Computation time-saving mirror image suppression method in Fourier-domain optical coherence tomography

Chiung-Ting Wu; Ting-Ta Chi; Yean-Woei Kiang; C. C. Yang

doi:10.1364/OE.20.008270

1. Introduction

In using the Fourier-domain optical coherence tomography (OCT) techniques, including spectral-domain OCT (SD-OCT) and swept-source OCT (SS-OCT), a two-dimensional (2-D) OCT image can be obtained after a Fourier transform of the acquired interference signals, which show a 2-D real number array, in the frequency domain (k). Due to the lack of the quadratic imaginary part of the OCT signal, the Fourier transform in the k domain leads to two mutually complex-conjugate components, which represent the real and mirror images in an OCT scanning result. The real and mirror images can be well separated if we arrange the interferometer such that the interfered signals of the sample structure are obtained away from the point of exact optical path matching (the zero-delay line). However, such an arrangement leads to the decrease of signal-to-noise ratio and the degradation of image resolution. For the advantages of high signal-to-noise ratio and image resolution, the interference around the zero-delay line is preferred. In this situation, the real and mirror images overlap. One of the images must be suppressed before a clear picture showing the sample structure can be obtained. Therefore, an effective and time-saving method for suppressing the mirror image is crucially important in the application of Fourier-domain OCT.

A phase-shift algorithm was first proposed for producing quadratic OCT signal [1]. Theoretically, as long as the quadratic component of the OCT signal can be obtained, an inverse Fourier transform can lead to mirror image suppression. The quadratic OCT signal can be obtained by scanning the sample multiple times with a piezoelectric transducer (PZT) carrying a reflecting mirror in the reference arm [2, 3]. However, the requirement of multiple scans for acquiring an OCT image leads to a lower imaging speed. Although an approach of tilting the reference wavefront was then proposed to reduce the scanning number, it resulted in lower system sensitivity [4]. Meanwhile, this approach can only be applied to an SD-OCT system. Actually, the quadratic OCT signal can be obtained by simply scanning the sample twice with a phase shift of 90 degrees [5]. Besides the aforementioned PZT, such a phase shift can be obtained by using an electro-optical phase modulator [6, 7]. Nevertheless, the procedure of double scans for one image still suffers from the problem of sample motion. To overcome this problem, several two-beam approaches have been demonstrated for realizing one-scan operation, including the uses of polarized light [8], beam splitter adjustment [9], and double reference paths [10]. However, such an approach requires two photodetectors in the OCT system and makes the system more complicated. The use of a 3 x 3 fiber coupler can also produce a phase shift for obtaining the quadratic signal and hence suppressing the mirror image [11–14]. However, such a complicated OCT system may suffer from the spectral limitation in broadband operation and the problem of temperature-caused instability. Similar to the phase shift method, the frequency shift approach was also proposed [15]. Such an approach can be implemented by using an electro-optical modulator [16], an acousto-optical modulator [17], or a dispersive optical delay line [18]. Nevertheless, to avoid the double-scan problem (the sample motion problem), this approach can only be applied to SS-OCT. The procedure of double scans is still needed when it is applied to SD-OCT [19]. Sinusoidal phase modulation can result in effective evaluation of the quadrature components based on an integrating-bucket acquisition algorithm [20] or a harmonic lockin detection technique [21, 22]. However, in such a method, either multiple A-mode scans are required or a long signal integration time is needed, leading to the reduction of imaging speed and the generation of motion artifact. Recently, a dispersion-encoded method was proposed [23–26]. In this method, numerical dispersion compensation was designed to blur the mirror image for effectively improving image quality. Nevertheless, the implementation of this method requires quite a long computation time.

Among various mirror image suppression methods, the BM-scan method based on a certain phase shift mechanism has been widely used [27–32]. In this method, the real spectral signals of a 2-D image are first Hilbert transformed along the B-mode scan direction to give the complex spectral signals, which can lead to mirror image suppression after a Fourier transform. In an equivalent scheme for saving computation time, the spectral OCT signals are first inverse Fourier transformed to give the real-space image. Then, the B-mode scan image signals are Fourier transformed to give the spatial-frequency spectra. After deleting the negative spatial-frequency components, an inverse Fourier transform leads to the real-space OCT image with the mirror image suppressed. In this computation scheme, the mirror image suppression procedure starts with the real-space image such that we can select the concerned depth range for processing and save the computation time [32]. It is noted that in an OCT system with a scanning galvanometer for lateral scanning, a system phase shift can be produced when the optical beam is aligned away from the galvanometer axis [33–35]. In this situation, no extra phase-shift device is needed.

