Abstract

Dispersion in optical coherence tomography (OCT) leads to a series of wavelength-dependent phase distortions, which cause degradation of axial resolution. Due to the lack of prior information or the complexity of an exhaustive search calculation, all-depth dispersion suppression can hardly be realized in practical cases, especially for high-speed processing and irregular-structure samples. This paper explores the understanding of the depth-dependent dispersion in the fractional Fourier domain (FRFD) and addresses a new method for dispersion correction based on the FRFD stepped detection algorithm that is able to adaptively compensate the dispersion at all depths of the sample. For the detection of each dispersion component, a coarse search followed by a localized fine search is presented in our algorithm to reduce the calculation complexity with high accuracy guaranteed. A signal separation method utilizing FRFD filtering is also designed to avoid the interference between the dispersion from different depths of the sample, which allows all-depth dispersion correction. The proposed algorithm is verified to be effective through the stratified media of ZnSe. The application of the proposed algorithm in OCT imaging of onion and human coronary artery also demonstrates the feasibility of our algorithm for dispersion correction in bio-tissues.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Optical Coherence Tomography (OCT) is an non-invasive interference imaging technique to obtain two-dimensional or three-dimensional structural images with high resolution and real-time imaging capabilities, which has been applied in many fields and has great potential [1]. Nowadays, the development of Fourier domain Optical Coherence Tomography (FD OCT) enables great improvements in sensitivity and imaging speed when compared with the time domain methods [2,3].

However, increasing axial resolution requires larger optical bandwidths, which makes OCT imaging more susceptible to dispersion. At present, the dispersion correction method in OCT can be generally divided into physical methods and numerical methods in terms of implementation. Physical methods are based on matching optical material [4] or phase control delay line [5] to compensate the dispersion between two interferometer arms. This two types of physical methods perform well in correcting the systematic dispersion, however, they can hardly compensate the dispersion from different depths of the sample. It is also difficult to readjust the devices and parameters when testing different kinds of samples.

Numerical methods are relatively more flexible and do not necessarily depend on the support of hardware. Some of them are based on automatic iterative method [6] and autofocus method [7], which require only a single dispersion correction term. Hence, they are also quite suitable for the systematic dispersion or the dispersion at one certain depth. However, they can hardly compensate the dispersion at all depths of the sample simultaneously.

In fact, the internal structure of most practical samples to be detected is unknown, whether or not there are multiple layers. When OCT is used to image such kind of samples, the dispersion will inevitably occur at multiple locations inside. Therefore, adaptive detection and automatic elimination of dispersion are expected to be completed before imaging. Several numerical methods have been proposed to deal with the problem of all-depth dispersion compensation. The correlation method [8] and the linear fitting method [9] take depth-based convolutional kernels or linear fitting functions as compensators. This two methods both require prior knowledge of the dispersion property, hence, they are not feasible for the case that the dispersion distribution at different depths is unknown or cannot be calculated accurately. In addition, a new method called the Conjugate Transform method [10], is recently published for full-depth dispersion compensation without any prior knowledge. However, it determines the optimal coefficients of dispersion by exhaustive search, which increases computational complexity.

All the previous studies on dispersion correction are based on the Fourier analysis. Note that recently a method using the fractional Fourier transform (FRFT) [11] can obtain compensated OCT images with directly imaging in one specific fractional Fourier domain (FRFD). However, this leads to the dispersion at all depths corrected by the same FRFD order. Only the dispersion at one certain depth will be exactly compensated, while those at other depths might be overcompensated or undercompensated. We note that the study in Ref. [11] also gives an inspiring proof-of-principle demonstration of the depth-dependent dispersion compensation based on numerical simulations and illustrates how the order $\alpha$ corresponds to the depth in the physical sense. But it seems more challenging to compensate the dispersion at all depths of real samples. Also, the exhaustive search process of the optimal FRFT order may limit its application in real cases. Here in our work, we provides a new understanding on the depth-dependent dispersion based on fractional Fourier analysis. We propose a new method for dispersion correction based on the FRFD Stepped Detection Algorithm, which is able to detect and compensate the dispersion at all depths of the sample adaptively. Herein, the mathematical model of dispersion in FD OCT is established and the relationship between the depth-dependent dispersion and the FRFT is analysed. Then the proposed FRFD Stepped Detection Algorithm and the process of dispersion correction are presented. The theoretical analysis of the proposed algorithm is also illustrated in simulations. Furthermore, the experiment on the stratified media of ZnSe is conducted to verify the effectiveness of our algorithm. Samples of onion and human coronary artery are taken to demonstrate the feasibility of the proposed algorithm for dispersion correction in bio-tissues.

2. FRFD representation of dispersion in OCT

2.1 Mathematical model of depth-dependent dispersion in OCT

In a FD OCT system, the backscatter light from various depths of the sample interferes with the reflected light from the reference arm. The interference signal in FD OCT is usually given by [12]

$$S(\omega ) = 2\textrm{{Re}}\{ \sum_n {\sqrt {{I_n}(\omega ){I_r}(\omega )} \exp [j\theta (\omega , {\tau _n})]} \}$$
Here, $\omega$ is the angular optical frequency, ${{I_n}(\omega )}$ and ${{I_r}(\omega )}$ describe the spectrum of the light reflected from the n-th layer of the sample and from the reference arm, respectively. $\theta (\omega , {\tau _n})$ is the phase difference between the reflected light at the n-th layer and the reference light, which denotes as
$$\theta (\omega , {\tau _n}) = \omega {\tau _n} + \phi (\omega , {\tau _n}).$$
${\tau _n}$ is the group delay of the spectrum from the n-th reflective layer, which corresponds to the depth of the sample in the OCT image. $\phi (\omega ,{\tau _n})$ describes the dispersive phase terms caused by the frequency-dependent propagation constants $\beta (\omega )$. For light with low coherence, the propagation constant at the n-th layer ${\beta _n}(\omega )$ may denote as a Taylor expansion around the center frequency $\omega _0$, as
$${\beta _n}(\omega ) = {\beta _{n0}}({\omega _0}) + {\beta _{n1}}({\omega _0})(\omega - {\omega _0}) + \frac{1}{2}{\beta _{n2}}({\omega _0}){(\omega - {\omega _0})^2} + \cdots$$
where the first three terms in the Taylor expansion are zero-order, first-order, and second-order dispersion (group-velocity dispersion, GVD), respectively. Since the third and higher order dispersion have less effects on the interference signal and will not significantly affect the quality of OCT imaging, these minuscule phase terms will be ignored [8,10,11,13]. After the constant terms merged, the residual interference signal can be rewritten as:
$$S(\omega ) = 2\textrm{{Re}}\{ \sum_n {I(\omega )\exp [j(\omega {\tau _n} + \frac{1}{2}{\beta _{n2}}\Delta {z_n}{{(\omega - {\omega _0})}^2})]} \}$$
$I(\omega )$ includes the spectrum of the sample-reflected light $I_n(\omega )$ and the spectrum of the reference-reflected light $I_r(\omega )$. ${\beta _{n2}}$ is the GVD coefficient of the n-th layer of the sample, and $\Delta {z_n}$ is the optical path difference between the n-th layer and the reference mirror.

Note that, the initial frequency of the detected dipersive signal at a certain sample layer can be extracted by taking the derivative of the phase term with respect to angular optical frequency $\omega$, as shown below:

$$\frac{{\partial \theta }}{{\partial \omega }} = {\tau _n} + {\beta _{n2}}\Delta {z_n}(\omega - {\omega _0})$$
which highlights the chirp (linear frequency modulation, LFM) nature of the group-velocity dispersion ($\beta _{n2}$). Thus, the interference signal from a single surface in FD OCT can be considered as a LFM signal with a single linear frequency rate. However, as for the whole signal which has multiple reflecting surfaces, the second-order dispersion term is a function of depth. Hence, for FD OCT, the detected interference signal $S(\omega )$ can be accordingly modeled as a multi-component LFM signal consisting of different linear frequency rates. In essence, the process of dispersion correction is to detect and eliminate all GVD terms ($\frac {1}{2}{\beta _{n2}}\Delta {z_n}{{(\omega - {\omega _0})}^2}$) from the interference signal $S(\omega )$, in another word, to find and remove the chirp terms from $S(\omega )$.

