Depth-dependent dispersion compensation for full-depth OCT image

Liuhua Pan; Xiangzhao Wang; Zhongliang Li; Xiangyang Zhang; Yang Bu; Nan Nan; Yan Chen; Xuan Wang; Fengzhao Dai

doi:10.1364/OE.25.010345

1. Introduction

Optical coherence tomography (OCT) is one of the most important noninvasive optical imaging modalities that captures micrometer-resolution, three-dimensional images within optical scattering media(e.g. biological tissue) [1, 2]. For clinical diagnosis and pathological study, OCT has been widely used in the medical fields such as ophthalmology, dermatology, cardiovascular and so on [3–6]. The axial resolution of OCT equals to the coherence length of the light source, which can be increased by using very wide-spectrum light sources [7, 8]. However, with the increasing of the spectral bandwidth, the dispersion effects caused by optical components and samples also increase. The result of dispersion causes chirping and broadening of the axial point spread function (PSF).

Generally, two strategies are developed to eliminate dispersion effect in OCT system. One method is termed hardware dispersion compensation, which consists in matching the optical materials in the two arms of the interferometer or which applies the grating-based phase delay scanners to compensate group velocity dispersion. This method works well in the OCT system for fixed samples, but they need more optoelectronic components and readjust the compensating devices for different samples [9–11]. The other method is termed software dispersion compensation, which compensates dispersion of the OCT system by numerical method. Numerical methods are more flexible and practically well suitable for compensating dispersion in OCT system, such as convolution with depth variant kernel [12], polynomial fitting algorithm [13], autofocus algorithm [14] and iterative algorithm [15]. The convolution algorithm based on numerical correlation of the depth scan signal with a depth variant kernel requires a priori knowledge of the dispersive properties of the medium. Polynomial fitting algorithm is utilized to fit the phase of spectral fringe signal and to remove the high-order dispersive phase, but it is not suitable for weak intensity signal reflected from greater depths in sample. The autofocus algorithm and iterative algorithm processes OCT images through compensating each high-order dispersion phase using a single dispersion compensation coefficient, which lead to under-compensation or over-compensation because the dispersions at different depths are different.

Assessing OCT image quality is an effective way to evaluate the effect of dispersion compensation. There are two assessing methods, i.e. subjective and objective assessment. Compared with the subjective assessment, the objective assessment is more convenient and could assess image quality dynamically with different algorithms and parameter settings. According to whether the original image is available, the objective assessment could be divided into full-reference, reduced-reference and no-reference [16]. No-reference image quality assessment is suitable for assessing OCT image. Sharpness as a metric is utilized to assess the image quality. Several no-reference objective sharpness metrics are proposed, such as transform-based approaches, image-gradient-based techniques, histogram-based techniques and so on [17].

In this paper, based on iterative algorithm, we proposed a depth-dependent dispersion compensation algorithm for enhancing full-depth OCT image quality. Using this algorithm, the second-order dispersion compensation coefficient at every depth is calculated and the second-order dispersion phase at corresponding depth is eliminated. We process the OCT images of the phantom and in vivo biological tissue with the depth-dependent dispersion compensation algorithm, and assess image quality by sharpness metrics based on variation coefficient. Using the depth-dependent dispersion compensation algorithm, the full-depth OCT image quality is improved.

2. Dispersion in spectral domain OCT

The backscattered light back from different depths of the sample interferes with light reflected back from the reference arm. Cross-sectional image of the spectral domain OCT is obtained by detecting broadband interference fringes. Subtracting the spectrum of the reference light, the interference signal is given by

\begin{array}{l} S_{int} (k) = 2 Re {\underset{n}{Σ} \sqrt{I_{n} (k) I_{r} (k)} \exp [i φ (k, Δ z_{n})]} \\ = 2 Re {\underset{n}{Σ} \sqrt{I_{n} (k) I_{r} (k)} \exp {i [k \cdot Δ z_{n} + Φ (k, Δ z_{n})]}}, \end{array}

where

k

is the wave number,

I_{n} (k)

is the intensity of light reflected from the n-th layer of the sample,

I_{r} (k)

is the intensity of light reflected from the reference arm,

Δ z_{n}

is the optical path length difference between the n-th layer reflection and the reference mirror,

φ (k, Δ z_{n})

is the phase difference of the n-th layer reflective light relative to the reference light, including the high-order dispersion phase

Φ (k, Δ z_{n})

. The purpose of dispersion compensation is to eliminate high-order dispersion phase which is the main reason for chirping and broadening of PSF of the OCT system . The phase

