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We show that recently introduced joint Spectral and Time domain Optical Coherence Tomography (STdOCT) can be used for simultaneous complex ambiguity removal and functional Spectral OCT images. This permits to take advantage of higher sensitivity achievable near the zero-path delay. The technique can be used with all Spectral OCT systems that are equipped with an optical delay line (ODL) and provide oversampled scanning patterns. High sensitivity provided by STdOCT allows this technique to be used in Spectral OCT setups with acquisition speed of 100 000 lines/s. We show that different imaging ranges and velocity ranges can be achieved by switching on/off the ODL and a small modification in the processing algorithm. Additionally, the relatively small computational burden of the technique allows for fast computations in the range of less than 5 minutes for 3D data set. We present application of proposed technique to full-range two- and three-dimensional imaging. Morphological and Doppler tomograms of human retina in-vivo are shown. Finally, we identify and discuss artifacts of the technique.
©2009 Optical Society of America
1. Introduction
Optical Coherence Tomography (OCT) [1] is one of the most rapidly developing techniques for imaging internal structure of semitransparent objects with micrometer resolution. OCT with Fourier domain detection (FdOCT) [2,3] enables rapid data acquisition with high imaging speed and sensitivity [4–6]. A spectrometer-based FdOCT system called Spectral OCT (SOCT) provides a direct access to spectral fringe signal, which makes it possible to analyze the evolution of amplitude and phase of light propagating inside the sample under investigation [7]. This makes SOCT very well suited for estimation of flow velocity of physiological fluids. Lately, flow velocity estimation attracts investigators’ attention. Reliable and efficient techniques for qualitative and quantitative estimation of the flow are explored. The most straightforward way of quantitative velocity estimation is to compare the evolution of phase in several SOCT spectral interferometric signals acquired consecutively in time. A measurement technique related to this idea is called phase-resolved Doppler Optical Coherence Tomography [8–11]. In this technique, the frequency of the spectral fringes encodes the axial position of the scattering particle, while the phase change is proportional to its velocity. In this approach the quantitative information of the velocity is well localized spatially as only few spectra are processed at once. More recently, several techniques have appeared which compute the Doppler frequency along the time axis directly by means of Fourier transformation instead of using the phase shift analysis [12–14]. Most of these techniques evolved from the BM-mode scanning method originally proposed by Yasuno et al. [15]. BM-mode OCT has been initially designed for complex ambiguity removal, where the Fourier transformation was applied along the “time”/”lateral position” axis over large amount of spectra. As a result, the computational burden of the algorithm is minimal but the quantitative information of flow velocity is lost and the low velocity flows are invisible. Therefore, the algorithms derivative to BM-mode focus mainly on segmentation of the vessel network, which is achieved by filtering out all static structural elements instead of measuring and calculating spatially distributed velocity values. Quantitative analysis of flow velocities is rendered very difficult in these approaches because the filtering of the static components of the sample usually causes a partial removal of information about low velocities. An attempt to perform a quantitative estimation of flow has been proposed using this technique but it requires filtering the tomogram several times with different filter settings, what can take few hours of calculations for a 3D data set [16]. Long calculation time makes the technique difficult to use in practice. Recently, Wang et al. [17] proposed a technique that combines phase-resolved approach with filtering in Fourier space. The algorithm is complex and requires some assumptions about the imaged tissue, but offers an increase in velocity detection sensitivity.
Another method that exploits Doppler effect to perform velocity measurements is joint Spectral and Time domain OCT (STdOCT) [18]. This technique uses only amplitudes of Fourier transformations to estimate the Doppler frequencies and, as compared to phase-resolved Doppler OCT, offers higher sensitivity and better reliability in quantitative velocity estimation in the entire velocity range. Additionally, it provides quantitative velocity values well localized spatially inside the sample and gives good velocity readings for low velocity values. The latter is caused by the fact that it does not use any thresholds that limit the velocity range. What is important, the technique does not require the data to be back Fourier transformed at any stage of the procedure. Consequently, its numerical complexity is only slightly increased compared to standard phase-resolved OCT imaging with A-scans averaging. Therefore, the data processing can be performed with the speeds of thousands lines per second. Recently, we also showed that its high sensitivity makes the technique well suited for segmentation of the vessel network in 3D data acquired at line-rates of 100 kHz [19].
