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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 14 — Jul. 2, 2012
  • pp: 15654–15668
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The effects of reduced bit depth on optical coherence tomography phase data

William A. Ling and Audrey K. Ellerbee  »View Author Affiliations


Optics Express, Vol. 20, Issue 14, pp. 15654-15668 (2012)
http://dx.doi.org/10.1364/OE.20.015654


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Abstract

Past studies of the effects of bit depth on OCT magnitude data concluded that 8 bits of digitizer resolution provided nearly the same image quality as a 14-bit digitizer. However, such studies did not assess the effects of bit depth on the accuracy of phase data. In this work, we show that the effects of bit depth on phase data and magnitude data can differ significantly. This finding has an important impact on the design of phase-resolved OCT systems, such as those measuring motion and the birefringence of samples, particularly as one begins to consider the tradeoff between bit depth and digitizer speed.

© 2012 OSA

1. Introduction

Optical coherence tomography (OCT) is a high-resolution, interferometric technique that is capable of imaging the structural properties of scattering samples. A single OCT interferogram captures the intensity of light reflected from the sample as a function of depth (along the direction of light propagation) at a single lateral position; this one-dimensional backscattering intensity profile is called an A-scan. Adjacent A-scans from different lateral positions of the sample can be organized into a 2D image or 3D volume. The speed of OCT image acquisition depends on the A-scan rate and the number of acquired A-scans. Several emerging applications of OCT - particularly for medical imaging - would benefit from higher scan rates (e.g., to reduce motion artifacts or patient scan times [1

1. V. J. Srinivasan, D. C. Adler, Y. L. Chen, I. Gorczynska, R. Huber, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Ultrahigh-speed optical coherence tomography for three-dimensional and en face imaging of the retina and optic nerve head,” Invest. Ophthalmol. Vis. Sci. 49(11), 5103–5110 (2008). [CrossRef] [PubMed]

]). Moreover, even in cases where scan rates are sufficiently fast, faster A-scan rates permit averaging of data to reduce noise and improve image quality.

The digitizing speed of the analog-to-digital converter (ADC) along with the on-board memory capacity limits the maximum data acquisition rate of the system. The rising popularity of faster OCT systems with higher A-scan rates has, however, necessitated a corresponding reduction of the resolution (bit depth) of the ADCs used for digitization of OCT data [2

2. W. Wieser, B. Biedermann, T. Klein, C. Eigenwillig, and R. Huber, “Multi-megahertz OCT: high quality 3D imaging at 20 million A-scans and 4.5 GVoxels per second,” Opt. Express 18(14), 14685–14704 (2010). [CrossRef] [PubMed]

], primarily because commercial digitizers force a trade-off between ADC resolution and speed. Even for applications that can tolerate lower imaging speeds, choosing the ADC with lower resolution could reduce the space needed for data storage and, thus, the overall time to process the data. Unfortunately, the loss of information that accompanies data reduction could lead to errors during the reconstruction of OCT images. This is of particular concern when the image data are used to make quantitative assessments, as is typical with many phase-sensitive functional extensions of OCT such as phase-resolved (PR) OCT [3

3. D. C. Adler, R. Huber, and J. G. Fujimoto, “Phase-sensitive optical coherence tomography at up to 370,000 lines per second using buffered Fourier domain mode-locked lasers,” Opt. Lett. 32(6), 626–628 (2007). [CrossRef] [PubMed]

, 4

4. C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. 30(16), 2131–2133 (2005). [CrossRef] [PubMed]

], Doppler (D) OCT [5

5. G. Liu, M. Rubinstein, A. Saidi, W. Qi, A. Foulad, B. Wong, and Z. Chen, “Imaging vibrating vocal folds with a high speed 1050 nm swept source OCT and ODT,” Opt. Express 19(12), 11880–11889 (2011). [CrossRef] [PubMed]

], and polarization-sensitive (PS) OCT [6

6. J. Zhang, W. Jung, J. S. Nelson, and Z. Chen, “Full range polarization-sensitive Fourier domain optical coherence tomography,” Opt. Express 12(24), 6033–6039 (2004). [CrossRef] [PubMed]

, 7

7. E. Gotzinger, M. Pircher, and C. K. Hitzenberger, “High speed spectral domain polarization sensitive optical coherence tomography of the human retina,” Opt. Express 13(25), 10217–10229 (2005). [CrossRef] [PubMed]

]. The effects of reduced ADC resolution on phase is particularly germane to such applications.

Several groups have studied the effects of reduced ADC resolution on the quality of images generated by swept-source (SS) OCT systems and have concluded that 8-bit resolution is adequate for most purposes [2

2. W. Wieser, B. Biedermann, T. Klein, C. Eigenwillig, and R. Huber, “Multi-megahertz OCT: high quality 3D imaging at 20 million A-scans and 4.5 GVoxels per second,” Opt. Express 18(14), 14685–14704 (2010). [CrossRef] [PubMed]

, 8

8. B. D. Goldberg, B. J. Vakoc, W-Y Oh, M. J. Suter, S. Waxman, M. I. Freilich, B. E. Bouma, and G. J. Tearney, “Performance of reduced bit-depth acquisition for optical frequency domain imaging,” Opt. Express 17(19), 16957–16968 (2009). [CrossRef] [PubMed]

10

10. R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. 31(20), 2975–2977 (2006). [CrossRef] [PubMed]

]. They found that sensitivity decreased little (only ∼1 dB) until the ADC resolution neared seven bits; furthermore, images generated from data with reduced bit depths did not show any noticeable qualitative changes from those generated with 14 bits. These studies, however, are incomplete for two reasons. First they only considered integer numbers of bits, which is unrealistic (real ADCs always exhibit fractional numbers of bits due to noise and errors in quantization thresholds; hence a commercial 8-bit ADC could have an effective resolution of 7.5 bits). Second, they only considered the effect of the reduced ADC resolution on the magnitude of OCT data and failed to address the implications of the reduced bit depth on the accuracy of phase data and the quantitative measurements that derive from them.

