tau$ accounts for negative delays. Equations 9), (11), and (12) can then be used to calculate the expected value for the auto-correlation of Ch1 [26,27] as:

$$g_{11}^{(1)}(\tau )=\exp \left[iq{v}_z\tau \right]\cdot \exp [-\frac{(v_x^2+v_y^2){\tau }^2}{w_{xy}^2}]\cdot \exp [-\frac{v_z^2{\tau }^2}{w_z^2}]\cdot \exp [-q^{2}D\mid \tau \mid ].$$ (13)

The first term is the Doppler term, the second gives the decorrelation due to non-axial flow, the third is the decorrelation due to axial flow, and the last is the decorrelation due to diffusion. The cross-correlations of Ch1 with Ch2 or Ch3 are expressed by:

$$\begin{array}{ll}{g}_{12}^{(1)}(\tau )=& \exp \left[iq{v}_z\tau \right]\cdot 1.41\frac{v_x\tau }{w_{xy}}\cdot \exp \left[-1.04\frac{(v_x^2+v_y^2){\tau }^2}{w_{xy}^2}\right]\cdot \exp [-\frac{v_z^2{\tau }^2}{w_z^2}]\cdot \exp [-q^{2}D\mid \tau \mid ],\\ g_{13}^{(1)}(\tau )=& \exp \left[iq{v}_z\tau \right]\cdot 1.41\frac{v_y\tau }{w_{xy}}\cdot \exp \left[-1.04\frac{(v_x^2+v_y^2){\tau }^2}{w_{xy}^2}\right]\cdot \exp [-\frac{v_z^2{\tau }^2}{w_z^2}]\cdot \exp [-q^{2}D\mid \tau \mid ],\end{array}$$ (14)

where the factors $1.41$ and $1.04$ come from the constant $\xi$ in Eq. (7) once integrated in Eq. (9). More interestingly, we can calculate the ratios of the previous equations:

$$\begin{array}{ll}{\eta }_{12/11}(\tau )& =\frac{g_{12}^{(1)}(\tau )}{g_{11}^{(1)}(\tau )}=1.41\frac{v_x\tau }{w_{xy}}\exp \left[-0.04\frac{(v_x^2+v_y^2){\tau }^2}{w_{xy}^2}\right],\\ {\eta }_{13/11}(\tau )& =\frac{g_{13}^{(1)}(\tau )}{g_{11}^{(1)}(\tau )}=1.41\frac{v_y\tau }{w_{xy}}\exp \left[-0.04\frac{(v_x^2+v_y^2){\tau }^2}{w_{xy}^2}\right],\end{array}$$ (15)

and obtain a quasi-linear relationship that is independent of diffusion. A previous study demonstrated how AGV introduces additional terms to Eq. (13), resulting in an increased rate of decorrelation and, consequently, an overestimation in flow measurements [22]. A similar effect is expected for Eq. (14), though the exact expression remains unknown. However, using a correlation ratio cancels out such sources of error when the axial velocity is only determined by the axial position: $v_z(z)$ . In this case, the triple integral from Eq. (9) can be divided into a double integral in the $xy$ plane and a single integral for $v_z$ :

$$\begin{array}{ll}{g}_{AB}^{(1)}(\tau )=& {\iint }_{\mathrm{\infty }}^{\mathrm{\infty }}{h}_{xy,A}^{\ast }(x_0,y_0)\cdot h_{xy,B}(x_0+x(v_x\cdot \tau ),y(y_0+v_y\cdot \tau ))d{x}_{0}d{y}_0\\ & \times {\int }_{\mathrm{\infty }}^{\mathrm{\infty }}{h}_z^{\ast }(z(0))\cdot h_z(z_0+v_z(z)\cdot \tau )d{z}_0.\end{array}$$ (16)

The axial integral is invariant between channels and does not contribute to ${\eta }_{12/11}$ nor ${\eta }_{13/11}$ . Thus, an AGV solely defined in the axial direction, as is the case considered in this work, will not affect the flow speed measurements.

Similarly to calculating statistics such as the standard deviation of a population, the more independent measurements are used to calculate the correlation, the more the measured value will converge toward its expected value (either from Eq. (13) or Eq. (14)). Also, this uncertainty is higher at lower correlation values [16,47,48]. Since $g_{12}^{(1)}(\tau =0)=0$ , we can expect an unknown complex number offset d. Assuming this offset, Eq. (15) becomes:

$$\begin{array}{ll}{\eta }_{12/11}(\tau )& =\frac{g_{12}^{(1)}(\tau )}{g_{11}^{(1)}(\tau )}=1.41\frac{v_x\tau }{w_{xy}}\exp \left[-0.04\frac{(v_x^2+v_y^2){\tau }^2}{w_{xy}^2}\right]+\frac{d}{g_{11}^{(1)}(\tau )},\\ {\eta }_{13/11}(\tau )& =\frac{g_{13}^{(1)}(\tau )}{g_{11}^{(1)}(\tau )}=1.41\frac{v_y\tau }{w_{xy}}\exp \left[-0.04\frac{(v_x^2+v_y^2){\tau }^2}{w_{xy}^2}\right]+\frac{d}{g_{11}^{(1)}(\tau )}.\end{array}$$ (17)

