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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 1 — Jan. 5, 2009
  • pp: 7–24
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Dispersion encoded full range frequency domain optical coherence tomography

Bernd Hofer, Boris Považay, Boris Hermann, Angelika Unterhuber, Gerald Matz, and Wolfgang Drexler  »View Author Affiliations


Optics Express, Vol. 17, Issue 1, pp. 7-24 (2009)
http://dx.doi.org/10.1364/OE.17.000007


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Abstract

We propose an iterative algorithm that exploits the dispersion mismatch between reference and sample arm in frequency-domain optical coherence tomography (FD-OCT) to effectively cancel complex conjugate mirror terms in individual A-scans and thereby generate full range tomograms. The resulting scheme, termed dispersion encoded full range (DEFR) OCT, allows distinguishing real structures from complex conjugate mirror artifacts. Even though DEFR-OCT has higher post-processing complexity than conventional FD-OCT, acquisition speed is not compromised since no additional A-scans need to be measured, thereby rendering this technique robust against phase fluctuations. The algorithm uses numerical dispersion compensation and exhibits similar resolution as standard processing. The residual leakage of mirror terms is further reduced by incorporating additional knowledge such as the power spectrum of the light source. The suppression ratio of mirror signals is more than 50 dB and thus comparable to complex FD-OCT techniques which use multiple A-scans.

© 2009 Optical Society of America

1. Introduction

With standard frequency-domain optical coherence tomography (FD-OCT), only half of the available depth range is used due to the occurrence of so-called complex conjugate artifacts. The complex conjugate artifacts or mirrored signal components are caused by the symmetry properties of the Fourier transform of a real-valued spectrum. Essentially the negative depth range (-range) is equal to the positive range (+range), if the dispersion of the interferometer sample and reference arm is properly matched in the optical system, thereby leading to symmetries around the zero-delay position. The +range is normally truncated in the final tomograms but components outside the -range fold back into the tomogram and cannot be distinguished from actual structures. Hence objects are imaged in the +range, away from the zero-delay, where the sensitivity would be the highest.

Complex FD-OCT techniques have been developed to allow use of the full depth range. They measure the complex spectrum by stepping the reference mirror [1–4

1. M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. 27, 1415–1417 (2002). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-27-16-1415 [CrossRef]

], employing electro-optic phase modulators [5

5. J. Zhang, J. S. Nelson, and Z. P. Chen, “Removal of a mirror image and enhancement of the signal-to-noise ratio in Fourier-domain optical coherence tomography using an electro-optic phase modulator,” Opt. Lett. 30, 147–149 (2005). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-30-2-147 [CrossRef] [PubMed]

,6

6. E. Götzinger, M. Pircher, R. A. Leitgeb, and C. K. Hitzenberger, “High speed full range complex spectral domain optical coherence tomography,” Opt. Express 13, 583–594 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-2-583 [CrossRef] [PubMed]

], using 3×3 fibre couplers [7–9

7. M. A. Choma, C. Yang, and J. A. Izatt, “Instantaneous quadrature low-coherence interferometry with 3×3 fiber-optic couplers,” Opt. Lett. 28, 2162–2164 (2003). http://ol.osa.org/abstract.cfm?URI=ol-28-22-2162 [CrossRef] [PubMed]

], enforcing a sequential phase shift between consecutive A-lines [10–15

10. Y. Yasuno, S. Makita, T. Endo, G. Aoki, M. Itoh, and T. Yatagai, “Simultaneous B-M-mode scanning method for real-time full-range Fourier domain optical coherence tomography,” Appl. Opt. 45, 1861–1865 (2006). http://ao.osa.org/abstract.cfm?URI=ao-45-8-1861 [CrossRef] [PubMed]

] or frequency shifting [16–19

16. S. Yun, G. Tearney, J. de Boer, and B. Bouma, “Removing the depth-degeneracy in optical frequency domain imaging with frequency shifting,” Opt. Express 12, 4822–4828 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4822 [CrossRef] [PubMed]

] and polarization diversity [20

20. B. J. Vakoc, S. H. Yun, G. J. Tearney, and B. E. Bouma, “Elimination of depth degeneracy in optical frequency-domain imaging through polarization-based optical demodulation,” Opt. Lett. 31, 362–364 (2006). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-31-3-362 [CrossRef] [PubMed]

]. Each technique has its advantages and disadvantages regarding complexity and quality. Common to all the techniques so far is the requirement of at least two measurements in order to reconstruct the full imaging range, therefore either increasing system complexity or reducing acquisition speed and transversal scanning range.

Dispersion mismatch between the two interferometer pathways leads to blurring of structures in the tomogram. Satisfactory compensation of physical dispersion mismatch can be achieved numerically as reported in [21–23

21. B. Cense, N. A. Nassif, T. C. Chen, M. C. Pierce, S. H. Yun, B. H. Park, B. E. Bouma, G. J. Tearney, and J. F. de Boer, “Ultrahigh-resolution high-speed retinal imaging using spectral-domain optical coherence tomography,” Opt. Express 12, 2435–2447 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-11-2435 [CrossRef] [PubMed]

]. While the peaks corresponding to true signal components will become narrower, mirrored signal components are further blurred as a side effect of numeric dispersion compensation [24

24. K. E. O’Hara and M. Hacker, “Method to suppress artifacts in frequency-domain optical coherence tomograghy,” US7330270 (2008).

]. This effect allows further suppression of residual mirrored signal components in complex FD-OCT [15

15. S. Makita, T. Fabritius, and Y. Yasuno, “Full-range, high-speed, high-resolution 1-μm spectral-domain optical coherence tomography using BM-scan for volumetric imaging of the human posterior eye,” Opt. Express 16, 8406–8420 (2008). http://www.opticsexpress.org/abstract.cfm?URI=oe-16-12-8406 [CrossRef] [PubMed]

]. Numerical dispersion compensation is also applied to the retinal tomograms in this paper by multiplying the measurements with a counter-dispersive complex phase term prior to inverse Fourier transformation.

In this manuscript, we propose to exploit the blurring effect by numeric dispersion compensation to reconstruct the full-range image. To this end, we consider the dispersion mismatch to actually encode the spectral phase and thereby position of individual complex signal components within the +range or -range. The resulting dispersion encoded full range (DEFR) algorithm decodes the position and strength of true structures via an iterative procedure that requires only the actual dispersive phase as prior knowledge. Based on a simple physical model, we provide an intuitive implementation and block diagram for DEFR, describing in detail the additionally required signal processing tasks. We show that reconstruction of the full image range from single measurements is possible using a novel scheme and test the algorithm on signals from a simple free-space interferometer. We also demonstrate the applicability to in vivo retinal imaging. In contrast to complex FD-OCT techniques that use more complicated measurement setups, DEFR uses standard signal acquisition and shifts the complexity increase to post-processing. This suggests that our approach is well suited for ultra-high resolution OCT imaging.