In order to take advantage of the highest sensitivity achievable in Spectral OCT near the zero optical path-delay position, the complex conjugate image (“mirror” image) of the sample should be removed. This would also facilitate imaging of the thick structures such as the retina in the proximity of the optic nerve head, where large retinal veins and arteries can appear simultaneously with choroidal vessels. To get rid of the undesired “mirror” images several techniques such as multi-frame methods [15,20–26] and variants of the BM-mode technique [15,24,27,28] have been presented to date. Unfortunately, in all of these methods the phase information associated to the sample and its complex counterpart interfere and quantitative velocity estimation is no longer possible.
Information encoded in the phase of spectral fringes is used in both velocity estimation and complex ambiguity removal techniques. These both methods require to use multiple spectral fringe signals to create one line of either structural or functional image. The spectra used in the calculations should be acquired at the same lateral position of the scanning beam in order to obtain well spatially localized phase information. It can be achieved by the lateral scanning either with a discrete step-like or a continuous ramp driving signal. The latter approach is more practical due to strong limitations in the settling time of galvanometric scanners. However, to achieve well defined phase information the continuous lateral scanning has to be performed along with the high sampling density. Additionally, another straightforward advantage of using several spectra to create one line of the tomogram is a sensitivity increase of the structural imaging. This is achieved simply by averaging backscattered intensities of multiple A-scans.
High sampling density is exploited in a modified joint Spectral and Time domain OCT, which we demonstrate in this contribution. The presented technique allows for the complex ambiguity removal simultaneously with quantitative estimation of velocity values in the entire imaging range. The only required modification in the setup is an optical delay line (ODL) in the reference arm that allows for introduction of constant velocity offset during data acquisition. As a result this method is able to work with data acquired with commonly used high density scanning protocols. One of the main advantages of this technique is that the calculation time does not exceed a few seconds for a single cross-sectional image. It has to be noted that doubling the imaging range causes the velocity range to be halved. However, the flexibility of the technique allows for changing from full imaging range to full velocity range by simply switching off the ODL.
We believe that the simplicity of the algorithm, ease of implementation in existing SOCT systems, high sensitivity and numerical efficiency make this new method a practical and reliable tool for retinal flow analysis.
2. Method
In the first step all spectra have to undergo standard SOCT data preprocessing in order to transform them from wavelength domain to wavenumber domain (k domain). Next, uncompensated dispersion is numerically removed and a numerical shaping of the spectral fringes is introduced [29]. One of the joint Spectral and Time domain OCT technique variants (described in the following subsections) is applied to the subset of acquired spectra used to create a single line of structural and velocity tomograms. The procedure is applied several times to different sets of spectral fringes until all lines of the tomogram are created. The subsequent subsections describe numerical processing applied to the subset of preprocessed spectra to create a single line of structural and velocity tomograms.
2.1. Standard STdOCT
The spectral fringe signals selected for calculations can be considered as a two-dimensional interferogram and expressed in the following way:
The above signal results from interference between light reflected from a stationary reference mirror (reflectivity coefficient Rr), and backscattered on several interfaces in the sample (reflectivity coefficients Rs). The wavelength dependent spectral envelope is denoted by S (k). The phase of the interferometric signal depends on the positions zs of the scattering interfaces as well as on the projections νs = Vs cos (α) of velocity Vs of the scattering particle in the sample moving in the direction inclined to the probing beam at angle α.
Due to the introduced oversampling, we assume that the lateral distance between consecutive positions of the probing beam is much smaller than the beam diameter. Because of this, in Eq. (1) the spectra can be regarded as having been acquired at the same position of the sample. Therefore, the dependence on the lateral position can be neglected. As a result, the beat frequency ωs = 2νsk that arises for each wavenumber k along the time axis encodes the laterally localized velocity of the s-th interface inside the sample. Simultaneously, the frequency of fringes along the wavenumber axis encodes the axial position of the s-th interface. It needs to be pointed out that the frequency of fringe pattern carries information along both spectral and time axes. The observation that the interferogram can be processed in similar way along both dimensions led to development of the joint Spectral and Time OCT (STdOCT).