The contribution of this offset can be canceled by calculating the variation around $\tau =0$ , while the exponential can also be neglected since its value is very close to 1. In this case, the variations of $\eta$ around $\tau =0$ can be defined by:

$$\begin{array}{ll}\mathrm{\Delta }{\eta }_{12/11}(\tau )& ={\eta }_{12/11}(\tau )-{\eta }_{12/11}(-\tau )=2\cdot 1.41\frac{v_x\tau }{w_{xy}},\text{and}\\ \mathrm{\Delta }{\eta }_{13/11}(\tau )& ={\eta }_{13/11}(\tau )-{\eta }_{13/11}(-\tau )=2\cdot 1.41\frac{v_y\tau }{w_{xy}}.\end{array}$$ (18)

Thus, the flow speed in the x and y directions can be calculated using the following equations:

$$\begin{array}{ll}{v}_x& =\frac{w_{xy}}{1.41\cdot 2\tau }|{\eta }_{12/11}(-\tau )-{\eta }_{12/11}(\tau )|,\\ v_y& =\frac{w_{xy}}{1.41\cdot 2\tau }|{\eta }_{13/11}(-\tau )-{\eta }_{13/11}(\tau )|.\end{array}$$ (19)

The application of the modulus operator is necessary since complex contributions remain after the subtraction. Because of this, Eq. (19) only gives us the amplitude of the flow speed. It cannot differentiate between a positive or negative value of $v_{\text{x}}$ or $v_{\text{y}}$ . Finally, the total flow speed can be calculated from its components such as:

$$v_{\text{tot}}=\sqrt{v_x^2+v_y^2+v_z^2},$$ (20)

where $v_z$ is the axial component of the velocity obtained through Doppler OCT [11]:

$$v_z=⟨\mathrm{\angle }{E}_1(\tau )-\mathrm{\angle }{E}_1(0)⟩\cdot \frac{1}{2k_{0}n\cdot \tau },$$ (21)

where $\mathrm{\angle }{E}_1(\tau )$ is the phase of the signal at a time $\tau$ . Alternatively, the ratio variation can be expressed independently of the flow direction as follows:

$$\mathrm{\Delta }{\eta }_{\text{tot}}=\sqrt{\mathrm{\Delta }{\eta }_{12/11}^2+\mathrm{\Delta }{\eta }_{13/11}^2}.$$ (22)

3. Material and methods

A few-mode OCT system was designed using a three-mode, mode-selective photonic lantern to quantify the accuracy of the correlation ratio approach to estimate flow speed.

3.1. FM-OCT system and experiments

The FM-OCT system comprises a swept-source laser (SL1310V1, Thorlabs, NJ, USA) and two balanced detectors (PDB460C, Thorlabs, NJ, USA). Figure 1(a) shows illumination through Ch1 of the MSPL, and interferometric detection of the classical OCT image (Ch1 using balanced detector 1) and the higher-order modes (Ch2 and Ch3, using balanced detector 2). While the MSPL has three channels, and the DAQ board only two analog inputs, both higher-order modes are collected with the same balanced detector using a 50:50 fiber splitter and an adjustable free-space delay line to offset their signals. This combination of signals decreases the SNR but allows acquisition with simpler electronics. The MSPL was held in place with a 3D-printed holder, and an aspheric lens was positioned to collimate its LP01 and LP11 beams. A 2-galvanometer-mounted mirror set and a scan lens (LSM02, Thorlabs, NJ, USA) completed the imaging head. Figure 1(b, c, and d) show the lateral CSFs for Ch1, Ch2, and Ch3 of the MSPL, respectively.

Fig. 1.

A few-mode optical coherence tomography system acquiring interferograms from all three channels of a mode-selective photonic lantern using two reference arms, one for LP01 and one for LP11 modes. Circulators (C1, C2, and C3), polarization controllers (PC) and a variable free-space delay line allow improved visibility and concurrent acquisitions, respectively. Theoretical coherent spread functions for Channels 1 (Ch1), 2 (Ch2), and 3 (Ch3) are shown in (b), (c), and (d), respectively.

The MSPL specifications were measured as explained in [33]. It has an operational wavelength range of 500 nm centered around 1300 nm, with a modal isolation of 32 dB, and an excess loss of less than 0.48 dB.