2.2 The relationship between the depth-dependent dispersion and the FRFT

The fractional Fourier transform (FRFT) is a generalized form of traditional Fourier transform. Since Almeida analyzed the relationship between the FRFT and the Wigner-Ville distribution (WVD), and illustrated the FRFT as a rotation of the time-frequency plane [14], FRFT became a especially suitable tool for processing chirp-like signals. The FRFT of signal $x(t)$ is defined as [15]

$$X_{\alpha}(u)=F^{p}[x(t)]=\int_{-\infty}^{\infty} x(t)K_{\alpha}(t,u)dt$$
where a real number $p$ represents the order of FRFT, $\alpha =p\pi /2$ and $u$ is the frequency in fractional Fourier domain (FRFD). $K_{\alpha }(t,u)$ is the transform kernel:
$${K_\alpha }(t,u) = \left\{ {\begin{array}{ll} {{A_\alpha }{e^{j[({t^2} + {u^2})\frac{{\textrm{{cot}}\alpha }}{2} - ut\textrm{{csc}}\alpha ]}},} & {\alpha \ne n\pi }\\ {\delta [t - {{( - 1)}^n}u],} & {\alpha = n\pi } \end{array}} \right.$$
where
$${A_\alpha } = \sqrt {\frac{{1 - j\cot \alpha }}{{2\pi }}}.$$
Similarly, the inverse FRFT is expressed as
$$x(t)=F^{-p}[x(t)]=\int_{-\infty}^{\infty}X(u)K_{-\alpha}(t,u)dt.$$
Equation (9) shows that the fractional Fourier transform (FRFT) has its natural advantages in analyzing non-stationary signals because its decomposition basis is a set of chirp signals with linearly time-varying frequencies. According to the definition of FRFT, a single-component LFM signal with finite length is able to be transformed into a peak function in a proper FRFD. Similarly, for each component of a multi-component LFM signal, a matched domain in which the component concentrates into a narrow peak can be found as the FRFD order changes. Figure 1 shows a sketch of the energy distribution of a three-component LFM signal in different FRFDs. It illustrates that the multiple components with different chirp rates $\mu _n$ are narrower in band in their own matched domains than in the FD or in the mismatched domains. The relationship between the chirp rate $\mu _n$ and its matched FRFD order $\alpha _n$ denotes as [15]
$$\mu_n = - \cot \alpha_n.$$
Hence, it is quite effective to estimated the chirp rate $\mu _n$ through the search of the optimal FRFD order $\alpha _n$.

 figure: Fig. 1.

Fig. 1. A sketch of the distribution of a three-component LFM signal in different FRFDs.

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From this perspective, it can be considered that when OCT is performed on the structure with multiple layers, a multi-component chirp signal will appear in the A-scan coherent signal caused by the GVD. The initial frequency $u$ of the multi-component LFM signal corresponds to the depth $\tau _n$ at which the dispersion is located, and the chirp rate $\mu _n$ corresponds to the coefficient of the GVD term $\frac {1}{2}{\beta _{n2}}\Delta {z_n}$. In essence, the initial frequency of each component is different, and the chirp rate is also different. According to the FRFT, the process of the parameter estimation of the chirp signal always detects the chirp rate corresponding to the FRFD with the largest peak of the power spectrum, and then derives the initial frequency according to the peak position of the power spectrum. Compared to the direct exhaustive search of the GVD coefficient, which usually requires the prior knowledge to determine the range of the exhaustive search, when the FRFT is used to estimate the matching order corresponding to the GVD coefficient, the search range has been mapped to [0, 2].

For dispersion correction in FD OCT, what we need to do is to detect all the chirp rates $\mu _n$ of the interference signal in the FRFD, and then the reconstructed signal can be obtained after removing these chirp terms in the time domain. Moreover, if the optical path difference $\Delta {z_n}$ is known as prior information, the chirp rate $\mu _n$ can directly represent the GVD coefficient $\beta _{n2}$. Note that although one of the multiple components is able to be focused in a particular domain, the energy of other components in this domain will interfere with it and affect the detection. Therefore, the suppression of such interference is also very important for the parameter detection in the signal from FD OCT.

3. Principle of the proposed FRFD stepped detection algorithm

For most practical cases, due to the unknown internal structure of the sample, it is hard to calculate the dispersion at different depths. Thus, for the interference signal from each A-scan in FD OCT, the method adopted here for dispersion detection is to search for the power spectrum peak of the signal as the FRFD order $\alpha$ varies from 0 to 2, acquiring a location map with the coordinate of ($u$, $\alpha$). Then the parameter detection of dispersion could be achieved through the peak searching in different FRFDs. We firstly consider the interference signal from a single reflective layer in FD OCT, then the target of dispersion detection can be described as

$$\{ {{\hat u}_0},{{\hat \alpha }_0}\} = \arg \mathop {\max }_{u,\alpha } \{ {\left| {{X_\alpha }(u)} \right|^2}\}$$
where $\hat u _0$ and $\hat \alpha _0$ are the initial estimation of parameters ($u$, $\alpha$). ${\left | {{X_\alpha }(u)} \right |^2}$ is the $\alpha$-order FRFD power spectrum of the detected interference signal from FD OCT.

Equation (11) illustrates the process of the two-dimensional search. However, if the traditional exhaustive method, which utilizes a point-by-point search, is implemented to search for the peak of the power spectrum ${\left | {{X_\alpha }(u)} \right |^2}$, an extensive computation will be introduced. Especially when high detection accuracy is required, the size of the search step should be select as small as possible, which might lead to much more calculation time. In order to reconcile the contradiction between the accuracy and the complexity of the algorithm, a Stepped Detection Algorithm is proposed in this paper. First, a large search step for a coarse search is select, and we look for the peak of the power spectrum ${\left | {{X_\alpha }(u)} \right |^2}$ in these selected domains. The peak of ${\left | {{X_\alpha }(u)} \right |^2}$ from the coarse search process corresponds to the approximate FRFD order $\hat \alpha _{co}$ where the dispersion relatively concentrates. This large size of search step can effectively reduce the FRFT times in the search process, but it is hard to directly obtain the exact FRFD order where the dispersion concentrates into a compact impulse. Thus an effective fine search is necessarily introduced to find the accurate peak value of the power spectrum in FRFD around the approximate order $\hat \alpha _{co}$. Then the exact FRFD order can be derived from the coordinate of the accurate peak of the power spectrum ${\left | {{X_\alpha }(u)} \right |^2}$. Here the quasi-Newton method [16] for the iterative fine search process is employed, which denotes as

$$\left[ {\begin{array}{l} {{{\hat u}_{i + 1}}}\\ {{{\hat \alpha }_{i + 1}}} \end{array}} \right] = \left[ {\begin{array}{l} {{{\hat u}_i}}\\ {{{\hat \alpha }_i}} \end{array}} \right] - {\lambda _i}{\textbf{{H}}_\textbf{{i}}}{\left. {\left[ {\begin{array}{l} {\frac{{\partial {{\left| {{X_\alpha }(u)} \right|}^2}}}{{\partial \alpha }}}\\ {\frac{{\partial \left| {{X_\alpha }(u)} \right|}^2}{{\partial u}}} \end{array}} \right]} \right|_{u = {{\hat u}_i},\alpha = {{\hat \alpha }_i}}},$$
where ${\lambda _i}$ is the length of the search step at $i$ times iteration, and ${\textbf {{H}}_\textbf {{i}}}$ is the estimation of the Hessian of ${{{\left | {{X_{{\alpha _{}}}}({u_{}})} \right |}^2}}$ when $\alpha$ = ${\hat \alpha _i}$, and $u$ = ${\hat u_i}$. The coordinates of the peak in the fine search process are noted as $\hat u_n$ and $\hat \alpha _n$. $\hat u_n$ corresponds to the depth $\hat \tau _n$ where the dispersion is located and $\hat \alpha _n$ corresponds to the coefficient of the GVD term $\frac {1}{2}{\beta _{n2}}\Delta {z_n}$ at the depth $\hat \tau _n$.

This model of single-component interference signal is quite suitable and efficient for the compensation of systematic dispersion caused by the mismatch of the two interference arm during the OCT imaging [11]. However, as for most practical samples with stratified media or irregular internal structure, the dispersion of the sample is often depth-dependent. The interference signal from a single A-scan presents as a superposition of multiple chirp components as described in Eq. (4). Thus it is necessary to detect the matched order of each chirp component of the signal. Since the dispersion detection method in this paper is based on the peak search in FRFD, there is always one specific domain where the energy of the chirp component concentrates. However, the FRFD orders which correspond to the GVD coefficients from different depths of the sample, are generally close and within a certain range. As a result, for the domain that one chirp component is concentrated as a peak function, other components in the signal also have some energy in this relatively unfocused domain, albeit the peak not that sharp. Hence the multiple components of the signal from FD OCT might interfere with each other in the peak search process. Hereby, we employ a signal separation method to suppress this interference. Specifically, in the first round of parameter detection which includes a coarse search followed by a fine search, the strongest peak is concentrated and its corresponding FRFD order is detected. Then it is blocked out using FRFD filtering method and the next round goes on. By this iterative process, all the FRFD orders corresponding to the GVD coefficients from each depth of the signal can be detected until the energy left is below a certain threshold.

4. Process of dispersion correction and complexity analysis

4.1 Process of dispersion correction

The process of the proposed dispersion detection and correction is presented and consists of three parts as shown in Fig. 2: I) Analytical Signal Construction; II) Dispersion Detection ; III) Spectrum Reconstruction. The specific processing procedures are discussed below. It is worth noting that the sampling theorem and dimensional normalization must be satisfied in each process of parameter calculation [17].

 figure: Fig. 2.

Fig. 2. Flowchart of the proposed dispersion correction algorithm.

