Open Access
Add to BibSonomy
Abstract
A common processing approach for optical coherence tomography (OCT) uses a window function (e.g., Hann or rectangular window) for spectral shaping prior to calculating the Fourier transform. Here we build on a multi-window approach [Opt. Express 8, 5267 (2017) [CrossRef] ] that enables improved resolution while still suppressing side-lobe intensity. The shape of the window function defines the trade-off between main-lobe width (resolution) and side-lobe intensity. We have extended the approach to include the interferometric phase for phase-sensitive applications like vibrometry and Doppler OCT. Using the Hann window as a reference, we show that 11 Taylor windows are sufficient to achieve 50% improvement in axial resolution, -31 dB side-lobe intensity, and 20% improvement in phase sensitivity with low computational cost.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical coherence tomography (OCT) has become a widely used tool for clinical and basic scientific research in ophthalmology [1,2], otolaryngology [3], cardiology [4], gastroenterology [4], and more [5]. The ability to non-invasively interrogate tissues with high resolution and speed has supported the growth of biomedical applications for OCT. In 2003, several groups [6–8] recognized that spectral interferometry provided better sensitivity than time-domain interferometry for OCT. Since then, spectral interferometry, sometimes called Fourier-domain or spectral-domain OCT, has become the dominant approach. Thus, a line image (A-line) requires a discrete inverse Fourier transform of the acquired OCT signal as one of the processing steps. A window function is typically applied in the spectral-domain (k-space) prior to the Fourier transform to shape the spectrum and tune the line-spread function in the space-domain (z). The choice of window function is a compromise between main-lobe bandwidth (resolution) and side-lobe intensity.
To illustrate this point, we first simulate the results of two different window functions (rectangular and Hann) on a theoretical single reflector in Fig. 1. We model this theoretical interferometric OCT signal as a cosine wave, H(k) = cos(2kΔz) with additive Gaussian white noise with a standard deviation of 0.75. We then multiply this theoretical OCT signal by a window function, W(k), before applying the inverse fast Fourier transform (iFFT) to the product to compute the line-spread function for both the rectangular (Rect) window and the Hann window. In this simulation, the Rect window results in ∼40% better resolution and ∼20% lower noise compared to the Hann window. However, the Rect window also produces strong side-lobes with the first side-lobe being -13 dB, which is within the dynamic range of most OCT images. The Hann window results in a wider line-spread function (i.e., worse axial resolution) than the Rect window but produces a lower first order side-lobe at -31 dB, which is nearly as low as the noise level in most OCT images. The signal-to-noise ratios (SNRs) are 34.5 dB and 32.8 dB for the Rect and Hann windows, respectively. The first few side-lobes are visible in the Rect trace, but the side-lobes are in the noise for the Hann trace. Ideally, we would like the high resolution and improved noise statistics of the Rect window with the superior side-lobe suppression of the Hann window for OCT imaging.
Fig. 1. Magnitude of the iFFT of simulated signal windowed with a rectangular window (Rect) and Hann window. Simulated signal, H(k) = cos(2kΔz) + nse, where nse is additive Gaussian white noise with a standard deviation of 0.75. Vertical axis is dB SNR relative to the peak. Horizontal axis is Δz in digital frequency, where the Nyquist frequency is normalized to 1.
Recent work by Fraser and coworkers [9] demonstrated a “multi-shaping technique” to achieve high side-lobe suppression while maintaining a narrow point-spread function (PSF). First, they created a matrix of rectangular windows with various widths, W(k,n), where n is the number of window functions. This was multiplied by a square matrix with the spectral interferogram, H(k), along the diagonal, i.e.
The 1-D inverse Fourier transform was taken of WH(k,n) to yield Wh(z,n), i.e.
They showed that this method could suppress the strong side-lobes normally associated with a rectangular window while maintaining the narrow PSF. Johnson, et al. [10] built on this work, using window functions that are naturally parameterized, such as the Kaiser-Bessel window, instead of the rectangular window to achieve similar results. The two papers use “multi-shaping” and “multi-windowing” to describe similar algorithms. The term multi-window is more descriptive of the process, hence going forward we will exclusively use this term.
In our work, we experimented with a range of windows, seeking to minimize width of the PSF while suppressing side-lobes using the minimum number of window functions possible. The former optimizes image resolution and SNR while the latter reduces the computational load of the approach. We also extended the multi-window approach to include the interferometric phase, which underpins many extensions of OCT (e.g., phase-sensitive OCT, Doppler OCT, phase-variance OCT, OCT vibrometry).