In STdOCT two dimensional Fourier transformation is applied to the subset of spectra. Such a two dimensional Fourier transformation converts the data set from wavenumber-time domain (kt domain) to “in-depth position”-“beat frequency” domain (zω domain). Two dimensional Fourier transformation can be calculated using two one-dimensional Fourier transformations conducted consecutively along the wavenumber axis and time axis or in opposite order. In order to emphasize this possibility it is useful to plot a STdOCT diagram that shows all possible Fourier transformations that can be applied to a two-dimensional interferogram, as it is presented in Fig. 1.
Fig. 1. STdOCT diagrams. Vertical transitions are accomplished by Fourier transformation along wavenumber axis, horizontal – by Fourier transformation along time axis. Amplitude of the complex signal is displayed for visualization purposes. In the zω-domain complex conjugate images are symmetrical with respect to the central point of the plot (zero position, zero velocity). (a) moving mirror experiment, two points (red arrows) represent two complex conjugate images of the mirror; each of the points gives simultaneously information about position and velocity of the moving mirror with respect to the reference mirror. (b) Laminar flow of Intralipid solution in a glass capillary. Two complex conjugate images of parabolic flow distribution are visible.
In the zω-domain each moving scattering interface is represented by two signal peaks in well determined in-depth position and beat frequency. The two points are positioned symmetrically with respect to the zero-delay and zero-velocity position due to the complex ambiguity problem. This is clearly visible in the case of SOCT signals coming from a moving mirror Fig. 1(a). The fact that the quantitative distributions of velocity versus depth position can be directly observed in the zω-domain is even better visualized in an experiment with Intralipid solution flowing in a capillary, Fig. 1(b). Here we can observe the parabolic flow velocity distribution indicating a laminar flow.
In our previous work [18] we showed that the signal amplitude is higher in the zω-domain compared to the average signal amplitude in the zt -domain, while the noise floor level remains unchanged. Due to these facts the signal-to-noise ratio increases by the factor proportional to the square root of the number of the averaged spectra. Therefore, we propose to use the zω-domain to retrieve both structural and velocity information. The comparison of the STdOCT and standard SOCT data processing schemes is presented on STdOCT diagrams in Fig. 2. In STdOCT for each in-depth position along the beat frequency axis the signal with maximal amplitude is found. Squared magnitudes of the highest peaks are proportional to the reflectivity of the scattering interfaces. Positions of highest peaks along beat frequency axis are proportional to their velocity. The distribution of the peaks positions as a function of depth gives a velocity A-scan, while the distribution of squared magnitudes as a function of depth forms a structural A-scan. In order to exploit information from all acquired spectra in the standard SOCT procedure, a final tomogram line (A-scan) is created by averaging amplitudes of all Fourier transformed spectra from the zt -domain along the time axis.
Fig. 2. Schematic STdOCT diagrams showing data flow in numerical processing from a subset of raw spectral fringe signals to the final A-scan. Grey areas indicate portions of the data from a given signal space that are used in calculations at each stage of the procedure; (a) standard SOCT data processing; (b) STdOCT data processing.
It has to be noted here that since the Fourier transformation is a linear operation, the sequence of operations transforming data from kt -domain to zω-domain does not have any importance. However, in structural imaging without complex conjugate removal only a half of the in-depth data carries useful information. This fact is depicted in diagrams in Fig. 2, where grey color marks these parts of signal spaces that are used in further processing. Therefore, it is advantageous to perform the transformation in a sequence starting from the transformation along wavenumber axis as depicted in Fig. 2(b). This allows the transformation along time axis to be performed only on one half of the zt -domain thus reducing the calculation burden by a factor of two.
The possibility of providing information of velocity and morphology simultaneously in one signal space has to be highlighted as an advantage of the STdOCT over other techniques using spatial filtering since it is the sole Fourier based technique that does not require the data to be back transformed at any stage of the procedure. As a result the computation time required for tomogram reconstruction is only a little bit longer than that for the standard SOCT method. However, in order to increase the velocity resolution it is required to perform zero-padding before Fourier transforming the data from zt -domain to the zω -domain, which impacts on the computation time.
