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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 Optical Society of America
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]). 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], 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, 4], Doppler (D) OCT [5], and polarization-sensitive (PS) OCT [6, 7]. 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, 8–10]. 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:
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 = 2n(Δz + δz(t)). Note that for simplicity, Eq. (1) omits the DC terms because they do not contribute to the interferometric signal of interest.Minute deviations in the position of a reflector in the sample may be detected by tracking the phase, rather than the magnitude of the A-scan data, even for motion well below the diffraction limit. For example, it can be shown that changes in δz(t) of a moving sample can be calculated from the phase of I(2nΔz,t). In particular,
where k0 is the center wavenumber of a source with center wavelength λ0 = 2π/k0 [12]. Note that the phase measured here corresponds to structural information about the sample; it is not related to the optical phase of the measured light.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, 12–15]. 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:
For convenience, we lump shot, RIN, dark current, and read-out noise into a single noise term ; 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], used extensively in the study of ADCs [19]:
- QN is an additive, stationary white-noise process.
- 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, .
- QN is uncorrelated with the input sequence to the quantizer.
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]. Hence, an ideal B-bit quantizer and a real B-bit ADC have noise characteristics given by Eq. (3) and Eq. (4), respectively, where is the variance of an additional error source that encompasses the effects of all non-idealities in real ADCs.
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]. 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)).
2.3. Effects of noise on the measurement of OCT phase
Consider an interferogram comprising a signal i(k) and additive noise n(k) such that y(k) = i(k) + n(k). After applying the discrete Fourier transform (DFT), we obtain transformed signal and noise vectors I(z) and N(z), respectively. Previous attempts to derive an analytical expression relating n(k) to the noise in ∠Y(z) employed a semi-deterministic model: additive shot noise was assumed to be the dominant noise source [12]; the DFT of N(z) was treated as a complex stochastic process with constant, deterministic magnitude (i.e., the standard deviation of the photoelectron shot noise) and random phase. The standard deviation of phase error was assumed to be the largest phase deviation that could be imparted to the signal by the noise variable of constant magnitude:
where σδθ is the standard deviation of the phase change caused by noise. In contrast with [12], we present a more general noise model that considers cases where quantization noise may dominate. Moreover, our model accounts for the varying (i.e., not deterministic) magnitude of the complex noise in addition to its random 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, , where 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).
Defining SNR as |I|/σN, we see that Eq. (9) differs from Eq. (7) by a factor of . 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 [20,21]. While the choice of resampling method has been shown to affect the accuracy of the measured phase data [22], the NDFT has been found to be accurate at least for magnitude reconstruction [21]; 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, 9] 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, 9], 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] 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]. 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, and , 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):
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 from Eq. (10), leading to the modified equation:
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, 9]. 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]:
- 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).
- 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.
- 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.
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 [11,14], 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.
To demonstrate the effects of vibration noise in PR-OCT, we added a piezoelectric transducer (PZT)-mounted mirror behind the coverslip (Fig. 5). The PZT was driven with a 250-mV peak-to-peak sinusoidal waveform using a standard waveform generator. The lack of a rigid connection between the reference and sample reflectors was a new source of noise (vibration) that had been highly suppressed in the previous experiment using the coverslip alone. The back surface of the coverslip served as a reference to minimize dispersion mismatch.
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 . Thus, by Eq. (9), standard deviation of the phase noise is
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 and . The values of the Ki depend on a number of system parameters such as source power spectral density, spectrometer efficiency, and responsivity [16, 17].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]). 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.
Previous studies investigating the effects of quantization on SNR concluded that 8 bits was adequate to keep the SNR to within 1 dB of the SNR obtained at 14 bits [8, 9]. We note that such results are not necessarily obtained in systems where non-ADC noise sources are comparatively small. In our CP setup, the DC power received by our camera was relatively low due to the weakly reflecting reference, resulting in low values of shot noise and RIN. Consequently, QN is a larger portion of the total noise and quickly becomes dominant as ENOB decreases. We thus observed a higher ENOB at which SNR drops by 1 dB, as shown in Fig. 6(d). Here we see that 10 bits are required for our system to maintain the SNR to within 1 dB of the original 12-bit SNR. A typical SS-OCT system utilizes balanced detectors to eliminate the received DC power and thus can operate with larger reference powers than equivalent SD-OCT systems. Although the DC power is cancelled, the accompanying shot noise is not cancelled, and RIN is only partially cancelled. The optimal reference power in SS-OCT systems is typically larger than what SD-OCT systems can handle, leading to more shot noise and potentially higher uncancelled RIN in SS-OCT systems. Thus, quantization noise is a smaller fraction of the total additive noise in such systems and only becomes dominant at lower ENOBs.
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 PR-OCT imaging of biological samples, the accuracy of the phase information impacts the inferred 3D structure, optical properties, or motion of the sample. While there are several potential benefits to using reduced ADC resolutions (e.g., faster scanning, reduced data storage, faster processing, higher Doppler velocity thresholds), reducing ADC resolution will always have a slight (although possibly insignificant) effect on the accuracy of extracted information. This study provides quantitative assessment of the effects of ADC resolution on the accuracy of phase-based measurements. We have introduced a new theoretical model for phase noise due to additive noise sources that makes fewer assumptions than the model of [12] and yields predictions that closely match experimental results. Moreover, we have introduced a new method to accurately simulate fractional ENOBs that better reflects the actual performance of ADCs than previous models that considered only integer ENOBs.
While previous studies on magnitude images in SS-OCT concluded that 8 bits was sufficient for accurate image reconstruction with minimal degradation, our work shows that this conclusion does not necessarily extend to phase measurements. In general, phase noise increases with decreasing reflectivity and decreasing ENOB, and quantization noise contributes more to phase noise at low reflectivities, such as those one might expect to find in biological samples. Thus the minimum ENOB needed to achieve a particular degree of phase accuracy depends on several things including: 1) sample reflectivity, because higher sample reflectivities are more insensitive to quantization noise; 2) the magnitude of all additive noise sources excluding quantization, because other additive noise sources have similar effects to reducing bit depth; and 3. the magnitude of unfiltered vibration noise, whose presence can mask the effects of additive noise sources like quantization. The following systematic method may be used to determine the minimum acceptable ENOB to achieve the desired absolute phase sensitivity for any PR-OCT system:
- Determine the maximum acceptable standard deviation σpn of the phase noise. This will depend on the minimum displacement, velocity, or change in optic axis one may wish to measure in the case of SDPM, Doppler, or PS-OCT.
- Determine the reflectivity of the sample to be imaged. This allows for determination of the interferometric signal strength |I| in Eq. (9).
- Measure the complex noise produced by all additive noise sources excluding the ADC, as described in [8]. Let the standard deviation of this noise be σa.
- Measure the vibration noise of the system. Vibration noise tends to have most of its power in frequencies related to mechanical resonances and may be reduced through filtering of the phase data. Let the remaining unfiltered vibration noise have standard deviation σv.
- Let σq be the standard deviation of the effective quantization noise from the ADC. In the transform domain, the quantization noise variance will be scaled by P, the length of the DFT (the number of pixels in the detector). The total phase noise of the system, including vibration noise, will have variance , from which one can solve for as shown below.
- Use Eq. (5) to determine the corresponding ENOB, where any logarithm base can be used:
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.
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