2. Methods
2.1 Single reflector
Simulation: All simulations were performed in MATLAB (MathWorks, R2021a). The spectral interferogram was modeled as a cosine function,
where k is wavenumber, Δz is the pathlength difference and ϕ is an arbitrary phase typically set to 0. Noise was simulated as additive Gaussian white noise using the randn keyword and scaling to get the desired signal to noise ratio (SNR).Experiments: Several experiments were done to validate our simulation findings. A commercial Mach-Zehnder interferometer (MZI, Thorlabs INT-MZI-1300) was used to produce a stable spectral interferogram free of fixed pattern noise with a single peak (i.e., an experimental realization of Eq. (1)). We used an Insight Akinetic swept laser source (SLE101) centered at 1310 nm with a 91.1 nm bandwidth, sweeping at 100 kHz. The spectral interferograms were recorded using an AlazarTech A/D converter (ATS9373) operating at 400 MHz and custom software written in python, C, and CudaC, as described previously [11]. Raw spectral interferograms were preprocessed in the following way before using the multi-window algorithm. First, any residual DC offset was removed by subtracting the mean of the signal. Then the instantaneous amplitude (S(k)) was calculated from the magnitude of the Hilbert transform and used to correct the envelope of the signal, making it flat top and normalized. The result for a set of Gaussian windows and the Hann window are shown in Fig. 2. This is an average of 2000 spectral interferograms, hence the high SNR.
Fig. 2. a) Set of 8 Gaussian windows, W(k,n), compared to the Hann window. Each was individually normalized so that their integral was 1. b) Signal from a reflector located at 0.5 for each Gaussian window in (a) and the computed rank-order minimum, $\textrm{mi}{\textrm{n}_n}\{{|{\textrm{Wh}({\textrm{z},\textrm{n}} )} |} \}$. Amplitude is in dB relative to the peak of the main lobe. c) Comparison of the computed rank-order minimum to results generated with a Hann and Rectangular window. The green line indicates the -30 dB level set as the criterion for the strongest side-lobe.
2.2 Two reflector and vibrometry
Simulation: Two reflectors were modeled as a sum of cosines, i.e. $H(k )= {H_1}(k )+ {H_2}(k )$ where the component ${H_n}(k ) $ follow Eq. (1). Noise was generated in the same way as for the single reflector. For modeling periodic vibrations, the phase modulation Δϕ, was modeled as a sine, i.e.
where ${\omega _v}$ is the angular frequency of the vibration, t is time, and $\varphi $ is the phase of the vibration.Experiment: The two-reflector experiment was performed by imaging a proximal pellicle beam splitter (PBS) and a distal mirror that was attached to a vibrating piezo, with a custom-built Mach-Zehnder interferometer. The proximal PBS was attached to an electromechanical linear translation stage (Zaber) to provide consistent and accurate movement in 3.81 µm increments. The negative frequency (-z) peak from the PBS overlapped with the positive frequency (+z) peak from the mirror to allow a close-proximity investigation of the two peaks without requiring the two samples to be physically touching. Once the peaks overlapped, a single-point M-scan was taken of the samples. The negative frequency peak of the PBS was then moved away from the mirror peak incrementally, and the M-scan protocol was repeated at each distance until the peaks were separated by the noise floor.
2.3 Flow phantom angiography
To observe the effect of multi-windowing on phase-resolved measurements, we used Doppler OCT (D-OCT) to measure the flow rate of a scattering solution and compared it with the known flow rate. We constructed a flow phantom using transparent tubing (Tygon S3, 0.89 mm ID) connected to a disposable syringe (3 mL, BD) via an 18-gauge needle. The syringe was filled with a light-scattering solution (100 mg/mL Nestlé Coffee-Mate Original dissolved in water) and driven using a syringe pump (Harvard Apparatus Model 11 Plus) at known flow rates through the tubing. We fixed the flow phantom angle using a goniometer and recorded a volumetric OCT image of the tubing to precisely determine the Doppler angle (73.4°). The volumetric flow rate from the pump (0-500 µl/min) was divided by the nominal cross-sectional area of the tubing (0.490 mm2) to calculate the average flow velocity in the tubing.
We recorded the motion of scatterers in the flow phantom using a BM scan. In this scanning mode, the laser beam momentarily pauses at each position, recording 2000 consecutive spectral interferograms (an M-scan) before moving to the next X-position in space (B scan). Thus, we record a cross-sectional image of the flow phantom and sample the axial motion of scatterers inside the flow phantom at the laser sweep frequency (100 kHz).
After Hann or multi-window processing, we extracted and unwrapped the OCT phase and used it to quantify the flow velocity (${V_{flow}}$) at each pixel using the following equation [12,13]:
We also computed the phase variance (PV) map [14,15], using each set of windows (see Table 1), as another way of obtaining OCT angiography. We used the contrast-to-noise ratio (CNR) to observe if potential improvements in phase noise due to multi-window processing could impact the CNR of these phase-based angiography methods.
We computed the CNR for both D-OCT and PV using the mean flow signal (${\textrm{S}_{\textrm{ROI}}}$) in a small ROI at the center of the flow phantom. The background signal (${\textrm{S}_{\textrm{BG}}}$) and noise ($\mathrm{\sigma }$) were calculated from a non-flowing region that still had OCT signal amplitude. In this non-flowing region, ${\textrm{S}_{\textrm{BG}}}$ was calculated as the mean flow signal, and $\mathrm{\sigma }$ was calculated as the standard deviation of the flow signal.
Table 1. Multi-window Parameters and Results on Single Reflectora
2.4 In-vivo OCT imaging
To test the multi-windowing functions for improved axial resolution and SNR, we used OCT images of the cochlea of an anesthetized CBA/CaJ mouse. This mouse strain is often used in auditory research [16]. After anesthetization, the mouse was surgically prepared using a previously reported protocol that enabled direct OCT imaging of the intact in-vivo cochlea [17]. All animal procedures were performed under a protocol approved by the University of Southern California Institutional Animal Care and Use Committee.