2.2. Complex ambiguity free STdOCT
In this subsection we show that a small modification to the previously described STdOCT technique allows for simultaneous velocity estimation and reconstruction of structural A-scans free from the complex conjugate problem.
Let us now assume that a constant change of optical path difference (OPD) is introduced between the reference and the sample arm of the interferometer. In the case of the OPD changes with the velocity νref the total time-dependent spectral fringe signal I(k,t) can be described by an expression similar to the one shown in Eq. (1):
In the zω-domain the complex conjugate images are positioned on the opposite sides of both zero-delay line (in case of zs ≠ 0 ) and on opposite sides of zero beat frequency line (in case of νs ≠ 0 ). It is important to note that by the introduction of the additional velocity νref ≠ 0 one shifts the image and its complex conjugate counterpart in opposite directions along the beat frequency axis.
Fig. 3. STdOCT diagrams of Intralipid flow in a glass capillary: (a) without additional velocity; (b) with additional velocity νref (marked by red arrow) introduced in the reference arm of the interferometer.
If the ODL velocity is large enough, the two images together with all the inner velocity components are completely separated along the ω axis. This effect is clearly visualized in Fig. 3(b), which presents data from the same experimental setup as data from Fig. 3(a) but with additionally introduced velocity νref. The conjugate velocity distributions of Intralipid solution are placed on “positive” and “negative” sides of the beat frequency axis.
Modification of the STdOCT algorithm from the previous section is simply based on the rejection of the half of the kω-domain placed on one side of the zero-velocity position, Fig. 4. Now the resulting beat frequency is proportional to (νs + νref ), so the velocity of inner motion of the sample can be easily extracted.
By analogy to the maximal axial depth-range z ±max = π/2Δk, which depends on the sampling period Δk in the spectral domain, the maximal velocity ν detectable without the 2π ambiguity is determined by the acquisition time of spectral fringe signal Δt:
If the velocity of optical path change is beyond this value it is aliased to the opposite side of the velocity range and gives misleading results. If no velocity offset is introduced in one of the interferometer arms (νref = 0), Eq. (4) determines the maximal velocity detectable inside the sample without the aliasing:
In such a case the sample velocity range is equal to the maximal velocity range. The spatial imaging range is halved in order to avoid an overlap of the complex conjugate images. The entire imaged structure has to fit into one half of the complex space. Conjugate images have to be separated in the way that each of them is placed on opposite side of the zero-path delay position of the imaging range.
The velocity of reference mirror νref ≠ 0 introduced to the system shifts the complex conjugate images along the beat-frequency axis in opposite directions (Fig. 3(b)). If all velocity components of one of the images fit in one half of the velocity range, then the complex conjugate image is placed on the opposite side of the beat-frequency axis. In such a case the beat frequency value ω = 0 separates completely the complex conjugate images in the kω-domain. The two images do not overlap if the following conditions are fulfilled:
Assuming that maximal and minimal velocities inside the sample have equal modules, the maximal velocity values that can be observed with no ambiguities inside the sample are obtained for νref = νmax/2 and are equal:
Now in order to reconstruct structural and velocity A-scan only the half of the zω domain located above or below the ω = 0 should be processed to extract the structural and velocity A-scans. If the expected velocity distribution is asymmetrical, it is necessary to change the velocity offset νref to fulfill the conditions given by formulas (5) and (6).
If the high velocity resolution is necessary, zero-padding along the time axis is performed in both standard and modified STdOCT, before applying the Fourier transformation, Fig. 2(b) and Fig. 4(a). It is therefore advantageous to transform the data from kt -domain to zt -domain first, and perform the second transformation from zt -domain to zω -domain. In the case of standard STdOCT the latter transformations can be performed only for z ≥ 0, Fig. 2(b), as half of the zt and zω -domains carries redundant information.