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Step 1: Analytical Signal Construction. The interference signal from FD OCT is obtained as $S(\omega )$, and the analytical form of $S(\omega )$ is given by

$${S_{a}}(\omega ) = S(\omega ) + j\hat S(\omega ) ,$$
where $\hat S(\omega )$ is the Hilbert transformation of the interference signal $S(\omega )$.

Step 2: Coarse Search. Select several FRFD orders $\alpha$ within a range of [0, 2]. For each selected $\alpha$, the corresponding FRFD spectrum $X_{\alpha }(u)$ is obtained by the $\alpha$-order FRFT of the analytical signal ${S_{a}}(\omega )$. According to Eq. (11), search for the peak of the power spectrum ${{{\left | {{X_{{\alpha _{}}}}({u_{}})} \right |}^2}}$ in this FRFD. Select the maximum among all the searched peaks, and the coordinate of this maximum value is obtained as $\hat u_{co}$ with its corresponding FRFD order as $\hat \alpha _{co}$.

Step 3: Fine Search. Search for the FRFD order $\hat \alpha _n$ that maximizes ${{{\left | {{X_{{\alpha _{}}}}({u_{}})} \right |}^2}}$ near the $\hat \alpha _{co}$-order FRFD utilizing the quasi-Newton iteration. The peak value of ${{{\left | {{X_{{\alpha _{}}}}({u_{}})} \right |}^2}}$ is given as $A_n$.

Step 4: Peak Detection. Compare the amplitude of the peak $A_n$ with the selected threshold $\gamma$ as shown in Eq. (14), and determine whether there is a distinct dispersion.

$$\rho = \left| {{A_n}} \right|\mathop {\mathop > ^{{\textrm{H}_\textrm{1}}} }_{\mathop <\limits _{{{{\textrm{H}}}_\textrm{0}}} } \gamma$$
where $\rho$ is the test statistic.Here the threshold $\gamma$ could be selected by means of Constant False Alarm Rate (CFAR) [18], which does not require any prior distribution of interference but a given false alarm rate. $\rm {H_1}$ and $\rm {H_0}$ represent the assumptions that $A_n$ exceeds the threshold $\gamma$ or not.

The coordinate corresponding to $A_n$ which outstrip the threshold $\gamma$ is acquired as $\hat u _n$.

Step 5: Signal Separation. Block out the detected peak by

$${{X'}_{{\hat \alpha _n}}}(u) = {X_{{ \hat \alpha _n}}}(u) \cdot H(u,{\hat \alpha _n}).$$
Here a FRFD band-pass filter $H(u, {\hat \alpha _n})$ is designed with a central instantaneous frequency $\hat u_n$ and is given by
$$H({u,\hat \alpha_n}) = \left\{ \begin{array}{l} 0, {\hat u_n} - {w_n} \le u \le {\hat u_n} + {w_n} \\ 1, \textrm{{otherwise}} \end{array} \right.$$
where ${w_n} = 2\pi /({T_{ns}}\csc {\hat \alpha _n})$ with $T_{ns}$ the sampling interval in FRFD at the n-th reflective layer of the sample [19].

Step 6: Repeat Step $2$ to $5$ until there is no detected peak that exceeds the threshold $\gamma$. Then the orders ${\hat \alpha _n}$ ($n=1,2,\ldots$) that were detected previously can be used as parameters for dispersion correction.

Step 7: Dispersion Correction. Reconstruct the interference signal from FD OCT by a phase correction term

$${{\bar \theta }_n}(\omega ) = - \mathop \sum _n {k_n}{(\omega - {\omega _0})^2}$$
where ${k_n} = \pi \cot {\hat \alpha _n}$ is the adjustment coefficient at the n-th layer of the signal to correct the GVD unbalance. For each A-scan signal obtained from the OCT system, the reconstructed interference signal is obtained through conducting Step 1 to Step 7 individually.

Step 8: FFT imaging. The compensated FD OCT image is obtained by the FFT of the reconstructed interference signals from all A-scans.

It should be emphasized that our target is not to image the signal collected by OCT system in one specific FRFD, but to estimate the parameters of the dispersion through the detection of the LFM signal in different FRFDs, and then to correct the spectral phase of the signal in the time domain. This allows the GVD from different layers to be fully corrected without causing overcompensation due to the change of imaging domains.

If the optical path difference $\Delta {z_n}$ is given as prior information, the corresponding parameters of group delay $\tau _n$ and group-velocity dispersion ${\beta _{n2}}$ can be derived from the relationship

$$\left\{ \begin{array}{l} {{\hat \beta }_{n2}} = \frac{{ - 2\pi \cot {{\hat \alpha }_n}}}{{\Delta {z_n}}}\\ {{\hat \tau }_n} = 2\pi {{\hat u}_n}\csc {{\hat \alpha }_n}. \end{array} \right.$$
Since we can directly obtain the adjustment coefficients $k_n$ through the FRFD Stepped Detection Algorithm for phase correction, it is not necessary to calculate the corresponding GVD coefficients ${\beta _{n2}}$ separately. But it is still feasible to extract ${\beta _{n2}}$ using Eq. (18).

4.2 Complexity analysis

The computation complexity of the dispersion detection algorithm mainly depends on two aspects: transformation and the search interval of the parameters. The algorithm in this paper is mainly based on the FRFT, which has a variety of fast algorithms to reduce the computing time. Considering the computational efficiency in our work, the FRFT in the dispersion detection process is carried out by the sampling type fast algorithm [20], which can be implemented by only a few times of FFT computations. This algorithm takes a complexity of $2N + \frac {3}{2}N{\log _2}N$ for a single FRFT where $N$ is the signal length.

In addition, most of the existing algorithms for dispersion correction are based on exhaustive search methods [10,11]. In this section, the single-round search process is considered to compare the complexity of the dispersion detection. In order to guarantee the accuracy of the dispersion detection, the size of the search step $Q$ in those exhaustive methods should be as small as possible. However, the small step size may lead to an additional computation complexity and increase the difficulty of the hardware implementation. In our work a coarse search with a large step size is firstly carried out to search for the approximate FRFD order where the dispersion relatively concentrates, and then the quasi-Newton method is implemented to find the exact FRFD order where the dispersion concentrates into a compact impulse (see in Step 2 -4). According to Eq. (12), the quasi-Newton iteration only contains several times of derivations and complex additions, which are quite limited relative to the calculation in the coarse search process. Thus the total complexity of the presented dispersion detection is mainly determined by the complexity of the coarse search process (Step 2). Compared with those algorithms of exhaustive search methods [10,11], the step size $Q$ in the proposed algorithm is greatly reduced to $R(R \ll Q)$, and the total complexity is estimated as $O(RN{\log _2}N)$.

The complexity comparison in Table 1 is given in the case of single-component dispersion detection. The total computation complexity of the dispersion detection in the proposed algorithm is compared with two other existing algorithms [10,11], which reveals the proposed FRFD Stepped Detection Algorithm enjoys the lowest complexity and outstrips the other two existing methods in the ability for dispersion compensation. Moreover, if the dispersion is present at multiple depths, the total calculation complexity for all-depth dispersion detection will depend on the amount of dispersion components, but is still lower than those exhaustive methods [10,11].

Tables Icon

Table 1. Comparison of complexity for different dispersion correction algorithms

5. Simulations and experiments

5.1 Simulations

In order to verify the validity of the proposed dispersion correction method, a simulated interference signal from FD OCT, which has four hypothetical reflective layers at 200 $\mu$m, 400 $\mu$m, 600 $\mu$m and 800 $\mu$m respectively, is adopted in the following simulations of this section. All the simulations are set up at a central wavelength of 1310 nm with a bandwidth of 170 nm at -10 dB. The GVD coefficients at each layer of the simulated sample are 50 $\textrm{f}{\textrm{s}^\textrm{2}}$/m, 70 $\textrm{f}{\textrm{s}^\textrm{2}}$/m, 100 $\textrm{f}{\textrm{s}^\textrm{2}}$/m, and 200 $\textrm{f}{\textrm{s}^\textrm{2}}$/m, respectively. In addition, all the signal processing procedures satisfy the sampling theorem and the dimensional normalization.

For an interference signal along A-scan in FD OCT, the all-depth dispersion detection is to search for the matched FRFDs and the coordinates where the peak value of the power spectrum exceeds the selected threshold. Figure 3 shows the relationship between the dispersion in an A-scan and its two parameters, that is, the position of the energy of each dispersion and the FRFD in which it is located.

 figure: Fig. 3.

Fig. 3. The energy distribution of the dispersion of a simulated sample with four reflective layers. The horizontal axis represents the FRFT order, and the vertical axis represents the depth of the sample.

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The coordinate of each dispersion component in Fig. 3 give the depth $\tau _n$ and the FRFD order $\alpha _n$ where this dispersion occurs. Moreover, it can be intuitively seen from Fig. 3 that the dispersion varies at different depths, and so does the matched FRFDs where the dispersion concentrates. In general, the dispersion increases with the depth because of the increased delay during the propagation of the light inside the sample. When the dispersion occurs at different depths in an A-scan, the matching FRFT order corresponding to the dispersion from the surface to the deep layer changes sequentially from large to small between 0 and 1.