To ensure the images processed with the Hann and various multi-window sets were presented on the same scale, we normalized each of the output OCT images so that the background levels were set to 0 dB and the original dynamic range of the images was preserved. We did this by first determining the mean background signal in a region far from potential sidelobes and dividing the image by that value, SBG. We then took the logarithm of that quotient to compute the normalized signal, which is relative to the background in dB. We calculated this as:
where S and ${S_{BG}}$ are the original OCT image and its mean background signal in an empty region of the image, respectively.3. Results and discussion
Our criteria for finding an optimal set of windows were as follows: 1) side-lobe suppression of at least -30 dB, like the Hann window (-31 dB), and 2) main-lobe bandwidth less than or equal to the rectangular window. These were to be achieved with a minimal set of windows to minimize computational time. Given the large parameter space, we settled on the following approach. For each window function, we created a set of windows where the widest bandwidth (in k) had the same full-width half maximum (FWHM) as a Hann window and the narrowest bandwidth approximated the full rectangular window. The windows between those extremes were generated by varying the parameter that controls bandwidth for the window function using either linear or log spacing depending on the window type. This was done using the single reflector model (Eq. (4)).
3.1 Single reflector
The relevant parameter and parameter space are listed with the windows in Table 1. The Taylor window has two parameters: the number of nearly constant-level sidelobes adjacent to the main-lobe (NBAR) and the maximum side-lobe level relative to the main-lobe peak (SLL). We found that the results varied with SLL, hence we only show results for the MATLAB default, SLL = 4 and a larger value, SLL = 10. The larger SLL gave superior resolution but at the cost of additional windows (i.e., computation time).
A set of 8 Gaussian windows are shown in Fig. 2 as an example of typical results. Panel (a) shows the Hann window (black) for reference along with the Gaussian windows (cyan).
Each window has been normalized such that the integral of the window is 1. Panel (b) shows the magnitude of the iFFT of a signal with Δz = 0.5 on a scale where the Nyquist Δz is equal to 1. The result from each of the 8 windows is shown in cyan, with the rank-order minimum filter shown in blue. The green horizontal line in panel (c) indicates -30 dB. The maximum side-lobe was -30.7 dB. The FWHM of the PSF was identical to the Rect window at 0.6 of the Hann window, i.e. the resolution was 40% better than with the Hann window. The results from this simulation, along with those for the other window sets tested are provided in Table 1. The first column indicates the window type and keyword in MATLAB. The second column describes the lower and upper bounds of the parameter space investigated. Within the parameter space, the values were spaced linearly or logarithmically. Both types of spacing were tested on all window types with linear working better for all except the Taylor window where log spacing was more efficient. The number of windows required to achieve at least -30 dB side-lobes is shown in column 3 along with the maximum side-lobe intensity in column 5 (labeled sim). The Chebyshev window is the only window with which we were unable to achieve -30 dB side-lobe suppression. This prompted us to retry the simulation with many windows (N). Column 4 (labeled sim100) shows the result from using 100 windows. The Gaussian, Rectangular, and Chebyshev only show a small improvement (< 1 dB) in side-lobe suppression. The Kaiser-Bessel window shows a modest 5 dB improvement. The Taylor and Tukey windows show much larger improvement, 13-16 dB and 20 dB, respectively. Finally, column 8 is the ratio of the FWHM resolution relative to the Hann window. The Gaussian, Rectangular, and Kaiser-Bessel window sets achieve the resolution of a single rectangular window, 0.60. The related Chebyshev and Taylor windows show improved resolution in the 0.50-0.54 range.
The columns labeled exp in Table 1, contain results from experimental measurements using a commercial Mach-Zehnder interferometer (MZI), as described in the methods. The spectral interferograms were processed and then averaged to provide a high SNR A-line where the side-lobe intensity and main-lobe bandwidth could easily be measured.
Experimental results for each window set are shown in Supplement 1 Fig. S1. The Hann window result has been plotted along with the multi-window result for easy visual comparison. The green line represents the -30 dB threshold. In Table 1, the maximum side-lobe intensity is shown in column 6, next to the simulation result. Both the simulated and experimental resolutions are given in column 8 relative to the Hann window. The experimental maximum side-lobe intensity agrees with the simulation to within 2 dB. The only exception is the Chebyshev multi-window, which differs by 8.5 dB from simulation. Similarly, the relative resolution agrees with the simulation for all but the Chebyshev window. As can be seen in Fig. S1(e), the main lobe of the PSF from the Chebyshev window has unresolved side-lobes on each side at -8.5 dB and -11 dB on the low and high frequency side. All the other windows have PSF main lobes that smoothly drop-off on either side to at least -30 dB. While the FWHM of the Chebyshev multi-window is close to the simulation, the PSF gets significantly broader below that point due to the shoulders. This can be seen in comparison to the Tukey window in Fig. S1(d) which has poorer resolution at -3 dB than the Chebyshev but is much narrower if we consider the width of the PSF between -3 and -30 dB. Nominally, we consider there to be good agreement between simulation and experiment for each multi-window set except the Chebyshev window set.