First-order correlation measurements require a phase-stable OCT system. Our system was stabilized using the method suggested in a previous study [49]. As illustrated in Fig. 1(a), a 99/1 coupler was added to provide 1% of the signal reflected off a mirror. The calibration signal was measured by Ch1 and numerically applied on the three channels. The calibration A-line was placed above the sample instead of below to address the laser source’s phase fluctuations. Cross-correlation required the precise co-registration of the different channels. While the MSPL guarantees transverse co-registration, axial co-registration was achieved numerically. Signal acquisitions were averaged in intensity instead of the complex field to obtain a single speckle-free A-line. A-lines from Ch2 and Ch3 were then shifted in the Fourier domain [50–52] to maximize the second-order correlation function [15] with Ch1. The measured offset was then applied to co-register Ch2 and Ch3 with Ch1.

Three different experiments were performed. Initially, a C-scan of a homogenous piece of plastic was acquired to calibrate the system on a static sample. Secondly, M-mode flow scans were acquired to demonstrate the validity of the correlation ratio method and compare its performance with existing correlation techniques. Finally, flow speed was measured on B-scans to demonstrate the method with an image.

3.2. Calibration

A C-scan of a piece of plastic was acquired to provide correlation values for a known displacement in the absence of diffusion and flow. The measurement was used to validate the prediction of the correlations and the ratio values and to calibrate our system by the measurement of $w_{xy}$ . Data from all channels were recorded simultaneously. The oversampled scan consisted of a 300 by 300 A-lines, resulting in a 0.15 mm by 0.17 mm scan region. Values of $g_{11}^{(1)},g_{12}^{(1)}$ and $g_{13}^{(1)}$ were calculated on each en-face image as follows:

$$g_{AB}^{(1)}({\mathrm{\Delta }}_x,{\mathrm{\Delta }}_y)=\frac{⟨E_A^{\ast }(0,0)\odot E_B({\mathrm{\Delta }}_x,{\mathrm{\Delta }}_y)⟩}{\sqrt{⟨{|E_A(0,0)|}^2⟩\cdot ⟨{|E_B({\mathrm{\Delta }}_x,{\mathrm{\Delta }}_y)|}^2⟩}},$$ (23)

where $E_A(0)$ refers to a 200 by 200 pixels sub-image centered from the original 300 by 300 pixels en-face image and $E_B({\mathrm{\Delta }}_x,{\mathrm{\Delta }}_y)$ refers to the sub-image with an offset of ${\mathrm{\Delta }}_x$ and ${\mathrm{\Delta }}_y$ on the x and y axis, respectively. The correlation was calculated on 100 en-face images inside the plastic piece, and then the absolute value was averaged axially, resulting in a single 101 by 101 pixels correlation map. Then, a fit was used to extract $w_{xy}$ , and $\mathrm{\Delta }{\eta }_{12/11}$ and $\mathrm{\Delta }{\eta }_{13/11}$ were calculated for a known displacement, allowing later flow speed calculations.

3.3. M-mode scans

A syringe pump connected to a plastic tube was used to emulate flow as shown in Fig. 2. Diluted milk, one part 2% milk for three parts water, was used as a sample for its availability and low absorption. A-lines were acquired in the middle of the tube with an integration time of 1 second. To study the precision of the methods, the 1-second data was subdivided into shorter integration time segments. The flow speed was measured for each segment, and the standard deviation between them was calculated, giving us information about measurement variability. This was done for integration time of 10, 20, 30, 40, and 50 ms.

Fig. 2.

Setup used to emulate blood flow. A tube containing flowing scatterers was put at the focus of the FM-OCT imaging head and tilted in plane (four values of $\theta$ ), and out of plane (one value of $\varphi$ ).

The transverse angle of the tube $\theta$ was varied to 0°, 30°, 60°, and 90°  to test the angular measurement capability. The tube was also tilted axially at an angle $\varphi =15°\hspace{0.17em}$ to test the effect of AGV. Due to refraction at the air-tube interface, the tilt resulted in an effective angle $\varphi =10°\hspace{0.17em}$ between the flow direction and the transverse plane. While the tube’s curvature induced optical distortion, those distortions were not on the same axis as the flow and did not contribute to it. A previous study also showed no impact of tube curvature on flow measurement [22].

The non-axial flow $(xy$ plane) was measured using three distinct correlation-based methods: our proposed method $\left(v_{\text{tot}}\right)$ , a fitting approach in logarithmic scale $\left(v_{\text{fit}}\right)$ , and a single sampled point $\left(v_{\text{sp}}\right)$ , as described [10,15,27]. The correlation was calculated for time delays ranging from −1 ms to 1 ms for each axial position. Noise correction was applied as described in a previous study [53]. The noise floor was measured and was similar for all depths. It was thus estimated by averaging the signal amplitude in a region without sample. To measure $v_{\text{fit}}$ , a fit was applied on the $\log (|g_{11}^{(1)}|)$ curve using Eq. (13). The lateral velocity $\left(\sqrt{v_x^2+v_y^2}\right)$ was determined using the fit coefficients. The axial velocity (z direction) was added from the Doppler measurement afterward. In parallel, $v_{\text{sp}}$ was calculated using the value of $g_{11}^{(1)}$ at $\tau =1\text{ms}$ by isolating the lateral velocity from Eq. (13). A moving average of 11 temporal points was applied on the $g_{11}^{(1)}$ curve to increase precision. The delay was selected post hoc to optimize the accuracy. The diffusion coefficient needed for this method was measured by fitting $g_{11}^{(1)}$ in the absence of flow and extracting the value from the fit coefficients. To measure $v_{\text{tot}}$ , another moving average of 11 points was first applied on $\mathrm{\Delta }\eta$ before calculating $v_x$ and $v_y$ with Eq. (19), which also improved precision. The calculated $v_x,v_y$ and $v_z$ were then summed to obtain $v_{\text{tot}}$ .