2. Background and theory

2.1. OCT signal with dispersion

According to [23

23. M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12, 2404–2422 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-11-2404 [CrossRef] [PubMed]

], let S(ω) = S̅(ω) + S̃(ω) be the detected spectrum dependent on optical frequency ω. The background signal S̅(ω) = |Er(ω)|2 + |Es(ω)|2 and the interference signal S̃(ω) = 2ℜ{E * r(ω)Es(ω)}both depend on the time averaged electric fields Er(ω) and Es(ω) of the reference and sample arm, respectively. To evaluate the influence of bulk chromatic dispersion mismatch between Er and Es on S̃(ω) we consider a free-space Michelson interferometer (FSI) with a dispersion free reference arm of length zr. The sample arm consists of multiple reflecting layers at geometrical positions z (n) s that are embedded in dispersive material which starts at position zd and has refraction index n(ω). The beam splitter surface is positioned at depth z = 0 as in [25

25. J. A. Izatt and M. A. Choma, Optical Coherence Tomography Technology and Applications (Springer, 2008), Vol. XXIX, chap. 2, “Theory of Optical Coherence Tomography”, pp. 47–72. http://www.springer.com/medicine/radiology/book/978-3-540-77549-2?detailsPage=samplePages [CrossRef]

]. With this model the plane waves Er and Es can be written as

Er(ω)=Ir(ω)exp(2zrc0)
(1)

and

Es(ω)=nIs(n)(ω)exp(2zdc0)exp(jωn(ω)2(zs(n)zd)c0).
(2)

The splitting ratio of the beam splitter, power spectrum of the light source, polarization effects and detector efficiency are incorporated in the frequency dependent intensities Ir(ω) and I (n) s(ω) which also account for the individual reflectivities. Assuming a thin sample we can make the approximation z (n) s - zdz (1) s -zd = d. This assumption very well applies to retinal imaging, corresponding to the dispersive vitreous of the eye of thickness d which is followed by the relatively thin retina comprised by layered structures [23

23. M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12, 2404–2422 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-11-2404 [CrossRef] [PubMed]

]. Furthermore using the dispersion relation k(ω) = n(ω)ω/c 0, (2) can be rewritten as

Es(ω)nIs(n)(ω)exp(2(zs(n)d)c0)exp(jk(ω)2d),
(3)

and the interference signal is than given by

S˜(ω)=2n{Ir(ωIs(n)(ω)exp(2(zs(n)zr)c0)exp(j(k(ω)ωc0)2d)}.
(4)

The Taylor series expansion of k(ω) around the center frequency ω 0, which reads [26

26. A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Numerical dispersion compensation for partial coherence interferometry and optical coherence tomography,” Opt. Express 9, 610–615 (2001). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-12-610 [CrossRef] [PubMed]

]

k(ω)=i=0ai(ωω0)i,ai=1i!dik(ω)dωiω0,
(5)

exp((ω))=exp(ji=2a˜i(ωω0)i),
(6)

with the dispersion coefficients ãi = 2dai. Using (6), the reference position in the dispersive system z̃r = zr + (1 - a 1 c 0)d and the dispersion shifted intensity I(n)(ω)=2Ir(ω)Is(n)(ω)exp(j(k(ω0)a1ω0)2d) in (4), we obtain the interference signal

S˜(ω)=n{I(n)(ω)exp(2(zs(n)z˜r)c0)exp((ω))}.
(7)

With the optical delays τn=2(zs(n)z˜r)c0 this finally results in the dispersed interference signal

S˜(ω)=nI(n)(ω)eτne(ω)+nI(n)*(ω)eτne(ω).
(8)

Dispersion compensation can be achieved by multiplying (8) with a compensating phase term e -(ω). Thus we obtain the dispersion compensated signal

S˜c(ω)=S˜(ω)e(ω)=nI(n)(ω)eτnS˜d(ω)+nI(n)*(ω)eτnej2ϕ(ω)S˜m(ω),
(9)

which is a superposition of the desired full range signal S̃d(ω) and its “doubly-dispersed” conjugate mirror components S̃m(ω).

2.2. DEFR algorithm

Let d ∈ ℂN be the complex vector corresponding to the sampled desired full range signal d(ω) in (9), with sample points uniformly spaced in optical frequency ω. This signal can be reasonably modeled as a linear combination of K < N vectors from the Fourier basis matrix Ψ := [ψ 1, ψ 2,…,ψ N]∈ ℂN×N, i.e.,

d=l=1Ktnlψnl=Ψt.
(10)

Here, the length-N vector t has only K non-zero elements tnl, i.e., t is a sparse vector. The parameter K is in the order of the number of reflecting layers from the physical model above, since with FD-OCT data the Fourier transform of the intensities I (n)(ω) is usually very narrow.

Employing an N × N measurement matrix Φ, a measurement vector f ∈ 𡄝N that corresponds to uniform samples of the interference signal (ω) can be calculated as

f=Φd+Φ*d*+w,
(11)

where w models measurement noise which will be considered in the discussion of Section 4. This is consistent with our physical OCT model in (8) when choosing Φ = diag(ϕ) with ϕ ∈ ℂN corresponding to uniform samples of the bulk dispersive phase e (ω). By inserting (10) into (11) we obtain

f=2{ΦΨt}+w.
(12)

Our goal is to recover the sparse vector t from the measured interference signal f. It has been shown recently that problems of this kind can be phrased as ℓ1 optimization, i.e. (cf [27

27. M. Duarte, M. Davenport, M. Wakin, and R. Baraniuk, “Sparse Signal Detection from Incoherent Projections,” in Proc. Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP)3, III305–308 (2006). http://dx.doi.org/doi:10.1109/ICASSP.2006.1660651

])

t̂=argmintt1subjectto{Vt}=f.
(13)

Here, ∥t1 = ΣN n=1 |tn| and V = 2ΦΨ is the dispersive basis. Our proposed DEFR reconstruction searches for a sparse representation of the measurement vector f in the dictionary {v i} consisting of the column vectors of the dispersive basis V (cf. [27

27. M. Duarte, M. Davenport, M. Wakin, and R. Baraniuk, “Sparse Signal Detection from Incoherent Projections,” in Proc. Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP)3, III305–308 (2006). http://dx.doi.org/doi:10.1109/ICASSP.2006.1660651

]). To this end, we adapt the greedy matching pursuit (MP) algorithm proposed in [27

27. M. Duarte, M. Davenport, M. Wakin, and R. Baraniuk, “Sparse Signal Detection from Incoherent Projections,” in Proc. Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP)3, III305–308 (2006). http://dx.doi.org/doi:10.1109/ICASSP.2006.1660651

] for the preceding signal model in order to iteratively find locally optimum solutions of (13). The algorithm requires specification of the maximum number IK of iterations and of the fraction e of the energy of f that may remain in the residual, which usually depends on the noise level and modeling errors (cf. [27

27. M. Duarte, M. Davenport, M. Wakin, and R. Baraniuk, “Sparse Signal Detection from Incoherent Projections,” in Proc. Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP)3, III305–308 (2006). http://dx.doi.org/doi:10.1109/ICASSP.2006.1660651

]).