Recently, we have briefly demonstrated a novel approach for mirror image suppression, with which the computation time can be significantly reduced [36]. It is noted that the phase shifts between two neighboring A-mode scans of the real- and mirror-image signals are mutually reversed. In this approach, we utilize this property of reversed phase shift for differentiating the real image from the mirror image. Although we demonstrate the approach based on galvanometer scanning for producing a system phase shift along the B-mode scan, this approach can be applied to a Fourier-domain OCT system with any other phase shift mechanism. It has the advantage of significantly shorter computation time, when compared with the aforementioned BM-scan method. Also, because this mirror image suppression process is a spatially localized operation of OCT signals, we can select any concerned A- and B-mode scan ranges for process to further save computation time. In other words, if we are particularly interested in a small portion of the image, we can simply process this portion of signal with our approach because it is a spatially localized method. However, in other methods, such as the BM-scan method, normally a certain range of B-mode scan signal must be included in the mirror image suppression process. In this situation, the computation time cannot be reduced. Therefore, our spatially localized method has the advantage of further reducing the process time when only a small portion of image is concerned. In this paper, we provide more detailed demonstrations about this approach, including the theory behind the operation and the use of iteration process for further improving the image quality. Also, the effects of parameter choices, including the scan phase shift, process phase shift, iteration number, and B-mode scan pixel size, for mirror image suppression process on suppression ratio are illustrated. Meanwhile, the results based on our approach are compared with those obtained from the widely used BM-scan technique. In section 2 of this paper, the theory with formulations, including that of the iteration operation, of this approach is demonstrated. The mirror image suppression results with the iteration operation are illustrated in section 3. Then, the variations of the mirror image suppression ratio under various process conditions are shown in section 4. Finally, the conclusions are drawn in section 5.

2. Theory

Our mirror image suppression method is based on the solution of two equations, which describe the relationships of the real- and mirror-image signals of the same A-mode scan pixel between the two neighboring A-mode scans. Suppose r_n and m_n represent the complex OCT signals of the real and mirror images, respectively, in a certain A-mode scan pixel of the n^th A-mode scan. Also, r_n+₁exp(iθ) and m_n+lexp(-iθ) represent the complex OCT signals of the real and mirror images, respectively, in the same A-mode scan pixel of the (n + 1)^th A-mode scan. Here, θ denotes the system phase shift between the two neighboring A-mode scans. This phase shift can be obtained from any mechanism of phase modulation, such as that by aligning the optical beam away from the galvanometer axis when a scanning galvanometer is used for lateral scanning. It is noted that in writing the expressions above, we have used the property of reversed phase shift between the real and mirror images.

The summation of the complex signals of the real and mirror images is equal to the measured OCT signal after a k-space Fourier transform of the acquired spectral signal. Suppose that S_n and S_n+₁ stand for measured OCT signals in the designated depth pixel of the n^th and (n + 1)^th A-mode scans, respectively. We have the following two equations:

S_{n} = r_{n} + m_{n}

and

S_{n + 1} = r_{n + 1} \exp (i θ) + m_{n + 1} \exp (- i θ) .

In our mirror image suppression algorithm, we assume that except the system phase shift θ, the complex OCT signals in the same depth pixel of two neighboring A-mode scans are the same, i.e., r_n = r_n+₁ and m_n = m_n+₁ for any integer n. With this assumption, for a given θ value, the two complex unknowns in Eqs. (1) and (2) can be solved to give

{\tilde{r}}_{n}

and

{\tilde{m}}_{n}

, which represent the solved real- and mirror-image signals, respectively, as

{\tilde{r}}_{n} = \frac{S_{n} - S_{n + 1} \exp (i θ)}{1 - \exp (i 2 θ)}

and

{\tilde{m}}_{n} = \frac{S_{n} - S_{n + 1} \exp (- i θ)}{1 - \exp (- i 2 θ)} .

