## Abstract

We report on a novel method to produce B-scan images in spectral domain optical coherence tomography (SD-OCT). The method proceeds in two steps. In the first step, using a mirror in the sample arm of the interferometer, channelled spectra are acquired for different values of the optical path difference (OPD) and stored as masks. In the second step, the mirror is replaced with an object and the captured channelled spectrum is correlated with each mask, providing the interference strength from the OPD value used to collect the respective mask. Such a procedure does not require data organized in equal frequency slots, and therefore there is no need for resampling as practiced in the conventional fast Fourier transform (FFT)-based SD-OCT technology. We show that the sensitivity drop-off versus OPD and the quality of B-scan images of the novel method are similar to those obtained in the conventional FFT-based SD-OCT, using spectral data linearly organized in frequency.

© 2014 Optical Society of America

Spectral domain optical coherence tomography (SD-OCT) has shown beyond doubt its superiority over conventional time-domain OCT. Both SD-OCT implementations, spectrometer based (Sp)-OCT and swept source (SS)-OCT, produce images with higher speed and sensitivity [1] than time-domain OCT. However, both spectral domain methods present the disadvantage of requiring recalibration of the interference signals in order to provide data evenly spaced in the frequency domain prior to being subject to fast Fourier transform (FFT). The origin of this drawback in Sp-OCT is the inherent noneven distribution of the optical frequencies onto the linear array by the dispersive element (diffraction grating), while in SS-OCT, it is the nonlinear sweep of the SS.

So far, several techniques have been demonstrated to successfully calibrate the interferometric data for both SD-OCT implementations. Thus, for the Sp-OCT method implemented in this Letter, calibration techniques such as using an additional light source that produces several spectral lines in the region of interest of the spectrometer, parametric iteration methods [2], phase linearization techniques [3], and automatic calibrations [4] have been proposed. Unfortunately, all of these methods require either additional expensive equipment and/or are computationally expensive and limit the real-time operation of the OCT systems.

In a previous report [5], we introduced master/slave interferometry (MSI), a method based on comparing the channelled spectrum (CS) at the interferometer output with previously acquired channelled spectra. In [5], we insisted on the capability of the MSI method of producing *en-face* OCT images by tuning a SS. In this Letter, we develop the MSI method more and show how it can be used to produce A-scan profiles and B-scan OCT images. The MSI principle can be applied to any SD-OCT technology. This is illustrated here on a traditional Sp-OCT setup, based on cross-correlating the CS provided by the linear camera when an object is placed in the interferometer arm of the OCT with channelled spectra (masks) previously recorded using a mirror instead of the object. The MSI method is inspired by the adaptive filtering technique, where the shape of the CS for each depth value is “recognized” by correlation with a set of masks built before the measurement/imaging process. Correlation is evaluated between shapes produced by the same setup. Obviously, the correlation process does not need data organized in equally spaced frequency slots, so there is no need for calibration of data. Denomination of method as master/slave reflects that a slave interferometer selects signals from that depth value in the object determined by the optical path difference (OPD) in a master interferometer. This makes the principle ideally suited to be used in constructing *en-face* OCT images, and this was the main objective of [5]. Here, we prove that the same principle can be used to generate B-scan OCT images. The master interferometers, as explained in [5], are replaced with the stored masks.

The Sp-OCT setup is presented in Fig. 1. As a broadband optical source, a superluminescent diode (Amonics, Hong Kong) with a central wavelength $\lambda =1050\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ and bandwidth $\mathrm{\Delta}\lambda =30.0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ was employed. The spectrometer incorporates a linear camera (LSC, model SU-LDH, Goodrich-SUI, Princeton, New Jersey).

