Phase fluctuations in a two-transverse-dimensional scanning Fourier domain optical coherence tomography (FDOCT) seriously affect in vivo phase related applications. The phase difference between two A-scans sampled at the same scanning position or adjacent scanning position is acquired by matching of the phase distribution characteristics on the surface of two A-scans. Finger and palm scanning experiments are performed and defocused images of finger and palm are recovered based on Fresnel scalar diffraction algorithm by using phase compensated OCT complex signals. To further prove the performance of the proposed method, human eye scanning experiments are also performed and blood flow images of retina are extracted from the phase registration results. The accurate, fast and simple phase compensation method is critical for in vivo phase related applications.
© 2013 OSA
Fourier domain optical coherence tomography (FDOCT) [1,2] has recently become preferred to the original time domain coherence tomography (TDOCT)  due to its sensitivity and high speed advantages and has facilitated the development of real-time in vivo tissue imaging  and also three-dimensional volumetric imaging . Phase stability is a primary requirement to many OCT studies that depend on the measurement of complex signals. For instance, phase stable measurements are required for Doppler OCT , phase microscopy [7,8], polarization sensitive OCT , coherent averaging , spectroscopic OCT [11,12], and defocused image recovery [13–17]. The inherent phase fluctuations of FDOCT systems are caused by thermal drifts, galvanometer positioning accuracy and system mechanical jitter, which is much lower than the phase fluctuations in TDOCT systems because the reference arm in the FDOCT system is fixed. Another phase error appearance is the bulk motion artifacts, from sample motion and environmental vibrations, which are an order of magnitude larger than those associated with the inherent phase instability of the system and can obscure true motion in the sample if uncorrected . The effect of environmental vibrations on the phase-resolved techniques can be eliminated by the using of the vibration isolation platform. However, for in vivo imaging applications, it is difficult to maintain phase stability over a whole two-transverse-dimensional scanning because of the phase fluctuations caused by the respiratory motion, cardiac motion and the involuntary movement of the subject. There physiological fluctuations are turned out to be much bigger than other fluctuations . Therefore, the phase fluctuations caused by the sample motion is the most significant one of them all and contributes more as a noise, and this has been testified by our experiment results that the en face out-of-focus images of static sample (such as fresh onion) can be digitally focused without phase compensation, however the out-of-focus images of in vivo tissues (such as human finger, palm) acquired from the same FDOCT system cannot be digitally focused if phase compensation is not performed.
A number of methods were investigated to overcome the phase stable problem. In general, phase fluctuations can be mitigated by development of ultrahigh speed swept laser sources, higher speed acquisition, vibration isolation, and minimizing fiber lengths to reduce thermal drifts. Besides, free-space or common path design [7,8,18] or a feedback control loop in conjunction with a fiber stretcher or piezo-electric modulator  are adopted to compensate for phase fluctuations. To eliminate the interferometric noise, Yaqoob et al.  used the line illumination beam and taken the phase measured at a part of the beam illuminating outside of the sample as a reference phase to remove the common-mode noise. However, these sophisticated and expensive hardware compensators may only be moved with limited speed and accuracy.