MP ALGORITHM FOR DEFR FD-OCT SIGNAL RECONSTRUCTION

  • Initialization: residual r 0 = f; approximation t̂ = 0; iteration counter i = 1.
  • Determine the dictionary vector which is maximally collinear with the residual,

    ni=argmaxn=1Nci,n,withci,n=vn,ri1vn,
    (14)

  • Update the estimate of the corresponding signal component and the residual:

    t̂ni=t̂ni+2ci,ni,
    (15)

    ri=ri12{ci,nivnivni}=ri1ci,nivnivnici,ni*vni*vni.
    (16)

  • Increment i. If i < I and ∥r i|2 > εf2, then continue iterating with Step 2; otherwise proceed with Step 5.
  • Obtain the estimate for the desired full range complex FD-OCT signal:

    d̂=Ψt̂.
    (17)

    We note that the factor of 2 in (16) and (15) is required to preserve signal energy.

3. Methods

3.1. Experimental setup

For in vivo imaging an ophthalmic FD-OCT system operating at 800nm was used, which employed a 90/10 fiber-based interferometer interfaced to a fundus camera and a spectrometer [28

28. B. Povazay, B. Hofer, B. Hermann, A. Unterhuber, J. E. Morgan, C. Glittenberg, S. Binder, and W. Drexler, “Minimum distance mapping using three-dimensional optical coherence tomography for glaucoma diagnosis,” J. Biomed. Opt. 12, 041 204 (2007). http://link.aip.org/link/?JBO/12/041204/1 [CrossRef]

]. The light source was a broadband Ti:sapphire laser (140nm full width half maximum (FWHM)) at 800nm (FEMTOLASER, Integral). The detection system utilized a 2048 pixel silicon CCD-camera (AVIIVA, Atmel M2 CL2014-BAO) based spectrometer and imaging was performed at a line rate of 20 kHz. Residual dispersion of the system and of the eye’s vitreous, which is equivalent to ~25 mm of water, was not compensated. However, polarization mismatch between the two arms was corrected by fiber-optic polarization paddles in both arms and optimized for wide spectral throughput. The optical power at the cornea did not exceed 800 μW when imaging normal subjects at the fovea in a typical 8°(2.3 mm) wide horizontal scan. Scans were performed at different zero delay positions. Strongest overlapping of complex conjugate artifacts was found when setting the zero delay position in the middle of the retina at the outer nuclear layer, around 150–200μm optical path length away from the retinal surface. The real time preview image was numerically corrected online for dispersion, which permitted to adjust the focus and to align the subject for position and tilt.

To calibrate the spectrometer and test the proposed scheme, a simple Michelson-type free-space interferometer (FSI) was set up and interfaced to the light source and the spectrometer. Based on a non-polarizing 50/50 prism-cube beam splitter and a reference arm, mounted on a translation-stage with fiber-optic input and output collimators the device permitted to balance power in both arms by a reflective continuous neutral density filter wheel.

To test the algorithms dependency on the amount of dispersion mismatch, different dispersive materials were alternatively inserted into the sample arm of the free space interferometer. A 25 mm deep glass-cuvette filled with water (“H2O 25mm”); an 8mm thick achromatic triplet (“Achromatizer”) consisting of two materials, 3 mm F2 and 5 mm N-SK4, used for axial chromatic aberration correction of the human eye [29

29. E. J. Fernández, A. Unterhuber, B. Považay, B. Hermann, P. Artal, and W. Drexler, “Chromatic aberration correction of the human eye for retinal imaging in the near infrared,” Opt. Express 14, 6213–6225 (2006). http://www.opticsexpress.org/abstract.cfm?URI=oe-14-13-6213 [CrossRef] [PubMed]

]; two BK7 1 mm microscope slides (“Glass 2 mm”) and a single BK7 1mm microscope slide (“Glass 1mm”) were used. The optical delay was compensated for by adjusting the reference mirror position accordingly and the intensity across the full spectral bandwidth was set to be flat with a similar quasi-Gaussian spectral profile. Due to the simple free-space setup the polarization difference between both arms was negligible.

Fig. 1. Block diagram of the dispersion encoded full range (DEFR) algorithm and associated pre-processing steps. The iterative procedure is indicated by iteration index i. The peak detector is denoted as PD; further notations: measured raw signal s, interference signal , measurement vector f, dispersive phase term ϕ(ω) and corresponding vector ϕ, fast Fourier transform F, residual signal r, intermediate spatial signal c; and finally the complex full range tomogram line t̂ ∈ ℂN. The diagram parts associated with equations (14), (16) and (15) are high-lighted in green, red and blue, respectively.

3.2. Signal processing

Figure 1 shows the block diagram for the proposed algorithm. Also the associated preprocessing steps are included. Data from both experimental setups were treated in almost the same manner, differences will be indicated where applicable. Let s ∈ ℝN be the measured raw signal corresponding to the non-uniformly sampled detected spectrum S(ωp) with sample frequencies ωp. Autocorrelation terms and fixed pattern noise are removed from s via background subtraction. The resulting corrected spectrum s̃ ∈ ℝN corresponds to the sampled interference signal (ωp). The measurement vector f whose elements correspond to uniform samples of (ω) is then obtained via resampling of s̃. The sample frequencies ωp depend on the pixel indices p ∈ {1,…,N} of the CCD linescan camera employed in the spectrometer (in our setup, N = 2048). Specifically, the frequencies ωp correspond to approximately uniformly spaced samples in the wavelength domain, i.e., ωλ -1. We describe below how to determine the actual mapping ωp = g(p). In addition to f, the DEFR algorithm as shown in Fig. 1 requires the vector ϕ of uniform samples of the frequency-dependent dispersive phase term e (ω) as input. The estimation of ϕ from f will be discussed below. The output of the algorithm consists of a complex full range tomogram line t̂ ∈ ℂN.