In this work, we present the first investigation of the effects of quantization on phase data generated with OCT. Our investigation presents a new framework that allows consideration of fractional ADC bit depths. We also provide a new focus on a spectral domain (SD) rather than a swept-source (SS) system, noting that the balanced detectors typically used with SS-OCT result in different sensitivities to ADC noise compared to SD-OCT because of the potential for higher DC power levels at the receiver. We limit our discussion to the impact of quantization on measured displacements, but our analysis has implications for understanding the effects of reduced ADC resolutions on refractive index changes, birefringence, and Doppler velocities measured with functional OCT systems, whether SD-OCT or SS-OCT. Particularly where these systems are used to assess biological data - as in imaging birefringence of the retina - it is important to avoid errors in the phase that can lead to misinterpretation of the biological phenomenon under observation.

2. Background/Theory

2.1. The origin of phase in OCT data

Raw OCT data consists of an interferogram resulting from the interference of broadband, backscattered light from a sample and reference reflector. The interferometric term of the interferogram acquired at time t, once resampled to be linear in wavenumber, bears the following form:
i(k,t)=A(k)RRRScos(2kn(Δz+δz(t))),
(1)
where A(k) accounts for the source spectrum, spectrometer efficiency, and line camera pixel non-uniformity; RR and RS are the reflectivities of the reference and sample, respectively; n is the spatially averaged index of refraction (assumed constant in time); and Δz is the displacement of the sample reflector from the position of zero optical pathlength relative to the position of the reference reflector to within a multiplicative constant of the pixel spacing in the z domain. The sum of Δz and δz(t) yields the exact position of the sample reflector. The Fourier transform (FT) of the interferogram yields a complex number encoding the position (in depth) and intensity (i.e., refractive index contrast) of reflectors within the sample. The A-scan is calculated from the magnitude of I(z,t), the FT of i(k,t), centered at z = 2nz + δz(t)). Note that for simplicity, Eq. (1) omits the DC terms because they do not contribute to the interferometric signal of interest.

Phase-resolved (PR) OCT systems make use of this phase information to make quantitative measurements. Several PR-OCT techniques have been developed to detect sub-diffraction-limited displacements of objects [3

3. D. C. Adler, R. Huber, and J. G. Fujimoto, “Phase-sensitive optical coherence tomography at up to 370,000 lines per second using buffered Fourier domain mode-locked lasers,” Opt. Lett. 32(6), 626–628 (2007). [CrossRef] [PubMed]

, 12

12. M. A. Choma, A. K. Ellerbee, C. Yang, T. Creazzo, and J. A. Izatt, “Spectral-domain phase microscopy,” Opt. Lett. 30(10), 1162–1164 (2005). [CrossRef] [PubMed]

15

15. A. K. Ellerbee and J. A. Izatt, “Phase retrieval in low-coherence interferometric microscopy,” Opt. Lett. 32(4), 388–390 (2007). [CrossRef] [PubMed]

]. Equation (2) can also be used to detect changes in the refractive index δn if the position or thickness of the object remains stationary.

2.2. Sources of noise in OCT systems

SD-OCT system noise originates from many sources: the light source contributes to shot noise and relative intensity noise (RIN); the spectrometer electronics introduce dark current noise and read-out noise. Digitization of the originally analog voltages also introduces noise. We can lump all such sources into a single additive noise term n(k), measured by the single pixel receiving wavenumber k. The measured signal is y(k) = i(k) + n(k).

We define a digitizer as a complete electronic module consisting of an ADC and additional auxiliary electronics and cable ports to interface between analog signals and a computer. By digitizer noise, we are referring to the total noise produced by such a component, and thus digitizer noise derives from the ADC circuitry as well as from the coupling of external noise in the interconnects of the digitizer. Here, we focus on the ADC noise component, which is the dominant noise source of a well-designed digitizer.

The physical origins of these additive noise sources are assumed to be uncorrelated. Thus their variances can be summed:
σn2=σshot2+σRIN2+σdark2+σamp2+σDAQ2=σdet2+σDAQ2.
For convenience, we lump shot, RIN, dark current, and read-out noise into a single noise term σdet2; σDAQ2 is the digitizer noise.

The ADC introduces primarily quantization noise (QN), defined as the error added to an analog signal due to representation of samples with a limited number of bits. Throughout this paper, we assume the following white-noise model for quantization noise [18

18. A. Oppenheim and R. Schafer, Discrete-time Signal Processing, 3rd ed. (Prentice Hall, 2009).

], used extensively in the study of ADCs [19

19. W. Kester, The Data Conversion Handbook (Elsevier, 2005).

]:
  1. QN is an additive, stationary white-noise process.
  2. QN is uniformly distributed on (−Δ/2, Δ/2] at all times, where Δ is the quantization step size. For a B-bit quantizer with a full-scale input voltage range of VFS, Δ=VFS2B.
  3. QN is uncorrelated with the input sequence to the quantizer.
It can be shown that such QN has a variance of σqn2=Δ212 [18

18. A. Oppenheim and R. Schafer, Discrete-time Signal Processing, 3rd ed. (Prentice Hall, 2009).

]. We note that two cases in which the white-noise model for ADC behavior is inaccurate are 1) when the signal fails to frequently cross quantization thresholds (for instance, a DC signal with fluctuations that are small enough that the quantizer output is constant) and 2) the input signal is correlated with the quantization as occurs, for example, when there is feedback from the quantizer output back to its input. The most extreme example of the latter scenario is when the output of a quantizer is requantized using the same quantizer, in which case no additional quantization noise is added. The first case is not of concern given the large dynamic range of the interferogram and the second case does not occur here.