Therefore, the real-image signal can be differentiated from that of the mirror image. It is noted that

{\tilde{r}}_{n}

and

{\tilde{m}}_{n}

can also be obtained by using the acquired OCT signals in the (n-1)^th and n^th A-mode scans, i.e., S_n_-1 and S_n, respectively. With

S_{n - 1} = r_{n - 1} \exp (- i θ) + m_{n - 1} \exp (i θ),

we can obtain another set of solved real- and mirror-image signals,

{\hat{r}}_{n}

and

{\hat{m}}_{n}

, respectively, as

{\hat{r}}_{n} = \frac{S_{n - 1} \exp (i θ) - S_{n} \exp (i 2 θ)}{1 - \exp (i 2 θ)}

and

{\hat{m}}_{n} = \frac{S_{n - 1} \exp (- i θ) - S_{n} \exp (- i 2 θ)}{1 - \exp (- i 2 θ)} .

The average of

{\tilde{r}}_{n}

and

{\hat{r}}_{n}

(

{\tilde{m}}_{n}

and

{\hat{m}}_{n}

) can be used to improve the mirror image suppression effect. With the average process, we have

{r^{'}}_{n} \equiv \frac{{\tilde{r}}_{n} + {\hat{r}}_{n}}{2} = \frac{(S_{n - 1} - S_{n + 1}) \exp (i θ) + S_{n} [1 - \exp (i 2 θ)]}{2 [1 - \exp (i 2 θ)]}

and

{m^{'}}_{n} \equiv \frac{{\tilde{m}}_{n} + {\hat{m}}_{n}}{2} = \frac{(S_{n - 1} - S_{n + 1}) \exp (- i θ) + S_{n} [1 - \exp (- i 2 θ)]}{2 [1 - \exp (- i 2 θ)]} .

When θ = 90 degrees, Eqs. (3) and (4) become

{\tilde{r}}_{n} = \frac{S_{n} - i S_{n + 1}}{2} = \frac{r_{n} + r_{n + 1}}{2} + \frac{m_{n} - m_{n + 1}}{2}

and

{\tilde{m}}_{n} = \frac{S_{n} + i S_{n + 1}}{2} = \frac{r_{n} - r_{n + 1}}{2} + \frac{m_{n} + m_{n + 1}}{2} .

Also, Eqs. (8) and (9) become

{r^{'}}_{n} = \frac{2 S_{n} + i (S_{n - 1} - S_{n + 1})}{4} = \frac{r_{n - 1} + 2 r_{n} + r_{n + 1}}{4} - \frac{m_{n - 1} - 2 m_{n} + m_{n + 1}}{4}

and

{m^{'}}_{n} = \frac{2 S_{n} - i (S_{n - 1} - S_{n + 1})}{4} = - \frac{r_{n - 1} - 2 r_{n} + r_{n + 1}}{4} + \frac{m_{n - 1} + 2 m_{n} + m_{n + 1}}{4} .

It is noted that r_n stands for the accurate real-image signal and

{\tilde{r}}_{n}

and

{r^{'}}_{n}

represent the approximated real-image signals obtained based on our mirror image suppression method. From Eq. (10), one can see that with our method, the solved real-image signal is the combination of the average of the two neighboring accurate real-image signals with the one-half difference of the two corresponding mirror-image signals. With the average procedure, as shown in Eq. (12), the solved real-image signal is the combination of a weighted average of the three neighboring accurate real-image signals with a weighted difference of the three corresponding mirror-image signals. The approximation used in our method is based on the assumption of the equal real- and mirror-image signals between two neighboring A-mode scans. Therefore, this method will lead to a poor result when there is a rapid variation in image signal along the B-mode scan. Such a situation can be encountered when the B-mode scan pixel size is large or when a steeply slant bright-line feature appears in the image. Certain tissue sample structures can cause strong backscattering to form bright-line features in an OCT image, such as the surface or an interface of a tissue sample. In this situation, the weighted difference in the second term of Eq. (12) between the mirror-image signals of three neighboring A-mode scans can lead to the more effective suppression of mirror image. For instance, if the mirror-image signal varies linearly along the B-mode scan, this mirror-image component in Eq. (12) can be completely cancelled. The weighted average in the first term of Eq. (12) among the corresponding real-image signals may degrade the lateral resolution of the processed image. However, as long as the total lateral range of the involved A-mode scans is smaller than the lateral resolution, the process of weighted average does not affect the lateral resolution. This point will be discussed in detail later.