3.3. DEFR algorithm implementation

What follows is the determination of the update of the residual (cf. (16)). To this end, the signal component ci, ni is subtracted from the intermediate spatial domain signal c i, which then undergoes FFT and inverse dispersion compensation. This amounts to removing the true signal component in the spectral domain. To cancel mirror artifacts (corresponding to the complex conjugate term in the right-hand side of (16), the inverse dispersive phase is applied again and another inverse FFT provides an intermediate spatial domain signal c̃i. The complex conjugate of the true signal component ci, ni is then subtracted from c̃i at the mirror position |N - ni|. Finally, computing the FFT of the “cleaned” intermediate spatial domain signal and undoing the inverse dispersion yields the new residual signal r i. This procedure continues in an iterative manner, adding further signal components to the output signal t̂ until the residual signal contains only noise, i.e., ∥r i2 < εf2, or a maximum number of iterations I is reached.

3.4. Background subtraction

Let b ∈ ℝN be the background signal corresponding to the non-uniformly sampled background spectrum S̅(ωp). To determine b, we employ two different strategies:

  • For in vivo imaging we obtain b as the mean value of the detected signals s x,x = 1,…,Nx, from a whole tomogram, i.e., b=1Nxx=1Nxsx (here, x is a spatial index determining the transversal position of an A-scan on the tomogram consisting of Nx = 1024 lines). This is reasonable when imaging biological samples, since their interference spectra Ŝx(ωp) will average out over a suitable transversal scan range due to structural independence and phase fluctuations [30

    30. R. K. Wang and Z. Ma, “A practical approach to eliminate autocorrelation artefacts for volume-rate spectral domain optical coherence tomography,” Phys. Med. Biol. 51, 3231–3239 (2006).http://www.iop.org/EJ/abstract/0031-9155/51/12/015/ [CrossRef] [PubMed]

    ].
  • For artificial samples such as the translation stage experiment, we determine b as a linear combination b = b s + b r - b d. Here, b s is the background signal from the sample arm when the reference arm is blocked, b r is the background signal from the reference arm with blocked sample arm, and b d is the dark background signal due to a fixed pattern offset on the CCD when both interferometer arms are blocked. To reduce the influence of measurement noise all signals are averaged over Nx = 1024 sequential measurements.

Different background correction strategies have been previously described for FD-OCT Multiplicative background suppression is proposed in [31

31. N. A. Nassif, B. Cense, B. H. Park, M. C. Pierce, S. H. Yun, B. E. Bouma, G. J. Tear-ney, T. C. Chen, and J. F. de Boer, “In vivo high-resolution video-rate spectral-domain optical coherence tomography of the human retina and optic nerve,” Opt. Express 12, 367–376 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-3-367 [CrossRef] [PubMed]

] and background subtraction is briefly described in [23

23. M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12, 2404–2422 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-11-2404 [CrossRef] [PubMed]

] and in more detail in [30

30. R. K. Wang and Z. Ma, “A practical approach to eliminate autocorrelation artefacts for volume-rate spectral domain optical coherence tomography,” Phys. Med. Biol. 51, 3231–3239 (2006).http://www.iop.org/EJ/abstract/0031-9155/51/12/015/ [CrossRef] [PubMed]

]. In our system, the strength of the background signal fluctuates over successive tomogram lines, which is probably caused by timing issues within the CCD-electronics of the employed camera. We therefore use a subtractive background correction that additionally adjusts for the strength of the background signal via the normalized inner product ⟨b, s⟩/∥b2, i.e.,

s˜=sb,sb2b.
(18)

This procedure effectively suppresses shaded horizontal lines otherwise visible on in vivo tomograms.

3.5. Spectrometer calibration and resampling

Exact calibration of the spectrometer sampling points ωp is essential for achieving depth independent high resolution in the spatial domain as has been shown in [32

32. C. Dorrer, N. Belabas, J.-P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17, 1795–1802 (2000). http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-17-10-1795 [CrossRef]

]. A parametric approach is to choose ωpp -1 [33

33. M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463 (2002). http://dx.doi.org/10.1117/1.1482379 [CrossRef] [PubMed]

,34

34. M. A. Choma, M. V. Sarunic, C. H. Yang, and J. A. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11, 2183–2189 (2003). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-18-2183 [CrossRef] [PubMed]

] as a first approximation that can be improved by taking the grating equation into account [31

31. N. A. Nassif, B. Cense, B. H. Park, M. C. Pierce, S. H. Yun, B. E. Bouma, G. J. Tear-ney, T. C. Chen, and J. F. de Boer, “In vivo high-resolution video-rate spectral-domain optical coherence tomography of the human retina and optic nerve,” Opt. Express 12, 367–376 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-3-367 [CrossRef] [PubMed]

, 35

35. B. H. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. J. Tearney, B. E. Bouma, and J. F. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 μm,” Opt. Express 13, 3931–3944 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-3931 [CrossRef] [PubMed]

]. Non-parametric approaches aim to use calibration targets within an interferometric setup. Here, ωp is estimated by local phase analysis of the spectral data from single reflections [36

36. R. A. Leitgeb, W. Drexler, A. Unterhuber, B. Hermann, T. Bajraszewski, T. Le, A. Stingl, and A. F. Fercher, “Ultrahigh resolution Fourier domain optical coherence tomography,” Opt. Express 12, 2156–2165 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-10-2156 [CrossRef] [PubMed]

–38 ]. To account for the entanglement with the phase induced by residual dispersion mismatch, a second measurement at different optical path length difference (OPD) can be used [15

15. S. Makita, T. Fabritius, and Y. Yasuno, “Full-range, high-speed, high-resolution 1-μm spectral-domain optical coherence tomography using BM-scan for volumetric imaging of the human posterior eye,” Opt. Express 16, 8406–8420 (2008). http://www.opticsexpress.org/abstract.cfm?URI=oe-16-12-8406 [CrossRef] [PubMed]

]. The non-parametric approaches are especially suited for systems employing broad-band lasers since each pixel can in principle be characterized individually as opposed to calibration light sources which only provide a limited number of spectral reference lines. Slight deviations from the true ωp will lead to depth dependent resolution loss. Thus, we use a phase analysis based calibration procedure as follows.

