Adaptive incremental method for strain estimation in phase-sensitive optical coherence elastography

Yulei Bai; Shuyin Cai; Shengli Xie; Bo Dong

doi:10.1364/OE.433245

1. Introduction

Optical coherence tomography (OCT) is a non-destructive and non-invasive imaging modality that measures internal structures of transparent and semi-transparent objects, e.g. tissues, polymers, and ceramics, using low-coherence interferometry [1]. The depth resolution and depth range of OCT are between ultrasound imaging [2] and confocal microscopy [3], making OCT irreplaceable in both clinical diagnoses and industrial inspections [4–7]. To study the microscopic deformation of objects under compressive stress, a combination of OCT and elastography termed optical coherence elastography (OCE) was developed [8,9]. The original displacement estimation technique in OCE is the speckle tracking [10,11]. By using the equivalence of speckle pattern’s shift and sample displacement, image cross-correlation approaches, e. g. digital image correlation, were introduced to track the speckle motion between successive B-scans OCT images. The speckle tracking is capable of measuring sample internal displacements as small as a few microns owing to the level of OCT depth resolution [11]. However, it has been already shown that the dynamic range of the speckle tracking is insufficient. An alternative approach which utilizes the phase information of interference spectrum was proposed to extend the dynamic range [12,13]. By using a linear relationship between optical path difference and interference phase, the evaluation of phase difference from interference spectrums before and after sample deformation is performed rather than the cross-correlation operation. This approach for depth-resolved displacement estimation is known as the phase-sensitive optical coherence elastography (PhS-OCE) [13]. Compared to the speckle tracing in OCE, the displacement dynamic range of PhS-OCE is enhanced to be more than 20-fold greater since the phase detection equips with nanometer sensitivity [9]. This advantage of high dynamic range enables the PhS-OCE becoming an effective solution of investigating mechanical behaviors inside tissues in recent years, e.g. corneal disease diagnose [14] and cancer assessment [15]. Besides biomedical applications, it has been shown that the PhS-OCE is also capable of characterizing thermal expansions and visualizing polymerization shrinkages inside polymers [16–18], featuring great potential in even solid mechanics and material science.

In a PhS-OCE measurement, the estimation of strain distribution from phase difference map plays an important role in quantifying sample internal mechanical properties. It is well known that the strain is essentially a phase gradient and the two methods, i.e. least squares method (LSM) [19] and vector method (VM) [20], are generally used for determining the phase gradient. In LSM, the strain is calculated by fitting of the phase difference slope. To suppress the noise amplitude, the LSM is updated by introducing the amplitude weighting [21]. In VM, the strain is calculated by performing the vector averaging for complex-valued OCT signals. Subsequently, the VM is improved to estimate the laterally inhomogeneous strains [22]. It is worth noting that, in some large deformed regions, speckle decorrelation [23] always result in missing phase-difference values, making the large strain distribution difficult to be estimated. To address this problem, an incremental method that can estimate the large strains is firstly proposed [24]. After that, the incremental method is applied to visualize strain dynamics in laser-irradiated corneal samples [25–27] and to obtain nonlinear stress-strain curves in a large range of strains [28,29]. Those experimental results show the effectiveness of incremental method in large strains estimation. Considering the incremental method itself, the issues of poor signal-to-noise ratio (SNR) and low computation efficiency are raised, owing to the strain estimations for each two neighboring phase frame indexes during the accumulation operation. Reduction of the cumulative number is a key to address the side effects of the incremental method. Zaitsev et al. proposed a general criterion for optimally determining the maximal interframe interval and therefore the cumulative number for strain estimations is reduced a lot [30]. But in that research, the optimized interframe interval remains unchanged, which makes the effect limited in processing the nonlinear change of strains, e. g. measuring nonlinear stress-strain relationships [28,29] and visualizing curing behaviors [18]. Thus, an incremental method that can automatically search the maximal interframe interval for strain estimation is highly demanded.