All physically realizable ADCs will exhibit a slight departure from the ideal quantizer: that is, no commercial B-bit digitizer is exactly B bits due to errors in the quantization thresholding circuitry and the inherent noise present in electronics [19

19. W. Kester, The Data Conversion Handbook (Elsevier, 2005).

]. Hence, an ideal B-bit quantizer and a real B-bit ADC have noise characteristics given by Eq. (3) and Eq. (4), respectively, where σe2 is the variance of an additional error source that encompasses the effects of all non-idealities in real ADCs.
σqn2=Δ212=VFS212(22B)
(3)
σeff2=σqn2+σe2.
(4)

In the characterization of ADCs, one often employs the concept of an effective number of bits (ENOB) to describe the performance of an ADC [19

19. W. Kester, The Data Conversion Handbook (Elsevier, 2005).

]. The ENOB is defined as a fractional number (Eq. (5)). Since we focus on ADC noise as the dominant noise source from the digitizer, as explained previously, we equate the DAQ noise to the effective noise (Eq. (6)).
VFS212(22ENOB)=σeff2
(5)
σDAQ2=σeff2
(6)

2.3. Effects of noise on the measurement of OCT phase

To understand the effects of N(z) on the phase, consider Fig. 1, where we represent the transformed noise as a sum of components N(z) = NI(z) + NQ(z) that are either in phase or in quadrature with the direction of the signal vector I(z). Our goal is to accurately measure θ, but N(z) causes the measured phase to deviate from the desired phase by δθ. Such noise and resulting phase error limits the sensitivity of the measurement of small changes in phase. Since a white noise source n(k) gives rise to an N(z) with uncorrelated real and imaginary parts having the same variance, the variance of the in-phase and quadrature components of N(z) are also equal: that is, σI2=σQ2=σN2/2, where σN2 is the variance of N(z). For the case in which |I| >> σN, we can make the approximations shown in Eq. (8), while the standard deviation of the phase deviation is given by Eq. (9).
δθtan(δθ)=NQ|I+NI|NQ|I|
(8)
σδθ=σN2|I|
(9)
Defining SNR as |I|/σN, we see that Eq. (9) differs from Eq. (7) by a factor of 2. Hence the theoretically achievable noise floor is seen to be lower than that given by Eq. (7), and sets a new lower limit on the accuracy of the measured phase. Thus, to the extent that ADC resolution contributes to additive noise, it affects the accuracy of the phase measurement.

Fig. 1 Vector plot of the measured OCT data (z-domain). I = signal vector, N = noise vector, and Y = the resultant. Components of the noise vector in quadrature with the signal vector lead to error δθ in the measured phase compared to the actual phase θ.

3. Methods

3.1. Simulation of reduced ADC resolution

To simulate the effects of lower ADC resolution, we requantized native 12-bit data to achieve ENOBs ranging from 7 – 11 bits in increments of 0.25 bits, and we compared the resulting phase measurements with the original data. To enable simulation of fractional ENOBs, we developed a novel method for simulating reduced ADC resolutions that accounts for fractional bits (fractional bits have not been considered in other works). Figure 2 illustrates the steps involved. The OCT interferograms were converted to A-scans using the non-uniform DFT (NDFT) described in [20

20. D. Hillmann, G. Huttmann, and P. Koch, “Using nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE 7372. 73720R (2009). [CrossRef]

,21

21. S. Vergnole, D. Lvesque, and G. Lamouche, “Experimental validation of an optimized signal processing method to handle non-linearity in swept-source optical coherence tomography,” Opt. Express 18(10), 10446–10461 (2010). [CrossRef] [PubMed]

]. While the choice of resampling method has been shown to affect the accuracy of the measured phase data [22

22. C. Copeland and A. K. Ellerbee, “The effects of different gold standards on the assessment of the accuracy of different resampling techniques for optical coherence tomography,” Proc. SPIE . 8225–8237 (2012).

], the NDFT has been found to be accurate at least for magnitude reconstruction [21

21. S. Vergnole, D. Lvesque, and G. Lamouche, “Experimental validation of an optimized signal processing method to handle non-linearity in swept-source optical coherence tomography,” Opt. Express 18(10), 10446–10461 (2010). [CrossRef] [PubMed]

]; because it calculates the complex DFT coefficients, its magnitude and phase should be simultaneously accurate.

Fig. 2 Method to simulate reduced ADC resolution, including fractional bits. An unresampled interferogram iraw(k) is captured with a 12-bit ADC and modified in post-processing to produce an N + F-bit output, where 0 < F < 1. The NDFT (non-uniform DFT) combines resampling and Fourier transformation into a single operation.

The simple integer quantization algorithms used in [8

8. B. D. Goldberg, B. J. Vakoc, W-Y Oh, M. J. Suter, S. Waxman, M. I. Freilich, B. E. Bouma, and G. J. Tearney, “Performance of reduced bit-depth acquisition for optical frequency domain imaging,” Opt. Express 17(19), 16957–16968 (2009). [CrossRef] [PubMed]

, 9

9. Z. Lu, D. K. Kasaragod, and S. J. Matcher, “Performance comparison between 8- and 14-bit- depth imaging in polarization-sensitive swept-source optical coherence tomography,” Biomed. Opt. Express 4(2), 794–804 (2011). [CrossRef]

] convert 12-bit outputs from the ADC to N-bit numbers, where N is an integer, by first dividing the 12-bit number by 212−N and then rounding the fractional part to the nearest integer. This method, however, produces an output with an ENOB less than N: the 12-bit input already contains QN, and rounding adds additional QN that makes the total noise greater than that of a true N-bit sequence. To accurately round from 12 bits to N bits, one must account for both the noise added by the N-bit quantizer and the QN of a 12-bit sequence, so that the sum of the original 12-bit QN and the newly added QN yields the desired QN of a true N-bit sequence. In [8

8. B. D. Goldberg, B. J. Vakoc, W-Y Oh, M. J. Suter, S. Waxman, M. I. Freilich, B. E. Bouma, and G. J. Tearney, “Performance of reduced bit-depth acquisition for optical frequency domain imaging,” Opt. Express 17(19), 16957–16968 (2009). [CrossRef] [PubMed]