The method proceeds in two steps. In a preparation step 1, switches ${K}_{1}$ and ${K}_{2}$ are placed in position 1. A highly reflective mirror is used as the object and conjugate versions of the FFT-channelled spectra ${M}_{p}$, for $p=\overline{1,P}$, corresponding to a set of optical path differences ${\mathrm{OPD}}_{p}$ measured between the reference and object arm lengths of the interferometer, are recorded [${R}_{p}=\mathrm{FFT}*({M}_{p})$] and placed in the “memory block.” Normally, signals ${R}_{p}$ should be recorded at OPD values separated by half of the coherence length of the optical source or denser. In the measurement step 2, switches ${K}_{1}$ and ${K}_{2}$ are placed in position 2, the object is placed in the object arm of the interferometer, and the acquired CS is correlated with all masks ${M}_{p}$. Correlation with each mask ${M}_{p}$ provides an output signal of amplitude ${A}_{p}$ as a point in the A-scan for the OPD value used to create that mask. The processing block shown in Fig. 1 illustrates how the amplitude of the signal originating from a certain depth in the sample is calculated. Correlation of the current CS with a mask ${M}_{p}$ is calculated as

The correlation signal is high-pass filtered to remove the dc component, and rectified. The maximum is placed around $k=0$, but depends on the phase between the mask and the current CS acquired. Therefore, an average is executed over the ${\text{Corr}}_{p}$ results, within a window of size $W=2w-1$ points around $k=0$:

Data resampling is required when producing FFT-based A-scans using the traditional SD-OCT methods. The use of the MSI principle to produce discrete points of the A-scan at $P$ OPD values, as determined by $P$ masks, does not need resampling/calibration. The masks ${M}_{p}$ are themselves chirped, but this is not a problem, as the same chirping law is present in both terms being compared (CS and ${M}_{p}$). Maximum signal is obtained when there is a shape within the CS similar to the ${M}_{p}$.

To demonstrate that the MSI method is immune to noncalibrated data, we produced correlation-based MSI and FFT-based A-scans for both calibrated and noncalibrated CS, and similar results to those reported in [5] were obtained. The full width at half maximum of the A-scan peaks corresponding to the OPD of the mirror position used as object defines the depth resolution of the system. When the conventional FFT method is used, the achievable depth resolution depends on the quality of calibration technique used to linearize the data. In our case, the resampling points were obtained using a procedure based on the fact that the phase of the interferometric signal should vary linearly with the frequency [6]. A depth resolution of 21 μm was achieved when data were resampled, which can be as large as 140 μm when no resampling is performed. On the other hand, the MSI technique is completely immune to the way in which data are sampled; the depth resolution in both situations was the same, 21 μm.

To enhance the strength of the signal, the correlation result is rectified and then integrated within a window of a certain width around $k=0$, according to Eq. (2). However, this will lead to some deterioration of depth resolution. The larger the window width, the wider the profile of the output signal versus OPD. This effect has been documented in [5]. Similarly here, for a narrow window size ($<5$ points), a depth resolution of 21 μm could be achieved. In our case, a number $M=512$ sampling points are used to digitize the signal corresponding to the spectral range of the optical source. As a consequence, the maximum width of the window, which can be applied to the correlation signal, is $2M-1=1023$, in which case the depth resolution worsened to 34 μm.

In order to evaluate the effect of the window on the signal strength, a low back-reflective sample (paper) was used, and the signal-to-noise ratio (SNR) was measured for a value of $\mathrm{OPD}=0.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$, obtaining a similar dependence to $W$ in Fig. 2(a) as that reported in [5]. The SNR increases with the width of the window $W$, and for $W$ larger than 3 points, the SNR becomes larger than its corresponding value calculated using the FFT-based OCT method. Therefore, a good choice for $W$ is $W=20\u201330$ points, where the depth resolution achieved is around 22 μm, a value close to its best measured value (21 μm), together with a SNR 4.5 dB better than that offered by the FFT-based technique. By averaging FFTs, better SNR would be achievable in the conventional FFT-based OCT method too. However, averaging is practiced here over the same calculated ${\text{Corr}}_{p}$ function, while FFT averaging would require repetition of data acquisition.

The sensitivity drop-off was evaluated experimentally. For 12 OPD values, we acquired and processed the measured CS using the traditional FFT-based OCT method and the MSI method. For the MSI case, we correlated the CS with 12 masks, ${M}_{p}$, $p=\overline{1,12}$, previously acquired at OPD values separated by 0.2 mm. Figure 2(b) shows the sensitivity drop-off profiles for both methods with data resampled or not. As can be noticed, when data are resampled, the sensitivity drop-off of the FFT-based method is quite similar to that of the MSI method. However, when data are not resampled, the sensitivity drop-off profile of the conventional FFT-based OCT method is much faster.