φ (k, Δ z_{n})

is given by

φ (k, Δ z_{n}) = β_{n} (k) \cdot Δ z_{n},

where

β_{n} (k)

is the propagation constant of light wave in the n-th layer of the sample. For any given material, the

β_{n} (k)

can be expanded as a Taylor series in powers of

k - k_{0}

,

k_{0}

is the center wave number.

\begin{array}{l} φ (k, Δ z_{n}) = β_{n} (k) \cdot Δ z_{n} \\ = [n_{n} (k_{0}) \cdot k_{0} + n_{g, n} (k_{0}) (k - k_{0}) + β_{n}^{''} (k_{0}) \frac{{(k - k_{0})}^{2}}{2!} + β_{n}^{'''} (k_{0}) \frac{{(k - k_{0})}^{3}}{3!} + ...] \cdot Δ z_{n} \\ = n_{n} (k_{0}) \cdot k_{0} \cdot Δ z_{n} + n_{g, n} (k_{0}) \cdot Δ z_{n} \cdot (k - k_{0}) + \frac{β_{n}^{''} (k_{0})}{2!} \cdot Δ z_{n} \cdot (^{k - k_{0}) 2} + \frac{β_{n}^{'''} (k_{0})}{3!} \cdot Δ z_{n} \cdot (^{k - k_{0}) 3} + ... \\ = n_{n} (k_{0}) \cdot k_{0} \cdot Δ z_{n} + n_{g, n} (k_{0}) \cdot Δ z_{n} \cdot (k - k_{0}) + a_{2} \cdot (^{k - k_{0}) 2} + a_{3} \cdot (^{k - k_{0}) 3} + ..., \end{array}

where

n_{n}

is the effective refractive index of the n-th layer of the sample,

n_{g, n}

is the effective group refractive index,

β_{n}^{''}

and

β_{n}^{'''}

are the second-order and third-order effective dispersion coefficient, respectively.

a_{2}

and

a_{3}

are the second-order and third-order effective dispersion compensation coefficient, respectively.

a_{2} \cdot {(k - k_{0})}^{2}

and

a_{3} \cdot {(k - k_{0})}^{3}

are the second-order and third-order dispersion phase, respectively.

3. Depth-dependent dispersion compensation algorithm

In the OCT system, high-order dispersion reduces the axial resolution. In particular, the second-order dispersion broadens the envelope and reduces the intensity of A-line signal. The main purpose of dispersion compensation is to eliminate the effect of second-order dispersion. The $β_{n}^{''} (k_{0})$ is given by [18]

β_{n}^{''} (k_{0}) = \frac{λ_{0}^{3}}{2 π} (\frac{d^{2} n_{n}}{d λ^{2}}) .

Equation (4) shows that $β_{n}^{''} (k_{0})$ is only related to the refractive index of the sample. The maximum axial depth of OCT image in biological tissue is about 3 mm [19] and the refractive index of biological tissue in this range does not change too much. Therefore the $β_{n}^{''} (k_{0})$ variation in the sample could be ignored and the second-order dispersion compensation coefficient $a_{2}$ is proportionate to depth of the image. So the second-order dispersion in the tissue at different depths can be compensated numerically with the relationship between the second-order dispersion coefficient and imaging depth.

The flow chart of depth-dependent dispersion compensation algorithm is shown in Fig. 1. Raw data is acquired by OCT system. Some imaging depths which include the structure information of sample are chosen and the second-order dispersion compensation coefficients at these imaging depths are obtained through iterative method [15]. Linear fitting the second-order dispersion compensation coefficients, we obtain an analytical formula between second-order dispersion compensation coefficient and imaging depth. Using this formula, we can eliminate the effect of the dispersion in an A-line signal. The prior procedure is repeated for each A-line signal, then the dispersion of a B-scan image is compensated.

Fig. 1 Flow chart of depth-dependent dispersion compensation algorithm.

Download Full Size | PDF

4. Sharpness metric based on variation coefficient

One of the evaluating parameters of image quality assessment is sharpness, which is determined by the high spatial frequencies of the light intensity distribution. The more high spatial frequencies there is, the clearer the image is [20]. The gray values of OCT image pixels are discrete. The more dispersion in the OCT system, the lower discrete degree of gray values of OCT image pixels. Generally, the metrics such as the standard deviation, the FWHM of a thin structure or the slope of an edge in the image are used for accessing the image quality. However, the gray values of OCT image pixels change after dispersion compensation. When the mean value of data is changing, the standard deviation is not suitable for measuring the discrete degree of the data. Meanwhile, the FWHM or the slope of an edge is adequate for a thin structure or an image with bright edge. When we access the quality of an OCT image, there would be many structures in the image and the edge of structure may be not bright and clear enough because of the diffuse reflection of the phantom and biological sample. Variation coefficient is ratio of standard deviation to mean, and is suitable for measuring the discrete degree of the gray values of OCT image pixels. We take the variation coefficient as the sharpness metric, which is given by

C .V= \frac{\sqrt{\frac{1}{N} \sum_{x} \sum_{y} {[f (x, y) - μ]}^{2}}}{μ},

where

N

is the number of image pixels,

f (x, y)

is the gray value of pixel,

μ

is the mean gray value of pixels. The higher the variation coefficient is, the clearer the image will be.