The Gaussian window set required the fewest windows (N = 8), to achieve -30 dB side-lobes and resolution 40% better than the Hann window. Using the Taylor window set with 11 windows and SLL = 10, we were able to achieve similar side-lobe suppression but with improved resolution, 50% better than the Hann window, i.e. a factor of two. We explored increasing the SLL to further improve the resolution and found the limit to be 0.48, or 52% improvement over the Hann window. As can be seen in Figure S1, every window set except Tukey has poorer side-lobe suppression further from the main-lobe compared to the Hann window. The Tukey set (supplemental Fig. S1(d)) achieves similar side-lobe suppression as the Hann window over a wide range of Δz. Stronger suppression of side-lobes could be useful when there is a strong reflector in the image.
We also considered the effect of multi-windowing on the noise statistics of the above simulated and experimental M-scan data. We added Gaussian white noise to the simulated data to get similar SNR as was measured in the single-reflector MZI experiment. The SNR was calculated by taking the ratio of the mean and standard deviation at the peak of the main lobe, for each of the 2000 A-lines in the M-scan data. Expressed in dB, the average SNR from multiple trials for the simulation and experiment using a Hann window was 74 dB and 79 dB, respectively. The second two columns of Table 2 show the SNR relative to the Hann window for each set of windows investigated. This is expressed as the difference in multi-window SNR and Hann window SNR, i.e. ΔSNR = SNRMW – SNRHann. Generally, we observed good agreement between simulation and experiment. As noted in the introduction, we expect better SNR with the rectangular window compared to the Hann and this is born out here with a 4.79 dB and 4.39 dB improvement shown for the simulation and experiment, respectively. The Tukey window set also shows a significant, ∼2 dB improvement. The other window sets are within 1 dB of the Hann window except for the Chebyshev set, where we see a SNR reduction of 3.39 and 3.96 dB for simulation and experiment, respectively.
Table 2. Experimental and simulation results relative to the Hann windowa
The mean and standard deviation of the phase noise in the frequency domain was calculated using the entire frequency range, 0 to 50 kHz. For the Hann window the mean (µ) ± standard deviation (σ) was 15 ± 8.0 µrad and 18 ± 9.4 µrad for simulation and experiment, respectively. We suspected that using a rank-order minimum (min) filter on the magnitude signal to reduce side-lobes may result in inaccurate phase estimation. Therefore, we also considered the use of rank-order mean and median filtering to see if they result in lower phase noise. To summarize, the multi-window phase was calculated using three methods: rank-order min, mean, and median filtering. The rank-order min filter is described as:
Generally, the median gave the best results with an improvement over that Hann window of ∼10-20% that was consistent between simulation and experiment. The mean gave similar although typically slightly worse results than median. The results using the min filter were worst and sometimes dramatically different than the median and mean. For instance, the rectangular window set had 0.80 ± 0.80 and 0.85 ± 0.84 for the median and mean, respectively, but 6.54 ± 6.00 for the min. All the window sets except the Chebyshev set, provided better phase noise than the Hann window using the median or mean.
3.2 Two-reflector and vibrometry
In the Single Reflector case described above we reported simulation and experimental findings side-by-side because the simulation conditions were very similar to what we were able to achieve experimentally. In the case of two-reflectors, we were not able to achieve this same level of agreement between simulation and experimental conditions. Thus, we first cover the simulation results, then the experimental results.
Simulation: In the two-reflector simulation, we varied the distance between the reflectors to quantify contrast and looked for phase leakage between the two peaks in the A-line. The two reflectors had equal intensity with one vibrating sinusoidally at 10 nm amplitude while the other was stationary. Figure 3 shows representative data from the Gaussian and Taylor 2 window sets. A complete set for all investigated windows is provided in the Supplement 1 Fig. S2 and S3. The contrast was quantified by calculating one minus the ratio of the center point between the peaks and the peak amplitude. When the peaks are overlapped, this ratio is 1 and the contrast is therefore 0. At the opposite limit when the peaks are well separated, this ratio will tend toward 0 and the contrast is therefore 1. The top row in Fig. 3 is the contrast as a function of peak separation. The Hann result (black) is plotted with the multi-window result (blue) where the vertical dotted lines indicate the FWHM from the single reflector simulations (Table 1). As in the plots, the contrast does not monotonically increase with peak separation. There are local max and min due to overlap of side-lobes from the two peaks as the contrast converges toward 1 at large reflector separation. The first local maximum is at 5 × 10−4 where the contrast reaches ∼75% for all windows (Fig. S2). This drops back to 0 for the Hann window but stays well above 0 for all multi-window results. For the Hann window this local maximum comes at exactly half of the FWHM resolution (black vertical line) reported in Table 1. For the multi-window results this local maximum is close to the FWHM resolution (blue vertical line) but only coinciding exactly for the Taylor 2 set of windows (Fig. 3(b)).
Fig. 3. a) Multi-window Gaussian results. b) Multi-window Taylor 2 results. Top row) contrast as a function of separation between equal reflectors. Bottom rows) absolute error of the vibrational amplitude as a function of the separation between two equal amplitude reflectors. Peak 1 was stationary while peak 2 vibrated with 10 nm amplitude.