When there is a rapid variation in image signal along the B-mode scan, an iteration procedure can further improve the mirror image suppression result. Based on the results of the first iteration shown in Eqs. (8) and (9), for the second iteration, we use the solutions of the real image, ${r^{'}}_{n - 1}$ , ${r^{'}}_{n}$ , and ${r^{'}}_{n + 1}$ , as the new OCT signals, i.e., ${S^{'}}_{n - 1} = {r^{'}}_{n - 1}$ , ${S^{'}}_{n} = {r^{'}}_{n}$ , and ${S^{'}}_{n + 1} = {r^{'}}_{n + 1}$ , respectively, and repeat the solution procedures (including the average procedure) above to obtain the new real-image solution. After two iterations, when θ = 90 degrees, the real-image solution, ${r^{″}}_{n}$ , is given by

\begin{array}{l} {r^{″}}_{n} = \frac{2 {S^{'}}_{n} + i ({S^{'}}_{n - 1} - {S^{'}}_{n + 1})}{4} = \frac{1}{16} (- S_{n - 2} + 4 i S_{n - 1} + 6 S_{n} - 4 i S_{n + 1} - S_{n + 2}) \\ = \frac{1}{16} (r_{n - 2} + 4 r_{n - 1} + 6 r_{n} + 4 r_{n + 1} + r_{n + 2}) + \frac{1}{16} (m_{n - 2} - 4 m_{n - 1} + 6 m_{n} - 4 m_{n + 1} + m_{n + 2}) . \end{array}

Here, again we can see the contribution of weighted average of real-image signals among neighboring A-mode scans and that of weighted difference between the corresponding mirror-image signals. With two iterations, the contribution from the mirror-image signals to the real-image solution is expected to be further reduced. Therefore, the process of more iterations is useful for further improving the real image quality.

3. Mirror image suppression demonstration

To demonstrate the mirror image suppression effect based on our approach, we use an SD-OCT system with a broadband super-luminescent diode system (Superlum, BroadLighter Q810) as the light source. The multiple-peak spectrum of the light source covers a full-width at half-maximum range from 767 to 967 nm. Figure 1 shows the setup of the SD-OCT system, which consists of a Michelson interferometer based on a broadband 50/50 fiber coupler (Thorlabs, FC850-40-50-APC). In the detection arm, a diffraction grating of 1200 lines/mm (Thorlabs, GR50-1208), a focusing lens (Thorlabs, AC127-019-B-ML), and a one-dimensional CCD (Basler, spL8192-70km) are used to form a spectrometer. In the sample arm, a scanning galvanometer (Cambridge Technology, 6350) is used for laterally scanning a sample. To obtain a system phase shift, θ, between two neighboring A-mode scans, the optical beam is aligned to be a distance, d, away from the rotation axis of the galvanometer. For instance, d = 3.6 mm corresponds to θ = 90 degrees [36]. The axial resolution of the OCT system is ~1.7 μm in free space. The A-mode scanning rate is ~39 kHz. The phase shift between two neighboring A-mode scans is evaluated based on the OCT scanning of a sample consisting of five Scotch tape layers on a 3M post-it note [36]. For scanning a sample, a focusing lens with the focal length of 19 mm is used to produce a pupil diameter of 5 mm and a beam spot diameter at the focus of 8.06 μm. The theoretical lateral resolution is evaluated to give 4.21 μm.

Fig. 1 Setup of the SD-OCT system for sample scanning.