5. Experiment and results

The schematic diagram of the OCT experiment system is shown in Fig. 2. A super-luminescent diode (SLD) (T850, Superlum Ltd, Russia) was used as the illumination source. The center wavelength of the light source is 835 nm and bandwidth is about 52 nm [Fig. 3(a)]. The output light was divided into reference light and sample light by fiber coupler. The reference light was reflected from the surface of mirror through an achromatic lens (focal length $f$ = 30 mm). The sample light was scanned by an x-y galvanometer, and was finally reflected from the sample through an achromatic lens ( $f$ = 30 mm). The reflected reference light and the reflected sample light were recombined in the fiber coupler and the interference fringes were detected by a spectrometer. The spectrometer consisted in a grating (grating period of 1200 lp/mm), a lens ( $f$ = 100mm) and a high-speed line-scan CCD (Aviiva-EM4-CL-2014, 2048 pixels operating in 12-bit mode, e2V). A data acquisition equipment (USB 6259, National Instruments) generated pulse signals, which controlled the scanner and CCD synchronously. The CCD signal was gathered by an image acquisition card (IMAQ PCI-1427, National Instruments) and transformed into a tomography image of the sample by a data processing. Figure 3(b) is the point spread function of the OCT system in which the path length difference is 0.5 mm. Due to the spectral calibration error with CCD, an axial resolution of about 8 μm in air is obtained.

Fig. 2 Schematic of the OCT system. L1~L2: Lens; C1~C2: Collimator; ND: Neutral Density Filter; PC: Polarization Controller.

Download Full Size | PDF

Fig. 3 (a)Spectrum from the light source. (b) Point spread function of OCT system.

Download Full Size | PDF

5.1 Experiment of phantom

White tape and transparent tape were alternatively overlaid as a phantom, which is shown in Fig. 4(a). To validate the depth-dependent dispersion compensation algorithm, the phantom image data is acquired by the OCT system. As above algorithm, some depths with structure information in the B-scan image were chosen [(Fig. 4(b)], and where the second-order dispersion compensation coefficient were calculated with the iterative algorithm. Then the second-order dispersion compensation coefficients were fitted as a linear function of the image depth, which was shown in Fig. 5 ( $R^{2} = 0.974$ ). The analytical formula of dispersion compensation coefficient was obtained,

a_{2} = 138.8 \times d e p t h - 41.7,

here the

d e p t h

is the imaging depth and its unit is millimeter. Because the optical elements in the sample arm and the reference arm are not identical, the dispersion in the sample arm which do not include the sample and the reference sample are not exactly the same. Therefore the analytical formula of dispersion compensation coefficient has a non-zero offset. The negative offset means the dispersion in the reference arm is greater than the dispersion in the sample arm which do not include the sample. Equation (6) is utilized to compensate depth-dependent compensation for the phantom image.

Fig. 4 (a)Schematic and (b) B-scan image of phantom.

Download Full Size | PDF

Fig. 5 Fitting line between the second-order dispersion compensation coefficient and imaging depth in phantom.

Download Full Size | PDF

Figure 6 shows the B-scan images of phantom without dispersion compensation and with three different dispersion compensation methods. In Fig. 6(b), dispersion was compensated with constant second-order dispersion compensation coefficient of 82. In Fig. 6(c), dispersion was compensated with polynomial fitting algorithm. In Fig. 6(d), dispersion was compensated with depth-dependent dispersion compensation algorithm. Three regions of image at different depths were chosen from Fig. 6(a) for image quality assessment. The same regions in Figs. 6(b)-6(d) were also chosen and the variation coefficient of all the chosen regions were shown in Fig. 7(b). After processing by constant coefficient or polynomial fitting algorithm, some regions in the B-scan images of phantom became clearer, but also other regions were blurring, such as the region I processed with constant coefficient algorithm or the region III processed with polynomial fitting algorithm. The constant coefficient algorithm and the polynomial fitting algorithm could not compensate dispersion at all depths. The image processed with depth-dependent algorithm was clear at all depths and the variation coefficient at different regions was high, which were shown in Figs. 6(d) and 7(b), respectively. It is indicated that depth-dependent dispersion compensation algorithm could improve full-depth OCT image quality of phantom.