We summarized the contrast results in Table 3 by recording the peak separation relative to the Hann window at three contrast levels, 50%, 25%, and 5%. We picked the points on the curve where the indicated contrast level never falls below that level. These are marked by asterisks in Fig. 3 (a) and (b). For the multi-window sets, since the contrast never falls below 25% after the first local maximum at 5 × 10−4 peak separation, the 5% and 25% contrast points are located on the rising edge of this first maximum for all windows. For all windows except the Taylor sets, the 50% contrast level is on the rising edge of the second maximum. For the Hann window all three contrast levels fall on the rising edge of the second maximum.
Table 3. Contrast relative to peak separationa
Because of the local maximum at 5 × 10−4 peak separation, all multi-window sets perform significantly better than the Hann window with improvement of 64-75% at 5% and 25% contrast levels. This improvement persists at the 50% contrast level for both Taylor sets of windows with an improvement of 60% and 65%. For the other windows, the improvement drops back down to 40% as predicted by the FWHM measurements in Table 1. The Chebyshev window set, again, is somewhat of an outlier. It performs the best at the 5% and 25% contrast levels but worse than predicted by the FWHM at the 50% contrast level.
Next, we measured the phase leakage between the two peaks. We monitored the vibrational amplitude as a function of peak separation, using the min, median, and mean of the phase. One reflector was stationary while the other vibrated with an amplitude of 10 nm. We plotted the absolute error versus peak separation results for the Gaussian and Taylor 2 window sets in Fig. 3. The same measurements for all investigated window sets are in Supplement 1 Fig. S3.
For each windowing simulation, the absolute error in the vibration measurement was ∼5 nm when the peaks were perfectly overlapped. For the Hann window the vibrational error dropped to between ∼1-3 nm as the two peaks began to separate (more contrast). The error rose back to 4-5 nm as the side-lobes overlapped perfectly and the contrast dropped to 0. Continued separation of the reflectors (high contrast between peaks) resulted in low absolute error in the vibration measurement, which approached a noise floor of ∼0.5 nm. The vertical dashed black line represents the 5% contrast peak separation for the Hann window. This threshold is a good indicator for when the absolute error has dropped to the noise floor indicating low phase-leakage between the two peaks.
The multi-window results initially showed similar behavior when the two reflector peaks overlapped. The error dropped rapidly to the noise floor as the contrast reached its first maximum. Further peak separation showed some minor increases in absolute error (due to overlapping side-lobes) before the contrast reached its second maximum. The vertical blue dashed line marks the 5% contrast point for the multi-window result and again appears to be a good indicator for when the absolute error has dropped to the noise floor indicating no or weak phase-leakage between the two peaks.
The median and mean perform alike in terms of the absolute error, however the min is worse with larger error between the first two maxima in the contrast plot. The Gaussian, Kaiser-Bessel and both Taylor sets of windows perform similarly well with little increase in the error between the first and second maxima of the contrast. The Rectangular and Chebyshev window sets performed the worst with relatively larger oscillations in the error between the first two maxima of the contrast. Again, the Chebyshev window set is somewhat of an outlier with very large errors, especially in the min trace. Note the vertical scale on the Chebyshev results in Supplement 1 Fig. 3 is twice that of all others to accommodate these large errors.
These contrast results provide further evidence that the multi-window approach provided better resolution for OCT vibrometry. The phase noise was shown to be lower when using the median or mean of the phase along the window dimension. These two features combine to provide lower phase leakage, such that correct amplitude of a vibrating reflector could be reported with significantly smaller peak separations than with the Hann window. Our simulation shows that the 5% contrast level is a reliable threshold for low phase leakage.
Experiment: As detailed in the methods, the experimental system was comprised of a piezo actuator and a pellicle beam splitter. The negative frequency (-z) image of the pellicle beam splitter was made to overlap with the image of the piezo. The separation was then increased in 3.81 µm steps.
First, we considered the contrast between the peaks. The pellicle beamsplitter appears as a doublet (front and back surface). For contrast calculations, we only considered the strongest peak that was made to overlap with the single peak due to the piezo. The results are shown in Fig. 4(a), where peak 1 was due to the piezo and peak 2 was due to the pellicle beamsplitter. Results for the single Hann window, and Gaussian, and Taylor 2 window sets are shown. As we would expect based on simulation, the multi-window approach shows better contrast results than the Hann window. The Gaussian and Taylor 2 windows gave nearly identical results.
Fig. 4. Experimental results measured on phantom with two reflectors. a) Contrast for two peaks. b) Absolute error of vibrational amplitude.
We also measured the vibrational response from both peaks. While the pellicle beam splitter was not actively driven in any way, we could not avoid mechanical coupling to the vibrating piezo. Therefore, unlike the simulation, both peaks are vibrating. The vibrational amplitude of the piezo and pellicle beam splitter tended toward 6.3 nm and 4.1 nm at large separations. We used these values as the true vibrational amplitudes to calculate the absolute error as a function of peak separation, plotted in Fig. 4(b). As in the simulation, we observed that the multi-window approach more quickly approaches the true vibrational amplitude as a function of peak separation than the Hann window. Similarly, the absolute error in vibrational amplitude closely follows the contrast (i.e., the absolute error drops rapidly as the contrast rises above 0).