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In Figs. 2 and 3 , we show the mirror image suppression results based on our method and their comparisons with those obtained from the BM-scan technique when the system phase shift θ is 90 degrees. In Fig. 2(a), the unprocessed OCT image of scanning the human skin on a finger of a volunteer is shown. With the pixel sizes of 2 and 0.56 μm in the A- and B-mode scans, respectively, the image in Fig. 2(a) consists of 680 and 1800 pixels in the A- and B-mode scans, respectively. By using our mirror image suppression method of one iteration described above with θ = 90 degrees (see Eq. (8)), we can obtain the result shown in Fig. 2(b). In this figure, the mirror image is effectively suppressed to show clearly the skin structure and the two sweat ducts. However, as indicated by the thick arrow, the bright skin surface of the mirror image is still visible. The residual mirror image can be suppressed by employing the second iteration described above. The resultant real image is shown in Fig. 2(c). Here, the residual skin surface feature in the mirror image is completely deleted. Also, the quality of the rest of the image is unchanged. For comparison, we use the time-saving algorithm of the BM-scan method [32] for suppressing the mirror image to show Fig. 2(d). One can see that the image qualities of Figs. 2(c) and 2(d) are about the same, indicating that after two iterations in our mirror image suppression method, the image quality is comparable to that based on the BM-scan method. However, the computation time of our method can be significantly shortened. From the complex OCT signals for Fig. 2(a), it takes 64.4 (one iteration), 119.4 (two iterations), and 250.9 ms (the BM-scan method) in computation time to obtain Figs. 2(b), 2(c), and 2(d), respectively. Therefore, our method can significantly save the image process time, particularly when a large three-dimensional image is to be processed. It is noted that the OCT scanning results are processed for suppressing the mirror images with a personal computer, which is equipped with Intel(R) Core(TM) i7 CPU 860 (8 central process units) of 2.8 GHz in processing speed. The memory of DDR3 with 3.24 GB in size is used. Also, the software of LabVIEW 2009 is utilized for processing the signals. The zero-delay line (the central horizontal line) in each part of Fig. 2 can be easily removed by deleting a few pixels in each A-mode scan of the image. This procedure affects little the image quality.

Fig. 2 OCT images of human skin on a finger, including (a) the un-processed image with the overlapping real and mirror images, (b) the image after one iteration of our mirror image suppression method, (c) the image after two iterations of our mirror image suppression method, and (d) the image processed with the BM-scan method. Both scan and process phase shifts are 90 degrees.

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Fig. 3 OCT images of hamster pouch mucosa, including (a) the un-processed image with the overlapping real and mirror images, (b) the image after one iteration of our mirror image suppression method, (c) the image after two iterations of our mirror image suppression method, and (d) the image processed with the BM-scan method. Both scan and process phase shifts are 90 degrees.

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In Figs. 3(a)–3(d), we show the similar comparisons to those in Figs. 2(a)–2(d) with another OCT scanning sample. Here, to demonstrate the mirror image suppression result of a more complicated tissue structure, we scan the pouch mucosa of a healthy hamster (see Fig. 3(a)). With the pixel sizes of 2 and 0.56 μm in the A- and B-mode scans, respectively, the image in Fig. 3(a) consists of 1000 and 1800 pixels in the A- and B-mode scans, respectively. Again, after the one-iteration operation of our mirror image suppression method, the bright sharp mucosa surface in the mirror image can still be seen even though the mirror image is essentially suppressed (see Fig. 3(b)). This residual feature disappears after the operation of the second iteration (see Fig. 3(c)). The image quality after two iterations in our method is comparable to that of the BM-scan method (see Fig. 3(d)). From the complex OCT signals for Fig. 3(a), it takes 90.6 (one iteration), 178.1 (two iterations), and 374.1 ms (the BM-scan method) in computation time to obtain Figs. 3(b), 3(c), and 3(d), respectively. The computation times based on our approach are significantly shorter.

The used average and iteration procedures correspond to the weighted average effects over the successive A-mode scans. Such an average effect may reduce the lateral spatial resolution of the processed image. To examine the possible degradation of lateral spatial resolution after the average and iteration procedures, we use the SD-OCT system to scan the USAF 1951 test target (R3L3S1N) along the vertical (pink) line shown in Fig. 4(a) . In the B-mode scan, the pixel size is 0.56 μm. The un-processed OCT scanning image is shown in Fig. 4(b), in which the concerned image portion is arranged to be away from the zero delay line. Because the scan phase shift is set at 90 degrees, the line-shaped image of a flat surface shown in Fig. 4(b) is tilted. Here, one can see five groups of slit, with three identical slits in each group. The slit widths range from 6.96 through 4.39 μm, as labeled in the figure. In each group, a short (white) bar is plotted to roughly indicate the location of the central slit. From this image, one can see that the lateral spatial resolution of the used OCT system is around 4.5 μm. Figures 4(c)–4(e) show the processed results of mirror image suppression with one, two, and three iterations, respectively. In each iteration procedure, the average process is included. Here, one can see that after 1-3 iterations of image process, the images are essentially unchanged, indicating that the average and iteration procedures in the process of mirror image suppression do not significantly affect the lateral resolution. This is so because the lateral resolution (4.5 μm) is about 8 times the pixel size (0.56 μm). The process of three iterations involves only seven A-mode scans. As long as the lateral range of the involved A-mode scans is smaller than the lateral resolution, the used average and iteration procedures do not blur the image. Because the mirror image suppression qualities of both the BM-scan method and our approach rely on the small pixel size of B-mode scan, as to be discussed later, the comparison between the two techniques based on the OCT scanning results of small B-mode-scan pixel sizes is reasonable.

