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Abstract
Optical coherence tomography (OCT) is a powerful imaging technique that is capable of imaging cross-sectional structures with micrometer resolution. After combining with phase-sensitive detection, it can sense small changes in the physical quantities inside an object. In OCT, axial resolution is generally improved by expanding the bandwidth of the light source. However, when the bandwidth is expanded discontinuously, the wavelength gap induces abnormal sidelobes when estimating OCT signals in the depth domain. This problem can lead to poor axial resolution. Herein, we present a method based on a real-valued iterative adaptive approach (RIAA) to achieve a high axial resolution under a discontinuous bandwidth condition. The method uses a weighted matrix to suppress the abnormal sidelobes caused by the wavelength gap and, therefore, can realize high-resolution measurements. A single-reflector OCT spectrum was first measured for validation, and its amplitude in the depth domain was estimated using different methods. The results indicate that the RIAA had the best capability of suppressing abnormal sidelobes, thereby achieving a high axial resolution. In addition, cross-sectional images and phase-difference maps of three different samples were measured. A comparison of the results validated the practical value of this method.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical coherence tomography (OCT) is a volumetric imaging technique that is capable of noninvasively visualizing microscopic cross-sectional structures inside transparent and semitransparent objects [1,2]. Owing to its micrometer resolution and millimeter range, OCT has been widely used in medical applications and has served as an “optical biopsy” for disease diagnosis [3]. In particular, the introduction of Fourier-domain OCT, with its superior signal-to-noise ratio (SNR) and high speed, has broadened its applications [4,5]. In addition to medical applications, OCT has gained attention from many other fields, including material science [6], artwork examination [7], and lens manufacturing [8], and it has been employed as an effective tool for nondestructive testing. Recently, it has been discovered that the phase information of OCT signals can deliver small changes in physical quantities [9]. Therefore, the OCT phase can be used to obtain functional information. Specifically, phase-sensitive OCT (PhS-OCT) methods can reveal the blood flow velocity fields of blood vessels [10,11], map elastic fields within tissues [12,13], and visualize curing behaviors inside polymers [14,15]. It has already been shown that PhS-OCT has become a hot topic in OCT studies.
Axial imaging of OCT is computed through Fourier transform (FT), which is different from lateral imaging through a lens. Consequently, the axial resolution is restricted by the mechanism of the FT. Specifically, the axial resolution is inversely proportional to the main lobe width of the OCT signal in the depth domain [16], as shown in Figs. 1(a)–(b). The broadening of the width of the main lobe arises from limited bandwidth. After determining the restriction of the axial resolution, a series of methods for enlarging the bandwidth of OCT have been introduced to improve its axial resolution. For example, Barrick et al. used a supercontinuum light source to realize a 2 µm axial resolution [17]. Graf et al. achieved a 1.22 µm axial resolution with the assistance of an ultrabroad bandwidth thermal light source [18]. However, owing to hardware limitations such as the mode hops of semiconductor lasers or the restricted bandwidth of the amplified spontaneous emission spectrum from semiconductor optical amplifiers [19], a large bandwidth with a continuous distribution cannot be acquired in some specific situations. Thus, only OCT spectrum under the condition of discontinuous bandwidth, as shown in Fig. 1(c), can be acquired to retrieve high-resolution axial information. However, estimating OCT signals in the depth domain under a discontinuous bandwidth condition result in abnormal sidelobes. These abnormal sidelobes can cause severe spectral crosstalk, which limits the realization of high axial resolution. Therefore, an alternative algorithm that can overcome the abnormal sidelobes in FT is required for OCT data processing with spectral gaps.
Fig. 1. Illustration of a single-reflector OCT spectrum and its amplitude in the depth domain. (a) Narrow continuous bandwidth. (b) Wide continuous bandwidth. (c) Wide discontinuous bandwidth.