The expected flow velocity in the tube $\left(v_{\text{th}}\right)$ was calculated from the known inner diameter of the tube and the pump-controlled volumetric flow velocity, assuming laminar flow resulting in a parabolic distribution of the flow speed from the center [54]. We compared the accuracy of the measurements of the different methods by calculating the root mean square error (RMSE). For the two existing methods, the RMSE was calculated over 1 mm in the center of the tube to avoid erroneous contributions of the tube edges. Due to image overlap and limited SNR, Ch2 and Ch3 were imaged sequentially instead of simultaneously, as for the calibration measurements. Ch1 was recorded in both of these measurements to calculate the simultaneous cross-correlations. Only the first measurement was used for $v_{\text{sp}}$ and $v_{\text{fit}}$ .

3.4. B-scan

The flow speed was quantified with 200 B-scans, similar to the previous section. Scans were acquired at 200 Hz and consisted of 500 A-lines covering roughly a region of 1.6 mm. Thus, the shortest delay between two A-lines acquired at the same position was 5 ms. Using this delay prevented the estimation of $v_{\text{fit}}$ as curve fitting required multiple points along the correlation curve, and given the long minimum delay, most points were entirely decorrelated and thus did not contribute to the fit. For the same reason, a moving average was not used to calculate $v_{\text{sp}}$ or $v_{\text{tot}}$ . Both variables were measured with $\tau =5\text{ms}$. The noise floor was corrected as in the previous section. The average speed versus depth was calculated over 11 A-lines in the center of the tube to facilitate comparison.

Previously, we calculated correlation by comparing a single pixel in the A-line over time. Here, given a 2D image in time, we correlated a 3× 3 pixels in time, providing higher accuracy at the cost of diminished spatial resolution. Furthermore, to highlight only the region with flow, two conditions were applied: the SNR had to be higher than 2 and $g_{11}^{(1)}>0.1$ . These restrictions allowed the quantification of flow measurement in the absence of a sample or where the signal decorrelated too rapidly. The velocity was measured at $\theta =90°$ , corresponding to the fast scanning axis, and was measured both without and with a tilt at the same angle as before (i.e., $\varphi =15°$ ).

4. Results

Figure 3 shows the experimental (top row, a-c) and theoretical (bottom row, d-f) auto- and cross-correlation images between OCT channels. Figure 3(a) shows the experimental first-order auto-correlation of Ch1, while (b) and (c) show the cross-correlations of Ch1 with Ch2 and Ch3, respectively. This pattern is repeated for theoretical values in Fig. 3(d) to (f). Minor differences in the experimental shape and background level are observed, but overall the system acts as theorized. These small discrepancies are primarily due to imperfections of the custom-built MSPL.

Fig. 3.

Speckle correlation of an en-face image of a piece of plastic. The auto-correlation amplitude of Ch1 is shown in (a), while the cross-correlation amplitude of Ch1 and Ch2, and Ch1 and Ch3 are shown in (b-c), respectively. The second row (d-f) shows the respective theoretical values.

Figure 4 presents the experimental cross-correlation and auto-correlation ratios obtained for a static object. Figure 4(a-b) shows the ratios ${\eta }_{12/11}$ and ${\eta }_{13/11}$ , respectively. In Fig. 4(d-e), the offset $d$ was subtracted using Eq. (18) to produce $\mathrm{\Delta }{\eta }_{12/11}$ and $\mathrm{\Delta }{\eta }_{13/11}$ images showing the linear trend between correlation ratios and delays. Figure 4(f) shows $\mathrm{\Delta }{\eta }_{\text{tot}}$ . As expected, it produces a cone with linearity highlighted in the two orthogonal sections showed in Fig. 4(c). The minor astigmatism observed in Fig. 4(c) was taken into account when calculating $v_x$ and $v_y$ . Random fluctuations on the contour are due to low correlation values in those regions, as seen in Fig. 3. The fluctuations are present only where the correlation is close to 0, and thus measurements for such displacement are inaccurate. Compared to ratios $\eta$ (Fig. 4(a–b)), the variation of ratios $\mathrm{\Delta }\eta$ (Fig. 4(d–e)) shows increased linearity, suggesting that the latter is more suitable for flow measurement.

Fig. 4.