The experimental trends we observed were like the simulation results. However, we were not able to experimentally differentiate between the Gaussian and Taylor 2 windows since the results were essentially identical. This is likely a consequence of the challenging nature of these experiments. While nearby and overlapping vibrating structures are common in our work [11], it was difficult to come up with a sample where we could reliably vary the distance between two reflectors starting from a sub-resolution separation. We tried several approaches and settled on this one because it gave the most flexibility for varying peak separation. We used the pellicle beamsplitter because the thin membrane introduces a relatively small amount of dispersion. Despite our efforts to minimize residual dispersion in the interferometer, all peaks were broadened, with the pellicle beamsplitter peaks more strongly broadened. The fact that we were overlapping the positive image (+z) of the piezo with the negative image (-z) of the pellicle beamsplitter precluded the use of our typical numerical correction for dispersion. Accordingly, we had competition between dispersion induced broadening and the line width due to the window. The system was good enough to clearly distinguish between the Hann window and the multi-window approaches where the expected FWHM difference is 40-50%. However, we were not able to differentiate between the two multi-window approaches where the expected FWHM difference is only 10%.
3.3 Angiography
Flow phantom: We tested the various multi-window algorithms for angiography using Doppler OCT (D-OCT) measurements of flow in a phantom (Fig. 5(a)). Using the different window sets, we found that they did not appreciably change the flow rates measured by Doppler OCT. (Figure 5(b)). D-OCT flow rates calculated using multi-window spectral shaping deviated from that of Hann windowing by between 0.7% to 7.1% (for Gauss, rectangular, Kaiser-Bessel, Tukey, Taylor 1, and Taylor 2 multi-window sets) (Fig. 5(c)). However, the Chebyshev multi-window set resulted in a much larger 15.2% deviation in flow rates measured by D-OCT. The averaged B-scans (n = 500 frames) showed that each multi-window set resulted in a more finely resolved speckle pattern compared to Hann windowing (Fig. 5(d)–5(g)), which was consistent with the single-reflector simulation and experiment.
Fig. 5. Phase-resolved Doppler OCT flow measurements. (a) Representative Doppler OCT (D-OCT) cross section of a flow phantom using Hann windowing. (b) Parabolic fits of flow velocity measured using D-OCT. Baseline values for each trace are offset by 5 mm/s from one another for clarity. c) Linear fits of measured D-OCT flow velocities vs known syringe pump flow velocities. d-g) Cross-sectional OCT B-scans of the flow phantom processed using Hann, rectangular, Gaussian, and Taylor 2 multi-windowing (averaged, n = 2000).
Many OCT angiography algorithms, including Doppler OCT and phase variance (PV), rely on the OCT interferometric phase. As a practical test of the effects of multi-windowing on the OCT phase, we computed the contrast-to-noise ratio (CNR) for Doppler OCT and phase variance for each flow rate. Welch’s one-way analysis of variance (tolerant of unequal variances) showed no significant differences in Doppler CNR or phase variance CNR across the different windowing methods for all flow rates investigated (0-17.0 mm/s). This suggested that multi-window spectral shaping does not significantly change the result of phase-based OCT angiography methods.
3.4 In vivo and phantom imaging
We then applied multi-window processing to the OCT images of a mouse cochlea to see if the improved axial resolution and SNR could enhance the small features in this biological sample. After Hann or multi-window processing, we normalized each OCT B-scan image to its background level (mean value of an empty region in the top left corner) to show each image relative to its background level (Fig. 6). Except for the Chebyshev multi-window set (36.8 dB), all other multi-window sets increased the image dynamic range compared to the Hann window (46.8 dB). The Gaussian and Taylor 2 multi-window sets provided a modest increase (+0.8 and +0.2 dB, respectively) in dynamic range over the Hann window. The images processed using the Taylor 1, Kaiser, Tukey, and Rect multi-window sets (Fig. S5) showed additional gains in dynamic range (+1, + 1.1, + 2.2, and +4.1 dB). The trend of increased dynamic range for the multi-window sets nearly matched the improvements in SNR found in the single-reflector MZI results in Table 2.
Fig. 6. Averaged (n = 500) OCT B-scans of a mouse cochlea in vivo. Images were generated from the same dataset using (a) Hann window, (b) Gaussian multi-window, and (c) Taylor 2 multi-window processing. Black arrows point to the boundary between the otic capsule and stria vascularis. Red arrows point to Reissner’s membrane. Both features appear to benefit from the improved axial resolution of b) and c). Images were normalized relative to their mean background levels.
A notable feature in these cochlea images that demonstrates the improved axial resolution of multi-window processing was a boundary layer between the stria vascularis and the otic capsule (white arrows). This boundary was relatively thick for the Hann window (Fig. 6(a)) compared to the Gaussian and Taylor 2 multi-window sets (Fig. 6(b),(c)). The apparent thickness of Reissner’s membrane (green arrows) in the left cochlear duct was thinner for Gaussian and thinnest for Taylor 2 multi-window processing, which was consistent with the single-reflector results in Table 1. Overall, we were pleased that the improved axial resolution and SNR from multi-window processing provided noticeable benefits for in-vivo images of the mouse cochlea.