In the similar way redundant information is present in one half of the kω- and zω-domains in the case of modified STdOCT processing. Therefore, the total number of Fourier transformations in complex ambiguity free STdOCT can be halved by the proper choice of order of Fourier transformations if no high resolution in velocity is required. In such a case, after transforming data from kt -domain to kω-domain, the following transition to zω-domain can be completed for ω ≥ 0 only, Fig. 4(b). This is especially useful for complex ambiguity removal in a real-time tomogram creation for purposes of alignment of the sample. The latter can be also used in the case where no velocity information is required and one is interested only in complex ambiguity removal. This technique has been used by our group for complex ambiguity removal in imaging of whole anterior chamber of the human eye in vivo along with both surfaces of the lens [30].
Fig. 4. Schematic STdOCT diagram showing data flow in the modified STdOCT algorithm allowing recovery of velocity distribution as a function of depth in the whole imaging range: (a) slower version of the algorithm with high velocity resolution provided by implementation of zero-padding procedure before the t→ω Fourier transformation; (b) faster version of the algorithm without zero-padding.
It is easy to see that both variants of the STdOCT technique differ only by choice of parts of zω-domain where the procedure of A-scan creation is performed. As a result, the trade-off between velocity range and structural imaging range is different for the regular and the modified technique. Standard STdOCT gives halved structural imaging range but the velocity detection range is full. On the contrary, the complex ambiguity free STdOCT provides full structural imaging range, but the velocity range is halved. In order to separate complex conjugate images in both techniques the images have to be placed on opposite sides of z-axis (in case of standard STdOCT) or ω-axis (in case of modified STdOCT). It has to be noted that if the SOCT setup allows for offsets’ introduction in both z and ω directions, the transition between the two modalities of STdOCT is straightforward, and allows the user to choose the velocity and imaging range according to current needs.
In principle there is exactly the same SNR advantage for both the standard and complex ambiguity free STdOCT techniques. However, in practice introduction of the constant change of optical path difference (OPD) will cause the fringe wash-out effect, which will decrease the spectral fringe visibility and the same the sensitivity of our technique [31,32]. We can easily calculate the magnitude of the fringe wash-out in the case of our optical delay line, since it is set to the constant velocity value (Eq. (7)) corresponding to the phase shift of λ/8 between consecutive spectral fringe signals. According to the theoretical analysis presented by Bachmann et al. [32] the sensitivity drop due to such phase shift is only 1dB.
3. Experimental set-up
3.1. SOCT device
The SOCT system based on fiber-optic Michelson interferometer configuration is shown in Fig. 5.
Fig. 5. SOCT set-up used in experiments: FC – fiber coupler (splitting ratio 90:10); PC – polarization controller; L1–L4 – lenses; NDF – neutral density filter; DC – dispersion compensation; ODL – optical delay line; SM-Z – galvanometer scanner (z-scanning); RM – reference mirror; SM-XY – galvanometer scanners (xy-scanning); DG – holographic volume diffraction grating; CMOS – line-scan camera; FG – frame-grabber; COMP – personal computer; IOC – input-output card.
The light source is a femtosecond Ti:Sapphire laser (Fusion, Femtolasers, Δλ = 160nm, central wavelength 810 nm). Light from the laser is launched to an optical single mode fiber interferometer (AC Photonics Inc., USA) that splits 90% of light energy into the reference arm and the rest into the object arm. The reference arm is equipped with a neutral density filter to adjust light intensity as well as with a dispersion controller to minimize the dispersion mismatch between both arms of the interferometer. Optical delay line (ODL) using galvanometer-based scanner (Cambridge Technology Inc., USA), collimating achromat lens (f1 = 19mm) and another achromat lens L2 (f2 = 40mm) is placed in the reference arm. Detailed description of the ODL is given in the next subsection. The detection arm consists of a custom designed spectrometer containing an achromatic collimating objective (Schneider Optics, USA), a volume holographic diffraction grating (1200 lpmm; Wasatch Photonics, USA) and a f-theta objective (Sill Optics, Germany) imaging the spectral interference fringes at the 12-bit line scan CMOS camera with 4096 pixels (Sprint, Basler AG, Germany). CMOS camera was set to acquire 2048 pixels from 4096 available photosensitive elements. Transversal scanning of the sample is provided by a set of two galvanometer-based optical scanners (Cambridge Technology Inc., USA). Triggers to all scanners and the camera are delivered in analog form by an analog input/output card (National Instruments, USA). The experimental data registration is realized by a framegrabber (National Instruments, USA).