, 9

9. Z. Lu, D. K. Kasaragod, and S. J. Matcher, “Performance comparison between 8- and 14-bit- depth imaging in polarization-sensitive swept-source optical coherence tomography,” Biomed. Opt. Express 4(2), 794–804 (2011). [CrossRef]

], neglect of the QN present in the original 14-bit data had little effect on the accuracy of subsequent quantization operations, since these rounding operations added far more QN than was originally present in the 14-bit data: the QN increases by 6 dB per bit rounded away [19

19. W. Kester, The Data Conversion Handbook (Elsevier, 2005).

] and their bit depths of interest were near 8 bits, making the 14-bit QN negligible. However, in this work, our methods for simulating fractional-bit resolutions can easily account for the QN present in our original 12-bit data, so there is no need to reduce the accuracy of our results. In fact, as we will later show, we obtain relevant results at bit depths close to the original 12 bits, so we cannot neglect the 12-bit noise.

In this work, we are interested in quantizers capable of generating a non-integer ENOB N + F, where F ∈ (0, 1). Throughout this work, we assume for simplicity that an ADC with a nominal resolution of N + 1 bits will have an ENOB between N and N + 1. Not all ADCs satisfy this requirement, but our analysis can be straightforwardly extended for those other cases.

Because a true (N + 1)-bit ADC is not an ideal (N + 1)-bit quantizer (i.e., an (N + 1)-bit rounding operation), various error sources combine to reduce the true resolution of the ADC. The effects of these error sources may be modeled as an additive noise source ne, placed before the ideal quantizer [19

19. W. Kester, The Data Conversion Handbook (Elsevier, 2005).

]. As a result, the ENOB of the ADC is B = N + F bits, where N is an integer and F ∈ (0, 1).

The standard additive noise model of an (N + 1)-bit ADC receiving an unquantized input is shown in two equivalent forms in Fig. 3. The system on the right replaces the ideal quantizer with its white-noise model equivalent, where nq is the quantization noise of the (N + 1)-bit quantizer. As mentioned previously, we must choose ne such that the total added noise, ne + nq, has a variance corresponding to an ENOB of N + F. Since ne and nq are uncorrelated, the total noise variance of the ADC is the sum of the individual variances of ne and nq, σe2 and σN+12, respectively. Thus, the variance of ne may be determined by the difference between the variance of the (N + F)-bit ADC output and that of its (N + 1)-bit internal quantizer, given by Eq. (5):
σN+F2=VFS212(22(N+F)),σN+12=VFS212(22(N+1)),σe2=σN+F2σN+12=VFS212(22(N+F))VFS212(22(N+1)).
(10)

Fig. 3 Behavior of a non-ideal (N + 1)-bit ADC. ne accounts for internal noise and thresholding errors of the ADC. Left: ideal (N + 1)-bit quantizer (i.e., a perfect rounding operation); the resulting ADC output has a resolution < N + 1 due to ADC imperfections described by ne. Right: the ideal quantizer can be replaced with an equivalent white-noise model in which QN is an uncorrelated additive noise.

Note that the approach described above assumes the input to the ADC is unquantized. To account for noise due to prior quantization events that occurred in the physical digitizer used to generate our 12-bit data (we assumed that the ENOB of the raw data was very close to 12 bits), we must additionally subtract VFS212(22(12)) from Eq. (10), leading to the modified equation:
σe,122=[VFS212(22(N+F))VFS212(22(N+1))]VFS212(22(12)).
(11)
To simulate the non-ideal ADC, we first generated a Gaussian pseudorandom sequence ne of the appropriate variance and added it to the raw data. An (N + 1)-bit quantizer was then implemented using rounding as in [8

8. B. D. Goldberg, B. J. Vakoc, W-Y Oh, M. J. Suter, S. Waxman, M. I. Freilich, B. E. Bouma, and G. J. Tearney, “Performance of reduced bit-depth acquisition for optical frequency domain imaging,” Opt. Express 17(19), 16957–16968 (2009). [CrossRef] [PubMed]

, 9

9. Z. Lu, D. K. Kasaragod, and S. J. Matcher, “Performance comparison between 8- and 14-bit- depth imaging in polarization-sensitive swept-source optical coherence tomography,” Biomed. Opt. Express 4(2), 794–804 (2011). [CrossRef]

]. As an example, Fig. 4 illustrates the two-step operation of our simulated ADC for achieving fractional quantization of a 4-bit quantized sinusoidal input signal (gray, N = 3) based on Fig. 2. In Fig. 4(a), a floating-point pseudorandom noise sequence (blue) is added to the signal, resulting in a new signal (red). Next, we apply a 4-bit integer quantizer to the noisy sinusoid; the residual errors (blue) are seen to be commensurate with the desired output ENOB of 3.2 (Fig. 4(b)). Figures 4(c) and 4(d) illustrate the same two steps for the case of a desired ENOB of 1.5.

Fig. 4 Demonstration of the two-step fractional requantizer from Fig. 2 to generate ENOBs of 3.2 (a and b) and 1.5 (c and d) bits from a 4-bit sinusoidal input sampled 32 times per period (gray). Simulated ADC signals appear in red. Step i (a and c): apply a pseudorandom noise sequence (blue) of the appropriate variance to the input to account for ADC non-idealities. Step ii (b and d): Quantize the resulting noisy sequence (red) to yield an output with the appropriate ENOB.