As shown in Fig. 4, the signals of different color represent the compensated signal compensated by different correction orders of the same A-scan. The A-scan signal corrected by the order 1 stands for the FFT of the original interference signal. At the same depth, different correction order used for dispersion correction result in different dispersion compensation effect. It can also be found that if a single FRFD order is used to compensate the dispersion for an A-scan signal, it can be accurately compensated at a specific depth, but at the same time there are both under- or over-compensation at other depths of the A-scan. It can be seen that as the depth increases, the spectral spread caused by the dispersion becomes more severe, which leads to a deterioration in image quality, for example, blurring the boundaries of the OCT image. In view of the fact that the depths and the FRFT orders of the four dispersion components are different in this simulation experiment, it is certainly impossible to compensate the four dispersion components with only one FRFT order, as shown in Fig. 4, the red, yellow, purple, and green lines are respectively the results of dispersion compensation in this A-scan with four different $\alpha _j$. It can also be clearly seen from Fig. 4 that the dispersion can be precisely eliminated only when the chirp rate of the dispersion component is completely matched with the FRFT order used in the correction. Whether the dispersion component is overcompensated (the correction order greater than its matching order) or undercompensated (the correction order less than its matching order), there are GVD terms left in the interference signal, that is, the dispersion still exists.

 figure: Fig. 4.

Fig. 4. Compensation results with the same correction order in each simulated interference signal from FD OCT. The signal in blue is the original interference signal without compensation and is obtained by FFT imaging(its correction order equals to 1). The other signals are compensated by four correction orders $\alpha _j=$ 0.998 (red line), 0.996 (yellow line), 0.994 (purple line), 0.992 (green line) respectively. The dispersion at depth $\tau _j =$ 200, 400, 600, 800 microns are precisely compensated using four $\alpha _j$ respectively, while that at other depths undercompensated or overcompensated.

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We demonstrate the capabilities of the proposed algorithm using the simulated four-layer dispersive signal in Fig. 5. Instead of imaging in a certain FRFD or correcting the dispersion at different depths with the same FRFT order, in this paper we detect the parameters of the dispersion in FRFD and reconstruct the signal in the time domain, acquiring dispersion compensated at all depths simultaneously. To illustrate this, we compare the dispersion compensation by a single correction order with that by the proposed FRFD Stepped Detection Algorithm in Fig. 5. Figures 5(b) and 5(c) show the compensated signal when the same correction order is adopted as the depth of 200 and 800 respectively. In both cases, the dispersion is corrected at only one certain depth. Specifically, in Fig. 5(b), the dispersion at depths above 200 is undercompensated, while in Fig. 5(c) the dispersion at depths below 800 is overcompensated. Instead, with the proposed FRFD Stepped Detection Algorithm, the dispersion is corrected at all depths and presents four compact peaks along the depth scan.

 figure: Fig. 5.

Fig. 5. Comparison of the simulated interference signal from FD OCT which is compensated by single FRFT order and by the proposed FRFD Stepped Detection Algorithm. (a) is obtained without dispersion compensation. (b) and (c) are the compensated signals obtained by using the same correction order as the depths of 200 and 800, respectively. (d) is the compensated signal obtained by using the proposed FRFD Stepped Detection Algorithm.

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5.2 Experiments

5.2.1 Stratified ZnSe experiment

To illustrate the effect of the proposed algorithm on dispersion suppression and the improvement of imaging quality, a dispersive sample of Zinc Selenide (ZnSe) which has a stratified structure, is imaged in a FD OCT system. The sample consists of two layers of 1 mm thick ZnSe plates stacked. We use a swept source with a center wavelength of 1310 nm and a bandwidth of 136.04 nm at -10 dB. All images presented in the following experimental analysis use the same intensity scale so as to highlight the visual differences of the dispersion dynamic range.

Figure 6 shows the OCT images of the ZnSe sample. The three yellow lines in Figs. 6(a)–6(d) display the three reflective layers of the ZnSe sample. The top yellow line corresponds to the top layer of the first ZnSe plate. The yellow line in the middle corresponds to the interface between the two plates. The bottom yellow line corresponds to the bottom layer of the second ZnSe plate.

 figure: Fig. 6.

Fig. 6. Cross-sectional view, B-scan, of a ZnSe sample. (a) Images without dispersion compensation. (b) Images corrected by PSF Peak Search method. (c) Images corrected by Conjugate Transform. (d) Images corrected by the proposed FRFD Stepped Detection Algorithm. (e) The enlarged and overlapped version of the middle layer signal along the dashed white line from (a)-(d) (at 170 microns) .

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Since the dispersion leads to an increase of axial bandwidth in OCT imaging, The three reflective surfaces shown by the yellow line in Fig. 6(a) appear blurred, resulting in a broadening of the boundary that should be clear. In Fig. 6(b), after the dispersion compensated by the PSF Peak Search method [11], the bottom layer become clearer, whereas the top layer become even more blurred than the uncompensated signal in Fig. 6(a). As analyzed in previous simulations in Section 5.1, this is mainly because the iterative process of the PSF search takes a single round and thus only the biggest peak could be obtained. Consequently, the dispersion in other regions are compensated using the same FRFT order as the order obtained by PSF search method and thus cannot be precisely corrected. We note that the Conjugate Transform method [10] in Fig. 6(c) can compensate the dispersion at all depths in some degrees. But it leads to a heavy cost of calculation due to the exhaustive search process of both the second-order dispersion $\beta _{n2}$ and the third-order dispersion $\beta _{n3}$. Figure 6(e) presents the improvement of the axial resolution using three dispersion compensation methods, which illustrates that the proposed FRFD Stepped Detection Algorithm is able to correct the dispersion at all depths. Compared to the other two methods [10,11], the highest axial resolution of 4 microns (purple line in Fig. 6(e)) is achieved using the proposed algorithm. The proposed algorithm with the amount of computation is almost 1/26 of the Conjugate Transform, and 1/18 of the PSF Peak Search method, for each A-scan.

It is worth mentioning that for all the three dispersion correction methods mentioned in Fig. 6, the accuracy of dispersion detection largely depends on the step size selected in the coarse search process. Since the image of the sample collected from OCT setup are usually accompanied by lower GVD and its depth-dependent acceleration, we just focus on a smaller range of the transform order in the proposed algorithm, that is, $0.9 \le \alpha \le 1.1$. In general, for most real samples, the step size of the coarse search can be set to one tenth of the total search range, and the quasi-Newton optimization method is applied to make the result converge faster in the iteration (only four iterations in Fig. 6(d)). The step size of the coarse search can be set smaller but at the expense of increased computational complexity. Even so, the proposed FRFD Stepped Detection Algorithm consumes much less time and computation than other methods under the same conditions.

5.2.2 Plant-tissue experiment

Even for biological tissue samples with more complex structures, the proposed algorithm still has a sound ability in sharpening the dispersion at all depths. Figure 7 presents B-scan images of an onion with different dispersion compensation methods. Two parts of each image are marked in the boxes and their zoom-in images are compared in Figs. 7(b1)–7(b8). It is concluded that using three dispersion compensation methods, the interference signal from FD OCT is compensated to different degrees. As shown in Fig. 7(a4), the result using the proposed FRFD Stepped Detection Algorithm obtains the highest image quality and signal-to-noise ration compared with the other dispersion compensation methods in Figs. 7(a1)–7(a3). Moreover, the depth scan comparison is given in Fig. 8, from which we can see the interference signal from the selected A-scan has four reflective layers. Figures 8(a2) and 8(a3) illustrate that both methods of the PSF Peak Search [11] and the Conjugate Transform [10] perform well when dealing with the dispersion at deep layer. However, a axial broadening of the signal at the top layer is introduced using the PSF Peak Search method [11]. It is found that the Conjugate Transform method [10] can compensate the dispersion at all depths, but the calculation time is heavily added for a single A-scan signal. Figures 8(b1) and 8(b2) present overlapped versions of the A-scan signal at two different layers, respectively. In Fig. 8(b1), comparing the bandwidth of the compensated signal by the PSF Peak Search method [11] (red line) and of the signal without compensation (blue line), the increase of axial bandwidth of the top layer signal can also be observed. As shown in Fig. 8(b2), the highest resolution of 5 microns is achieved at the deep layer using the FRFD Stepped Detection Algorithm (purple line), with a computing cost of 1/7 of the Conjugate Transform, and of 1/5 of the PSF Peak Search method, for a single A-scan. As for the whole OCT image dispersion compensation, this efficiency enhancement will enlarge and provide availability of real-time computing.

 figure: Fig. 7.

Fig. 7. Cross-sectional view, B-scan, of an onion. (a1) Images without dispersion compensation. (a2) Images corrected by the PSF Peak Search method. (a3) Images corrected by the Conjugate Transform method. (a4) Images corrected by the proposed FRFD Stepped Detection Algorithm. (b1)-(b8) Zoom-in images in the marked boxes from (a1-a4).