Measured axial resolution in air is equal to 3 μm what corresponds to 2.3 μm in tissue. Imaging range without the complex ambiguity removal technique equals 1 mm in tissue and is doubled to 2 mm in case the complex ambiguity removal technique is used. All measurements were performed with the power of light incident the cornea equal to 750μW at 810 nm.
3.2. Optical delay line
In order to introduce a velocity offset we equipped the reference arm with a simple optical delay line (ODL) based on galvanometer scanner that deflects the light beam towards the reference mirror (Fig. 6). The axis of rotation of the scanner is shifted from the point hit by the light beam, what causes the optical path length of the reference arm to be dependent on the instantaneous angle of the scanner.
The concept to use the ODL based on a scanner with off-axis illumination has been exploited recently by several groups [33–36], but they placed it in the sample arm and used the laterally scanning mirror to introduce the optical path delay. This idea is very elegant but we deliberately decided not to use it as it couples velocity offset with lateral scanning parameters. When ODL is placed in the reference arm, it is much easier to introduce an arbitrary velocity along with arbitrary lateral scanning protocol. It is also possible to switch the ODL off if necessary. This fact is important because of different trade-offs between velocity and imaging ranges in standard and complex ambiguity free STdOCT as mentioned in the method section. Switching off the delay line brings the velocity offset to zero and enables standard STdOCT to provide full velocity range with limited structural imaging range. With the aid of this ODL such transition is possible at any moment, for example during aiming and aligning of the sample.
Fig. 6. (a) Detailed scheme of the optical delay line introduced in the reference arm of the SOCT set-up used in experiments: L1, L2 – lenses; SM-Z – galvanometric scanner; RM -reference mirror; f – focal length of the lens L2; For clarity the neutral density filter and dispersion compensator are removed from the drawing. (b) Parameters used to calculate the optical path difference: δ – shift between the incoming beam and the axis of rotation of SM-Z; α – tilt angle of SM-Z; z – optical path distance introduced by scanner tilt. The part of the light beam marked in red is responsible for optical path difference between the two positions of the rotating mirror. See text for further details.
When the optical beam is deflected by the mirror attached to the galvanometric scanner SM-Z in the distance δ from the pivot, the optical path difference zδ (α) depends on the angle α between the direction perpendicular to the impinging beam and the surface of the scanner:
Taylor expansion of the above expression around an initial angle α 0 yields:
In this expression the quadratic term can be omitted, since it is usually much smaller than the linear part. This allows for calculation of the velocity introduced by the scanner rotating with angular frequency ω SM-Z = dα/dt:
The factor 2 is a consequence of the fact that the light travels the path zδ (α) twice.
In the configuration used in our experiments, the angle α 0 = 45°, therefore Eq. (10) simplifies to:
4. Results and discussion
4.1. Extinction ratio and scanning protocols for complex ambiguity free STdOCT
In order to measure the extinction ratio of the complex conjugate removal in STdOCT method, we performed experiment with one static mirror and one moving mirror in the interferometer arms. Measured ratio of the complex conjugate remaining signals to the peak corresponding to the mirror position was -22dB and -30dB for 16 and 32 spectral fringes used for STdOCT analysis.
The SOCT setup operates in two main modes. The preview mode displays two structural tomograms (each of 420 A-scans created from 2172 spectra) acquired perpendicularly to each other in horizontal and vertical directions. These two images are used for alignment of the head of the device (probing beam) with respect to the sample. Fast mode of complex ambiguity free STdOCT, Fig. 4(b), allows for real-time display of images at 5 frames/sec.
The imaging mode uses high velocity resolution mode of complex ambiguity free STdOCT, Fig. 4(a), and provides structural and velocity tomograms in two or three dimensions. As mentioned in previous sections in order to be able to perform quantitative velocity estimation lateral shift between the positions of the scanning beam has to be smaller than the beam diameter, which is assumed to be 20 μm at the retina. Spectra are acquired at 100 000 lines/s with the integration time of 8.7 μs. As a result, the maximal velocity range νmax is equal to 20.2 mm/s for standard imaging and 10.1 mm/s for complex ambiguity free imaging. Examples of scanning protocols that fulfill this condition are presented in Table 1. It can be seen that in all cases the lateral oversampling factor [37] which is the ratio of the sampling density to the beam width is not larger 0.2.