Fig. 6 B-scan images of phantom. (a)Image without dispersion compensation. (b) Image with constant coefficient algorithm. (c) Image with polynomial fitting algorithm. (d) Image with depth-dependent algorithm.

Download Full Size | PDF

Fig. 7 (a)Three regions for image assessment in the B-scan image of phantom. (b) Variation coefficients of different regions processed with different dispersion compensation algorithms.

Download Full Size | PDF

5.2 Experiment of fisheye in vivo

Figure 8(a) shows the live goldfish and the experiment part is marked by a red rectangular outline. During experiment, the goldfish was wrapped by wet facial tissue except eye to keep the goldfish alive. Drops of fresh water were applied to the fisheye every minute to prevent dehydration of fisheye.

Fig. 8 (a) Photograph of live goldfish. (b) B-scan image of live fisheye.

Download Full Size | PDF

Figure 8(b) shows the OCT image of live fisheye. The protective film, cornea, lens and iris were obvious. But owing to dispersion, some details of fisheye were blurring. We compensated the dispersion with depth-dependent dispersion compensation algorithm. As described above, some depths with structure information of fisheye were chosen and the corresponding second-order dispersion compensation coefficients were calculated with the iterative algorithm. The function relationship between the second-order dispersion compensation coefficient and the imaging depth for in vivo fisheye was fitted ( $R^{2} = 0.874$ ). The function relationship is expressed as

a_{2} = 130.4 \times d e p t h - 44.

Equation (7) is used for depth-dependent dispersion compensation in the imaging of fisheye.

Figure 9 shows the B-scan images of fisheye without dispersion compensation and with three different dispersion compensation algorithms which were described above. In Figs. 9(b)-9(d), the dispersion was compensated with constant coefficient algorithm, polynomial fitting algorithm and depth-dependent algorithm, respectively. Three regions of image at different depths were chosen from Fig. 10(a) for image quality assessment. The same regions in Figs. 9(b)-9(d) were also chosen and the variation coefficient of all the chosen regions were shown in Fig. 10(b). It was shown that the part of protective film and cornea in the region I became blurring after the image was processed with constant coefficient algorithm. The part of iris in the region III was also not clear when the image was processed with polynomial fitting algorithm. The constant coefficient algorithm and the polynomial fitting algorithm could enhance the quality of partial image of fisheye, but could not improve the quality of the whole image. After image processed with depth-dependent algorithm, all structure of fisheye were clear at all depth, and the variation coefficient of the chosen regions were high, which were shown in Figs. 9(d) and 10(b), respectively. It is concluded that the depth-dependent dispersion compensation could improve full-depth quality of the in vivo fisheye OCT image.

Fig. 9 B-scan images of fisheye. (a) Image without dispersion compensation. (b) Image with constant coefficient algorithm. (c) Image with polynomial fitting algorithm. (d) Image with depth-dependent algorithm.

Download Full Size | PDF

Fig. 10 (a)Three regions for image assessment in the B-scan image of in vivo fisheye. (b) Variation coefficients of different regions processed with different dispersion compensation algorithms.

Download Full Size | PDF

6. Conclusion

A depth-dependent dispersion compensation algorithm is proposed based on the variation of second-order dispersion compensation coefficient at different imaging depth. In this algorithm, an analytical formula is obtained to compensate the depth-dependent dispersion. After processing OCT image with depth-dependent dispersion compensation algorithm, the details of image of phantom and the fisheye are clear. We introduced the variation coefficient as a sharpness metric to assess the OCT image quality. Compared with the other dispersion compensation algorithms, the variation coefficients of the selected regions processed with depth-dependent algorithm are higher. The depth-dependent dispersion compensation could effectively improve the full-depth OCT image quality.

Funding

Innovation Action Plan of Science and Technology Commission of Shanghai Municipality (15441905600); the Open Fund of Key Laboratory of Optoelectronic Information Processing of University in Guangxi (KFJJ2016-04); National Natural Science Foundation of China (61405210).