In this work, we proposed an adaptive incremental method for strain estimation in PhS-OCE. The interframe interval can be automatically determined by counting the number of phase noise points and calculating the phase noise ratio. Then the cumulative strain estimation is performed in terms of the vector method. Real experiments of mapping polymer curing shrinkages were done to test whether the method can increase the computation efficiency and enhance the SNR, when considering the nonlinear change of strains. The remaining part of this paper is organized as follows. In section 2, an adaptive incremental method for strain estimation is presented in details. In section 3, the experimental results of strain estimation using the adaptive incremental method are discussed. Finally, the conclusion is provided in section 4.

2. Methodology

In a PhS-OCE measurement, the axial displacement field d (m, j) on the cross-section of an object can be acquired from the phase difference values ΔΦ(m, j) measured by using a Fourier-domain OCT system [18]

(1)$$\left\{ \begin{array}{l} \Delta \Phi (m,j)\textrm{ = ta}{\textrm{n}^{ - 1}}\left\{ {\frac{{{\mathop{\rm Im}\nolimits} [{{{\tilde{I}}_1}(m,j)} ]{\textrm{Re}} [{{{\tilde{I}}_2}(m,j)} ]- {\textrm{Re}} [{{{\tilde{I}}_2}(m,j)} ]{\mathop{\rm Im}\nolimits} [{{{\tilde{I}}_1}(m,j)} ]}}{{{\textrm{Re}} [{{{\tilde{I}}_1}(m,j)} ]{\textrm{Re}} [{{{\tilde{I}}_2}(m,j)} ]+ {\mathop{\rm Im}\nolimits} [{{{\tilde{I}}_2}(m,j)} ]{\mathop{\rm Im}\nolimits} [{{{\tilde{I}}_1}(m,j)} ]}}} \right\}\\ d(m,j) = \frac{{\textrm{Un}[{\Delta \Phi (m,j)} ]}}{{2 \cdot n \cdot {k_0}}} \end{array} \right.$$

where k₀ is the initial wavenumber and the operator “Un” represents two-dimensional phase unwrapping. n is the refractive index of the sample. “Im” and “Re” are the real and imaginary parts of complex number. The subscript number 1 and 2 denote the interference spectrums before and after sample deformation. The space coordinates in which m represents the depth index while j represents the horizontal index. The axial strain is estimated by performing the gradient of phases along the depth index, as shown in Eq. (2).

(2)$$\varepsilon (m,j) = \frac{{\textrm{Un}[{\Delta \Phi (m,j + 1)} ]- \textrm{Un}[{\Delta \Phi (m,j)} ]}}{{2{k_0} \cdot n \cdot [z(j + 1) - z(j)]}}$$

where z is the depth position of slice inside the sample. Practically, it is not suitable to use Eq. (2) for strain estimation, because the numerical difference in Eq. (2) can magnify the noise amplitude. Two methods, which refer to the VM and the LSM, are developed by employing statistical ideology to minimize the noise effect. It is noted that neither the two methods can completely estimate large strains owing to the missing information of wrapped phase difference induced by speckle decorrelation in some regions. An incremental method that employs the continuous nature of deformation changes is proposed to overcome the speckle decorrelation.

Supposing the time-dependent interference spectrums, e. g. ${\tilde{I}_1}({m,j} )$, ${\tilde{I}_2}({m,j} )$, …, ${\tilde{I}_P}({m,j} )$, are captured during the sample deformation, the cumulative strain estimation is estimated by an incremental method [30]:

(3)$$\varepsilon (m,j) = \sum\limits_{p = 1}^{P - 1} {{\varepsilon _p}(m,j)}$$

Here, the subscript p = 1, 2, …, P represents frames and ε _p (m, j) is the estimated strain from spectrums ${\tilde{I}_p}({m,j} )$ and ${\tilde{I}_{p + 1}}({m,j} )$.The issue of speckle decorrelation in estimating the ε _p (m, j) can be avoided as long as the frame per second (fps) of camera is fast enough. The Eq. (3) suggests that the strain estimations using LSM or VM need to be performed several times before the cumulative strain is obtained. Considering the algorithmic computation of LSM or VM, the cumulative strain estimation in Eq. (3) will give rise to the low computation efficiency. In addition, the noise is also accumulated several times. To minimize the negative effect of accumulation, every several frames, rather than the neighboring frames, are retained to estimate the cumulative strain. However, it is difficult for the unchanged interframe interval to find a balance between the high and low strain rates in nonlinear change of sample deformations. Designing an adaptive method, which can automatically choose the interframe interval with the change of strain rates, is a strategy to make the incremental method more practical for the complex scene.