Table 1. Scanning protocols used in experiments with complex ambiguity free STdOCT.
The signal driving the galvanometer scanner in the ODL is triangular. The angular frequency of the scanner is set to introduce reference velocity νref = νmax/2, which provides maximal velocity range, according to Eq. (7). The amplitude of the scanner driving signal remains constant. Exactly 543 spectra acquisitions are performed along each monotonic ramp of the signal. The triangular signal is synchronized with the beginning of every B-scan. In all cases one final A-scan is constructed from 16 spectral fringe signals. To create B-scans or velocity maps with a reasonable quality we divided our initial 543 spectral fringes into 105 partially redundant sets of 16 spectral fringes (105 =⌊ (543 -16)/5⌋). Separation between each set of 16 fringes is chosen to be 5, only to optimize the calculation time. This procedure introduces some kind of smoothing or interpolation into the velocity maps. Zero-padding (z-axis) up to 128 data points is performed before applying Fourier transformation, Fig. 2(b) and Fig. 4(a).
The calculations are performed with our software written in C and run on a personal portable computer (HP Compaq 8510w). The average computation time in all the above examples (in high velocity resolution mode) is about 1000 final tomogram lines per second. Total computation time depends on the protocol and is equal 6.7 seconds for the 2D_1 protocol, 9.5 seconds for 2D_2 protocol and 286 seconds (approx. 5 minutes) for 3D_1 protocol.
4.2. Complex ambiguity free STdOCT
In order to show the capability of the complex ambiguity free STdOCT technique to provide mirror free velocity and structural tomograms, we performed experiments of in-vitro flows in glass capillaries as well as in retinal vessels in-vivo.
Fig. 7. STdOCT measurement of two glass capillaries with Intralipid solution. (a) structural cross-sectional image obtained using standard STdOCT processing; (b) velocity map obtained using phase-resolved OCT ν max = 20.2 mm/s ; (c) structural cross-sectional image obtained using complex ambiguity free STdOCT; (d) velocity map obtained using complex ambiguity free STdOCT ν max = 10.1 mm/s.
In the first experiment water solution of Intralipid flowing inside two glass capillaries of 700 μm diameter was imaged. Laminar flow of the fluid was assured by a HPLC (High performance liquid chromatography) pump (Selko Industries, Poland). Scanning protocol 2D_1 described in Table 1 was utilized. Fig. 7(a) shows a cross-sectional image obtained by application of standard STdOCT algorithm with immobilized ODL. Fig. 7(b) depicts velocity tomogram obtained with aid of standard phase-resolved [11]. Two complex conjugate (mirror) images are visible in both tomograms. Figs. 7(c) and 7(d) present structural and velocity tomograms free from one of the mirror images. The data were obtained with the ODL turned on. In the following experiments data were acquired from human retina in-vivo using scanning protocol 2D_2, Table 1. Results are shown in Fig. 8. Sixteen spectra are used to create a single A-scan of the velocity map and complex ambiguity free cross-sectional image from the lateral range of 5 mm. As a result, the effective lateral resolution drops approximately by 50%, to 30 μm. The complex conjugate image is suppressed in both structural and velocity images.
Fig. 8. High density cross-sectional image of human optic disk in-vivo obtained using complex ambiguity free STdOCT: (a) structural tomogram; (b) velocity tomogram ν max =10.1 mm/s.
The scanning protocol 3D_1 was used to create 3D data set (Table 1). Because we are analyzing 5×5mm region the effective lateral resolution is 60 μm. Moreover, one can notice that the complex conjugate images are almost removed. The remaining parts of the vessels marked by red arrows in Fig. 9(b) and Fig. 9(d) are discussed in the next subsection.
Fig. 9. Two cross-sectional images of human retina in-vivo measured in the close proximity of optic nerve head from 3D data set. (a) and (c) structural tomogram obtained using complex ambiguity free STdOCT; (b) and (d) velocity map obtained using complex ambiguity free STdOCT ν max = 10.1 mm/s . Arrows indicate artifacts – remaining mirror images of vessels with high flow velocity.