The input fiber of the spectrometer is connected to the FSI. The calibration of ωp remains mechanically long-term stable (> 6 months) as long as the most critical component, the input fiber, stayed connected to the FC/APC connector on the collimator of the spectrometer. The FSI was adjusted such that virtually no dispersion mismatch between both interferometer arms occurred, which was verified during the phase analysis. Recordings for several OPDs where taken and according to (7) the spectrometer output interference signal from the simple free space setup with I(ωp) ∈ ℝ then reads

S˜(ωp)=I(ωp){exp(jωp2(zszr)c0)}=I(ωp){exp(jωpτz)}.
(19)

The position-dependent phase φp,z = ωp τz can thus be obtained as the phase of the the analytic signal of (19). For known positions τz the mapping function g(p) = ωp can be derived from φp,z and used for resampling onto the uniform grid

ωu=ω1+ωNω1N1(u+N2),ωu[ω1,ωN],
(20)

ωp(u)=b0+b1u+b2u2+b3u3+=b0+b1u+ω˜p(u).
(21)

Thereby b 0 and b 1 are readily determined as b0=ω1+N2b1,b1=ωNω1N1 and the higher order terms have been combined into ω˜ p(u). The mapping function g(p) can now be rewritten from (21) by using the discrete mapping function g̃(u), i.e.,

g(p)=ωp(u)=ω1+ωNω1N1(g˜(u)+N2),withg˜(u)=u+ω˜p(u)b1.
(22)

From Eqs. (20) and (22) we observe that resampling from ωp to ωu corresponds to resampling from (u) to u. Given the phase function φp,z = ω p(u) τz = b 0 τz + b 1 τzu + ω̃p(u) τz we can calculate (u) as g˜(u)=φp,zb0τzb1τz. If the discrete mapping function (u) is used for resampling, the position τz does not need to be known exactly. Only the combined terms b 0 τz and b 1 τz have to be estimated and can finally be obtained directly from φp,z via least-squares estimation:

[b0τzb1τz]=([1u]T[1u])−1[1u]Tφz,
(23)

where 1 ∈ ℝN, u=[N2,,N21]T and φz ∈ ℝN with elements φp,z.

Any dispersion mismatch would be visible as separation of the non-linear phase functions (u) - u from positive and negative OPDs. Since the dispersion was closely matched this effect is not visible. Measurement noise is reduced by averaging 1024 sequentially measured values of the s̃z corresponding to (ωp). Furthermore the accuracy of depth independence of the extracted mapping function (u) is increased by averaging over several positive and negative positions z. Only positions not effected by aliasing are considered, since the phase function is corrupted above 2/3 of the depth range due to aliasing around positive and negative end of depth (EOD). We also exclude positions close to the zero delay (ZD) position (within ±50μm) for which an overlap with the complex conjugate terms occurs. Finally influence of an internal reflection from the light source can be reduced by band-pass filtering the main peak. We used linear resampling of the signals s̃ from g̃(u) to u. Up-sampling by a factor 2 and sinc-interpolation filter with length 2N was used prior to the resampling to prevent additional aliasing components close to EOD and to reduce resampling errors of the fast linear procedure. Finally the resampled signals were decimated again to obtain the measurements f.

3.6. Dispersion compensation and estimation

A number of different approaches have been proposed for dispersion compensation and thus resolution enhancement for OCT. Proper dispersion management is essential when using broad bandwidth light sources and can be accomplished by hardware dispersion balancing of both interferometer arms [39

39. W. Drexler, U. Morgner, F. X. Kärtner, C. Pitris, S. A. Boppart, X. D. Li, E. P. Ippen, and J. G. Fu-jimoto, “In vivo ultrahigh-resolution optical coherence tomography,” Opt. Lett. 24, 1221–1223 (1999). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-24-17-1221 [CrossRef]

]. Software dispersion compensation by application of a correcting phase term in the frequency domain was already proposed in [26

26. A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Numerical dispersion compensation for partial coherence interferometry and optical coherence tomography,” Opt. Express 9, 610–615 (2001). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-12-610 [CrossRef] [PubMed]

,40

40. J. F. de Boer, C. E. Saxer, and J. S. Nelson, “Stable carrier generation and phase-resolved digital data processing in optical coherence tomography,” Appl. Opt. 40, 5787–5790 (2001). http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-40-31-5787 [CrossRef]

] and is well suited for FD-OCT [21

21. B. Cense, N. A. Nassif, T. C. Chen, M. C. Pierce, S. H. Yun, B. H. Park, B. E. Bouma, G. J. Tearney, and J. F. de Boer, “Ultrahigh-resolution high-speed retinal imaging using spectral-domain optical coherence tomography,” Opt. Express 12, 2435–2447 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-11-2435 [CrossRef] [PubMed]

,23

23. M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12, 2404–2422 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-11-2404 [CrossRef] [PubMed]

]. It was also recognized that phase adaptation can be realized via signal resampling [22

22. 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, 3038–3046 (2003). http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-42-16-3038 [CrossRef] [PubMed]

,41

41. D. L. Marks, A. L. Oldenburg, J. J. Reynolds, and S. A. Boppart, “Digital algorithm for dispersion correction in optical coherence tomography for homogeneous and stratified media,” Appl. Opt. 42, 204–217 (2003). http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-42-2-204 [CrossRef] [PubMed]

], which is also well applicable to swept-source FD-OCT [42

42. A. R. Tumlinson, B. Hofer, A. M. Winkler, B. Považay, W. Drexler, and J. K. Barton, “Inherent homogenous media dispersion compensation in frequency domain optical coherence tomography by accurate k-sampling,” Appl. Opt. 47, 687–693 (2008). http://ao.osa.org/abstract.cfm?URI=ao-47-5-687 [CrossRef] [PubMed]

]. As described above we implement dispersion compensation by applying the compensating phase term e -(ω) prior to inverse FFT. We estimate the frequency-dependent dispersive phase ϕ(ω) using information entropy of the spatial domain signal (on a linear scale) as sharpness metric R(·). This is equivalent to the procedure described in [43

43. Y. Yasuno, Y. Hong, S. Makita, M. Yamanari, M. Akiba, M. Miura, and T. Yatagai, “In vivo high-contrast imaging of deep posterior eye by 1-um swept source optical coherence tomography andscattering optical coherence angiography,” Opt. Express 15, 6121–6139 (2007). http://www.opticsexpress.org/abstract.cfm?URI=oe-15-10-6121 [CrossRef] [PubMed]

]. We only determine two parameters â2 and â3 corresponding to the second and third order dispersion coefficients ã2 and ã3 in (6), i.e., ϕ(ωu)=1N1(â2u2+â3u3).. This is sufficient in our measurements and results in nearly no deviation from the optimal free-space resolution as shown later. For data from the free-space interferometer we combined measurements at several OPDs to estimate dispersion from this combined OCT image and thereby ensured depth-independence of the estimated dispersion coefficients.