High strain rate is more likely to cause the speckle decorrelation and therefore the strain rate is positively associated with the level of phase noise. The self-adjustable interframe interval can be determined in terms of the phase noise quantification and therefore enable the incremental method to adapt different strain rates. Motivated by this idea, we introduce a localization method of phase noise points [31] to define a criterion for the chosen of interframe interval. In [31], the phase noise points are marked as 1 in a binary map. Then the level of phase noise is quantified by counting the number 1 and the self-adjustable interframe interval is determined by:

(4)$$\left\{ \begin{array}{l} \mathop {\max }\limits_{\Delta p \in Z} \Delta p\\ s.t.\textrm{ }\vartheta \textrm{ = }\frac{{\sum\limits_{m = 1}^M {\sum\limits_{j = 1}^J {|{{\eta_{\Delta p}}(m,j)} |} } }}{{M \cdot J}} \le \xi \end{array} \right.$$

where Z is the integer set; Δp is the interframe interval to be optimized; ϑ represents the phase noise ratio; ξ is the preset threshold value; M and J are the maximum pixels along the vertical and horizontal directions, respectively. η represents the binary map of phase noise and is evaluated as follows [31],

\begin{array}{l} {\eta _{\Delta p}}(m + \Delta m,j + \Delta j)\\ = \left\{ \begin{array}{ll} 1 - \left|{\left\lfloor {\frac{{\Delta {\Phi _{\Delta p}}(m + 1,j + 1) - \Delta {\Phi _{\Delta p}}(m + 1,j)}}{{2\pi }}} \right\rfloor + \left\lfloor {\frac{{\Delta {\Phi _{\Delta p}}(m,j + 1) - \Delta {\Phi _{\Delta p}}(m + 1,j + 1)}}{{2\pi }}} \right\rfloor + \left\lfloor {\frac{{\Delta {\Phi _{\Delta p}}(m + 1,j) - \Delta {\Phi _{\Delta p}}(m,j + 1)}}{{2\pi }}} \right\rfloor } \right|, &(\Delta m,\Delta j) = (0,0)\\ 1 - \left|{\left\lfloor {\frac{{\Delta {\Phi _{\Delta p}}(m,j) - \Delta {\Phi _{\Delta p}}(m,j + 1)}}{{2\pi }}} \right\rfloor + \left\lfloor {\frac{{\Delta {\Phi _{\Delta p}}(m,j + 1) - \Delta {\Phi _{\Delta p}}(m + 1,j + 1)}}{{2\pi }}} \right\rfloor + \left\lfloor {\frac{{\Delta {\Phi _{\Delta p}}(m + 1,j + 1) - \Delta {\Phi _{\Delta p}}(m,j)}}{{2\pi }}} \right\rfloor } \right|, &(\Delta m,\Delta j) = (1,0)\\ 1 - \left|{\left\lfloor {\frac{{\Delta {\Phi _{\Delta p}}(m,j) - \Delta {\Phi _{\Delta p}}(m,j + 1)}}{{2\pi }}} \right\rfloor + \left\lfloor {\frac{{\Delta {\Phi _{\Delta p}}(m + 1,j) - \Delta {\Phi _{\Delta p}}(m,j)}}{{2\pi }}} \right\rfloor + \left\lfloor {\frac{{\Delta {\Phi _{\Delta p}}(m,j + 1) - \Delta {\Phi _{\Delta p}}(m + 1,j)}}{{2\pi }}} \right\rfloor } \right|, &(\Delta m,\Delta j) = (1,1)\\ 1 - \left|{\left\lfloor {\frac{{\Delta {\Phi _{\Delta p}}(m + 1,j + 1) - \Delta {\Phi _{\Delta p}}(m + 1,j)}}{{2\pi }}} \right\rfloor + \left\lfloor {\frac{{\Delta {\Phi _{\Delta p}}(m + 1,j) - \Delta {\Phi _{\Delta p}}(m,j)}}{{2\pi }}} \right\rfloor + \left\lfloor {\frac{{\Delta {\Phi _{\Delta p}}(m,j) - \Delta {\Phi _{\Delta p}}(m + 1,j + 1)}}{{2\pi }}} \right\rfloor } \right|, &(\Delta m,\Delta j) = (0,1) \end{array} \right. \end{array}