4.3. Artifacts
Optimal configuration of the complex ambiguity free STdOCT requires setting the additional velocity value to reach level of the maximal velocity divided by 2. The additional velocity of the reference mirror causes the optical path difference (OPD) between consecutive spectral fringe signals of νrefΔt = λ 0/8, where λ 0 is the central wavelength of the light used in the experiments. Therefore, OPD between the two extreme positions of the ODL in case of 543 spectra acquisitions is as large as 55 μm. As a result, tomograms acquired with operating ODL suffer from geometrical distortion caused by the OPD changes during scanning, Fig. 10(a).
Fig. 10. Correction of artifact connected with optical path delay changes during data acquisition. (a) tomogram without correction; (b) tomogram with numerical correction of tomogram lines shifts.
This effect can be easily removed by a small modification in numerical dispersion compensation algorithm that is usually performed in the preprocessing stage of the data processing. Each spectrum in the interferogram, Eq. (2), has to be multiplied by the same phase component equal to exp[-i·2Δz·k], where Δz is the instantaneous OPD introduced by the ODL at the moment of acquisition of the central spectrum from the interferogram. Since the phase element is constant for each A-scan of the tomogram, it can be calculated once and tabularized. Therefore, it can be incorporated in numerical dispersion compensation algorithm at no additional computational cost. The result of the procedure is depicted in Fig. 10(b), where the geometrical distortion is no longer present. This procedure has been applied to all tomograms presented in this paper.
The second artifact appears when the velocity of the inner motion of the sample does not fulfill conditions formulated by Eq. (5) and Eq. (6) and it is larger than the velocity limits. In such a case the parts of the complex conjugate image are visible in the resulting image. This effect is clearly visible in Fig. 11 as well as in Figs. 9(b) and 9(d). The signal in zω-domain reveals the parabolic distribution of flow inside the vessel wrapped to the opposite side of the velocity range and thus entering on the opposite side of beat frequency domain, see dotted arrow in Fig. 11(c). Similar artifact has already been reported by Makita et al. [28] and is also the basis of the vessel segmentation technique proposed by Wang et al. [12]. The solution to this problem is to increase the available velocity range, what can be done by increasing the line-rate of the acquisition system. Obvious possibility is just to switch off the optical delay line and perform standard STdOCT imaging to double the velocity range but the cost to be paid in such a case is the limited imaging range.
Fig. 11. Artifacts appearing when actual flow velocity exceeds velocity range. (a) Structural tomogram with vessel from the complex conjugate image visible (arrows); (b) velocity tomogram of (a), ν max =10.1 mm/s ; (c) signal in zω-plane corresponding to the tomogram line (dotted line), parabolic velocity distribution cross the velocity range limit.
5. Conclusions
We showed that the joint Spectral and Time domain OCT is a straightforward and computationally efficient data processing scheme that can take advantage of the ultra-high speed OCT imaging of more than 100 000 lines/s. It can be used with any densely sampled scanning protocols and can be incorporated in standard Spectral OCT setups. Simple modification to the algorithm combined with an optical delay line in the reference arm can provide structural and functional complex ambiguity free cross-sectional images. High flexibility of the algorithm allows for rapid changes between standard STdOCT (full velocity range and halved imaging range) and complex ambiguity free STdOCT (halved velocity range and full imaging range). The computational burden is increased as compared to the standard SOCT processing but still a single tomogram can be calculated in few seconds and a 3D data set in no longer than 5 minutes.
Acknowledgments
Project supported EURYI grant/award funded by the European Heads of Research Councils (EuroHORCs) together with the European Science Foundation (ESF – EURYI 01/2007PL) and by Ventures Programme co-financed by the EU European Regional Development Fund both programs are operated within the Foundation for Polish Science. Maciej Szkulmowski and Anna Szkulmowska acknowledge additional support of Foundation for Polish Science (scholarship START 2009). We would also like to acknowledge FEMTOLASERS Produktions GmbH for their support and Prof. Bogusław Buszewski (Department of Environmental Chemistry and Ecoanalytics; Faculty of Chemistry; Nicolaus Copernicus University, Toruń) for providing the pump.
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