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 figure: Fig. 8.

Fig. 8. (a1-a4) A certain A-scan signal along the dashed white line in Fig. 7(a-d). (b1-b2) Overlapped signals of the top layer (onion epidermis) and the deep layer signal, respectively.

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5.2.3 Human coronary artery experiment

Dispersion that occurs at different depths or that is not specifically considered for elimination can affect the brightness and boundaries of the OCT image, which can lead to difficulty or error in determining tissue type and morphology. Because the light are able to be transmitted through the optical fiber, OCT has been introduced into blood vessels by embedding an optical fiber into the imaging catheter. As a medical imaging modality, Intravascular Optical Coherence Tomography (IVOCT) can display transverse section images of the coronary with high axial resolution. Nonetheless, quantitative analysis and further diagnosis can only be achieved by accurate imaging for vulnerable plaques. Since high resolution which requires broad optical bandwidths of the OCT system is sensitive to sample dispersion [3], all-depth dispersion correction must be attached importance in high-resolution IVOCT imaging. Herein, a human coronary artery is taken as a sample to verify the capability of the proposed FRFD Stepped Detection Algorithm for dispersion correction in real bio-tissues with more general structures.

In this experiment, a swept light source OCT system is used with the center wavelength being 1.7 ${\mu }$m. And a high speed scanning laser is taken as the swept source, which offers a sweeping range of 173.8 nm and a sweeping rate of 90 kHz. All these enable a large penetration depth and a high SNR, which contributes to the visualization of the whole plaque [21]. Fig. 9 displays IVOCT images of two arterial sites with different pathological characteristics, which demonstrates the improvement of image quality using the proposed FRFD Stepped Detection Algorithm. Figures 9(a1) and 9(b1) are obtained without dispersion compensation, while Figs. 9(a2) and 9(b2) are images corrected by the proposed FRFD Stepped Detection Algorithm. In Fig. 9(a1), a sparse region (marked by the blue arrow) is found, which indicates the feature of calcified plaque. As shown in Fig. 9(a2), after dispersion correction by the proposed algorithm, the calcified plaques in the IVOCT image can be more clearly identified. Similarly, in Fig. 9(b1), a bulky sparse region (marked by the blue arrow) indicates the feature of thick-cap fibroadenoma (ThCFA). After dispersion correction by our algorithm, the structure of the ThCFA are presented more accurately. This two kinds of plaques are verified by hematoxylin and eosin (H&E) histology in Fig. 3 Ie and IIe of Ref. [21].

 figure: Fig. 9.

Fig. 9. IVOCT images of a human coronary artery. (a1-a2) and (b1-b2) images obtained from two different sites. (a1) and (b1) Dispersion corrupted B-scan images of coronary artery obtained with IVOCT system. (a2) and (b2) B-scan images after sample dispersion correction by the proposed FRFD Stepped Detection Algorithm.

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For comparison, two groups of magnified images obtained by different dispersion compensation methods are illustrated in Fig. 10. Through analyzing the results improved by the three methods, it is clearly seen that images corrected by the FRFD Stepped Detection Algorithm in Figs. 10(a4) and 10(b4) have the highest image quality and the structure of plaques in the images are more feasible to differentiate. In addition, quantified improvement indexes of image quality and axial resolution are given in Table 2.

 figure: Fig. 10.

Fig. 10. Zoom-in images of a human coronary artery at two different sites using different dispersion compensation methods. (a1) and (b1) IVOCT images without dispersion compensation. (a2) and (b2) IVOCT images corrected by the PSF peak search method. (a3) and (b3) IVOCT images corrected by the Conjugate Transform method. (a4) and (b4) IVOCT images corrected by the proposed FRFD Stepped Detection Algorithm.

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Tables Icon

Table 2. Performance analysis of different dispersion correction algorithms.

Table 2 compares the enhancement indexes for different dispersion compensation methods, which are defined as $\textrm{EI = B/}{\textrm{B}_\textrm{0}}$. $\rm {B}$ and $\rm {B_0}$ describe the Full Width at Half Maximum (FWHM) of the signal from a certain reflective surface with and without dispersion correction, respectively. The results in Table 2 show that the proposed FRFD Stepped Detection Algorithm for dispersion compensation enhances the axial resolution of ZnSe as $\rm {EI} = 2.7272$. Even for the sample of human coronary artery with irregular structure, the resolution gets improved as $\rm {EI} = 1.3976$. Moreover, performance analysis on Peak Signal To Noise Ratio (PSNR) [22] is also shown in Table 2. The proposed algorithm yield a PSNR of 28.1229 for stratified ZnSe, outperforming the PSNR of 26.5706 achieved by the PSF Search method [11], of 26.7928 achieved by the Conjugate Transform method [10]. The improvement of PSNR, which represents the overall image quality enhancement, is not obvious in human coronary artery because of its more complex internal structure. However, for the area where the lesion tissue exists (see in Fig. 9), this degradation of axial bandwidth and the limited enhancement of PSNR are enough to improve the brightness and sharpness of the OCT image in this area for further plaque identification.

6. Conclusion

In this paper, we provide a new insight on the depth-dependent dispersion from FD OCT, which can be viewed as an multi-component LFM signal with two parameters considered: the modulation frequency (which can be derived by the order of the matching FRFD), and the initial frequency (which can be extracted by detecting the coordinates of the peak in the matching FRFD). A new dispersion compensation method based on the FRFD Stepped Detection Algorithm is presented in our work, which does not need any prior knowledge and can adaptively compensate the dispersion at all depths without any overcompensation or undercompensation. Since the focusing performance of the group-velocity dispersion varies in different FRFDs, the proposed algorithm is able to extract the parameters of the dispersion from their matching domains and then to correct the phase terms of the interference signals before imaging. We utilize a coarse search followed by a fine search in the dispersion detection process, in which the expansion of the search step size and the optimization of quasi-Newton method greatly improve the accuracy of dispersion detection and reduce the computation complexity simultaneously. In order to suppress the mutual interference between the dispersion from different depths of the sample, a signal separation method based on the FRFD filtering is employed. The algorithm presented in this paper provides a new approach to depth-dependent dispersion parameter detection and dispersion compensation, which can be used in other fields with similar requirements. The simulation results show the theoretical rationality of the proposed method in compensating depth-dependent dispersion of the signal in FD OCT. The applications to the dispersion compensation in ZnSe and onion verify the feasibility of our algorithm in both stratified media and bio-tissues. Moreover, implemented in IVOCT systems, the proposed FRFD Stepped Detection Algorithm can help obtain high-resolution images of human coronary arteries and better identifies the anatomical morphology and chemical composition of atherosclerotic plaques. In the future, the proposed method can be instructively applied to broader fields of dispersion detection, such as medical imaging, optical fiber communication and astronomy.

Funding

National Natural Science Foundation of China (61421001, 61701028, 61975017).

Acknowledgments

We are particularly grateful to Dr. Zhongping Chen and his research team in University of California, Irvine, for providing the raw data of human coronary artery.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991). [CrossRef]  

2. 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]  

3. 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]  

4. A. Fercher, C. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002). [CrossRef]  

5. G. J. Tearney, B. E. Bouma, and J. G. Fujimoto, “High-speed phase- and group-delay scanning with a grating-based phase control delay line,” Opt. Lett. 22(23), 1811–1813 (1997). [CrossRef]  

6. M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12(11), 2404–2422 (2004). [CrossRef]  

7. D. L. Marks, A. L. Oldenburg, J. J. Reynolds, and S. A. Boppart, “Autofocus algorithm for dispersion correction in optical coherence tomography,” Appl. Opt. 42(16), 3038–3046 (2003). [CrossRef]  

8. A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Numerical dispersion compensation for partial coherence interferometry and optical coherence tomography,” Opt. Express 9(12), 610–615 (2001). [CrossRef]  

9. L. Pan, X. Wang, Z. Li, X. Zhang, Y. Bu, N. Nan, Y. Chen, X. Wang, and F. Dai, “Depth-dependent dispersion compensation for full-depth oct image,” Opt. Express 25(9), 10345–10354 (2017). [CrossRef]  

10. W. Zhang, X. Zhang, C. Wang, W. Liao, S. Ai, Z. Peng, J. Hsieh, N. Zhang, B. Tang, and P. Xue, “Conjugate transformation for dispersion compensation in optical coherence tomography imaging,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–7 (2019). [CrossRef]  

11. N. Lippok, S. Coen, P. Nielsen, and F. Vanholsbeeck, “Dispersion compensation in fourier domain optical coherence tomography using the fractional fourier transform,” Opt. Express 20(21), 23398–23413 (2012). [CrossRef]  

12. A. Fercher, C. Hitzenberger, G. Kamp, and S. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117(1-2), 43–48 (1995). [CrossRef]  

13. A. Kohlhaas, C. Fromchen, and E. Brinkmeyer, “High-resolution ocdr for testing integrated-optical waveguides: dispersion-corrupted experimental data corrected by a numerical algorithm,” J. Lightwave Technol. 9(11), 1493–1502 (1991). [CrossRef]  