As a second, more controlled test on a scattering phantom, we imaged a stack of tape to visualize the expected resolution improvement provided by the multi-window approach. Here, we compare an image where the full spectrum was processed using a single Hann window with images where only the middle half of the spectrum was processed using a single Hann window, Taylor 2 and Tukey multi-window sets. The results are shown in Fig. 7. In Fig. 7(a), full-spectrum Hann, we can clearly see many of the layers within the tape stack. As expected, when we reduce the spectrum by half (Fig. 7(b)), the resolution is reduced, and the tape layers are harder to discern. In principle, using the Taylor 2 set of windows, we should be able to achieve twice the resolution as in Fig. 8(b) and similar resolution to the full-spectrum Hann image (Fig. 7(a)). The surface peak in Figs. 7(b)-(c) show a FWHM of 19.6, 37.6 and 18.8 µm, respectively (see Fig S7). As expected, the resolution is improved by a factor of 2 by using the Taylor 2 set of windows, i.e. the resolution was improved from 37.6 µm to 18.8 µm. The resolution for the Taylor 2 window set is slightly better than full-spectrum Hann, likely because of better performance of the preprocessing steps, e.g. background subtraction and dispersion compensation, through the narrower region of the spectrum. Even with the improved resolution of the Taylor 2 window set, it is not as easy to discern deep layers of tape in Fig. 7(c) as in Fig. 7(a). That may be due to lower speckle density in the image from using only half of the spectrum. It could also be the larger side-lobe intensity from the strong surface reflection in Fig. 7(c) reducing contrast for the deeper tape layers. To test this, we reprocessed the half-spectrum data using the Tukey set of windows because it has side-lobe suppression similar to that of the Hann window (See Fig S1d). The Taylor set of windows produced a surface peak with FWHM of 22.7 µm, very close to the expected 22.6 µm, predicted using the last column of Table 1. The image in Fig. 7(d) indeed shows better side-lobe suppression from the surface reflection and qualitatively better contrast between deeper layers of tape than Fig. 7(c), although the lower speckle density is still apparent when compared to Fig. 7(a).
Fig. 7. a) Image of tape stack using the full-spectrum and a Hann window. b-d) Same image using only half of the spectrum with a Hann window (b), multi-window Taylor 2 (c), and multi-window Tukey (d). Scale bar is 100 µm.
3.5 Discussion
Our experimental data agree with our simulations, which predict 40-50% better axial resolution while maintaining equivalent sidelobe suppression with the Hann window using only 8-10 windows. We also explored using mixed sets of windows, e.g. Gaussian and Taylor windows, but were not able to improve over the results using a single window type. The clear disadvantage to the multi-window approach is that it requires additional FFTs equal to the number of windows used. However, given the efficiency of FFTs on GPUs and their continuous performance improvement, it seems likely that the additional computational load would not significantly impact performance, even for real-time imaging applications.
In general, we saw small improvements in phase noise statistics while providing similar performance in quantitative phase measurements like Doppler flow. The small improvement in frequency domain phase noise found in the single reflector experiments, 10-20% compared to using the Hann window, did not translate to improvements in contrast-to-noise ratio in the phase variance angiography. It is not clear why, given the clear relationship. Our first thought was that maybe the improvements only hold for high SNR (74 dB) like in the single reflector experiments. We reran some of the simulations for 15 and 35 dB SNR reflectors and found that the 10-20% improvement persisted. It seems most likely that the flow statistics are larger than the improvements in phase noise we get from using multi-window spectral shaping. In other words, if we wanted to measure this relatively small effect, we would need to increase the number of trials. Practically, while the improved phase noise statistics would improve vibrometry, we did not see any improvement for angiography.
The most important benefit of this approach is the factor-of-two improvement in axial resolution. This could be used simply to improve image resolution for existing OCT systems. However, we could also use this approach to decrease the cost of subsequent OCT devices. One example would be to use multi-window spectral shaping to compensate for the lower resolution of a narrower bandwidth (less expensive) source. This could achieve the same axial resolution as an OCT system with an expensive broadband source that only uses standard single-window spectral shaping. Using the multi-window approach in this manner could reduce the cost of an OCT system and likely increase imaging speed in swept-source applications.
It would also be possible to compensate for some or all of the resolution loss with split spectrum techniques, for instance, reducing speckle using the dual window method (DW) [18] while maintaining the axial resolution of the system. Because speckle is decorrelated across the interferometric spectrum, half of the interferometric spectrum could be used to remove the decorrelated speckle by averaging it with the other half of the spectrum. Used in conjunction with split-spectrum amplitude-decorrelation angiography (SSADA) [19], the axial resolution of the OCT image could be partially recovered by using the multi-window approach to process each of the spectral “pieces”. Alternatively, the multi-window approach could be used with SSADA to increase the speckle decorrelation by splitting of the spectrum into more “pieces” at shorter bandwidths while maintaining a higher axial resolution than single-window spectral shaping. This could produce better contrast of angiography images while also delivering similar axial resolution to current SSADA methods that use fewer, wider-bandwidth pieces of the spectrum with single-window spectral shaping.