Fig. 4 OCT scanning and processed results of the USAF 1951 test target (R3L3S1N). (a): The image of the test target with the OCT scanning trace indicated by the vertical (pink) line; (b): The OCT scanning image before the process of mirror image suppression with the labeled slit widths. (c)-(e): The OCT images after the processes of mirror image suppression with 1, 2, and 3 iterations, respectively.

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4. Mirror image suppression ratio

To evaluate the mirror image suppression results under various process conditions, we use the SD-OCT system for scanning a mirror surface to obtain the unprocessed OCT image shown in the insert at the upper-left corner of Fig. 5 . This OCT scanning result is obtained by setting the phase shift θ equal to 90 degrees (d = 3.6 mm). By changing the d value, various θ values can be obtained. The condition of d = 0 corresponds to the case of θ = 0. This θ value will be called as the “scan phase shift” because it is the system phase shift used in OCT scanning. On the other hand, the phase shift used to process the image for mirror image suppression will be called as the “process phase shift”. In processing an OCT scanning result, we may not have the accurate information about the scan phase shift due to the imprecise calibration of the OCT system. In Fig. 5, we show the mirror image suppression ratios as functions of the scan phase shift under different process conditions when the process phase shift is always equal to the scan phase shift. The suppression ratio is defined as the B-mode average of the maximum OCT signal intensity in the lower half of the image (designated as the real image signal after mirror image suppression) over the OCT signal intensity at the symmetric point (designated as the residual mirror image signal after mirror image suppression) with respect to the zero-delay line in an A-mode scan. The insert at the lower-right corner of Fig. 5 shows an example of suppression ratio evaluation. Here, the suppression ratio, which is represented by the horizontal (red) line, corresponds to the average of the fluctuating OCT signal intensity ratios along the B-mode scan. In Fig. 5, the results of four process conditions are shown for comparison, including the process operations of one (1), two (2), and three (3) iterations based on our method, and the BM-scan (BM) method. Here, one can see that the suppression ratio is essentially symmetric with respect to the zero phase-shift point. It is noted that a negative suppression ratio in dB means the same result as the case of the positive suppression ratio with the same magnitude in dB. With a negative θ value, the roles of real and mirror images are interchanged. Generally speaking, the suppression ratio increases with the phase shift. It tends to saturate when the phase shift approaches 90 degrees. This saturation behavior implies that a slight inaccuracy in setting a phase shift around 90 degrees does not much affect the mirror image suppression result. With the process of only one iteration in our method, the suppression ratio is always smaller than that obtained from the BM-scan technique. However, after the processes of two or three iterations in our method, the suppression ratio becomes comparable to or even larger than the result based on the BM-scan technique. Nevertheless, further enhancement of suppression ratio in increasing the iteration number beyond two seems to gradually saturate. The suppression ratio can be as high as 26 dB when the phase shift is 90 degrees. It is noted that a higher signal-to-noise ratio (SNR) can lead to a higher mirror image suppression ratio. The weighted average process in our iteration procedure may improve the SNR and hence increase the suppression ratio. To see whether such a weighted average process can enhance the suppression ratio based on the BM-scan method, the weighted average process of three (five) neighboring A-mode-scan signals is applied before using the BM-scan method to give the curve labeled by BM(1) (BM(2)) in Fig. 5. The average weightings of the involved A-mode-scan signals are the same as those shown in Eqs. (12) and (14) for BM(1) and BM(2), respectively, except that only the positive real weighting coefficients are used (the minus sign and the factor i are not included). To reduce the curve number for clarity in this figure, curve BM(1) (BM(2)) is plotted only in the portion of negative (positive) phase shift. The results in the portions of positive and negative phase shifts are roughly symmetric. The results of curves BM(1) and BM(2) show that the weighted average process does not significantly change the mirror image suppression ratio based on the BM-scan method. In Fig. 5, it is noted that the suppression ratio result at zero phase shift is not available because the denominator becomes zero leading to a singularity in either Eq. (3) or (4). The condition of zero phase shift cannot result in a mirror image suppression effect in any similar method.