14. L. B. Almeida, “The fractional fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42(11), 3084–3091 (1994). [CrossRef]  

15. V. Namias, “The fractional order fourier transform and its application to quantum mechanics,” IMA J. Appl. Math. 25(3), 241–265 (1980). [CrossRef]  

16. J. E. Dennis Jr. and J. J. More, “Quasi-newton methods, motivation and theory,” SIAM Rev. 19(1), 46–89 (1977). [CrossRef]  

17. H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996). [CrossRef]  

18. I. S. Reed and X. Yu, “Adaptive multiple-band cfar detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust., Speech, Signal Process. 38(10), 1760–1770 (1990). [CrossRef]  

19. R. Tao, B. Deng, W. Zhang, and Y. Wang, “Sampling and sampling rate conversion of band limited signals in the fractional fourier transform domain,” IEEE Trans. Signal Process. 56(1), 158–171 (2008). [CrossRef]  

20. S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine fourier transforms,” IEEE Trans. Signal Process. 48(5), 1338–1353 (2000). [CrossRef]  

21. Y. Li, J. Jing, E. Heidari, J. Zhu, Y. Qu, and Z. Chen, “Intravascular optical coherence tomography for characterization of atherosclerosis with a 1.7 micron swept-source laser,” Sci. Rep. 7(1), 14525 (2017). [CrossRef]  

22. Q. Huynh-Thu and M. Ghanbari, “Scope of validity of psnr in image/video quality assessment,” Electron. Lett. 44(13), 800–801 (2008). [CrossRef]  

References

  • View by:

  1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
    [Crossref]
  2. 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]
  3. 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]
  4. A. Fercher, C. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
    [Crossref]
  5. G. J. Tearney, B. E. Bouma, and J. G. Fujimoto, “High-speed phase- and group-delay scanning with a grating-based phase control delay line,” Opt. Lett. 22(23), 1811–1813 (1997).
    [Crossref]
  6. M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12(11), 2404–2422 (2004).
    [Crossref]
  7. D. L. Marks, A. L. Oldenburg, J. J. Reynolds, and S. A. Boppart, “Autofocus algorithm for dispersion correction in optical coherence tomography,” Appl. Opt. 42(16), 3038–3046 (2003).
    [Crossref]
  8. A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Numerical dispersion compensation for partial coherence interferometry and optical coherence tomography,” Opt. Express 9(12), 610–615 (2001).
    [Crossref]
  9. L. Pan, X. Wang, Z. Li, X. Zhang, Y. Bu, N. Nan, Y. Chen, X. Wang, and F. Dai, “Depth-dependent dispersion compensation for full-depth oct image,” Opt. Express 25(9), 10345–10354 (2017).
    [Crossref]
  10. W. Zhang, X. Zhang, C. Wang, W. Liao, S. Ai, Z. Peng, J. Hsieh, N. Zhang, B. Tang, and P. Xue, “Conjugate transformation for dispersion compensation in optical coherence tomography imaging,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–7 (2019).
    [Crossref]
  11. N. Lippok, S. Coen, P. Nielsen, and F. Vanholsbeeck, “Dispersion compensation in fourier domain optical coherence tomography using the fractional fourier transform,” Opt. Express 20(21), 23398–23413 (2012).
    [Crossref]
  12. A. Fercher, C. Hitzenberger, G. Kamp, and S. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117(1-2), 43–48 (1995).
    [Crossref]
  13. A. Kohlhaas, C. Fromchen, and E. Brinkmeyer, “High-resolution ocdr for testing integrated-optical waveguides: dispersion-corrupted experimental data corrected by a numerical algorithm,” J. Lightwave Technol. 9(11), 1493–1502 (1991).
    [Crossref]
  14. L. B. Almeida, “The fractional fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42(11), 3084–3091 (1994).
    [Crossref]
  15. V. Namias, “The fractional order fourier transform and its application to quantum mechanics,” IMA J. Appl. Math. 25(3), 241–265 (1980).
    [Crossref]
  16. J. E. Dennis Jr. and J. J. More, “Quasi-newton methods, motivation and theory,” SIAM Rev. 19(1), 46–89 (1977).
    [Crossref]
  17. H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996).
    [Crossref]
  18. I. S. Reed and X. Yu, “Adaptive multiple-band cfar detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust., Speech, Signal Process. 38(10), 1760–1770 (1990).
    [Crossref]
  19. R. Tao, B. Deng, W. Zhang, and Y. Wang, “Sampling and sampling rate conversion of band limited signals in the fractional fourier transform domain,” IEEE Trans. Signal Process. 56(1), 158–171 (2008).
    [Crossref]
  20. S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine fourier transforms,” IEEE Trans. Signal Process. 48(5), 1338–1353 (2000).
    [Crossref]
  21. Y. Li, J. Jing, E. Heidari, J. Zhu, Y. Qu, and Z. Chen, “Intravascular optical coherence tomography for characterization of atherosclerosis with a 1.7 micron swept-source laser,” Sci. Rep. 7(1), 14525 (2017).
    [Crossref]
  22. Q. Huynh-Thu and M. Ghanbari, “Scope of validity of psnr in image/video quality assessment,” Electron. Lett. 44(13), 800–801 (2008).
    [Crossref]

2019 (1)

W. Zhang, X. Zhang, C. Wang, W. Liao, S. Ai, Z. Peng, J. Hsieh, N. Zhang, B. Tang, and P. Xue, “Conjugate transformation for dispersion compensation in optical coherence tomography imaging,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–7 (2019).
[Crossref]

2017 (2)

L. Pan, X. Wang, Z. Li, X. Zhang, Y. Bu, N. Nan, Y. Chen, X. Wang, and F. Dai, “Depth-dependent dispersion compensation for full-depth oct image,” Opt. Express 25(9), 10345–10354 (2017).
[Crossref]

Y. Li, J. Jing, E. Heidari, J. Zhu, Y. Qu, and Z. Chen, “Intravascular optical coherence tomography for characterization of atherosclerosis with a 1.7 micron swept-source laser,” Sci. Rep. 7(1), 14525 (2017).
[Crossref]

2012 (1)

2008 (2)

Q. Huynh-Thu and M. Ghanbari, “Scope of validity of psnr in image/video quality assessment,” Electron. Lett. 44(13), 800–801 (2008).
[Crossref]

R. Tao, B. Deng, W. Zhang, and Y. Wang, “Sampling and sampling rate conversion of band limited signals in the fractional fourier transform domain,” IEEE Trans. Signal Process. 56(1), 158–171 (2008).
[Crossref]

2004 (1)

2003 (3)

2002 (1)

A. Fercher, C. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

2001 (1)

2000 (1)

S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine fourier transforms,” IEEE Trans. Signal Process. 48(5), 1338–1353 (2000).
[Crossref]

1997 (1)

1996 (1)

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996).
[Crossref]

1995 (1)

A. Fercher, C. Hitzenberger, G. Kamp, and S. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117(1-2), 43–48 (1995).
[Crossref]

1994 (1)

L. B. Almeida, “The fractional fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42(11), 3084–3091 (1994).
[Crossref]

1991 (2)

A. Kohlhaas, C. Fromchen, and E. Brinkmeyer, “High-resolution ocdr for testing integrated-optical waveguides: dispersion-corrupted experimental data corrected by a numerical algorithm,” J. Lightwave Technol. 9(11), 1493–1502 (1991).
[Crossref]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

1990 (1)

I. S. Reed and X. Yu, “Adaptive multiple-band cfar detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust., Speech, Signal Process. 38(10), 1760–1770 (1990).
[Crossref]

1980 (1)

V. Namias, “The fractional order fourier transform and its application to quantum mechanics,” IMA J. Appl. Math. 25(3), 241–265 (1980).
[Crossref]

1977 (1)

J. E. Dennis Jr. and J. J. More, “Quasi-newton methods, motivation and theory,” SIAM Rev. 19(1), 46–89 (1977).
[Crossref]

Ai, S.

W. Zhang, X. Zhang, C. Wang, W. Liao, S. Ai, Z. Peng, J. Hsieh, N. Zhang, B. Tang, and P. Xue, “Conjugate transformation for dispersion compensation in optical coherence tomography imaging,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–7 (2019).
[Crossref]

Almeida, L. B.

L. B. Almeida, “The fractional fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42(11), 3084–3091 (1994).
[Crossref]

Arikan, O.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996).
[Crossref]

Boppart, S. A.

Bouma, B. E.

Bozdagt, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996).
[Crossref]

Brinkmeyer, E.

A. Kohlhaas, C. Fromchen, and E. Brinkmeyer, “High-resolution ocdr for testing integrated-optical waveguides: dispersion-corrupted experimental data corrected by a numerical algorithm,” J. Lightwave Technol. 9(11), 1493–1502 (1991).
[Crossref]

Bu, Y.

Cense, B.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Chen, Y.

Chen, Z.