4. Conclusions
We have minimized the computational load required for a multi-window approach to processing OCT data. The approach provides a two-fold improvement in axial resolution while maintaining the -30 dB side-lobe suppression of a Hann window. We extended the algorithm to include the interferometric phase and demonstrated small improvements in the phase noise statistics and reduced phase leakage. Furthermore, we showed that this multi-window approach does not impact quantitative phase-based measurements such as Doppler OCT.
Funding
National Institute on Deafness and Other Communication Disorders (R01DC014450, R01DC017741); National Institute of Biomedical Imaging and Bioengineering (R01EB027113).
Acknowledgements
We gratefully acknowledge Scott Fraser for valuable discussions during the course of the research reported here.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results in this work are not available publicly but are available from the authors upon request.
Supplemental document
See Supplement 1 for supporting content.
References
1. J. Fujimoto and E. Swanson, “The development, commercialization, and impact of optical coherence tomography,” Invest. Ophthalmol. Visual Sci. 57(9), OCT1–OCT13 (2016). [CrossRef]
2. A. H. Kashani, C. L. Chen, J. K. Gahm, F. Zheng, G. M. Richter, P. J. Rosenfeld, Y. Shi, and R. K. Wang, “Optical coherence tomography angiography: A comprehensive review of current methods and clinical applications,” Prog. Retinal Eye Res. 60, 66–100 (2017). [CrossRef]
3. T. J. Matthews and R. Adamson, “Optical coherence tomography: current and future clinical applications in otology,” Curr. Opin. Otolaryngol. Head Neck Surg. 28(5), 296–301 (2020). [CrossRef]
4. T. H. Tsai, C. L. Leggett, A. J. Trindade, A. Sethi, A. F. Swager, V. Joshi, J. J. Bergman, H. Mashimo, N. S. Nishioka, and E. Namati, “Optical coherence tomography in gastroenterology: a review and future outlook,” J. Biomed. Opt. 22(12), 1–17 (2017). [CrossRef]
5. A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12(5), 051403 (2007). [CrossRef]
6. M. A. Choma, M. V. Sarunic, C. H. Yang, and J. A. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11(18), 2183–2189 (2003). [CrossRef]
7. J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. 28(21), 2067–2069 (2003). [CrossRef]
8. R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of Fourier domain vs. time domain optical coherence tomography,” Opt. Express 11(8), 889–894 (2003). [CrossRef]
9. Y. Chen, J. Fingler, and S. E. Fraser, “Multi-shaping technique reduces sidelobe magnitude in optical coherence tomography,” Biomed. Opt. Express 8(11), 5267–5281 (2017). [CrossRef]
10. B. Johnson, T. Ford, S. Woo, W. Atia, J. Jabbour, N. Kemp, M. Arain, T. Schmoll, R. Williams, and P. Whitney, Achieving the Ideal Point Spread in Swept Source OCT (SPIE, 2020).
11. H. Y. Lee, P. D. Raphael, J. Park, A. K. Ellerbee, B. E. Applegate, and J. S. Oghalai, “Noninvasive in vivo imaging reveals differences between tectorial membrane and basilar membrane traveling waves in the mouse cochlea,” Proc. Natl. Acad. Sci. U. S. A. 112(10), 3128–3133 (2015). [CrossRef]
12. X. J. Wang, T. E. Milner, and J. S. Nelson, “Characterization of fluid flow velocity by optical Doppler tomography,” Opt. Lett. 20(11), 1337–1339 (1995). [CrossRef]
13. Z. Chen, T. E. Milner, S. Srinivas, X. Wang, A. Malekafzali, M. J. van Gemert, and J. S. Nelson, “Noninvasive imaging of in vivo blood flow velocity using optical Doppler tomography,” Opt. Lett. 22(14), 1119–1121 (1997). [CrossRef]
14. J. Fingler, D. Schwartz, C. Yang, and S. E. Fraser, “Mobility and transverse flow visualization using phase variance contrast with spectral domain optical coherence tomography,” Opt. Express 15(20), 12636–12653 (2007). [CrossRef]
15. A. Zhang, Q. Zhang, C. L. Chen, and R. K. Wang, “Methods and algorithms for optical coherence tomography-based angiography: a review and comparison,” J. Biomed. Opt. 20(10), 100901 (2015). [CrossRef]
16. K. K. Ohlemiller, S. M. Jones, and K. R. Johnson, “Application of mouse models to research in hearing and balance,” J. Assoc. Res. Otolaryngol. 17(6), 493–523 (2016). [CrossRef]
17. Z. Jawadi, B. E. Applegate, and J. S. Oghalai, “Optical coherence tomography to measure sound-induced motions within the mouse organ of corti in vivo,” Methods Mol. Biol. 1427, 449–462 (2016). [CrossRef]
18. Y. Zhao, K. K. Chu, W. J. Eldridge, E. T. Jelly, M. Crose, and A. Wax, “Real-time speckle reduction in optical coherence tomography using the dual window method,” Biomed. Opt. Express 9(2), 616–622 (2018). [CrossRef]
19. Y. Jia, O. Tan, J. Tokayer, B. Potsaid, Y. Wang, J. J. Liu, M. F. Kraus, H. Subhash, J. G. Fujimoto, J. Hornegger, and D. Huang, “Split-spectrum amplitude-decorrelation angiography with optical coherence tomography,” Opt. Express 20(4), 4710–4725 (2012). [CrossRef]