Recently, many advanced algorithms have been introduced into OCT, such as the random-sampling FT (RSFT) [20], Prony [21], autoregressive model [22], compressed sensing (CS) [23], and adaptive filter-bank (AFB) approach [24]. Among these algorithms, the RSFT can overcome the nonstructured window transform induced by spectral gaps during OCT signal sampling. Therefore, to some extent, the RSFT can restore normal sidelobes when estimating the OCT signal in the depth domain for a discontinuous bandwidth. Unfortunately, experimental results showed that the spectral gaps could not exceed 10% of the total bandwidth [20]. This restriction makes RSFT nearly impractical for gapped spectrum OCT. Researchers have proposed a compressed-sensing-based algorithm to address this limitation. CS estimates the OCT signal in the depth domain through sparsity optimization and, therefore, can suppress the amplitude of the sidelobe and magnify the amplitude of the main lobe under a discontinuous bandwidth condition. Both the amplitude and phase results using a mode-hoped laser output showed that the ability of sidelobe suppression using CS can be enhanced by nearly six times compared to the RSFT [23]. However, owing to a weak convergence at a low SNR, CS is only suitable for measuring smooth surfaces. Recently, Wang et al. presented the use of the AFB approach to estimate the missing part between spectral bands in OCT spectral data and produce images with improved axial resolution [24]. However, OCT phase imaging has not been investigated using AFB. More importantly, AFB indicated that the spectral gaps could not exceed 40% of the total bandwidth. Therefore, the problem of gapped spectrum restriction remains. Thus, a practical method that can effectively suppress abnormal sidelobes in gapped spectrum, is still under investigation. It is crucial to realize a high axial resolution under a discontinuous bandwidth condition.
The iterative adaptive approach (IAA) is a newly developed spectrum estimation algorithm with high-frequency resolution [25]. To reduce the computational cost induced by the processing of complex-valued numbers, a real-valued iterative adaptive approach (RIAA) was proposed [26]. Since IAA and RIAA do not require user-defined parameters, they have been used in many applications in recent studies. For example, Wit et al. used IAA to rebuild an OCT image with axial super-resolution [27]; Du et al. analyzed the Doppler spectrogram of human gait using IAA [28]; Pan et al. introduced IAA to multiple-output radar for range-azimuth processing [29]; and, Wang et al. used RIAA to eliminate the micro-Doppler effect in inverse synthetic aperture radar [30].
These successful applications also suggest that IAA and RIAA can effectively minimize the sidelobes when estimating signal spectra in the frequency domain. Motivated by their superior performance on sidelobe suppression, we demonstrated a method to achieve high axial resolution in PhS-OCT using a gapped spectrum and RIAA. In our approach, the OCT signal model was first established under a discontinuous bandwidth condition. A method for estimating OCT signals in the depth domain using RIAA is presented herein, and the mechanism of abnormal sidelobe suppression is discussed. A high axial resolution in PhS-OCT was achieved using a wide bandwidth with discontinuity. For validation, a single-reflector OCT spectrum was first measured, and its amplitude in the depth domain was estimated using different methods. The results indicated that RIAA had the best capability for abnormal sidelobe suppression. Subsequently, the cross-sectional images and phase-difference maps of three different samples were measured and compared, validating the functional performance of RIAA in gapped spectrum and showing the reconstruction of high axial resolution in PhS-OCT.
2. Theory
2.1 Optical coherence tomography signal model in gapped spectrum
The collected OCT spectrum can be represented by the sum of sine waves, whose frequencies and phases correspond to the depth locations of different slices inside a sample [10].
Here, ΛRm = zR − zLm is the optical path difference for the m-th slice inside the sample. The subscript j denotes the j-th separated spectral band, whose central wavelength and bandwidth are λCj and Δλj, respectively. The wavenumber k is discretized by n = 1, 2, N1, N1 + 1, …, NJ points, and the initial value N0 is set to 0. ε (x) is the jump function with a value of 1 for x ≥ 0 and 0 for x < 0. Noted that only a non-zero intensity is considered in Eq. (2).