In Fig. 2(d) it can be seen that prior to dispersion compensation the raw signals’ magnitude from positive OPDs (solid lines) and mirror terms from negative OPDs (dashed lines) cannot be differentiated and are equally blurred. The dispersion compensated signals over the positive depth range are shown in Fig. 2(e). Signals originating from the +range appear sharp, whereas the mirror terms in the -range are further blurred and can thus be clearly differentiated from the non-mirrored (true) signal components. This practically demonstrates the basis of the DEFR algorithm. Whereas here the mirror terms are 16 dB below the true signals, this separation is much lower for smaller dispersion mismatch (10 dB for “Achromatizer”, 5 dB “Glass 2mm”, 2.5 dB “Glass 1mm”).

Fig. 2. (a) Sharpness metric R as a function of dispersion parameter â2 with â3 = 0, (b) R as function of â3 with â2 = -0.083, i.e., the optimum value from (a), (c) combined optimization of â2 and â3 to find the global sharpness optimum, (d) mirror signals for various positions, (e) mirror signals after application of compensation phase; positive positions are plotted as solid lines, signals from negative mirror positions as thin dashed lines.

4. Results and discussion

Fig. 3 depicts results for mirror signals with various optical path delays measured with the FSI using the same source and spectrometer as for in vivo imaging. The FWHM output bandwidth of the setup was determined as 120 nm. Dispersion mismatch was obtained by introducing a 25mm water cell into the reference arm. Standard processing included dispersion compensation and calculation of the inverse FFT. Blurred mirror terms visible in Fig. 3(a) at positive positions are clearly cancelled by the DEFR algorithm (Fig. 3(b)). For the algorithm the number of iterations was set to the number of sample points I = N = 2048. The stopping parameter ε was set to 0 in all experiments, thus also the noise floor w was reconstructed (cf. (11)) which also confirms the algorithms stability since the algorithm did not diverge after all signal components have been found. We note that the noise w in an OCT system is typically comprised of residual coherent noise terms which have not been removed by background subtraction and might also not necessarily fit into the dispersive signal model proposed in Section 2, furthermore it includes detector shot noise as well as electronic noise. Since no further knowledge about w was used in the derivation of the algorithm (cf. (13)), the ability of the algorithm to reconstruct the noise floor of the system experimentally verifies this necessary condition for stability after convergence.

Fig. 3. (a) Mirror signals after dispersion compensation, (b) mirror signals after dispersion encoded full range (DEFR) algorithm. Signals at different positions are plotted using different colors.

Properties for DEFR reconstruction and influence of bandwidth and different dispersion levels on the algorithm’s performance were quantified and are shown in Fig. 4(a)–(c). To evaluate the influence of the dispersion level, different materials were inserted in the sample-arm of the free-space interferometer as described above. To verify the ability of the algorithm to perform well also on signals with standard resolution bandwidth, the raw signals were band-limited to a 55 nm FWHM Gaussian shaped spectrum. Since the bandwidth reduced signals have smaller power as compared to the full bandwidth signals, the 120nm and 55 nm signal loss curves differ by about 7 dB. The drop in signal power for the 55nm signals was compensated for determination of the suppression ratio to allow a direct comparison with respect to the different bandwidths and materials. Since the complex conjugate mirror terms after the DEFR algorithm were in the order of the noise components, the suppression ratio was the average value from 128 measurements. Following [12

12. R. A. Leitgeb, R. Michaely, T. Lasser, and S. C. Sekhar, “Complex ambiguity-free Fourier domain optical coherence tomography through transverse scanning,” Opt. Lett. 32, 3453–3455 (2007). http://ol.osa.org/abstract.cfm?URI=ol-32-23-3453 [CrossRef] [PubMed]

], the suppression ratio was defined as magnitude of the signal peak divided by the magnitude of the signal at the conjugate peak position.

The DEFR algorithm exhibits the same depth dependent signal loss as conventional processing (Fig. 4(a)), i.e., the curves for 120nm standard processing (SP) (solid green) and DEFR algorithm (red squares) nearly overlap as well as the curves for 55 nm SP (dashed cyan) and DEFR (blue squares). It is also apparent that inserting no dispersive material resulted in the highest signal. The optical elements 1mm glass, 2 mm glass and the achromatizer result in signal intensities reduced by only 1 dB, whereas with the 25mm water cell a 3 dB signal reduction due to absorption is observed. This suggests that for applications in non-ophthalmic imaging, optical elements with less absorption should be employed to cause a dispersion mismatch usable for DEFR. However, for retinal imaging the vitreous of the eye causes a natural dispersion mismatch and the observed signal reduction would be inherent.

Since mirror terms are suppressed down to the noise level (cf. Fig. 3(b)), accordingly the artifact suppression ratio exhibits a value of more than 50 dB (Fig. 4(b)). As a first impression the curves appear very similar with a shape dominated by the signal loss, i.e., the suppression ratio decreases towards EOD. This demonstrates the ability of the algorithm to cope with less bandwidth (55 nm) and less dispersion mismatch (1mm glass). As a trend the algorithm performs at 120 nm bandwidth slightly better than at 55nm for the individual materials and more dispersion provides a higher suppression ratio. Not taking into account values above ±1200μm, which are influenced by aliasing terms and signals from the subsequent ZDs, shows the expected behavior: the least dispersion mismatch and smallest bandwidth signal (1 mm glass, 55 nm) resulted in the weakest suppression ratio, high dispersion and high bandwidth provides best results (25 mm water cell, 120 nm) with a separation of around 7 dB. The result for the achromatizer and 120 nm slightly outperformed the 25 mm water cell. We believe this was due to the high asymmetric third order dispersion with this optical element, which suggests that materials or material combinations with increased higher order dispersion are interesting candidates to be used within the DEFR scheme. It has to be noted that broader bandwidth and higher dispersion mismatch results in more broadened complex conjugate artifacts of the standard processing Fig. 3(a). Thus the suppression ratio of the numeric dispersion compensation itself can be evaluated to be ~ 16dB for the 25mm water cell, ~ 10dB for achromatizer, ~ 5dB for 2mm glass, and ~ 2.5dB for 1mm glass at 120nm bandwidth. However, the algorithm aims to cancel the mirror terms rather than blurring them as can be seen from Fig. 3(b), increased dispersion thereby helps the algorithm to better confine the signal components from mirror terms.

Tomogram resolution (see Fig. 4(c)), exhibits similar values for different levels of dispersion. The theoretical axial resolution with a laser spectrally centered at 824 nm are 2.5μm and 5.5μm for 120nm and 55nm FWHM, respectively. The measured resolutions for a Gaussian shaped 55nm signal are closer to the theoretical value than the 120nm full bandwidth signals because the laser spectrum deviated from the theoretical Gaussian shape. As mentioned above, the depth-independence of the resolution indicates that our frequency resampling was sufficiently accurate.