Post-processing techniques were also developed to compensate for phase fluctuations through alignment of the complex A-scans. Tomlins et al.  calculated cross-correlation of the magnitudes of two A-scans and located its maximum peak corresponding to the offset between the two signals by Lorentzian function fitting method to identify the small path-length difference between the two scans. Then, each of the A-scans in a data sets were digitally stabilized by multiplying the spectrogram by a complex exponential, and spectrally averaged in order to enhance detection sensitivity. Because the phase of A-scan is much more sensitive to the change of optical path-length than the magnitude of A-scan, the magnitude cross-correlation method is not suitable for accurately detecting phase fluctuations. Yang et al.  prepared the sample on a covers lip and referenced the heterodyne phase of the light reflected from the sample to that of the light reflected from the cover slip, as a result, the phase fluctuations between reference and sample arms was mitigated, displacement and velocity sensitivities of 3.6 nm and 1 nm/s were achieved, enabling the observation of cell membrane dynamics. Ralston et al.  and Yu et al.  placed a covers lip above the specimen, located the reference object, and calculated phase and group delay values for each A-scan to compensate for the differential variations in optical path length. They used phase compensated results to ISAM (interferometric synthetic aperture microscopy) and angular spectrum algorithm respectively for recovery of out-of-focus OCT images. However, in most cases, using a phase reference cover slip is impractical or undesirable and need to correct residual phase fluctuations due to motion between the covers lip and the sample. To minimize the motion artifacts in optical micro-angiography (OMAG) flow image caused by the inevitable subject movement, providing volumetric vasculatural images in retina and choroids, An et al.  tried to compensate the bulk tissue motion by histogram analysing or averaging all the phase differences along A-scan signals. Although the histogram analysing method is more accurate than the averaging method, it is computationally inefficient. The common disadvantage of these two approaches is that the noise phase in A-scans has great effect on the measurement accuracy of phase differences.
In this paper, the phase registration principle and results based on matching of phase distribution characteristics is introduced firstly. Then, the digitally focused images of human finger and palm by using the phase compensated OCT complex signals are demonstrated, and the blood flow images of retina extracted from the phase registration results are provided. Finally, the influence of parameters on phase compensation method and basic rules to determine parameters are described.
2. Phase registration principle and results
2.1 FDOCT complex signal
The spectral intensity at the interferometer exit of FDOCT isEq. (1) can be removed by background subtraction method . We get
2.2 Phase registration principle
Typical 3D volume data of FDOCT is illustrated as Fig. 1. Each A-scan is digitalized to j sampling points (P1-Pj) in z direction. Each fast scanning in x direction yields a B-scan including n A-scans(A1-An). Slow scanning in y direction yields m B-scans(B1-Bm). Generally speaking, it is reasonable to assume that phases of j sampling points (P1-Pj) in a single A-scan is static because of the high speed of FDOCT. However, it is difficult to maintain phase stability over a whole two transverse dimensional scan because of the phase fluctuations caused by the movement of the sample. It is necessary to detect phase fluctuations between two A-scans sampled at the same or adjacent lateral scanning position, or reference phases of all A-scans in a volume data set to the phase of a single A-scan in the data set for phase information applications.
Adjacent A-scans are considered having the same structure and amplitude distribution and phase distribution along the depth when the horizontal distance between two scanning point is small enough. A small change of path length difference between the sample arm and the reference arm yields translation of sample complex signal as Eq. (6).
Therefore, phase fluctuation () between adjacent A-scans can be acquired by comparing phase distribution characteristics of adjacent A-scans.
Figure 2(a) displays a typical B-scan amplitude image of tomato. Figures 2(c) and 2(d) represents phase distribution curve of A-scanand A-scanrespectively from z = 125 to z = 145, and Fig. 2(e) is the phase difference between A-scan and A-scanin Fig. 2(a). It is obvious that the phase difference between these two adjacent A-scans is stable in the area where generate strong scattering signal (we named it as real scattering signal), however, phase difference is irregular in the area where scattering signal is weak or only has noise (we named it as noise signal). So, to distinguish real scattering signal from noise signal is critical for calculating the phase difference between adjacent A-scans accurately.