To simultaneously reconstruct the amplitude and phase information from the collected OCT spectrum I (n), as described in Eq. (2), the OCT signal in the depth domain $\tilde{I}(f )$ should be estimated first. Currently, the available methods for processing OCT data under a discontinuous bandwidth condition are RSFT, CS, and AFB. Since CS is only suitable for measuring smooth surfaces and AFB mainly focuses on OCT amplitude information, we herein discuss the RSFT.
Based on Eq. (4), the base function exp[-i·2πfq·k(n)] in the RSFT suffers from missing samples owing to the wavelength gaps between separated spectral bands. Therefore, when the wavelength gap increases, the periodicity of the OCT spectrum may be severely affected by the missing sample. In this case, the acquisition of OCT data cannot be simply treated as a nonlinear or random sampling process. Thus, RSFT still suffers from abnormal sidelobes.
2.2 Real-valued Iterative adaptive approach in gapped spectrum
The RIAA treats the spectral estimation as a weighted least-squares problem as follows [26]:
Here, vector $\boldsymbol{\mathrm{\theta}}(f)$ contains the information of the OCT signal in the depth domain and can be estimated in terms of linear regression.
Notably, $\tilde{I}(f )$ ideally exhibits sparsity owing to the quasi-periodicity of the OCT spectrum. Equation (6) suggests that the accumulation of sparse data enables the determinant of inverse matrix W to be large at the true depth (fRm = ΛRm / π). Hence, the amplitude at locations that are not identical to the true fRm is suppressed owing to the small determinant of W-1. This is key for the RIAA to effectively suppress abnormal sidelobes in the gapped spectrum. From Eqs. (5)–(7), the implementation of RIAA can be summarized in the following two steps: 1) initialize the weighted matrix W by submitting the RSFT of I(n) into Eq. (6), and 2) estimate vector $\boldsymbol{\mathrm{\theta}}(f)$ in terms of Eq. (7) and then update the weight matrix W. The inverse matrix of W should be calculated for each frequency (depth location) and, therefore, would be computationally intensive. To reduce the computational complexity, a simple calculation as described in [26] was implemented. In our study, the estimation of $\boldsymbol{\mathrm{\theta}}(f)$ was terminated after 15 iterations. Finally, the OCT signal in the depth domain with sidelobe suppression can be reconstructed from vector $\boldsymbol{\mathrm{\theta}}(f)$.
3. Experiment
A self-established line-field spectral-domain OCT system was employed in the experiment [31]. The resolution, exposure time, and acquisition rate of the system were 7.1 µm × 9.7 µm (axial × lateral), 80 ms, and 10 fps, respectively. Using this system, a single-reflector OCT spectrum was first measured, and its amplitudes in the depth domain estimated by different methods, that is, FT, RSFT, and RIAA, were compared. Subsequently, three different polymer samples—a light-cured polymer droplet, silicone rubber film, and human tooth with composite filling—were measured. The cross-sectional images and phase-difference maps of these samples were retrieved using FT, RSFT, and RIAA for comparison.
3.1 Validation of the axial resolution
An optical wedge was first measured using the established OCT system, and the acquired single-reflector OCT spectrum was analyzed using FT, RSFT, and RIAA, as shown in Fig. 2. For validation, the amplitude of the OCT signal in the depth domain under a 50 nm continuous bandwidth evaluated by FT was used as a reference. As shown in Fig. 2(a), the full width at half maximum (FWHM) of the OCT amplitude, also referred to as the axial resolution, was δΛFT = 8.5 µm. Figures 2(b)–(c) show the OCT amplitude of the 50 nm discontinuous bandwidth with a 5 nm small gap (10% of the total bandwidth), where Fig. 2(b) shows the result of RSFT and Fig. 2(c) shows the result of RIAA. The results indicate that both methods overcame the wavelength discontinuity owing to abnormal sidelobe suppression. In this case, the results of axial resolutions δΛRSFT = 9.2 µm and δΛRIAA = 4.6 µm were close to the reference values. In particular, owing to the energy concentration of the main lobe by the weighted matrix W, the result estimated by RIAA was better than that of the reference. Figures 2(d)–(e) show the OCT amplitude of the 50 nm discontinuous bandwidth with a 25 nm large gap (50% of the total bandwidth), where Fig. 2(d) shows the result of the RSFT and Fig. 2(e) shows the result of the RIAA. Because the wavelength gap between the two separated spectral bands increased to a relatively large level, the RSFT could no longer suppress the abnormal sidelobes [20]. Thus, severe spectral crosstalks resulted from the abnormal sidelobes, and the axial resolution deteriorated to δΛRSFT = 21.0 µm. The RIAA results show that the abnormal sidelobes could still be effectively suppressed, and the FWHM of the OCT amplitude, that is, δΛRIAA = 6.3 µm, was still better than the reference value. The comparison results indicate that RIAA could achieve excellent axial resolution under a gapped spectrum condition.