Experimental cross-correlation and auto-correlation ratios for (a) ${\eta }_{12/11}$ , and (b) ${\eta }_{13/11}$ obtained for a static object. Ratio variations for (d) $\mathrm{\Delta }{\eta }_{12/11}$ , and (e) $\mathrm{\Delta }{\eta }_{13/11}$ , and their combination (f) $\mathrm{\Delta }{\eta }_{\text{tot}}$ . Subfigure (c) plots cross-sections of $\mathrm{\Delta }{\eta }_{\text{tot}}$ shown in (f).

Figure 5 shows comparisons between the measurement of laminar flow in a plastic tube at different speeds and angles using our approach ( $v_{\text{tot}}$ , blue crosses) and the existing techniques described previously ( $v_{\text{sp}}$ and $v_{\text{fit}}$ in red and orange dots, respectively). The pump flow speeds in the center of the tube are 0.0 mm/s for sub-figures (a-d), 0.9 mm/s for (e-h), 2.7 mm/s for (i-l), 9.0 mm/s for (m-p), and 36.0 mm/s for (q-t). The flows measured by the techniques were shown for transverse angles of $\theta =0°$ (column 1), $\theta =30°$ (column 2), $\theta =60°$ (column 2), and $\theta =90°$ (column 4). An angle of $\theta =0°$ corresponds to a flow in the x axis, as illustrated in Fig. 2. Meanwhile, there is no axial tilt (i.e. $\phi =0$ ). The solid black curve shows the expected flow speed. Two discrepancies are observed with the $v_{\text{fit}}$ technique (yellow curve): in Fig. 5(m), flow measurement inaccuracy is observed for high values, while in Fig. 5 d), no flow was recorded. We hypothesize a pump and an algorithm failure, respectively. Table 1 compiles the RMSE of flow speeds measured in Fig. 5. Our method shows increased accuracy (i.e. lower RMSE) for all flow speeds.

Fig. 5.

Laminar flow as measured by $v_{\text{tot}}$ (blue crosses), $v_{\text{fit}}$ (orange dots), and $v_{\text{sp}}$ (yellow dots) as compared to theoretical values (solid black curve) at $\theta =0°$ (column 1), $\theta =30°$ (column 2), $\theta =60°$ (column 3), and $\theta =90°$ (column 4), and for flow speeds of 0.0 mm/s (a–d), 0.9 mm/s (e–h), 2.7 mm/s (i–l), 9.0 mm/s (m–p) and 36 mm/s (q–t).

Table 1. Root-mean-square error (RMSE) of flow speed measurements ( $v_{\text{tot}}$ compared to previous published techniques ( $v_{\text{sp}}$ and $v_{\text{fit}}$ ) for the flow speeds used in Fig. 5.

	max speed [mm/s]:	0.0	0.9	2.7	9.0	36.0
RMSE	$v_{\text{tot}}$	0.22	0.14	0.43	1.13	3.99
	$v_{\text{sp}}$	0.97	0.48	0.61	4.17	5.19
	$v_{\text{fit}}$	1.20	0.86	0.84	4.63	10.2

Figure 6 shows comparisons between the measurement of laminar flow in a plastic tube in the presence of an effective $\phi =10°$   axial tilt to quantify the effects of AGV. Flow measurements were performed at different speeds and at a transverse angle $\theta =90°$ using our approach ( $v_{\text{tot}}$ , blue crosses) and the existing techniques described previously ( $v_{\text{sp}}$ and $v_{\text{fit}}$ in red and orange dots, respectively). Figure 6 shows overestimated flow speeds generated by $v_{\text{sp}}$ and $v_{\text{tot}}$ compared to the planar case, particularly for (d-e). Table 2 shows the RMSE of flow speeds measured in Fig. 6 in the presence of an axial tilt $(\varphi =10°$ ). The correlation ratio method was shown to be more accurate (except at 12.6 mm/s in comparison to $v_{\text{fit}}$ ).

Fig. 6.

Laminar flow as measured by $v_{\text{tot}}$ (blue crosses), $v_{\text{fit}}$ (orange dots), and $v_{\text{sp}}$ (yellow dots) as compared to theoretical values (solid black curve) at the center of the tube for speeds of 0.0 mm/s (a), 0.9 mm/s (b), 2.7 mm/s (c), 9.0 mm/s (d), and 36 mm/s (e) with an effective axial tilt of $\varphi ={10}^{\circ }$ .

Table 2. Root-mean-square error (RMSE) of flow measurements with a tilt angle $\varphi =10°$ ( $v_{\text{tot}}$ ) compared to previously published techniques ( $v_{\text{sp}}$ and $v_{\text{fit}}$ ) for the flow speeds used in Fig. 6.

	max speed [mm/s]:	0.0	0.9	2.7	12.6	36
RMSE	$v_{\text{tot}}$	0.61	0.18	0.53	8.37	13.4
	$v_{\text{sp}}$	0.91	0.76	1.12	57.2	103
	$v_{\text{fit}}$	0.89	1.06	0.97	7.31	24.1

Figure 7(a) and (b) summarize the not tilted (Fig. 5) and tilted cases (Fig. 6), respectively, by plotting the measured velocities as a function of the expected velocities. Blue dots represent our approach $\left(v_{\text{tot}}\right)$ , while the existing techniques described previously ( $v_{\text{sp}}$ and $v_{\text{fit}})$ are depicted by red and orange dots, respectively. The black line shows a perfect match with the expected flow speed. To visualize all scales of flow speed, the figure is plotted on a log scale; therefore, flow speeds of zero are excluded.