Phase distribution characteristics is defined as a vector consisting of a sequence of phase differences between adjacent pixels
Sample scattering signals of two adjacent A-scans have similar phase distribution characteristics since they are scattered from similar internal structure. The judging function of the sample real scattering signal in I-th A-scan and (I-1)-th A-scan is defined as
2.3 Phase registration results
The OCT system used to achieve 3D volume data of finger and palm is similar to one of previous studies . The system used a super luminescent diode light source, with a central wavelength of 800 nm and a bandwidth of 120 nm that provided a 3μm axial resolution. The light is split into two paths in a fiber based Michelson interferometer. One beam is coupled onto a stationary reference mirror and the second beam is focused with a collimating lens and a focal lens at the sample with a theoretical lateral resolution of ~12μm. The spectral interferogram between the reference light and the light backscattered from the sample was sent to a homebuilt spectrometer via an optical circulator. The spectrometer consisted of a transmission grating (1200 lines/mm), a camera lens and a CMOS 1024 element line-scan camera (A-scan rate is 70 kHz, B-Scan rate is 130 frames per second).
Figures 3(a) and 3(b) show the phase differences between adjacent A-scans of first B-scan image of human finger and palm, respectively. From the results we can see that the phase fluctuations within one B-scan are small for both finger and palm because A-scan rate is much faster than the moving speed of the sample.
Figures 4(a) - 4(d) show the phase differences between adjacent A-scans of adjacent B-scans and , and , and, and of human finger, respectively. Figures 4(e)- 4(h) show the phase differences between adjacent A-scans of adjacent B-scans and , and , and, and of human palm, respectively. By comparing Fig. 3 with Fig. 4, we can find that the phase fluctuations between adjacent B-scans are much larger than the phase fluctuations within one B-scan. By comparing Figs. 4(a) - 4(d) with Figs. 4(e) - 4(h), we can find that the signal phase stability of the palm is poorer than that of the finger because it is difficult to keep the palm still.
Figures 5(a) and 5(e) show the phase differences of all A-scans to the phase of the first A-scan of the first B-scan of human finger expressed with positive angles (phase lead) and cyclic angle i.e. , respectively, Figs. 5(c) and 5(g) denote partially enlargement of Figs. 5(a) and Fig. 5(e), respectively. Figures 5(b), 5(d), 5(f) and 5(h) show the results of human palm. By comparison, the phase fluctuations of a human palm is much bigger than that of finger. Compensation phase functions in Figs. 5(e) and 5(f) can be expressed as and used to phase compensation of OCT complex signals of finger and palm, respectively for defocused image recovery.
3. Application of phase registration in FDOCT
In order to show the important role of phase registration in FDOCT in vivo applications, two OCT systems were used to scan human finger and palm, and human retina, respectively, and structure image of finger and palm and blood flow image of retina acquired by using phase compensated results are illustrated below.
3.1 Application of phase registration in digital focusing of FDOCT images
When the OCT system scans a tissue, the probe beam is typically focused into the sample by an objective lens. The lateral resolution of the OCT system is inversely proportional to the numerical aperture (NA) of the objective lens, therefore, it can be enhanced by the increase of numerical aperture. However, only the OCT image that falls within the depth of field (DOF) exhibits the desired lateral resolution, whereas the OCT image that falls outside the DOF region is blurred laterally and the increase of the numerical aperture leads to the reduction of DOF. In general, the higher lateral resolution and the longer DOF are two of the most wishful parameters for the most OCT imaging applications; however, they are reciprocally coupled. The relationship between the lateral resolution and DOF is schematically illustrated in Fig. 6(a) for low and Fig. 6(b) for high NAs.
To improve the lateral resolution by using digitally focusing method, phase fluctuations of the original FDOCT complex signals need to be compensated using the following equation
Here, is the defocused en face FDOCT complex signal outside the DOF region, is the focused complex signal, k is the wave number, z is the diffraction distance, are spatial frequencies, and denotes the Fourier transform.