Fig. 4. (a) Signal loss after DEFR algorithm, (b) artifact suppression ratio of the DEFR algorithm, (c) resolution with no dispersion (black curve) and after dispersion compensation for different dispersive materials in sample arm. Results for 120 nm and 55 nm bandwidth are plotted as solid and dashed lines, respectively. Results for standard processing are plotted in green (120 nm) and cyan (55 nm), DEFR algorithm in red (120 nm) and blue (55 nm).

Results for in vivo imaging are depicted in Fig. 5. Dispersion compensation was applied to the standard FD-OCT images. The dispersion parameters were determined on the data set of Fig. 5(b) and used for all tomograms. As shown in Fig. 5(a)–(d), the blurred complex conjugate artifacts are clearly visible in the retinal tomograms after standard processing. The image without dispersion compensation features symmetries around the zero-delay position indicated by the yellow line. Typically, the +range is omitted in the final tomograms as indicated in Fig. 5(d). The normal imaging range thus is 1320μm and corresponds to the -range. From Fig. 5(c) and (d) it is also visible that structures exceeding the positive end of the +range fold back into the tomogram and cannot be distinguished from the original signal other than they appear blurred due to the numerical dispersion compensation. The artifact suppression resulting from our DEFR scheme can be seen in Fig. 5(e)–(h). After applying the DEFR algorithm with I = 256 iterations, the full range tomogram (comprised of N × Nx = 2048×1024 pixels) can be reconstructed, which exhibits twice the depth range compared to conventional processing (2640μm instead of 1320μm). Currently, some artifact components remain visible in the final tomogram, e.g. in the -range of Fig. 5(h). Considering Fig. 5(e), it appears that mirrored aliasing terms affecting Fig. 5(a) around -1200μm are reduced by DEFR reconstruction.

Fig. 5. (a)–(d) Retinal tomograms (fovea) obtained by conventional processing; (e)–(h) to-mograms after 256 iterations of dispersion encoded full range (DEFR) algorithm.

Figure 6 shows a more detailed view of the full range tomograms with structures imaged around the zero delay position. At this position the complex conjugate artifacts are most disturbing after conventional processing (see Fig. 6(a)). The tomogram resulting with the DEFR algorithm (shown in Fig. 6(b)) exhibits strongly reduced artifacts. Better suppression can be achieved by incorporating additional prior knowledge into the DEFR algorithm. The improved DEFR algorithm employs a global estimate of the average light source power spectrum [43

43. Y. Yasuno, Y. Hong, S. Makita, M. Yamanari, M. Akiba, M. Miura, and T. Yatagai, “In vivo high-contrast imaging of deep posterior eye by 1-um swept source optical coherence tomography andscattering optical coherence angiography,” Opt. Express 15, 6121–6139 (2007). http://www.opticsexpress.org/abstract.cfm?URI=oe-15-10-6121 [CrossRef] [PubMed]

] which is used for pulse shaping of detected true signal components. More specifically the shape of the spectrum ĥ ∈ ℝN (with elements ĥu) is first estimated as the mean square value of the resampled interference spectra f x ∈ ℝN (with elements fx,u), i.e.,

Fig. 6. Tomograms with sample close to zero delay (ZD), (a) conventional processing with dispersion compensation, (b) dispersion encoded full range (DEFR) algorithm, (c) improved DEFR algorithm.
ĥu=1Nxx=1Nxfx,u2,
(24)

where x is the spatial index determining the transversal position of an A-scan on the tomogram consisting of Nx = 1024 lines and u=[N2,,N21] is the discrete frequency index. Smoothing of ĥ with a zero phase filter and using a Gaussian kernel of length 256 results in ĥ. Finally the normalized pulse spectrum is obtained as

h=h˜h˜N.
(25)

For pulse shaping, detected signal components ci,ni can then be convolved with the inverse Fourier transform of h. Thereby, the leakage of symmetric terms is further reduced as can be seen in Fig. 6(c). Additional improvement is also expected if the greedy algorithm is evolved to more sophisticated optimization methods. Finally we note that the same intensity range and colorbar has been used for all images. No further image processing steps such as contrast enhancement were applied to the images in Fig. 5 and Fig. 6.

In the presented implementation with results as shown in Fig. 5 one iteration of the algorithm requires 4 times the calculation of a FFT as compared to one FFT for standard processing. Furthermore the total processing time depends on the number of iterations, i.e., the processing time for a single scan line is about 4I times more complicated than standard processing (with I = 256 processing of a single scan is as costly as standard processing for a tomogram comprised of Nx = 1024 A-scans). The asymptotic computational complexity of the algorithm for a single A-scan is about 𝓞(4IN(1+logN)), whereas standard processing with numeric dispersion compensation obeys O(N(1 + logN)). A dictionary for the signal components could be used and thereby complexity for one iteration would be in the order of conventional processing. Furthermore multiple non-overlapping components may be detected within each iteration, thus reducing the total number of iterations required. Accordingly neighboring depth-scans could be incorporated to utilize structural similarity. However complexity reduction and speed improvements are beyond the scope of this proof of principle.

5. Conclusion

The iterative dispersion encoded full range (DEFR) algorithm for FD-OCT allows numerical reconstruction of full range tomograms by successive interference cancelation along a single scan line. Whereas existing complex FD-OCT techniques increase the complexity of the measurement setup and require at least two measurements, DEFR needs only one measurement and increases only post-processing complexity. Mirror terms are clearly canceled on retinal tomograms but the algorithm exhibits some leakage on in vivo signals with low dynamic range. The quantified artifact suppression ratio of more than 50 dB is promising as it exhibits a value well comparable to the most advanced multiline techniques.

Acknowledgments

The authors thank Cristiano Torti, Alex Tumlinson and Christoph Meier for constructive discussions and two anonymous reviewers for their excellent comments which helped to improve the manuscript. This work was supported in part by Cardiff University; FP6-IST-NMP-2 STREPT (NANOUB - 017128), Action Medical Research (AP1110), DTI OMICRON (1544C), FP7 FunOCT, Grant N10606 of the Austrian Science Fund (NFN SISE), and Carl Zeiss Meditec Inc., Dublin, CA, US.

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A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Numerical dispersion compensation for partial coherence interferometry and optical coherence tomography,” Opt. Express 9, 610–615 (2001). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-12-610 [CrossRef] [PubMed]

27.