Fig. 7.

Laminar flow as measured by $v_{\text{tot}}$ (blue dots), $v_{\text{fit}}$ (orange dots), and $v_{\text{sp}}$ (yellow dots) as compared to theoretical values (solid black curve) without an axial tilt (a) and with an effective axial tilt of $\varphi ={10}^{\circ }$ (b).

Figure 8 shows the standard deviation of the flow speed measurements for an integration time of 50 ms at varying speeds (a) and in 21 pixels at the center region of the tube for a flow speed of 3 mm/s for varying integration times for $v_{\text{tot}}$ (blue), $v_{\text{fit}}$ (orange), and $v_{\text{sp}}$ (yellow).

Fig. 8.

Standard deviation on laminar flow measurements done by $v_{\text{tot}}$ (blue), $v_{\text{fit}}$ (orange), and $v_{\text{sp}}$ (yellow) for varying expected flow speed and an integration time of 50 ms (a) and for an expected flow speed of 3 mm/s and varying integration times.

Figure 9(a-d) shows individual flow velocity components, including the axial z-direction (dark blue), transverse x-direction (light blue), and y-direction (medium blue). The total flow speed measured is also displayed ( $v_{\text{tot}}$ , blue crosses) with the expected total flow speed in a solid black line $\left(v_{\text{th}}\right)$ . From these components, the orientation is calculated and shown in Fig. 9(e-h) with the measured angle (green dots) as well as its average value (dark green dotted line). The measured difference from the expected angle is shown in (i) for all measurements from Fig. 5.

Fig. 9.

Laminar flow as measured by the correlation ratio method in the $x$ (light blue dots), $y$ (blue dots) and $z$ (dark blue dots) axes for transverse flow orientations of $\theta =0^{\circ }$ (a), $\theta ={30}^{\circ }$ (b), $\theta ={60}^{\circ }$ (c), and $\theta ={90}^{\circ }$ (d), with $v_{\text{tot}}$ (blue crosses) and expected flow speed (solid black line). The corresponding measured angle is shown below in (e–h) respectively. The absolute difference from the expected angles is shown in (i) for all measures made in Fig. 5.

Figure 10 shows B-scans of flow speeds of the tube measured using $v_{\text{tot}}$ (a-d) $v_{\text{sp}}$ (e-h). Figure 10(i-l) displays flow speeds in the vertical section at the center of the B-scan measured by $v_{\text{tot}}$ (dark blue) and $v_{\text{sp}}$ (orange), as well as the expected value ( $v_{\text{th}}$ , solid black line). The flow speed is displayed for a maximum pump flow speed at the center of the tube of 0 mm/s (column 1), 0.9 mm/s (column 2), 1.8 mm/s (column 3) and 2.7 mm/s (column 4). The white regions correspond to situations where either the flow could not be measured due to poor SNR $(\text{SNR}<2)$ or complete B-scans decorrelation $(g_{11}^{(1)}<0.1)$ . Given a delay of 0.005 seconds between each B-scan, the maximum measurable speed was 1.8 mm/s. The correlation ratio method is more accurate on the edges, where the flow speed is lower.

Fig. 10.

Laminar flow measured in a tube by $v_{\text{tot}}$ (a-d) compared to $v_{\text{sp}}$ (e-h) for flow speed of 0 mm/s (column 1), 0.9 mm/s (column 2), 1.8 mm/s (column 3) and 2.7 mm/s (column 4). The flow in the vertical axis of the tube is displayed in (i-l) with $v_{\text{tot}}$ (blue), $v_{\text{sp}}$ (orange), and the expected result (black solid line).

Figure 11 shows similar flow speed measurements than Fig. 10 but with the tube tilted axially at an effective angle of $\varphi =10°$ . Again, (a-d) shows flow speed as measured by $v_{\text{tot}}$ compared to (e-h), where $v_{\text{sp}}$ is used, and (i-l) displays the flow speed in the center of the tube compared to the expected result in black. The results for both methods are less accurate than with no tilt.

Fig. 11.

Laminar flow measured in a tube by $v_{\text{tot}}$ (a-d) compared to using $v_{\text{sp}}$ (e-h) for flow speed at the center of 0 mm/s (a,e,i), 0.9 mm/s (b,f,j), 1.8 mm/s (c,g,k) and 2.7 mm/s (d,h,l). The tube was tilted by an effective angle of $\varphi =10°$ . The flow in the center axis of the tube is displayed in (i-l) with $v_{\text{tot}}$ (blue), $v_{\text{sp}}$ (orange), and the expected result (black solid line).