Y. Li, J. Jing, E. Heidari, J. Zhu, Y. Qu, and Z. Chen, “Intravascular optical coherence tomography for characterization of atherosclerosis with a 1.7 micron swept-source laser,” Sci. Rep. 7(1), 14525 (2017).
[Crossref]

Coen, S.

Dai, F.

de Boer, J. F.

Deng, B.

R. Tao, B. Deng, W. Zhang, and Y. Wang, “Sampling and sampling rate conversion of band limited signals in the fractional fourier transform domain,” IEEE Trans. Signal Process. 56(1), 158–171 (2008).
[Crossref]

Dennis Jr., J. E.

J. E. Dennis Jr. and J. J. More, “Quasi-newton methods, motivation and theory,” SIAM Rev. 19(1), 46–89 (1977).
[Crossref]

Ding, J.-J.

S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine fourier transforms,” IEEE Trans. Signal Process. 48(5), 1338–1353 (2000).
[Crossref]

Duker, J. S.

El-Zaiat, S.

A. Fercher, C. Hitzenberger, G. Kamp, and S. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117(1-2), 43–48 (1995).
[Crossref]

Fercher, A.

A. Fercher, C. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

A. Fercher, C. Hitzenberger, G. Kamp, and S. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117(1-2), 43–48 (1995).
[Crossref]

Fercher, A. F.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Fromchen, C.

A. Kohlhaas, C. Fromchen, and E. Brinkmeyer, “High-resolution ocdr for testing integrated-optical waveguides: dispersion-corrupted experimental data corrected by a numerical algorithm,” J. Lightwave Technol. 9(11), 1493–1502 (1991).
[Crossref]

Fujimoto, J. G.

Ghanbari, M.

Q. Huynh-Thu and M. Ghanbari, “Scope of validity of psnr in image/video quality assessment,” Electron. Lett. 44(13), 800–801 (2008).
[Crossref]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Heidari, E.

Y. Li, J. Jing, E. Heidari, J. Zhu, Y. Qu, and Z. Chen, “Intravascular optical coherence tomography for characterization of atherosclerosis with a 1.7 micron swept-source laser,” Sci. Rep. 7(1), 14525 (2017).
[Crossref]

Hitzenberger, C.

A. Fercher, C. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

A. Fercher, C. Hitzenberger, G. Kamp, and S. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117(1-2), 43–48 (1995).
[Crossref]

Hitzenberger, C. K.

Hsieh, J.

W. Zhang, X. Zhang, C. Wang, W. Liao, S. Ai, Z. Peng, J. Hsieh, N. Zhang, B. Tang, and P. Xue, “Conjugate transformation for dispersion compensation in optical coherence tomography imaging,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–7 (2019).
[Crossref]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Huynh-Thu, Q.

Q. Huynh-Thu and M. Ghanbari, “Scope of validity of psnr in image/video quality assessment,” Electron. Lett. 44(13), 800–801 (2008).
[Crossref]

Jing, J.

Y. Li, J. Jing, E. Heidari, J. Zhu, Y. Qu, and Z. Chen, “Intravascular optical coherence tomography for characterization of atherosclerosis with a 1.7 micron swept-source laser,” Sci. Rep. 7(1), 14525 (2017).
[Crossref]

Kamp, G.

A. Fercher, C. Hitzenberger, G. Kamp, and S. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117(1-2), 43–48 (1995).
[Crossref]

Karamata, B.

A. Fercher, C. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Numerical dispersion compensation for partial coherence interferometry and optical coherence tomography,” Opt. Express 9(12), 610–615 (2001).
[Crossref]

Ko, T. H.

Kohlhaas, A.

A. Kohlhaas, C. Fromchen, and E. Brinkmeyer, “High-resolution ocdr for testing integrated-optical waveguides: dispersion-corrupted experimental data corrected by a numerical algorithm,” J. Lightwave Technol. 9(11), 1493–1502 (1991).
[Crossref]

Kowalczyk, A.

Kutay, M. A.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996).
[Crossref]

Lasser, T.

A. Fercher, C. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Numerical dispersion compensation for partial coherence interferometry and optical coherence tomography,” Opt. Express 9(12), 610–615 (2001).
[Crossref]

Leitgeb, R.

Li, Y.

Y. Li, J. Jing, E. Heidari, J. Zhu, Y. Qu, and Z. Chen, “Intravascular optical coherence tomography for characterization of atherosclerosis with a 1.7 micron swept-source laser,” Sci. Rep. 7(1), 14525 (2017).
[Crossref]

Li, Z.

Liao, W.

W. Zhang, X. Zhang, C. Wang, W. Liao, S. Ai, Z. Peng, J. Hsieh, N. Zhang, B. Tang, and P. Xue, “Conjugate transformation for dispersion compensation in optical coherence tomography imaging,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–7 (2019).
[Crossref]

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Lippok, N.

Marks, D. L.

More, J. J.

J. E. Dennis Jr. and J. J. More, “Quasi-newton methods, motivation and theory,” SIAM Rev. 19(1), 46–89 (1977).
[Crossref]

Namias, V.

V. Namias, “The fractional order fourier transform and its application to quantum mechanics,” IMA J. Appl. Math. 25(3), 241–265 (1980).
[Crossref]

Nan, N.

Nielsen, P.

Oldenburg, A. L.

Ozaktas, H. M.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996).
[Crossref]

Pan, L.

Park, B. H.

Pei, S.-C.

S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine fourier transforms,” IEEE Trans. Signal Process. 48(5), 1338–1353 (2000).
[Crossref]

Peng, Z.

W. Zhang, X. Zhang, C. Wang, W. Liao, S. Ai, Z. Peng, J. Hsieh, N. Zhang, B. Tang, and P. Xue, “Conjugate transformation for dispersion compensation in optical coherence tomography imaging,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–7 (2019).
[Crossref]

Pierce, M. C.

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Qu, Y.

Y. Li, J. Jing, E. Heidari, J. Zhu, Y. Qu, and Z. Chen, “Intravascular optical coherence tomography for characterization of atherosclerosis with a 1.7 micron swept-source laser,” Sci. Rep. 7(1), 14525 (2017).
[Crossref]

Reed, I. S.

I. S. Reed and X. Yu, “Adaptive multiple-band cfar detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust., Speech, Signal Process. 38(10), 1760–1770 (1990).
[Crossref]

Reynolds, J. J.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Srinivasan, V. J.

Sticker, M.

A. Fercher, C. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Numerical dispersion compensation for partial coherence interferometry and optical coherence tomography,” Opt. Express 9(12), 610–615 (2001).
[Crossref]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Tang, B.

W. Zhang, X. Zhang, C. Wang, W. Liao, S. Ai, Z. Peng, J. Hsieh, N. Zhang, B. Tang, and P. Xue, “Conjugate transformation for dispersion compensation in optical coherence tomography imaging,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–7 (2019).
[Crossref]

Tao, R.

R. Tao, B. Deng, W. Zhang, and Y. Wang, “Sampling and sampling rate conversion of band limited signals in the fractional fourier transform domain,” IEEE Trans. Signal Process. 56(1), 158–171 (2008).
[Crossref]

Tearney, G. J.

Vanholsbeeck, F.

Wang, C.

W. Zhang, X. Zhang, C. Wang, W. Liao, S. Ai, Z. Peng, J. Hsieh, N. Zhang, B. Tang, and P. Xue, “Conjugate transformation for dispersion compensation in optical coherence tomography imaging,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–7 (2019).
[Crossref]

Wang, X.

Wang, Y.

R. Tao, B. Deng, W. Zhang, and Y. Wang, “Sampling and sampling rate conversion of band limited signals in the fractional fourier transform domain,” IEEE Trans. Signal Process. 56(1), 158–171 (2008).
[Crossref]

Wojtkowski, M.

Xue, P.

W. Zhang, X. Zhang, C. Wang, W. Liao, S. Ai, Z. Peng, J. Hsieh, N. Zhang, B. Tang, and P. Xue, “Conjugate transformation for dispersion compensation in optical coherence tomography imaging,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–7 (2019).
[Crossref]

Yu, X.

I. S. Reed and X. Yu, “Adaptive multiple-band cfar detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust., Speech, Signal Process. 38(10), 1760–1770 (1990).
[Crossref]

Zawadzki, R.

A. Fercher, C. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Numerical dispersion compensation for partial coherence interferometry and optical coherence tomography,” Opt. Express 9(12), 610–615 (2001).
[Crossref]

Zhang, N.

W. Zhang, X. Zhang, C. Wang, W. Liao, S. Ai, Z. Peng, J. Hsieh, N. Zhang, B. Tang, and P. Xue, “Conjugate transformation for dispersion compensation in optical coherence tomography imaging,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–7 (2019).
[Crossref]

Zhang, W.

W. Zhang, X. Zhang, C. Wang, W. Liao, S. Ai, Z. Peng, J. Hsieh, N. Zhang, B. Tang, and P. Xue, “Conjugate transformation for dispersion compensation in optical coherence tomography imaging,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–7 (2019).
[Crossref]

R. Tao, B. Deng, W. Zhang, and Y. Wang, “Sampling and sampling rate conversion of band limited signals in the fractional fourier transform domain,” IEEE Trans. Signal Process. 56(1), 158–171 (2008).
[Crossref]

Zhang, X.