To demonstrate the effectiveness of our method for modeling ADC behavior, we simulated the output of a fractional-bit ADC for both unquantized (floating-point) and 12-bit quantized sinusoidal inputs. The pseudorandom sequences added to each sinusoid had variances given by Eq. (10) or Eq. (11), respectively. The ENOB of the output was determined using the standard method for measuring the ENOB of an ADC [19

19. W. Kester, The Data Conversion Handbook (Elsevier, 2005).

]:
  1. Generate the spectrum of the ADC output using the DFT. (Note that the length of the DFT is set to an integer multiple of the period of the sinusoid to eliminate windowing effects).
  2. Calculate the SNR of the ADC output. Determine the total noise energy by summing the squared magnitude of all DFT coefficients excluding the signal and DC.
  3. Calculate the ENOB using ENOB = (SNR − 1.76)/6.02, where SNR is in dB.

Table 1 summarizes our comparisons of the simulated to desired ENOBs for unquantized (Case 1) and quantized (Case 2) sinusoidal inputs. The method is shown to be highly accurate even for Case 2, which describes our experimental setup.

Table 1. Accuracy of the fractional requantizer in generating desired ENOBs for unquantized (Case 1) and quantized (Case 2) sinusoidal inputs.

table-icon
View This Table

3.2. Experimental design

Our primary objective was to characterize the phase sensitivity of an SD-OCT system strictly as a function of the resolution of the digitizer. Thus, we first measured the phase variance of a rigid, stationary sample to minimize contributions from other noise sources. The surface of the coverslip closer to the incident light (i.e., the top) acted as the reference reflector and the bottom surface was the sample reflector, forming a common-path (CP) system. The elimination of common-mode noise afforded by CP systems leads to lower phase noise and higher phase sensitivity [11

11. A. B. Vakhtin, D. J. Kane, W. R. Wood, and K. A. Peterson, “Common-path interferometer for frequency-domain optical coherence tomography,” App. Opt. 42(34), 6953–6958 (2003). [CrossRef]

,14

14. A. K. Ellerbee, T. L. Creazzo, and J. A. Izatt, “Investigating nanoscale cellular dynamics with cross-sectional spectral domain phase microscopy,” Opt. Express 15(13), 8115–8124 (2007). [CrossRef] [PubMed]

], as the effects of vibration noise and thermal fluctuations are significantly reduced. In our case, the perfectly rigid connection between the surfaces of the coverslip allowed us to characterize the phase variance due only to the noise sources of interest.

We used a commercial SD-OCT system (Telesto, ThorLabs) with a broadband light source (λ0 = 1325 nm, ΔλFWHM = 150 nm). The lateral resolution of the system was 12 μm (NA = 0.08). All interferograms were captured with a 12-bit digitizer operating at an A-scan rate of 5.5 kHz (high-sensitivity mode). At this A-scan rate and in the CP configuration, the sensitivity is 90 dB. The system was operated in CP mode by closing the aperture of the reference arm. A schematic of the system is shown in Fig. 5. We acquired 8000 A-scans and calculated the phase of the DFT for the pixel at depth 2nΔz, corresponding to the position of the bottom (sample) surface. This experimental protocol was repeated after changing the reflectivity of the coverslip by adding a liquid droplet of water or index-matching gel to its bottom surface. All reported phase sensitivities are given by the standard deviation of 8000 measurements for a given condition. Our choice of 8000 was arbitrary but satisfactory, as we observed little phase drift over the course of the experiments.

Fig. 5 Common-path OCT system used in this work. The sample comprised either a coverslip or a piezo-mounted mirror imaged from behind the coverslip. The reflectivity of the coverslip was changed by adding liquid droplets of varying refractive index.

4. Results and discussion

Figures 6(a)–6(c) shows the standard deviation of the measured thickness of a coverslip (i.e., displacement sensitivity) as a function of simulated ENOB for air-glass, water-glass, and gel-glass interfaces. The refractive indices were 1.50, 1.33, and 1.45 for the coverslip glass, distilled water, and gel (G608N index-matching gel, Thorlabs), respectively. The air-glass A-scan peak was 3.5 dB below the upper limit of the 45-dB dynamic range for the CP configuration. Also shown are two theoretical curves corresponding to the predicted standard deviations given by Eq. (7) and the newly proposed Eq. (9) derived in this work. It is clear that our new equation provides a better fit to the experimental data, suggesting that the prior equation tends to overestimate phase noise. The residual mismatch between the experimental and theoretical curves may be explained by the possible effects of lateral vibrations for a coverslip whose thickness may vary even on the scale of λ/10. Such vibrations could result in time-varying thicknesses that cause direct rotation of the complex signal vector - which is different from the effect of additive noise - leading to additional phase noise.

Fig. 6 Displacement sensitivity vs ENOB for (a) glass-air, (b) glass-water, and (c) glass-gel interfaces. Error bars indicate one standard deviation of variation between trials. Theoretical curves are based on Eq. (7) and Eq. (9); the latter is clearly a better fit to the data. Experimental data and theoretical predictions converge with decreasing sample reflectivity or ENOB, as predicted. The arrows in (a)–(c) indicate the point at which the experimental sensitivity worsened by 3 dB. Note that plot axes differ to improve visibility of the data. The dotted black lines in (d) indicate the point at which SNR dropped by 1 dB relative to the SNR at 12 bits.

As seen in Eq. (1), the magnitude of the interferometric term is proportional to RS. Thus, by Eq. (9), standard deviation of the phase noise is
σδθ=P(VFS212(22ENOB)+K1(RS+RR)+K2(RS+RR)2+σdark2+σamp2)K32RS
(12)
where P is the DFT length (typically the number of camera pixels) and the Ki are constants. K1(RS + RR) is the variance of the shot noise and K2(RS + RR)2 is the variance of the RIN; K3 is the proportionality factor relating RS and |I|:|I|=K3RS. The values of the Ki depend on a number of system parameters such as source power spectral density, spectrometer efficiency, and responsivity [16

16. S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftimia, and B. E. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11(22), 2953–2963 (2003). [CrossRef] [PubMed]

, 17

17. R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express 11(8), 889–894 (2003). [CrossRef] [PubMed]

].