Figure 7 shows a typical en-face image (amplitude image of ) and recovered clear image (amplitude image of ) of human finger. Figure 8(a) is a typical cross-section (y-z plane) image of an original 3D volume data of human palm. Figure 8(b) is the recovered image without using phase compensation, which is acquired by digitally focusing en face images in different depths of 3D volume data according to Eq. (11) using original complex signals, and extracting the y-z plane image with the same x coordinate as Fig. 8(a). Figure 8(c) is the digitally focused image using phase compensated complex signals, which is acquired also according to Eq. (11), but the original complex signals must be phase compensated using Eq. (10) before digitally focusing. It is obvious that the sweat gland in Fig. 8(b) is blurred compared with the sweat gland in Fig. 8(a) because of the influence of phase fluctuations, however, the sweat gland in Fig. 8(c) is focused and clear since the phase fluctuations are compensated. In order to further demonstrate the effectiveness of the phase compensation, we plotted the gray value of the line marked z = zi in Figs. 8(a)-8(c) as displayed in Figs. 9(a), 9(b) and 9(c), respectively. It is easy to find that the width of the intensity profile of the sweat gland in Fig. 9(c) is the smallest among three intensity profiles, therefore, the lateral resolution of the FDOCT system is improved by digitally focusing using the phase compensated complex signals.
3.2 Application of phase registration in retina blood flow imaging
The OCT system used to scan human retina is similar to one of previous studies . The system used a super luminescent diode light source, with a central wavelength of 842 nm and a bandwidth of 46 nm that provided a ~8 µm axial resolution. The light was delivered to a 20/80 fiber coupler and split into the sample arm (20% light) and the reference arm (80% light). A 2cm water chamber was used in the reference arm to pre-compensate the wavelength dispersion effect caused by the human eye. In the sample arm, the beam is focused with a collimating lens, an objective lens and an ocular lens at the posterior part of the eye with a theoretical lateral resolution of ~16μm. The lights returning from the reference arm and the eye were sent to a homebuilt spectrometer, consisting of a transmission grating (1200 lines/mm), an achromatic focusing lens with a 100 mm focal length and a high speed line scan CMOS camera (Basler, Sprint spL4096-140 k). To acquire original 3D volume data, we used two galvo-scanners to raster-scan the focused beam spot with 240 kHz A-scan rate, 400 B-scan rate across the sample with a scanning area of 2.5mm x 2.0 mm. A saw tooth waveform was used to drive the fast B-scan, and a step function waveform was used to drive the slow C-scan. For a B-scan cross sectional image, 500 A-lines were captured with 5 um spatial intervals between adjacent A-lines. For the C-scan, the 2mm scan range was evenly divided into 200 steps with a 10μm spatial interval between them. In each step, at list two B-scan frames were captured and processed to extract one B-scan cross-sectional flow image.
Each B-scan complex signal can be written as
The B-scan cross-sectional blood flow image is acquired by using Eq. (14) when the effects of sample motion are eliminated by applying the phase registration method.
From Fig. 10, we can observe that the blood flow image can't be extracted effectively from original OCT signals [Figs. 10(a) and 10(b)] without phase registration. Only the phase difference [Fig. 10(d)] between A-scans of two original B-Scans is accurately detected and compensated by the proposed method, the high quality two dimension blood flow image [Fig. 10(e)] can be acquired. The three-dimension blood flow image of retina [Fig. 10(f)], consisted of 200 B-scan blood flow images, shows the structure of blood vessels clearly.
4. Discussion on influence of parameters M and ε
Parameters M and ε in Eq. (7) and Eq. (8), corresponding to a number of elements in phase distribution vector and the threshold value, respectively, have direct influence on the quality of phase registration method. We conducted a large number of experiment researches on the changes of phase difference measurement results, digitally focused image quality and retina blood flow image quality with parameters M and ε.
Figure 11 shows a typical B-scan image of human finger and the change of searching results of strongly scattering points and phase difference measurement results of two adjacent A-scans with parameters M and ε. From the B-scan image and the phase difference curves in Fig. 11, we can see that pixels from z = 30 to z = 36 are real strongly scattering points on the surface of two A-scans. The red horizontal lines denote calculated phase differences, and “ο” are used to label the found strongly scattering points. The phase difference between two adjacent A-scans cannot be detected accurately when M is very small (M = 2, ε = 0.5, 1, 1.5, 2; M = 3, ε = 1, 2). Conversely, strongly scattering points cannot be found and phase registration procedure fail when M is very big (M = 6, ε = 0.5; M = 7, ε = 1; M = 8, ε = 1.5, 2). Similarly, when ε is small, the phase difference measurement results is accurate, however, it is easy to cause phase registration fail. Conversely, when ε is bigger, the phase difference measurement results is inaccurate, however, phase registration fail can be avoided.