M. Duarte, M. Davenport, M. Wakin, and R. Baraniuk, “Sparse Signal Detection from Incoherent Projections,” in Proc. Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP)3, III305–308 (2006). http://dx.doi.org/doi:10.1109/ICASSP.2006.1660651

28.

B. Povazay, B. Hofer, B. Hermann, A. Unterhuber, J. E. Morgan, C. Glittenberg, S. Binder, and W. Drexler, “Minimum distance mapping using three-dimensional optical coherence tomography for glaucoma diagnosis,” J. Biomed. Opt. 12, 041 204 (2007). http://link.aip.org/link/?JBO/12/041204/1 [CrossRef]

29.

E. J. Fernández, A. Unterhuber, B. Považay, B. Hermann, P. Artal, and W. Drexler, “Chromatic aberration correction of the human eye for retinal imaging in the near infrared,” Opt. Express 14, 6213–6225 (2006). http://www.opticsexpress.org/abstract.cfm?URI=oe-14-13-6213 [CrossRef] [PubMed]

30.

R. K. Wang and Z. Ma, “A practical approach to eliminate autocorrelation artefacts for volume-rate spectral domain optical coherence tomography,” Phys. Med. Biol. 51, 3231–3239 (2006).http://www.iop.org/EJ/abstract/0031-9155/51/12/015/ [CrossRef] [PubMed]

31.

N. A. Nassif, B. Cense, B. H. Park, M. C. Pierce, S. H. Yun, B. E. Bouma, G. J. Tear-ney, T. C. Chen, and J. F. de Boer, “In vivo high-resolution video-rate spectral-domain optical coherence tomography of the human retina and optic nerve,” Opt. Express 12, 367–376 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-3-367 [CrossRef] [PubMed]

32.

C. Dorrer, N. Belabas, J.-P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17, 1795–1802 (2000). http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-17-10-1795 [CrossRef]

33.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, “In vivo human retinal imaging by Fourier domain optical coherence tomography,” J. Biomed. Opt. 7, 457–463 (2002). http://dx.doi.org/10.1117/1.1482379 [CrossRef] [PubMed]

34.

M. A. Choma, M. V. Sarunic, C. H. Yang, and J. A. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11, 2183–2189 (2003). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-18-2183 [CrossRef] [PubMed]

35.

B. H. Park, M. C. Pierce, B. Cense, S. H. Yun, M. Mujat, G. J. Tearney, B. E. Bouma, and J. F. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 μm,” Opt. Express 13, 3931–3944 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-3931 [CrossRef] [PubMed]

36.

R. A. Leitgeb, W. Drexler, A. Unterhuber, B. Hermann, T. Bajraszewski, T. Le, A. Stingl, and A. F. Fercher, “Ultrahigh resolution Fourier domain optical coherence tomography,” Opt. Express 12, 2156–2165 (2004). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-10-2156 [CrossRef] [PubMed]

37.

Y. Yasuno, V. D. Madjarova, S. Makita, M. Akiba, A. Morosawa, C. Chong, T. Sakai, K. P. Chan, M. Itoh, and T. Yatagai, “Three-dimensional and high-speed swept-source optical coherence tomography for in vivo investigation of human anterior eye segments,” Opt. Express 13, 10652–10664 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-26-10652 [CrossRef] [PubMed]

38.

M. Mujat, B. H. Park, B. Cense, T. C. Chen, and J. F. de Boer, “Autocalibration of spectral-domain optical coherence tomography spectrometers for in vivo quantitative retinal nerve fiber layer birefringence determination,” J. Biomed. Opt. 12, 041 205 (2007). http://link.aip.org/link/?JBO/12/041205/1 [CrossRef]

39.

W. Drexler, U. Morgner, F. X. Kärtner, C. Pitris, S. A. Boppart, X. D. Li, E. P. Ippen, and J. G. Fu-jimoto, “In vivo ultrahigh-resolution optical coherence tomography,” Opt. Lett. 24, 1221–1223 (1999). http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-24-17-1221 [CrossRef]

40.

J. F. de Boer, C. E. Saxer, and J. S. Nelson, “Stable carrier generation and phase-resolved digital data processing in optical coherence tomography,” Appl. Opt. 40, 5787–5790 (2001). http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-40-31-5787 [CrossRef]

41.

D. L. Marks, A. L. Oldenburg, J. J. Reynolds, and S. A. Boppart, “Digital algorithm for dispersion correction in optical coherence tomography for homogeneous and stratified media,” Appl. Opt. 42, 204–217 (2003). http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-42-2-204 [CrossRef] [PubMed]

42.

A. R. Tumlinson, B. Hofer, A. M. Winkler, B. Považay, W. Drexler, and J. K. Barton, “Inherent homogenous media dispersion compensation in frequency domain optical coherence tomography by accurate k-sampling,” Appl. Opt. 47, 687–693 (2008). http://ao.osa.org/abstract.cfm?URI=ao-47-5-687 [CrossRef] [PubMed]

43.

Y. Yasuno, Y. Hong, S. Makita, M. Yamanari, M. Akiba, M. Miura, and T. Yatagai, “In vivo high-contrast imaging of deep posterior eye by 1-um swept source optical coherence tomography andscattering optical coherence angiography,” Opt. Express 15, 6121–6139 (2007). http://www.opticsexpress.org/abstract.cfm?URI=oe-15-10-6121 [CrossRef] [PubMed]

OCIS Codes
(100.3010) Image processing : Image reconstruction techniques
(100.3020) Image processing : Image reconstruction-restoration
(100.5070) Image processing : Phase retrieval
(110.4500) Imaging systems : Optical coherence tomography
(170.4500) Medical optics and biotechnology : Optical coherence tomography
(300.6300) Spectroscopy : Spectroscopy, Fourier transforms

ToC Category:
Imaging Systems

History
Original Manuscript: November 13, 2008
Revised Manuscript: December 17, 2008
Manuscript Accepted: December 18, 2008
Published: December 22, 2008

Virtual Issues
Vol. 4, Iss. 3 Virtual Journal for Biomedical Optics

Citation
Bernd Hofer, Boris Považay, Boris Hermann, Angelika Unterhuber, Gerald Matz, and Wolfgang Drexler, "Dispersion encoded full range frequency domain optical coherence tomography," Opt. Express 17, 7-24 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-1-7


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References

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  41. D. L. Marks, A. L. Oldenburg, J. J. Reynolds, and S. A. Boppart, "Digital algorithm for dispersion correction in optical coherence tomography for homogeneous and stratified media," Appl. Opt. 42, 204-217 (2003). http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-42-2-204 [CrossRef] [PubMed]
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