When using the MSI method, there are two ways for generating B-scans: by assembling A-scans into a B-scan [Fig. 3(a)], and by assembling T-scans (transversal reflectivity profiles) into a B-scan [Fig. 3(b)]. First, masks ${R}_{p}$ are acquired for each depth pixel $p$ out of $P$ pixels along the depth coordinate. Second, for each position of GX, $h=\overline{1,H}$ along the T-scan, for each ${\mathrm{CS}}_{h}$ provided by the IMAQ board, ${U}_{h}=\mathrm{FFT}({\mathrm{CS}}_{h})$ is calculated and stored in the RAM. In Fig. 3(a), for each ${U}_{h}$ taken sequentially from the RAM, $P$ calculations are performed with the $P$ masks ${R}_{1-P}$ providing $P$ points along the depth, which can be assembled into an ${A}_{h}$-scan. Then the procedure is repeated for all $H$ pixels along the transversal coordinate, obtaining $H$ A-scans which finally are assembled into a B-scan. In Fig. 3(b), the RAM is read and all $H$ ${U}_{h}$ are multiplied with the same ${R}_{p}$ followed by ${\mathrm{FFT}}^{-1}$; hence, a succession of $P$ amplitude values corresponding to a T-scan ${T}_{p}$ are produced. Then the procedure is repeated for all $P$ masks, obtaining $P$ T-scans which finally are assembled into a B-scan. In both figures, $H\times P$ correlation products are evaluated, leading to a B-scan OCT image of $H\times P$ pixels.

Using a PC equipped with two Intel quad core processors (model E5345) and making use of the Labview’s vi-s, we evaluated the time required to produce a B-scan using the FFT-based Sp-OCT and the MSI method.

To keep the comparison simple, let us consider that all $H=200$ CS terms (the same number as in [5]) are initially acquired and stored in the RAM, and that the time to transfer data to and from storage can be ignored. The acquisition time of a single CS, as determined by the maximum speed of the camera, is ${t}_{\text{acq}}=21.28\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu s}$. The acquisition of all $H$ CS spectra requires a time ${T}_{\text{RAM}}=4.256\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ms}$. An FFT of the CS made of $M=512$ points takes ${t}_{\mathrm{FFT}}\sim 0.9\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu s}$. For a B-scan image made of $H=200$ A-scans, this requires ${T}_{B,\mathrm{FFT}}=H{t}_{a}=0.18\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ms}$. When resampling is needed, using a vi implementation of a cubic spline interpolation, the time to resample a single CS is ${t}_{s}=6.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu s}$; hence, ${t}_{A,\mathrm{FFT},S}={t}_{\mathrm{FFT}}+{t}_{s}=7.4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu s}$, and the time to produce a B-scan is ${T}_{B,\mathrm{FFT},S}=H{t}_{A,\mathrm{FFT},S}=1.48\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ms}$.

Let us now consider the MSI principle, where supplementary FFT(CS) are stored, which requires a ${T}_{\text{RAM}}=H({t}_{\text{acq}}+{t}_{\mathrm{FFT}})=4.656\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ms}$ (here, ${t}_{\mathrm{FFT}}\sim 2.0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu s}$, as each CS has to be zero padded to $2M$ points). For $M=512$ spectra, the maximum number of cycles in the CS is 256, so up to $P=256$ masks are acquired. A correlation process such as that used in [5], using the NI vi, takes ${t}_{c,3\mathrm{FFT}}=12.3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu s}$. The time to produce an A-scan using the MSI method by correlating the CS with $P=256$ stored masks comes to ${t}_{A,\mathrm{MSI},3}=P{t}_{c}=3.15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ms}$. For a B-scan image made from $H=200$ A-scans, the time needed to produce an MSI-based B-scan image is ${T}_{B,\mathrm{MSI},3}=H{t}_{A,\mathrm{MSI},3}=630\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ms}$.