Here, the operation “⌊ ⌋” denotes rounding to the nearest integer. The wrapped phase difference ΔΦ_Δ_p is evaluated from two spectrums ${\tilde{I}_p}({m,j} )$ and ${\tilde{I}_{p + p}}({m,j} )$. Figure 1(a) shows the framework of the adaptive incremental method in estimating the cumulative strain. To illustrate how the interframe interval adjusts itself, we start from p = 1 and present a schematic diagram as shown in Fig. 1(b). The phase noise maps corresponding to Δp = 1, 2, …, 30 are successively estimated. It is found that the ratios of phase noise points are smaller than the preset threshold value ξ, except for the interframe interval Δp = 30. Accordingly, the interframe interval is finally determined by maximizing integer series {1, 2, …, 29}. In the following step, the spectrum ${\tilde{I}_p}({m,j} )$ starts from p = 30 and the interframe interval remains Δp = 29. Then the phase noise, which corresponds to the wrapped phase difference evaluated from ${\tilde{I}_{30}}({m,j} )$ and ${\tilde{I}_{30 + 29}}({m,j} )$, is mapped and its ratio also can be calculated. Similarly, if the preset threshold value ξ is bigger, then the interframe interval Δp should be added by one until the ratios of phase noise points exceeds the preset threshold value. Otherwise, the interframe interval Δp should be decreased by one until the constraint condition in Eq. (4) is satisfied. It is seen that the proposed determination of interframe interval is adaptively adjusted in terms of the phase noise ratios. Therefore, the cumulative strain estimation for both high and low rates can be balanced. Owing to such an advantage, the adaptive incremental method can increase computation efficiency and reduce the cumulative noise, compared to the neighboring incremental method in application of nonlinear change of strains.

Fig. 1. Framework of the adaptive incremental method in estimating the cumulative strain, where (a) is the pseudocode and (b) is the schematic diagram

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3. Optical setup

Before the experiment, a self-established line-field spectral-domain OCT system was established as shown in Figs. 2(a) and 2(b). A cross-section in the y-z plane of the specimen S is firstly illuminated by the super-luminescent laser diode (SLD) light source (SLD-mCS-371-MP-SM-840, Superlum Diodes, Ltd., Ireland), and then the light scattered from the specimen interferes with the reference light. Finally, the interference forms a spectrum imaged on the CCD camera (Manta G-125B, Allied Vision Technologies, Germany). The focal lengths of L₁-L₄, and CL are 50 mm, 100 mm, 50 mm, 100 mm, and 75 mm, respectively; the reflective diffraction grating G is of 600 grooves/mm with a blaze wavelength of 850 nm; The bandpass region of the spectral filter SF is 800-900 nm. The spatial resolution, exposure time, acquisition rate, and displacement sensitivity of the system are 7.5μm × 3μm (axial × lateral), 80 ms, 10fps, and 10 nm, respectively.

Fig. 2. The self-established OCT system and measured polymer samples (a) photograph of the established OCT system, (b) schematic of the established OCT system, (c) cross-sectional image of the polymer sample without an internal defect, (d) cross-sectional image of the polymer sample with internal defect. SLD: Super-luminescent laser diode; L1-L4: lens; CL: cylindrical lens; S: sample; CBS: cube beam splitter; NDF: neutral density filter; R: reference plane; SF: spectral filter; G: diffraction grating; CCD: charge-coupled device camera.