Fig. 5 Mirror image suppression ratios as functions of the scan phase shift when the process phase shift is always equal to the scan phase shift under different process conditions, including the cases of one (1), two (2), and three (3) iterations based on our method, and the cases without the weighted average (BM), with the weighted averages of three A-mode scans (BM(1)) and five A-mode scans (BM(2)) based on the BM-scan technique. The results are obtained by scanning a mirror surface with the unprocessed OCT image shown in the insert at the upper-left corner. The insert at the lower-right corner shows an example of suppression ratio evaluation. The suppression ratio, which is represented by the horizontal (red) line, corresponds to the average of the fluctuating OCT signal intensity ratios along the B-mode scan.

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In Fig. 6 , we show the suppression ratios as functions of the process iteration number based on our method when the process phase shift is equal to the scan phase shift, which is labeled in the figure for each curve. Here, one can see that when the phase shift is between 80 and 90 degrees, the curves of suppression ratio essentially overlap except the process condition of one iteration. Also, their suppression ratio increases monotonically with iteration number. However, the increasing slope becomes significantly smaller beyond two iterations. When the phase shift is smaller than a certain value between 70 and 80 degrees, the suppression ratio first increases with iteration number to reach the individual maximum value at two or three in iteration number. Beyond this condition, the suppression ratio decreases monotonically with iteration number. From the results in Fig. 6, one can see that the operation of three iterations can be regarded the optimized condition for any phase shift value when the factor of saving computation time is considered in processing an OCT image.

Fig. 6 Suppression ratios as functions of iteration number in our method based on mirror surface scanning when the process phase shift is equal to the scan phase shift, which is labeled for each curve in the figure.

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In practical operation, we may not have the accurate information about the scan phase shift. It is useful to understand the effect of an inaccurate process phase shift on the mirror image suppression result. In Fig. 7 , we show the suppression ratios as functions of iteration number with various scan phase shift values, as labeled for the curves in the figure. All the results are obtained by using 90 degrees as the process phase shift. Here, with one-iteration operation, the suppression ratio is the largest when the scan phase shift is the same as the process phase shift at 90 degrees. Also, the suppression ratio generally decreases with the deviation between the scan and process phase shifts. However, when the iteration number is larger than one, the largest suppression ratio is obtained when the scan phase shift is 100 degrees. Roughly speaking, except the case of 80 degrees in scan phase shift, when the iteration number is large, the suppression ratio decreases with increasing deviation of the scan phase shift from 100 degrees. The curves of 80 and 90 degrees in scan phase shift approximately coincide when the iteration number is larger than one. In the case that the aforementioned deviation is smaller than ~15 degrees, the suppression ratio increases with iteration number when it is larger than two. When the deviation is larger than ~15 degrees, beyond the maximum level (at three or four in iteration number), the suppression ratio decreases monotonically with iteration number. From the results in Fig. 7, one can realize that a deviation between the scan and process phase shifts of <20 degrees will reduce the mirror image suppression ratio by less than 2 dB.

Fig. 7 Suppression ratios as functions of iteration number with various scan phase shift values, as labeled for the curves in the figure, based on mirror surface scanning. The results are obtained by using 90 degrees as the process phase shift in our method.