Fig. 2. Amplitude of the single-reflector OCT signal in the depth domain. (a) FT result for the 50 nm continuous bandwidth, (b) RSFT result for the 50 nm discontinuous bandwidth with a 5 nm wavelength gap, (c) RIAA result for the 50 nm discontinuous bandwidth with a 5 nm wavelength gap, (d) RSFT result for the 50 nm discontinuous bandwidth with a 25 nm wavelength gap, and (e) RIAA result for the 50 nm discontinuous bandwidth with a 25 nm wavelength gap.
To quantitatively investigate the performance of sidelobe suppression using the different methods, the sidelobe-suppression ratio (SSR) was calculated as follows:
where Aside represents the largest amplitude of the sidelobes and Amain represents the amplitude of the main lobe. Figure 2 shows that the SSR for RIAA was approximately -19.7 dB in a 50 nm discontinuous bandwidth with a 25 nm wavelength gap, which was higher than that for RSFT. Moreover, the SSR value was superior to that of the FT in a 50 nm continuous bandwidth. The above comparison results validate the axial resolution and SSR performance of the RIAA in the OCT-gapped spectrum.3.2 Cross-sectional imaging and phase-difference measurement
To further investigate the performance of RIAA in practical applications, three polymer samples—a light-cured polymer droplet, silicone rubber film, and human tooth with composite filling—were measured using a self-established line-field spectral-domain OCT system, and cross-sectional images and phase-difference maps of the samples were estimated using FT, RSFT, and RIAA, respectively. Figure 3 shows the light spectra with continuous and discontinuous bandwidths. The first column of Fig. 4 shows the reference cross-sectional images of the three samples estimated using FT for the OCT spectrum in a 50 nm continuous bandwidth. A partially enlarged figure of each result is shown in the bottom-left corner. The second column of Fig. 4 shows the cross-sectional images estimated using the RSFT for the OCT spectrum in a discontinuous bandwidth. Here, the bandwidth and wavelength gap were 50 nm and 25 nm, respectively. Since the RSFT cannot address a relatively large wavelength gap, the deterioration of axial resolution induced by the weak suppression of abnormal sidelobes can be observed from the results. The third column of Fig. 4 shows the cross-sectional images estimated using RIAA in a 50 nm discontinuous bandwidth with a 25 nm gap. Owing to the superior performance of RIAA for sidelobe suppression, cross-sectional images with a high axial resolution were successfully obtained, and the results were close to the reference values. These comparisons proved that RIAA can achieve a high axial resolution for the gapped spectrum, which is superior to the previously proposed RSFT.
Fig. 4. Cross-sectional images of the three samples using different methods, where the first column shows the results for the 50 nm continuous bandwidth estimated using FT, the second column shows the results for the 50 nm discontinuous bandwidth with a 25 nm gap estimated using RSFT, and the third column shows the results for the 50 nm discontinuous bandwidth with a 25 nm gap estimated using RIAA.