If we neglect the dependence of noise on RS (an accurate approximation when shot noise and RIN are small compared to all other additive noise sources), then by Eq. (12) we would expect the displacement sensitivity for the glass-water sample to be 3.33 times as large as for glass-air. The experimental results showed a factor of 2.64 at 12 bits that increased to 3.15 at 7 bits. Similarly, a simplified calculation for the gel sample would indicate a sensitivity difference between air and gel of 11.8, while the experimental result was 9.48 at 12 bits and increased to 11.4 at 7 bits. The discrepancy between the expected and actual values deceases with ENOB because QN, which is independent of RS, becomes more dominant in the numerator of Eq. (12).

That the theoretical and experimental curves of Fig. 6 converge at low ENOBs reflects the higher relative contribution of quantization noise over vibration noise in this regime; note that vibration noise is independent of ENOB. Note also that the theoretical prediction becomes more accurate as the sample reflectivity decreases, as seen in progressing from Fig. 6(a) to 6(b) to 6(c). Reducing the sample reflectivity increases the total contribution of phase noise due to additive noise sources, as seen in Eq. (9). Since vibration noise is independent of the sample reflectivity, its relative contribution to phase noise diminishes as additive noise sources become more dominant with decreasing reflectivity.

Although we have shown samples with relatively high reflectivities, these results extend to lower reflectivity samples as well (typical reflectivities for human skin are −30 to −40 dB and −60 to −70 dB for human retina [23

23. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254(5035), 1178–1181 (1991). [CrossRef] [PubMed]

]). Our CP configuration required larger sample reflectivities due to the low reference reflectivity. However, the only requirement for the accuracy of our results is meeting the trigonometric approximation of Eq. (8). This would be the case, for instance, when the sample reflectivity is −70 dB in a system with 100 dB sensitivity.

To further illustrate the importance of signal strength in determining the relative influence of bit depth in PR-OCT, we generated topographical images of the thickness of a coverslip and a 30-dB reflective neutral density filter (NDF). The top images of Fig. 7 show the topography of a coverslip (1 mm x 1 mm field of view) generated using phase data at 12, 8, and 5 bits. The image produced from 5-bit data differs from that of the 12-bit data by an RMSE of 0.3% of the 49-nm dynamic range of the 12-bit image (RMSE is calculated by assuming the 12-bit image to be perfect and finding the difference in other images). Even at 5 bits, the image is nearly indistinguishable from the original, demonstrating the minute effect of quantization noise on phase when the sample reflectivity is high. Note that although, according to the curve in Fig. 6(a), the displacement sensitivity has increased considerably in moving from 12 bits to 7 bits, it is still small (in the pm range) compared to the 49-nm dynamic range of the image. In contrast, quantization noise has a much more significant impact on phase noise when the reflectivity decreases. The lower images of Fig. 7 show the topography of the ND filter generated from the phase of 12, 8, and 5-bit data. The image produced by the 8-bit data differs from that of the 12-bit data by an RMSE of 0.4% of the 36-nm dynamic range of the 12-bit image, a difference nearly invisible to the eye, though greater than the same case for the coverslip at an even lower bit depth. For the NDF, the image at 5 bits differs from the original by an RMSE of 3% (10-fold worse than for the coverslip). Hence, the quality of the image, measured by RMSE, degrades more quickly for lower sample reflectivities. Places with significant degradation are highlighted by the boxes and arrows.

Fig. 7 1mm x 1mm views of the topography of a coverslip (CS) and neutral density filter (NDF) generated from phase using 12-bit, 8-bit, and 5-bit data. Due to the high reflectivity of the coverslip surface, quantization-induced phase noise is small, allowing accurate phase extraction even at very low bit-depths. Phase noise increases more rapidly with reduced ENOB when the reflectivity is lower, as for the NDF (the box shows an area of strong degradation with the arrow pointing to a ring that disappears in the 5-bit image). Note that scales are different in the top and bottom images.

To demonstrate the effects of vibration noise in PR-OCT, we imaged the motion of a mirror attached to a PZT stack driven with a 100-Hz sinusoidal signal. Figure 8(a) shows the measured displacement of the PZT obtained from the phase of 12-bit data. Figure 8(b) shows the error that would result if the displacement were instead calculated using 10, 7, and 6-bit data, with the 12-bit data taken as the reference for the error calculations. Error was calculated by point-wise subtraction of reduced-bit data from the 12-bit values (assumed “perfect”). The dashed line in Fig. 8(b) shows 0.5-nm RMS of vibration noise in the system, as measured with a stationary PZT. This vibration noise was due to the lack of a rigid support connecting the reference and sample reflectors. Only at 6 bits is the quantization-induced RMSE comparable to the RMS vibration noise. For higher bit depths, quantization noise is rendered insignificant due to the strong reflectivity of the sample.

Fig. 8 Displacement of a PZT driven at 100 Hz sinusoidal drive as calculated from 12-bit data. Fig. (b) shows the error in the displacement that would result if the waveform were instead generated using lower ENOBs (the waveform from 12-bit data is considered error-free and the waveforms from lower ENOBs are subtracted); the dashed horizontal lines show the RMS vibration noise (0.5 nm corresponds to 8% of the dynamic range). Except at very low bit depths, quantization noise is insignificant compared to vibration noise, since the sample reflectivity is large.

It is important to note that although vibration noise is significant in this phase measurement, its effects would be invisible in magnitude data, since nanometer-scale displacements have little effect on the position and magnitude of the A-scan peak. To the contrary, quantization noise associated with reduction of ENOB from 12 to 7 bits would be clearly visible in the A-scan magnitude, and corresponds to a 12-dB reduction in the signal, as seen in Fig. 6(d). Taken together, this suggests that quantization noise can have very different effects on magnitude and phase data.