5. Discussion

We have demonstrated a novel method for measuring transverse flow by exploiting the OCT signal obtained from three channels of an MSPL: the different channels resulting in distinct CSFs. The method uses the ratios between the cross-correlation of Ch1 with either Ch2 or Ch3 and the autocorrelation of Ch1. This approach results in a linear relation between the measured ratio and the transverse component of the measured flow speed, providing more accurate measurements without a priori assumption on the diffusion coefficient value. Furthermore, the method gives information on the angle of the transverse orientation of the flow, which would allow for direct measurement of blood flow orientation.

In the following paragraphs, experimental results are further discussed. Figure 3(a–c) shows higher background correlation than its theoretical values (d–f), which could either originate from the phase stabilization method [22], or from the statistical noise when calculating correlation around $\tau =0$ . As described in Sec. 3.2, the correlation map were first calculated for each depth and the absolute value was then averaged axially. Any negative correlation value would thus be considered as positive and created the observed background signal.

Figure 6 shows how accurately measuring flow with an axial tilt is much more challenging. This technical difficulty can be due to the faster decorrelation rates caused by the AGV and the statistical noise due to the lower correlation values. However, the local overestimation of $v_{\text{tot}}$ in (d) at depths around −0.5 mm and 0.5 mm greatly increased the RMSE, and remains unexplained. The slight overestimation in (e) seems to match the deviation observed in Fig. 5 and might be explained by pump instabilities. The underestimation of $v_{\text{fit}}$ in Fig. 6(d-e) is due to the algorithm’s inability to fit the correlation curve and return a null flow in the x and y directions. Speed is partially measured since the Doppler effect contributes to the axial direction speed flow, although some bias due to the AGV is expected [22]. Furthermore, contrary to Fig. 5(a–d), some flow was detected at a pump flow speed of 0 mm/s, possibly due to a minor leak in the pumping circuit. There is no reason why a tilt in the tubing should affect flow measurement when there is no flow. Nevertheless, Fig. 7 highlights how our proposed method is more accurate, particularly at lower flow speeds and in the presence of an axial tilt.

Figure 8(a) shows that the uncertainty decreases for lower flow speeds, allowing distinguishing small velocities from static tissues. Furthermore, as expected, precision increases inversely with longer integration times, but the exact order of the relation has yet to be determined.

Figure 11 highlights again challenge associated with measuring tilted flow. The maximum measurable speed decreases because of the signal decorrelating much faster due to AGV. Interestingly, Fig. 11(c,g) show a white ring for which flow cannot be measured, likely resulting from higher AGV values close to the edges. At the tube’s extremity, despite the AGV being at its highest value, since the flow is low, the signal does not decorrelate as fast as in the white ring region, and flow remains measurable.

The method has two fundamental advantages. Firstly, the linear relationship between speed and the correlation ratio (i.e., Eq. (19)) allows very low values of flow to be measurable, even while using small values of $\tau$ . Secondly, using the ratio causes cancellations of artifacts impacting both channels, such as diffusion axial motions, or laser instabilities. A delay as short as the A-line period can be used, opening the door to new scanning strategies for flow measurements.

In this work, we used different scanning strategies. The first consisted of taking multiple A-lines at the same position, allowing for the measurement of faster flow speeds from the smaller $\tau$ value. However, measuring an entire B-scan or C-scan in such a way would take a long time. Alternatively, A-lines were also correlated from one B-scan to the next. This allowed us to measure the flow on an entire B-scan in the time it would take for a single A-line, which greatly reduced the acquisition time. However, this has resulted in a decreased maximum measurable flow speed. Here, a 200 Hz scan rate resulted in a measurable maximum flow speed of ≈ 2 mm/s. Higher flow speed would necessitate a faster scanning system or a larger beam waist to avoid scatterers exiting the image field-of-view by the time of the next B-scan.

While a linear relationship was assumed in this work, a third-degree polynomial would have been slightly more accurate. Inspection of Fig. 4(c) reveals that a third-degree polynomial or a look-up table could have improved the accuracy.

This approach has limitations. Like other correlation methods, the correlation ratio approach still lacks precision, requiring significant integration times to be precise. The statistical nature of correlation requires multiple independent measurements to converge towards the expected value. However, this technique was more precise than previously published methods for lower flow speeds [10,15,27].

Some issues were observed with the phase stabilization of our system. The issue was highlighted by comparing second-order correlation function $\left(g^{(2)}\right)$ to the first-order correlation function $\left(g^{(1)}\right)$ . Light intensity is used to calculate $g^{(2)}$ , and thus, the phase is unnecessary. Both correlations may be compared using the Siegert relation: $g^{(2)}=|g^{(1)}{|}^2+1$ [55]. Since $g^{(1)}$ decorrelated much faster, we hypothesize some phase instability originating from the laser source.