Figure 12 shows a typical y-z plane image of human palm and the change of digitally focused results with parameters M and ε. The lateral resolution of the image was improved and clear sweat gland images were acquired when parameters M and ε were set properly, such as M = 3, ε = 0.5, 1, 1.5, 2; M = 4, ε = 1, 1.5, 2; M = 5, ε = 1.5, 2; M = 6, ε = 1.5, 2.
Figure 13 shows two B-scan frames of retina captured in the same B-scan cross section plane at different sampling time, extracted blood flow image without phase compensation (on the top right corner of Fig. 13), and extracted blood flow images obtained using Eq. (14) and Eq. (13) for different M and ε. The retina blood flow images were effectively extracted after phase registration process was performed with proper M and ε, such as M = 3, ε = 0.5; M = 4, ε = 1, 1.5, 2; M = 5, ε = 1.5, 2; M = 6, ε = 2.
Figure 14 shows the better choices of parameters M and ε for three different applications. “o” denotes preferred value for phase difference measurement of human finger, “Δ” denotes preferred value for image digitally focusing of human palm, and “﹡” denotes preferred value for retina blood flow imaging.
Parameters M and ε for phase registration of a specific sample can be selected according to experiment results as displayed in Fig. 14 or refer to the following basic rules:
First step is to fix M. M is the length of phase distribution vector, a number of pixels included in phase distribution computation, it should be smaller than the number of consecutive strongly scattering points in the sample, otherwise, inequality Eq. (8) is not satisfied for any points in a data set and phase registration procedure will fail. However, if M is very small, phase difference between two adjacent A-scans cannot be detected accurately. Generally speaking, M can be determined based on the inner structure characteristics of the sample and the sample variation between two A-scans, which is related to the lateral resolution of FDOCT system, lateral pixel size, sample surface roughness, galvanometer positioning accuracy, system mechanical jitter and especially the sample movement.
Second step is to fix the value of ε. Parameter ε is determined based on the selected value of M and the sample variation between two A-scans. If M is big, ε cannot be selected too small, since it is easy to cause phase registration procedure fail. If M is small, ε cannot be selected too big, since phase difference cannot be detected accurately.
Phase difference between two FDOCT A-scans sampled at the same or adjacent scanning position is detected by searching similar phase distribution characteristics on the surface of two A-scans. The performance of the proposed phase registration method was tested by the recovery of defocused images of human finger and palm and by the blood flow imaging of retina by using phase compensated complex signals. As displayed in Eq. (7) and Eq. (8), phase distribution characteristics is defined as a vector consisting of M elements and an inequality with a threshold value ε is used to find strongly scattered points on the sample surface for phase registration. Parameters M and ε have direct influence on the quality of phase registration method, improper value of M and ε may cause inaccurate phase registration or fail of phase registration procedure. Parameters M and ε for a specific sample can be determined according to systematic experiment results, the internal structure of the sample and the sample variation between two A-scans caused by sample surface roughness, system parameters, scanning parameters, system mechanical precision and especially the sample movement. Our future work will focus on parameters optimization methods and the joint registration of amplitude and phase. The proposed phase registration method is of great advantage due to the fact that it is accurate because the effect of noise is eliminated. Secondly it is fast since only the sample surface signal is matched. Moreover it is convenient because no hardware is needed. These afore advantages make it possible to achieve 3D phase stability in FDOCT for in vive applications.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 61240057 and 61108047)
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