Using the MSI procedure shown in Fig. 3(a), the time to perform the calculation according to Eq. (1) is ${t}_{c,2\mathrm{FFT}}=2.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu s}$. This leads to an improvement in the time required: ${t}_{A,\mathrm{MSI},2}=P{t}_{c}=0.64\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ms}$ and ${T}_{B,\mathrm{MSI},2}=H{t}_{A,\mathrm{MSI},2}=128\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ms}$. For the MSI principle implemented according to Fig. 3(b), the time required to produce a single T-scan by calculating Eq. (1) for each CS and the same ${R}_{p}$, where ${U}_{h}$ are acquired for $H=200$ positions of GX, is ${t}_{T}=H{t}_{c}=0.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ms}$; hence, the time for a B-scan image made of $P$ T-scans is ${T}_{T,\mathrm{MSI},2}=PH{t}_{c}=128\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ms}$. These evaluations show that an improvement by a factor of 4.9 is obtained by implementing the algorithm with two FFTs in Fig. 1 instead of three FFTs used in [5]. Even so, the time required by the MSI is much longer than that of the FFT-based technique, i.e., larger than ${T}_{\text{RAM}}$, and therefore cannot be practiced on the fly, while ${T}_{B,\mathrm{FFT},S}<{T}_{\text{RAM}}$, which allows the conventional technique to perform almost simultaneous display with acquisition.

In Fig. 4, we show B-scan images of an infrared (IR) card obtained using the two methods. In Fig. 4(a), we demonstrate images obtained via the conventional FFT method with data resampled, while Fig. 4(b) was produced using the MSI method with no resampled data. If the data were not resampled, all features in Fig. 4(a) would have been extended axially, and the larger their depth position, the wider their stretched appearance, specific for performing FFT of noncalibrated data. As a difference, when the MSI method is used, the images look sharp; the axial extension of features is similar to that in Fig. 4(a) and is not affected by not resampling the data.

In conclusion, we have demonstrated that the MSI method can be used to produce B-scans in Sp-OCT. The MSI method does not require resampling of the interference signal. While the MSI method exhibits similar sensitivity and depth resolution to that provided by the FFT-based technique, at the moment, the production of a B-scan OCT image requires more time than the time needed by the conventional FFT-based method, because calculation of correlations for each mask are evaluated sequentially.

The MSI method opens interesting parallel processing avenues which may allow several hardware- and software-specific methods to be developed in the near future with more research. One such avenue at hand is the utilization of the CUDA parallel computing architecture on GPU cards. In comparison with the conventional FFT-based OCT, only 196 KB extra memory is required in the PC for the masks ($M=512$ points acquired at 12 bits from $P=256$ axial positions). We do not need a lot of memory to store data to be processed either. The amount of data to produce a T-scan is 153 KB, while 39.3 MB is required for a B-scan. To produce the whole volume, 7.8 GB is needed. Larger image sizes may require more memory, but this is within the current capabilities and does not raise the cost of a MS-OCT system.

Let us say that, using the FFT-based OCT, all FFT and interpolation operations are performed in parallel for all H pixels, so ${T}_{B,\mathrm{FFT},S}={t}_{A,\mathrm{FFT},S}=7.4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu s}$. If $H$ processing blocks are used in parallel, a T-scan is obtained using the MSI technique in ${t}_{T}={t}_{c}=2.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu s}$. All such productions of T-scans can also be achieved in parallel for all $P$ depths, in the time for a T-scan, so ${T}_{T,\mathrm{MSI},2}={t}_{c}=2.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu s}$, with the only disadvantage being that $PH=51,200$ processors are needed for the MSI method instead of $H=200$ processors for FFT and interpolation operations for the FFT-based technique. $P=256$ more parallel processing steps may not represent a difficulty for the present capability of FPGAs and GPUs. Fast parallel processing may allow a B-scan on the fly, as is achievable with the current FFT-based OCT method. If all $P=256$ correlators could operate in the time required for a single calculation step ${t}_{c}+{t}_{\mathrm{FFT}}=4.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu s}$, then the MSI method would operate on the fly as well as for the example given; this is less than ${t}_{\text{acq}}=21.28\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu s}$.

The authors acknowledge the support of the European Research Council, advanced grant 249889. A. Podoleanu is also supported by the NIHR-BRC at Moorfields Eye Hospital and UCL Institute of Ophthalmology.

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