References and links

1. X. Guo, P. Bu, X. Z. Wang, O. Sasaki, N. Nan, B. J. Huang, and Z. L. Li, “Scattering imaging of skin in Fourier domain optical coherence tomography,” Opt. Commun. 305, 137–142 (2013). [CrossRef]

2. N. Nan, X. Wang, P. Bu, Z. Li, X. Guo, Y. Chen, X. Wang, F. Yuan, and O. Sasaki, “Full-range Fourier domain Doppler optical coherence tomography based on sinusoidal phase modulation,” Appl. Opt. 53(12), 2669–2676 (2014). [CrossRef] [PubMed]

3. 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 et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991). [CrossRef] [PubMed]

4. V. J. Srinivasan, M. Wojtkowski, J. G. Fujimoto, and J. S. Duker, “In vivo measurement of retinal physiology with high-speed ultrahigh-resolution optical coherence tomography,” Opt. Lett. 31(15), 2308–2310 (2006). [CrossRef] [PubMed]

5. X. Wen, S. L. Jacques, V. V. Tuchin, and D. Zhu, “Enhanced optical clearing of skin in vivo and optical coherence tomography in-depth imaging,” J. Biomed. Opt. 17(6), 066022 (2012). [CrossRef] [PubMed]

6. L. Liu, J. A. Gardecki, S. K. Nadkarni, J. D. Toussaint, Y. Yagi, B. E. Bouma, and G. J. Tearney, “Imaging the subcellular structure of human coronary atherosclerosis using micro-optical coherence tomography,” Nat. Med. 17(8), 1010–1014 (2011). [CrossRef] [PubMed]

7. A. Unterhuber, B. Povazay, B. Hermann, H. Sattmann, W. Drexler, V. Yakovlev, G. Tempea, C. Schubert, E. M. Anger, P. K. Ahnelt, M. Stur, J. E. Morgan, A. Cowey, G. Jung, T. Le, and A. Stingl, “Compact, low-cost Ti:Al2O3 laser for in vivo ultrahigh-resolution optical coherence tomography,” Opt. Lett. 28(11), 905–907 (2003). [CrossRef] [PubMed]

8. B. Povazay, K. Bizheva, A. Unterhuber, B. Hermann, H. Sattmann, A. F. Fercher, W. Drexler, A. Apolonski, W. J. Wadsworth, J. C. Knight, P. S. J. Russell, M. Vetterlein, and E. Scherzer, “Submicrometer axial resolution optical coherence tomography,” Opt. Lett. 27(20), 1800–1802 (2002). [CrossRef] [PubMed]

9. W. Drexler, U. Morgner, F. X. Kärtner, C. Pitris, S. A. Boppart, X. D. Li, E. P. Ippen, and J. G. Fujimoto, “In vivo ultrahigh-resolution optical coherence tomography,” Opt. Lett. 24(17), 1221–1223 (1999). [CrossRef] [PubMed]

10. L. Froehly, L. Furfaro, P. Sandoz, and P. Jeanningros, “Dispersion compensation properties of grating-based temporal-correlation optical coherence tomography systems,” Opt. Commun. 282(7), 1488–1495 (2009). [CrossRef]

11. C. C. Rosa, J. Rogers, and A. G. Podoleanu, “Fast scanning transmissive delay line for optical coherence tomography,” Opt. Lett. 30(24), 3263–3265 (2005). [CrossRef] [PubMed]

12. A. Fercher, C. 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] [PubMed]

13. B. J. Huang, P. Bu, X. Z. Wang, and N. Nan, “Optical coherence tomography based on depth resolved dispersion compensation,” Acta Opt. Sin. 32(2), 0217002 (2012). [CrossRef]

14. 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] [PubMed]

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

16. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004). [CrossRef] [PubMed]

17. R. Ferzli and L. J. Karam, “A no-reference objective image sharpness metric based on the notion of just noticeable blur (JNB),” IEEE Trans. Image Process. 18(4), 717–728 (2009). [CrossRef] [PubMed]

18. S. Diddams and J. C. Diels, “Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. B 13(6), 1120–1129 (1996). [CrossRef]

19. J. G. Fujimoto, “Optical coherence tomography for ultrahigh resolution in vivo imaging,” Nat. Biotechnol. 21(11), 1361–1367 (2003). [CrossRef] [PubMed]

20. N. Damera-Venkata, T. D. Kite, W. S. Geisler, B. L. Evans, and A. C. Bovik, “Image quality assessment based on a degradation model,” IEEE Trans. Image Process. 9(4), 636–650 (2000). [CrossRef] [PubMed]

Depth-dependent dispersion compensation for full-depth OCT image

Abstract

Corrections

1. Introduction

2. Dispersion in spectral domain OCT

3. Depth-dependent dispersion compensation algorithm

4. Sharpness metric based on variation coefficient

5. Experiment and results

5.1 Experiment of phantom

5.2 Experiment of fisheye in vivo

6. Conclusion

Funding

References and links

Cited By

Figures (10)

Equations (7)

Optics Express