Phase instabilities highlighted how the correlation ratio method was more robust regarding phase disturbances. Since the phase instabilities affect $g_{11}^{(1)}$ and $g_{12}^{(1)}$ in the same way, calculating the ratio removed the error originating from phase instabilities. Such issues mainly occured when calculating $v_{\text{sp}}$ and $v_{\text{fit}}$ .

Moreover, the absolute value in Eq. (19) prevents disambiguation of the velocity direction, i.e., the left or right direction of the flow cannot be distinguished. As a consequence, negative and positive angles could not be differentiated. Indeed, for $|v_x|=|v_y|$ , two flow direction exist: 45°  or −45°. This also explains why in Fig. 9(e, h), the measured angles were overestimated and underestimated, respectively. In this case, a 94° angle could be measured as 86°, leading to an underestimation. Something similar happened for the 0°  angle but with negative angles being measured as positive. Figure 5(e–h) also shows higher inaccuracies at the edges of the tube where flow speed approaches 0. Figure 9(i) illustrates it better in showing how the error increases as the flow speed goes below 1 mm/s. Theoretically, in Eq. (19), the phase variation inside the modulus operator should be indicative of the flow direction (left or right), but experiments contradict this. More work is needed to understand why and achieve complete measurement of flow direction.

From an experimental point of view, the most significant limitation originated from the reliability of the syringe pump, in which friction varied over time. While the pump provided reliable flows in initial experiments, it became less reliable as experiments went on.

The technique presented in this paper faces similar difficulties to other correlation-based methods already used in ophthalmology, from which solutions may be leveraged. Future work involves implementing the method for in vivo imaging and solving associated challenges. For instance, lower SNR is expected in vivo; however, previous work has shown that correlation measurements are accurate at very low SNRs [53]. While this method has not yet been tested using blood, previous work has shown that the shape of diffusers does not impact measurements [28], and our own approach intrinsically compensates for diffusion. Bulk-tissue motion, however, is a challenge, but one that has already been addressed in other works [52,56–59]. The method was successful in measuring flow through more than 1 mm of moving scatterers, but it remains to be seen whether this depth is enough for ophthalmology.

Measuring flow speeds higher than those demonstrated here is a challenge. Increasing scanning speeds, both spatially and spectrally, is a possible solution to expanding the range of measured velocities. Alternatively, using a coarser resolution could increase the maximum measurable speed as $w_{xy}$ is related to how fast the signal decorrelates. Indeed, using an M-mode approach permits the measurement of higher velocities in a single region, as it allows for shorter delay values. Since the minimum possible delay would be 500 times shorter than for the B-scan case, we could expect 500 times faster (≈ 1 m/s) measurable flows.

The biggest unknown is how the anatomical shape and the small size of the capillaries will affect measurement accuracy. Furthermore, the impact of varying flow speeds in arteries due to heartbeats is unknown and needs to be investigated. Theoretically, the measurement should return the average flow speed during the integration time.

While the MSPL holding an alignment was currently tricky in this work, another group [30] has successfully managed to splice the lantern tip to conventional connectors, removing the issue.

Furthermore, the analysis for AGV was limited to a single axial angle and only for a gradient in the axial direction. However, the higher accuracy on the sides of the tube in Fig. 11 suggests that our method is more robust for gradients in transverse directions. Further studies, both experimental and theoretical, are needed to determine with more accuracy how it affects our method.

6. Conclusion

In this work, we proposed a novel method for measuring flow speed using the ratios between the cross-correlation and auto-correlation of the signal acquired from two distinct CSFs through the use of a modally-specific photonic lantern. The method proved to be more accurate, particularly at low speeds, distinguishing flows as slow as 0.5 mm/s from static ones for 1 second integration time. The technique was less sensitive to errors induced by AGV in the axial component of the flow speed and did not require prior knowledge of the diffusion coefficient. Furthermore, our approach could measure the transverse angle of the flow direction, allowing, when combined with Doppler OCT, the knowledge of the 3D flow orientation.

The mathematical framework presented in this work could also be extended to other CSFs as demonstrated in a previous study [39]. Alternatively, such CSFs could be numerically assessed in the same way deconvolution can change a system’s CSF. For example, an image could be digitally shifted in the axial direction to artificially create a CSF similar to ${h_{xy}(x,y)|}_{\text{Ch2}}$ . The proposed ratio method could then be used to measure the axial component of the flow. More interestingly, the approach could be extended to full-field imaging devices, where the image is manipulated numerically to create an alternate CSF.

Funding

Natural Sciences and Engineering Research Council of Canada 10.13039/501100000038 ( RGPIN-2018-06151); Institut TransMedTech 10.13039/501100024255.

Disclosures

C.B.: Castor Optics, inc. (I,P). R.M.-T. : (P). The other authors have no conflicts of interest to disclose.

Data Availability

Imaging datasets acquired in this work are not publicly available at this time, but may be obtained from the authors upon reasonable request.