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Two UV-cured polymer specimens, one with an internal defect and the other without it, were prepared. The cross-sectional images of these two samples measured using the established OCT system are shown in Figs. 2(c) and 2(d), respectively. During the experiment, the samples were illuminated by the UV-light and the polymerization shrinkage of the samples were continuously measured by the OCT system. Since both refractive index and refractive index variation will not affect the adaptive incremental method, they have not been considered in the experiment for simplicity.

4. Results

We firstly investigate the strain estimation for the polymer without internal defects. The wrapped phase difference maps at time 0.5 second, 1.5 second, 5.1 second and 10.0 second are illustrated in Fig. 3(a). It is found that the fringe density of phase-difference maps increases with the degree of polymerization shrinkage. Besides, with the shrinkage degree increased, large deformation induced speckle decorrelation gradually appears at the right corner of the sample. As the section 1 introduced, both LSM and VM are available for the strain estimation in practical. Compared to the LSM, the VM does not require the phase unwrapping and has better performance for suppressing the phase noise. Accordingly, we employ the VM to estimate the strain distributions which corresponds to the wrapped phase difference in Fig. 3(a). The strain estimated results using the VM with a processing widow of 15 × 15 pixels are presented in Fig. 3(b). Observed from the results, it is shown that the strain values of some large deformation regions cannot be correctly estimated owing to the speckle decorrelation. As indicated by Fig. 1(d) in the Ref. [18], the curing rate presents a strong nonuniformity, resulting the nonlinear change of strains during the polymer curing. A neighboring incremental method is applied for estimating the cumulative strains, as shown in Fig. 3(c). All the incorrect strain values are successfully restored and therefore the effectiveness of incremental method for estimating large strain is validated. Figure 3(d) shows the cumulative strains estimated by the adaptive incremental method. From the results, there are slight differences between the adaptive and the neighboring incremental methods in estimating the cumulative strains. However, the computation time of the two different incremental methods is quite different. We implemented the two different incremental methods on a same operating environment (CPU: Intel i5-8500, Memory: 8Gb, MATLAB 2016b). The changes of computation time with the frame index increased is shown in Fig. 4. The results suggest that the consumption time of neighboring incremental method linearly increased. This is because the strain estimation needs to be performed once when each index of phase frame is added. In the first experiment, 100 interference spectrums are captured during the sample curing. If we wish to learn the final state of sample curing process, then the strain estimations will require to be performed 99 times. The consumption time of adaptive incremental method increased as a ladder-like distribution owing to the unnecessary strain estimation for neighboring frame index. Each a ladder is added, it means that the strain estimation is performed once. In addition, since the computation time of localizing phase noise is much less than that of the strain estimation, it can hardly be observed from the figure. In Fig. 4, 8 ladders are counted which suggests that the strains were only estimated 8 times for the adaptive incremental method. Therefore, the computation time can be effectively decreased (99-8)/8 ≈ 11 folds, when employing an incremental method to address the issue of speckle decorrelation.

Fig. 3. Results of the polymer sample without internal defects in curing process. (a) wrapped phase difference at time 0.5 second, 1.5 second, 5.1 second and 10.0 second, (b) strain results estimated using the non-incremental method for VM, (c) strain results estimated using the neighboring incremental method for VM, (d) strain results estimated using the adaptive incremental method for VM.

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Fig. 4. Computation time of the neighboring and the adaptive incremental methods in estimating the cumulative strains (homogeneous deformations case)

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Since the sample deformations in the first experiment present homogeneity, it is not evident to show the SNR enhancement of the cumulative strain estimation using the adaptive incremental method. Here, we performed the second experiment for the polymer with an internal defect and investigate the SNR performance of the adaptive incremental method as priority. The wrapped phase difference maps at time 0.6 second, 1.6 second, 2.1 second and 5.0 second are shown in Fig. 5(a). Owing to the presence of internal defect, the distributions of wrapped phase difference maps show inhomogeneity. Particularly, more regions introduce the problem of speckle decorrelation with the increase of curing time. Figures 5(b)–5(d) are the corresponding strain estimations using the non-incremental method, the neighboring incremental method and the adaptive incremental method. The results verified that the two different incremental methods still can effectively estimate the large strains for the inhomogeneous deformation distributions.