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Our method for mirror image suppression relies on the key assumption of equal OCT signals at the same depth pixel of two neighboring A-mode scans. The condition of a large difference in OCT signal between the two pixels of the same depth in two neighboring A-mode scans can often be encountered when a slant line feature of strong scattering appears in an OCT image. To evaluate the effect of such a slant line feature on mirror image suppression, we use a rough surface (the relatively flat portion of a coin) as the scanning sample and tilt its surface in the OCT scanning experiment to observe the variation of the suppression ratio. However, when we scan along the tilted surface (B-mode scan) of the sample, the SNR varies. Here, the SNR of a particular A-mode scan is defined as the ratio of the maximum signal intensity over the average intensity in an image area far away from the line-shaped signal feature. Under each tilt-angle condition, a least-mean-square fitting can provide us with a linearly decreasing variation of SNR along the B-mode scan. Because a higher SNR can lead to a higher mirror image suppression ratio, to obtain a fair comparison between the conditions of different tilt angles, in the OCT scanning signals of each tilt-angle condition, we select a B-mode scan range of 200 pixels with the fitted SNR in the range between 16 and 17 dB for evaluating the suppression ratio. In Fig. 8 , we demonstrate the suppression ratios as functions of the tilt angle of the sample surface under various process conditions when both scan and process phase shifts are 90 degrees. Here, because the SNR is fixed in the range between 16 and 17 dB, the effect of SNR on the mirror image suppression ratio is minimized. Since the variation of suppression ratio in each process case is always larger than 1 dB when the tilt angle increases from 0 to 40 degrees, the decreasing trend of suppression ratio with tilt angle in Fig. 8 can indeed illustrate the effect of tilted sample surface. Here, one can see that the suppression ratio decreases with tilt angle under all the process conditions. It is noted that with the rough surface as the scanning sample for Fig. 8, the backscattered intensity is weakened or the SNR in OCT scanning is decreased. In this situation, the suppression ratio is also reduced. Therefore, the suppression ratios at zero tilt-angle in Fig. 8 are lower than those of the similar cases in Figs. 5–7.

Fig. 8 Suppression ratios as functions of the tilt angle of the scanned rough surface under various process conditions, including one (1), two (2), and three (3) iterations based on our method, and the BM-scan (BM) technique, when both scan and process phase shifts are 90 degrees.

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The adequacy of the assumption of equal OCT signals at the same depth pixel of two neighboring A-mode scans also relies on the B-mode scan pixel size. In Fig. 9 , we show the suppression ratios as functions of the B-mode scan pixel size. To obtain those results, OCT scanning on a reflecting mirror with normal incidence is performed (the same scanning operation and sample as those for Figs. 5–7). Also, both scan and process phase shifts are set at 90 degrees. Here, under the process conditions of one and two iterations based on our method, the suppression ratio starts to significantly drop when the B-mode pixel size is larger than ~0.8 μm. However, under the process conditions of three iterations based on our method and the BM-scan technique, the suppression ratio does not significantly drop until the B-mode pixel size becomes larger than ~1.1 μm. For high suppression ratios in both our method and BM-scan technique, the B-mode pixel size needs to be smaller than ~1 μm.

Fig. 9 Suppression ratios as functions of the B-mode pixel size based on mirror surface scanning under various process conditions, including one (1), two (2), and three (3) iterations in our method, and the BM-scan (BM) technique, when both scan and process phase shifts are 90 degrees.

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5. Conclusions

In summary, we have demonstrated the theory and experimental results of a computation time-saving mirror image suppression method in Fourier-domain OCT, which utilized the property of reversed system phase shift between the real and mirror images, for differentiating one from the other. By solving a set of two equations based on a reasonable approximation, the real image signal could be obtained. In particular, we illustrated the theoretical backgrounds and the improved real image quality of the average and iteration operations in this method. Also, the mirror image suppression ratios under various process conditions, including different process iteration numbers and different system phase shifts between two neighboring A-mode scans, were evaluated. Meanwhile, the mirror image suppression results based on our method were compared with those obtained from the widely used BM-scan technique. It was found that when a process procedure of two iterations was used, the mirror image suppression quality based on our method could be higher than that obtained from the BM-scan technique. The computation time of our method is significantly shorter than that of the BM-scan technique.

Acknowledgment

This research was supported by National Science Council and National Health Research Institute, The Republic of China, under the grants of NSC 100-2218-E-002-003, NSC 99-3114-B-002-005, and NHRI-EX101-10043EI. Also, the preparation of hamster for OCT scanning by Professor Ivy Hsu’s group of Christian Chung Yuan University, Zhong-Li, Taiwan, is appreciated.

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Computation time-saving mirror image suppression method in Fourier-domain optical coherence tomography

Abstract

1. Introduction

2. Theory

3. Mirror image suppression demonstration

4. Mirror image suppression ratio

5. Conclusions

Acknowledgment

References and links

Cited By

Figures (9)

Equations (14)

Optics Express