Compared with the cross-sectional images acquired from the OCT amplitudes, the phase-difference maps obtained from the OCT phases were more critical for PhS-OCT measurements. Therefore, after investigating the effectiveness of RIAA in cross-sectional imaging, we studied the performance of the proposed method in retrieving high-resolution phase-difference maps. To implement the study, we chose the OCT spectrum captured before and after sample deformation to evaluate the phase-difference maps [31]. The results for the zoomed-in area are shown in Fig. 5. Notably, sample deformations were produced via polymer curing [31]. The first column of the figure shows the reference phase-difference maps estimated using FT in a 50 nm continuous bandwidth. Since the optical properties of these three samples were not the same, the phase-noise levels of these phase-difference results were different. The second and third columns of Fig. 5 show the phase-difference maps estimated in a 50 nm discontinuous bandwidth with a 25 nm wavelength gap, where the second column shows the RSFT results and the third column shows the RIAA results. After comparing these results, two observations were obtained. (i) The sensitivity of the OCT phase was extremely high, and it was more easily affected by spectral crosstalk. Therefore, because of the weak suppression of sidelobes by the RSFT for the large wavelength gaps, the phase estimation at the peaks of the neighboring main lobe were severely influenced by the abnormal sidelobes. The phase jitter induced by severe spectral crosstalk arises, resulting in a much worse effect for achieving a high axial resolution. Consequently, the issue of abnormal sidelobes became more important for OCT phase estimation. (ii) Benefiting from the capability of RIAA in suppressing sidelobes, the phase jitters caused by the abnormal sidelobes were eliminated, and the quality of the phase maps was close to the reference values. Therefore, the axial resolution of the phase-difference map was achieved in the gapped spectrum using RIAA. In addition, the high quality of phase-difference maps estimated using RIAA enabled phase gradient algorithms [32] to obtain a high SNR of strain field distributions during sample deformation.
Fig. 5. Phase-difference maps of the three samples at zoomed-in area using the various methods, where the first column shows the results for the 50 nm continuous bandwidth estimated using FT, the second column shows the results for the 50 nm discontinuous bandwidth with a 25 nm gap estimated using RSFT, the third column shows the results for the 50 nm discontinuous bandwidth with a 25 nm gap estimated using RIAA.
Notably, the ratio of the total bandwidth to the wavelength gap was 50% in this experiment. To consider more complex conditions of gapped spectrum, we further investigated the influence of wavelength gaps on the RSFT and RIAA approaches. Here, a sample of a human tooth with the composite filling was selected, and the results between RSFT and RIAA in terms of different wavelength gaps were estimated and compared. Figure 6 shows the light spectra, where (a) shows the results for the 30 nm continuous bandwidth, and (b)–(d) show the results for the 30 nm discontinuous bandwidths with different wavelength gaps. The ratios of the total bandwidth to these wavelength gaps were 50%, 100%, and 150%, respectively. Figure 7(a) shows the reference cross-sectional image for a 30 nm continuous bandwidth. Figures 7(b)–(d) show the estimated cross-sectional images corresponding to the different gaps using the RSFT. Since the amplitude of the abnormal sidelobe was amplified as the wavelength gap increased, the effect of the spectral crosstalk was more severe in the neighboring main lobe. In some pixels, the amplitudes of the main lobes were covered by abnormal sidelobes. Thus, the quality of cross-sectional images estimated using the RSFT gradually deteriorated with an increase in the wavelength gap, and the axial resolution has significantly worsened in the discontinuous bandwidth. Figures 7(e)–(g) show the estimated cross-sectional images using RIAA for different wavelength gaps. Compared to the reference results, the quality of the cross-sectional images estimated using the RIAA remained nearly unchanged even when the wavelength gap increased to 45 nm (150% of the total bandwidth). Accordingly, RIAA maintained a high axial resolution under more complex conditions of the gapped spectrum.
Fig. 6. (a) Light spectra with 30 nm continuous bandwidth. (b)–(d) Light spectra with 30 nm discontinuous bandwidth, where (b) shows the result for the 15 nm wavelength gap, (c) shows the result for the 30 nm wavelength gap, and (d) shows the result for the 45 nm wavelength gap.