Fig. 5. Results of the polymer sample with an internal defect in curing process. (a) wrapped phase difference at time 0.6 second, 1.6 second, 2.1 second and 5.0 second, (b) strain results estimated using the non-incremental method for VM, (c) strain results estimated using the neighboring incremental method for VM, (d) strain results estimated using the adaptive incremental method for VM.

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In order to compare the SNR for the estimated strain using the two different incremental methods, zoomed-in areas from Figs. 5(c-4) and 5(d-4) are shown in Fig. 6. It is seen that the strain results estimated by the adaptive incremental method are much smoother because the cumulative number are reduced a lot. The SNR of the result can be estimated by using the formula described in [21], and the estimated SNR values of the neighboring incremental result and the adaptive incremental result are 14.63 dB and 26.12 dB, respectively. Notably, the effect of SNR enhancement is similar to Zaitsev and others’ work [30]. However, the difference is that the cumulative number for the proposed method depends on the phase noise level and therefore the interframe interval is adaptively adjusted to be suitable so as to ensure that the corresponding phase noise remains at a relatively low level. This feature is the key to enable the proposed method better effective for SNR enhancement in nonlinear change of inhomogeneous deformations.

Fig. 6. Comparation of SNR on strain results using the two different incremental methods

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Finally, we also present the computation efficiency between the two different incremental methods for inhomogeneous deformations. As shown in Fig. 7, the strain estimations need to be performed 50 times for the neighboring incremental method, while the number of strain estimations is reduced to 11 times for the adaptive incremental method. The computation efficiency is improved (50-11)/11 ≈ 4 folds in inhomogeneous deformations. All the experimental results testify that the adaptive incremental method for cumulative strain estimation can improves the computing efficiency while also reducing the cumulative noise in nonlinear change of deformations, which features strong feasibility in more practical applications.

Fig. 7. Computation time of the neighboring and the adaptive incremental methods in estimating the cumulative strains (inhomogeneous deformations case).

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5. Conclusion

In PhS-OCE, strain estimation from wrapped phase difference may suffer from the issue of speckle decorrelation, resulting in erroneous strain results for the regions of large deformation. Incremental method is a powerful technique to overcome the speckle decorrelation by estimating the cumulative strain. When considering the nonlinear change of strains, the neighboring incremental method is introduced to estimate the cumulative strain. However, it gives rise to the problems of low computation efficiency and poor SNR. In this work, we proposed an adaptive incremental method to address these problems in both linear and nonlinear strain measurement situations. The method firstly counts the amount of phase noise points by mapping the binary noise map. After the noise threshold value is preset, the interframe interval is adaptively adjusted in terms of the phase noise ratio. Finally, the efficient estimation of cumulative strain is implemented by reducing the cumulative number. For validation, real experiments of visualizing polymerization shrinkage were performed, indicating that the adaptive incremental method for cumulative strain estimation can increase the computation efficiency 11folds in homogeneous deformation distribution, while increase it 4 folds in inhomogeneous deformation distribution. Moreover, the SNR of cumulative strain is enhanced, especially for the inhomogeneous deformation distribution. The proposed method expands the practicability of the incremental method, which can be used to estimate the large strain distributions with both high efficiency and SNR in more complex scenes.

In our next work, a method that can automatically determine the threshold value ξ by estimating the level of additive noises from neighboring phase frames will be further investigated.

Funding

Natural Science Foundation of Guangdong Province (2021A1515011945, 2021A1515012598); China Postdoctoral Science Foundation (2020M672531, 2021T140137); National Natural Science Foundation of China (11802008, 61705047, 61727810).

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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Adaptive incremental method for strain estimation in phase-sensitive optical coherence elastography

Abstract

1. Introduction

2. Methodology

3. Optical setup

4. Results

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (5)

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