W. Zhang, X. Zhang, C. Wang, W. Liao, S. Ai, Z. Peng, J. Hsieh, N. Zhang, B. Tang, and P. Xue, “Conjugate transformation for dispersion compensation in optical coherence tomography imaging,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–7 (2019).
[Crossref]

L. Pan, X. Wang, Z. Li, X. Zhang, Y. Bu, N. Nan, Y. Chen, X. Wang, and F. Dai, “Depth-dependent dispersion compensation for full-depth oct image,” Opt. Express 25(9), 10345–10354 (2017).
[Crossref]

Zhu, J.

Y. Li, J. Jing, E. Heidari, J. Zhu, Y. Qu, and Z. Chen, “Intravascular optical coherence tomography for characterization of atherosclerosis with a 1.7 micron swept-source laser,” Sci. Rep. 7(1), 14525 (2017).
[Crossref]

Appl. Opt. (1)

Electron. Lett. (1)

Q. Huynh-Thu and M. Ghanbari, “Scope of validity of psnr in image/video quality assessment,” Electron. Lett. 44(13), 800–801 (2008).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

W. Zhang, X. Zhang, C. Wang, W. Liao, S. Ai, Z. Peng, J. Hsieh, N. Zhang, B. Tang, and P. Xue, “Conjugate transformation for dispersion compensation in optical coherence tomography imaging,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–7 (2019).
[Crossref]

IEEE Trans. Acoust., Speech, Signal Process. (1)

I. S. Reed and X. Yu, “Adaptive multiple-band cfar detection of an optical pattern with unknown spectral distribution,” IEEE Trans. Acoust., Speech, Signal Process. 38(10), 1760–1770 (1990).
[Crossref]

IEEE Trans. Signal Process. (4)

R. Tao, B. Deng, W. Zhang, and Y. Wang, “Sampling and sampling rate conversion of band limited signals in the fractional fourier transform domain,” IEEE Trans. Signal Process. 56(1), 158–171 (2008).
[Crossref]

S.-C. Pei and J.-J. Ding, “Closed-form discrete fractional and affine fourier transforms,” IEEE Trans. Signal Process. 48(5), 1338–1353 (2000).
[Crossref]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996).
[Crossref]

L. B. Almeida, “The fractional fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42(11), 3084–3091 (1994).
[Crossref]

IMA J. Appl. Math. (1)

V. Namias, “The fractional order fourier transform and its application to quantum mechanics,” IMA J. Appl. Math. 25(3), 241–265 (1980).
[Crossref]

J. Lightwave Technol. (1)

A. Kohlhaas, C. Fromchen, and E. Brinkmeyer, “High-resolution ocdr for testing integrated-optical waveguides: dispersion-corrupted experimental data corrected by a numerical algorithm,” J. Lightwave Technol. 9(11), 1493–1502 (1991).
[Crossref]

Opt. Commun. (2)

A. Fercher, C. Hitzenberger, G. Kamp, and S. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117(1-2), 43–48 (1995).
[Crossref]

A. Fercher, C. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

Opt. Express (5)

Opt. Lett. (2)

Sci. Rep. (1)

Y. Li, J. Jing, E. Heidari, J. Zhu, Y. Qu, and Z. Chen, “Intravascular optical coherence tomography for characterization of atherosclerosis with a 1.7 micron swept-source laser,” Sci. Rep. 7(1), 14525 (2017).
[Crossref]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

SIAM Rev. (1)

J. E. Dennis Jr. and J. J. More, “Quasi-newton methods, motivation and theory,” SIAM Rev. 19(1), 46–89 (1977).
[Crossref]

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Figures (10)

Fig. 1.
Fig. 1. A sketch of the distribution of a three-component LFM signal in different FRFDs.
Fig. 2.
Fig. 2. Flowchart of the proposed dispersion correction algorithm.
Fig. 3.
Fig. 3. The energy distribution of the dispersion of a simulated sample with four reflective layers. The horizontal axis represents the FRFT order, and the vertical axis represents the depth of the sample.
Fig. 4.
Fig. 4. Compensation results with the same correction order in each simulated interference signal from FD OCT. The signal in blue is the original interference signal without compensation and is obtained by FFT imaging(its correction order equals to 1). The other signals are compensated by four correction orders $\alpha _j=$ 0.998 (red line), 0.996 (yellow line), 0.994 (purple line), 0.992 (green line) respectively. The dispersion at depth $\tau _j =$ 200, 400, 600, 800 microns are precisely compensated using four $\alpha _j$ respectively, while that at other depths undercompensated or overcompensated.
Fig. 5.
Fig. 5. Comparison of the simulated interference signal from FD OCT which is compensated by single FRFT order and by the proposed FRFD Stepped Detection Algorithm. (a) is obtained without dispersion compensation. (b) and (c) are the compensated signals obtained by using the same correction order as the depths of 200 and 800, respectively. (d) is the compensated signal obtained by using the proposed FRFD Stepped Detection Algorithm.
Fig. 6.
Fig. 6. Cross-sectional view, B-scan, of a ZnSe sample. (a) Images without dispersion compensation. (b) Images corrected by PSF Peak Search method. (c) Images corrected by Conjugate Transform. (d) Images corrected by the proposed FRFD Stepped Detection Algorithm. (e) The enlarged and overlapped version of the middle layer signal along the dashed white line from (a)-(d) (at 170 microns) .
Fig. 7.
Fig. 7. Cross-sectional view, B-scan, of an onion. (a1) Images without dispersion compensation. (a2) Images corrected by the PSF Peak Search method. (a3) Images corrected by the Conjugate Transform method. (a4) Images corrected by the proposed FRFD Stepped Detection Algorithm. (b1)-(b8) Zoom-in images in the marked boxes from (a1-a4).
Fig. 8.
Fig. 8. (a1-a4) A certain A-scan signal along the dashed white line in Fig. 7(a-d). (b1-b2) Overlapped signals of the top layer (onion epidermis) and the deep layer signal, respectively.
Fig. 9.
Fig. 9. IVOCT images of a human coronary artery. (a1-a2) and (b1-b2) images obtained from two different sites. (a1) and (b1) Dispersion corrupted B-scan images of coronary artery obtained with IVOCT system. (a2) and (b2) B-scan images after sample dispersion correction by the proposed FRFD Stepped Detection Algorithm.
Fig. 10.
Fig. 10. Zoom-in images of a human coronary artery at two different sites using different dispersion compensation methods. (a1) and (b1) IVOCT images without dispersion compensation. (a2) and (b2) IVOCT images corrected by the PSF peak search method. (a3) and (b3) IVOCT images corrected by the Conjugate Transform method. (a4) and (b4) IVOCT images corrected by the proposed FRFD Stepped Detection Algorithm.

Tables (2)

Tables Icon

Table 1. Comparison of complexity for different dispersion correction algorithms

Tables Icon

Table 2. Performance analysis of different dispersion correction algorithms.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

S ( ω ) = 2 {Re} { n I n ( ω ) I r ( ω ) exp [ j θ ( ω , τ n ) ] }
θ ( ω , τ n ) = ω τ n + ϕ ( ω , τ n ) .
β n ( ω ) = β n 0 ( ω 0 ) + β n 1 ( ω 0 ) ( ω ω 0 ) + 1 2 β n 2 ( ω 0 ) ( ω ω 0 ) 2 +
S ( ω ) = 2 {Re} { n I ( ω ) exp [ j ( ω τ n + 1 2 β n 2 Δ z n ( ω ω 0 ) 2 ) ] }
θ ω = τ n + β n 2 Δ z n ( ω ω 0 )
X α ( u ) = F p [ x ( t ) ] = x ( t ) K α ( t , u ) d t
K α ( t , u ) = { A α e j [ ( t 2 + u 2 ) {cot} α 2 u t {csc} α ] , α n π δ [ t ( 1 ) n u ] , α = n π
A α = 1 j cot α 2 π .
x ( t ) = F p [ x ( t ) ] = X ( u ) K α ( t , u ) d t .
μ n = cot α n .
{ u ^ 0 , α ^ 0 } = arg max u , α { | X α ( u ) | 2 }
[ u ^ i + 1 α ^ i + 1 ] = [ u ^ i α ^ i ] λ i {H} {i} [ | X α ( u ) | 2 α | X α ( u ) | 2 u ] | u = u ^ i , α = α ^ i ,
S a ( ω ) = S ( ω ) + j S ^ ( ω ) ,
ρ = | A n | > H 1 < H 0 γ
X α ^ n ( u ) = X α ^ n ( u ) H ( u , α ^ n ) .
H ( u , α ^ n ) = { 0 , u ^ n w n u u ^ n + w n 1 , {otherwise}
θ ¯ n ( ω ) = n k n ( ω ω 0 ) 2
{ β ^ n 2 = 2 π cot α ^ n Δ z n τ ^ n = 2 π u ^ n csc α ^ n .

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