Since the measured phase is limited to the range of (−π, π], phase data should be unwrapped in time to track changes over large displacements (each π phase corresponds to a displacement of λ0/4n). Hence, displacements that correspond to more than a π phase change within a single time interval cannot be accurately tracked. Moreover, phase wrapping will occur during the time step when the phase noise is sufficiently high, making the phase data ambiguous. To the extent that quantization noise affects phase noise, this can cause additional problems for accurately measuring phase data.

5. Conclusions

In cases where vibration noise is expected to dominate the phase error (as in traditional, non-common-path systems), reducing the ADC resolution will have significantly less effect on the accuracy of the phase data than the vibration noise, though the errors are complementary and cumulative. Keep in mind that vibration noise would have little effect on the magnitude image but enormously impacts phase. In contrast, in the absence of vibration noise (e.g., in CP systems), reducing the ADC resolution will cause the phase noise to increase rapidly. Future work should consider the effects of other noise sources (e.g., multiplicative effects of RIN) on OCT phase data and possible ways to mitigate or correct for the noise.

Acknowledgments

We gratefully acknowledge the help of Dierck Hillmann, Matthias Voelker, and other members of the Telesto system team at Thorlabs, GMBH for their assistance with software.

References and links

1.

V. J. Srinivasan, D. C. Adler, Y. L. Chen, I. Gorczynska, R. Huber, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Ultrahigh-speed optical coherence tomography for three-dimensional and en face imaging of the retina and optic nerve head,” Invest. Ophthalmol. Vis. Sci. 49(11), 5103–5110 (2008). [CrossRef] [PubMed]

2.

W. Wieser, B. Biedermann, T. Klein, C. Eigenwillig, and R. Huber, “Multi-megahertz OCT: high quality 3D imaging at 20 million A-scans and 4.5 GVoxels per second,” Opt. Express 18(14), 14685–14704 (2010). [CrossRef] [PubMed]

3.

D. C. Adler, R. Huber, and J. G. Fujimoto, “Phase-sensitive optical coherence tomography at up to 370,000 lines per second using buffered Fourier domain mode-locked lasers,” Opt. Lett. 32(6), 626–628 (2007). [CrossRef] [PubMed]

4.

C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. 30(16), 2131–2133 (2005). [CrossRef] [PubMed]

5.

G. Liu, M. Rubinstein, A. Saidi, W. Qi, A. Foulad, B. Wong, and Z. Chen, “Imaging vibrating vocal folds with a high speed 1050 nm swept source OCT and ODT,” Opt. Express 19(12), 11880–11889 (2011). [CrossRef] [PubMed]

6.

J. Zhang, W. Jung, J. S. Nelson, and Z. Chen, “Full range polarization-sensitive Fourier domain optical coherence tomography,” Opt. Express 12(24), 6033–6039 (2004). [CrossRef] [PubMed]

7.

E. Gotzinger, M. Pircher, and C. K. Hitzenberger, “High speed spectral domain polarization sensitive optical coherence tomography of the human retina,” Opt. Express 13(25), 10217–10229 (2005). [CrossRef] [PubMed]

8.

B. D. Goldberg, B. J. Vakoc, W-Y Oh, M. J. Suter, S. Waxman, M. I. Freilich, B. E. Bouma, and G. J. Tearney, “Performance of reduced bit-depth acquisition for optical frequency domain imaging,” Opt. Express 17(19), 16957–16968 (2009). [CrossRef] [PubMed]

9.

Z. Lu, D. K. Kasaragod, and S. J. Matcher, “Performance comparison between 8- and 14-bit- depth imaging in polarization-sensitive swept-source optical coherence tomography,” Biomed. Opt. Express 4(2), 794–804 (2011). [CrossRef]

10.

R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett. 31(20), 2975–2977 (2006). [CrossRef] [PubMed]

11.

A. B. Vakhtin, D. J. Kane, W. R. Wood, and K. A. Peterson, “Common-path interferometer for frequency-domain optical coherence tomography,” App. Opt. 42(34), 6953–6958 (2003). [CrossRef]

12.

M. A. Choma, A. K. Ellerbee, C. Yang, T. Creazzo, and J. A. Izatt, “Spectral-domain phase microscopy,” Opt. Lett. 30(10), 1162–1164 (2005). [CrossRef] [PubMed]

13.

M. V. Sarunic, S. Weinberg, and J. A. Izatt, “Full-field swept-source phase microscopy,” Opt. Lett. 31(10), 1462–1464 (2006). [CrossRef] [PubMed]

14.

A. K. Ellerbee, T. L. Creazzo, and J. A. Izatt, “Investigating nanoscale cellular dynamics with cross-sectional spectral domain phase microscopy,” Opt. Express 15(13), 8115–8124 (2007). [CrossRef] [PubMed]

15.

A. K. Ellerbee and J. A. Izatt, “Phase retrieval in low-coherence interferometric microscopy,” Opt. Lett. 32(4), 388–390 (2007). [CrossRef] [PubMed]

16.

S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftimia, and B. E. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11(22), 2953–2963 (2003). [CrossRef] [PubMed]

17.

R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express 11(8), 889–894 (2003). [CrossRef] [PubMed]

18.

A. Oppenheim and R. Schafer, Discrete-time Signal Processing, 3rd ed. (Prentice Hall, 2009).

19.

W. Kester, The Data Conversion Handbook (Elsevier, 2005).

20.

D. Hillmann, G. Huttmann, and P. Koch, “Using nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE 7372. 73720R (2009). [CrossRef]

21.

S. Vergnole, D. Lvesque, and G. Lamouche, “Experimental validation of an optimized signal processing method to handle non-linearity in swept-source optical coherence tomography,” Opt. Express 18(10), 10446–10461 (2010). [CrossRef] [PubMed]

22.

C. Copeland and A. K. Ellerbee, “The effects of different gold standards on the assessment of the accuracy of different resampling techniques for optical coherence tomography,” Proc. SPIE . 8225–8237 (2012).