Fig. 7. Cross-sectional images of a human tooth with composite filling. (a) Reference result for the 30 nm continuous bandwidth estimated using FT. (b)–(d) Results for the 30 nm discontinuous bandwidth with different wavelength gaps estimated using RSFT. (e)–(g) Results for the 30 nm discontinuous bandwidth with different wavelength gaps estimated using RIAA.
Figure 8(a) shows the reference phase-difference map for 30 nm continuous bandwidth. Figure 8(b)–(d) show the phase-difference maps for 30 nm discontinuous bandwidth with different wavelength gaps estimated using RSFT. Owing to the high sensitivity of the OCT phase, it was found that the deterioration of the phase-difference maps estimated using RSFT was more severe than that of the structural image. In particular, noises caused by the phase jitters led to a poor axial resolution for the 45 nm wavelength gap. Figures 8(e)–(f) show the phase-difference maps estimated using RIAA for the 30 nm discontinuous bandwidth with different wavelength gaps. The results show that the phase jitters were effectively removed by suppressing the abnormal sidelobes, and the quality of phase-difference maps remained almost unchanged as the wavelength gap increased. Importantly, all the phase-difference maps estimated using RIAA were close to the reference results, demonstrating the robust performance of RIAA in achieving high axial resolution under a gapped spectrum condition.
Fig. 8. Phase-difference maps of a human tooth with composite filling. (a) Reference result for the 30 nm continuous bandwidth estimated using FT. (b)–(d) Results for the 30 nm discontinuous bandwidth with different wavelength gaps estimated using RSFT. (e)–(g) Results for the 30 nm discontinuous bandwidth with different wavelength gaps estimated using RIAA.
Two points need to be discussed before the conclusion. The first point is the RIAA convergence. According to [27], the performance of the RIAA depends on the SNR. Our experimental results suggest that the SNR of the OCT spectrum collected inside the polymer was acceptable for RIAA. However, when acquiring an OCT spectrum with low-SNR condition, e.g., measuring the deformation inside a composite material, it requires the development of a new RIAA approach with improved noise immunity. This will be the focus of our future work. The second point is the computational cost. In our study, RIAA was implemented on a computer (CPU: Intel i7-11700F, memory: 16 GB) using MATLAB 2019a. The reconstruction of a B-scan OCT image (1292 × 964 pixels) using RIAA required 18 min. Another future work is to investigate the accelerated RIAA approach in terms of its fast implementation [27].
4. Conclusion
In this paper, a method employing a RIAA for achieving high-resolution OCT measurements under a gapped spectrum condition is presented. It was experimentally shown that the abnormal sidelobes generated by the wavelength gap were effectively suppressed using the RIAA approach. Consequently, the influence of the wavelength gap was removed, and a high axial resolution was achieved using a wide bandwidth with discontinuity. For validation, a single-reflector OCT spectrum was first measured, and its amplitude in the depth domain was estimated using different methods, that is, FT, RSFT, and RIAA. The results show that RIAA had the best abnormal sidelobe-suppression capability and achieved excellent axial resolution. In addition, the cross-sectional images and the phase-difference maps of three different samples—a light-cured polymer droplet, a silicone rubber film, and a human tooth with composite filling—were also measured and compared, validating the practical value of the RIAA in tomographic measurement for high axial resolution when using gapped spectrum with different wavelength gaps. Since the RIAA can retrieve both structural images and phase maps at a high axial resolution under continuous or discontinuous bandwidth, it is expected that the method can be developed as an effective tool for dealing with OCT signals and can be widely employed in future research.
Funding
Science and Technology Program of Guangzhou (202201010581); Natural Science Foundation of Guangdong Province (2019B1515120036, 2021A1515011945, 2021A1515012598, 501200069); National Natural Science Foundation of China (61705047, 61727810, 62171140, 62273105).
Acknowledgments
We would like to thank Editage (www.editage.cn) for English language editing.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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