23.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254(5035), 1178–1181 (1991). [CrossRef] [PubMed]

OCIS Codes
(030.4280) Coherence and statistical optics : Noise in imaging systems
(100.5070) Image processing : Phase retrieval
(170.3890) Medical optics and biotechnology : Medical optics instrumentation
(170.4500) Medical optics and biotechnology : Optical coherence tomography

ToC Category:
Medical Optics and Biotechnology

History
Original Manuscript: May 11, 2012
Revised Manuscript: June 18, 2012
Manuscript Accepted: June 19, 2012
Published: June 26, 2012

Virtual Issues
Vol. 7, Iss. 9 Virtual Journal for Biomedical Optics

Citation
William A. Ling and Audrey K. Ellerbee, "The effects of reduced bit depth on optical coherence tomography phase data," Opt. Express 20, 15654-15668 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15654


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References

  1. V. J. Srinivasan, D. C. Adler, Y. L. Chen, I. Gorczynska, R. Huber, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Ultrahigh-speed optical coherence tomography for three-dimensional and en face imaging of the retina and optic nerve head,” Invest. Ophthalmol. Vis. Sci.49(11), 5103–5110 (2008). [CrossRef] [PubMed]
  2. W. Wieser, B. Biedermann, T. Klein, C. Eigenwillig, and R. Huber, “Multi-megahertz OCT: high quality 3D imaging at 20 million A-scans and 4.5 GVoxels per second,” Opt. Express18(14), 14685–14704 (2010). [CrossRef] [PubMed]
  3. D. C. Adler, R. Huber, and J. G. Fujimoto, “Phase-sensitive optical coherence tomography at up to 370,000 lines per second using buffered Fourier domain mode-locked lasers,” Opt. Lett.32(6), 626–628 (2007). [CrossRef] [PubMed]
  4. C. Joo, T. Akkin, B. Cense, B. H. Park, and J. F. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett.30(16), 2131–2133 (2005). [CrossRef] [PubMed]
  5. G. Liu, M. Rubinstein, A. Saidi, W. Qi, A. Foulad, B. Wong, and Z. Chen, “Imaging vibrating vocal folds with a high speed 1050 nm swept source OCT and ODT,” Opt. Express19(12), 11880–11889 (2011). [CrossRef] [PubMed]
  6. J. Zhang, W. Jung, J. S. Nelson, and Z. Chen, “Full range polarization-sensitive Fourier domain optical coherence tomography,” Opt. Express12(24), 6033–6039 (2004). [CrossRef] [PubMed]
  7. E. Gotzinger, M. Pircher, and C. K. Hitzenberger, “High speed spectral domain polarization sensitive optical coherence tomography of the human retina,” Opt. Express13(25), 10217–10229 (2005). [CrossRef] [PubMed]
  8. B. D. Goldberg, B. J. Vakoc, W-Y Oh, M. J. Suter, S. Waxman, M. I. Freilich, B. E. Bouma, and G. J. Tearney, “Performance of reduced bit-depth acquisition for optical frequency domain imaging,” Opt. Express17(19), 16957–16968 (2009). [CrossRef] [PubMed]
  9. Z. Lu, D. K. Kasaragod, and S. J. Matcher, “Performance comparison between 8- and 14-bit- depth imaging in polarization-sensitive swept-source optical coherence tomography,” Biomed. Opt. Express4(2), 794–804 (2011). [CrossRef]
  10. R. Huber, D. C. Adler, and J. G. Fujimoto, “Buffered Fourier domain mode locking: unidirectional swept laser sources for optical coherence tomography imaging at 370,000 lines/s,” Opt. Lett.31(20), 2975–2977 (2006). [CrossRef] [PubMed]
  11. A. B. Vakhtin, D. J. Kane, W. R. Wood, and K. A. Peterson, “Common-path interferometer for frequency-domain optical coherence tomography,” App. Opt.42(34), 6953–6958 (2003). [CrossRef]
  12. M. A. Choma, A. K. Ellerbee, C. Yang, T. Creazzo, and J. A. Izatt, “Spectral-domain phase microscopy,” Opt. Lett.30(10), 1162–1164 (2005). [CrossRef] [PubMed]
  13. M. V. Sarunic, S. Weinberg, and J. A. Izatt, “Full-field swept-source phase microscopy,” Opt. Lett.31(10), 1462–1464 (2006). [CrossRef] [PubMed]
  14. A. K. Ellerbee, T. L. Creazzo, and J. A. Izatt, “Investigating nanoscale cellular dynamics with cross-sectional spectral domain phase microscopy,” Opt. Express15(13), 8115–8124 (2007). [CrossRef] [PubMed]
  15. A. K. Ellerbee and J. A. Izatt, “Phase retrieval in low-coherence interferometric microscopy,” Opt. Lett.32(4), 388–390 (2007). [CrossRef] [PubMed]
  16. S. H. Yun, G. J. Tearney, J. F. de Boer, N. Iftimia, and B. E. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express11(22), 2953–2963 (2003). [CrossRef] [PubMed]
  17. R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express11(8), 889–894 (2003). [CrossRef] [PubMed]
  18. A. Oppenheim and R. Schafer, Discrete-time Signal Processing, 3rd ed. (Prentice Hall, 2009).
  19. W. Kester, The Data Conversion Handbook (Elsevier, 2005).
  20. D. Hillmann, G. Huttmann, and P. Koch, “Using nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE7372. 73720R (2009). [CrossRef]
  21. S. Vergnole, D. Lvesque, and G. Lamouche, “Experimental validation of an optimized signal processing method to handle non-linearity in swept-source optical coherence tomography,” Opt. Express18(10), 10446–10461 (2010). [CrossRef] [PubMed]
  22. C. Copeland and A. K. Ellerbee, “The effects of different gold standards on the assessment of the accuracy of different resampling techniques for optical coherence tomography,” Proc. SPIE. 8225–8237 (2012).
  23. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science254(5035), 1178–1181 (1991). [CrossRef] [PubMed]

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