Abstract

The high acquisition speed of state-of-the-art optical coherence tomography (OCT) enables massive signal-to-noise ratio (SNR) improvements by signal averaging. Here, we investigate the performance of two commonly used approaches for OCT signal averaging. We present the theoretical SNR performance of (a) computing the average of OCT magnitude data and (b) averaging the complex phasors, and substantiate our findings with simulations and experimentally acquired OCT data. We show that the achieved SNR performance strongly depends on both the SNR of the input signals and the number of averaged signals when the signal bias caused by the noise floor is not accounted for. Therefore we also explore the SNR for the two averaging approaches after correcting for the noise bias and, provided that the phases of the phasors are accurately aligned prior to averaging, then find that complex phasor averaging always leads to higher SNR than magnitude averaging.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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2018 (5)

B. Tan, A. Wong, and K. Bizheva, “Enhancement of morphological and vascular features in OCT images using a modified Bayesian residual transform,” Biomed. Opt. Express 9(5), 2394–2406 (2018).
[Crossref]

B. Braaf, S. Donner, A. S. Nam, B. E. Bouma, and B. J. Vakoc, “Complex differential variance angiography with noise-bias correction for optical coherence tomography of the retina,” Biomed. Opt. Express 9(2), 486–506 (2018).
[Crossref]

B. Baumann, M. Augustin, A. Lichtenegger, D. J. Harper, M. Muck, P. Eugui, A. Wartak, M. Pircher, and C. K. Hitzenberger, “Polarization-sensitive optical coherence tomography imaging of the anterior mouse eye,” J. Biomed. Opt. 23(08), 1 (2018).
[Crossref]

T. B. Dubose, D. Cunefare, E. Cole, P. Milanfar, J. A. Izatt, and S. Farsiu, “Statistical models of signal and noise and fundamental limits of segmentation accuracy in retinal optical coherence tomography,” IEEE Trans. Med. Imaging 37(9), 1978–1988 (2018).
[Crossref]

M. Sugita, R. A. Brown, I. Popov, and A. Vitkin, “K-distribution three-dimensional mapping of biological tissues in optical coherence tomography,” J. Biophotonics 11(3), e201700055 (2018).
[Crossref]

2017 (7)

2016 (7)

S. Makita, K. Kurokawa, Y.-J. Hong, M. Miura, and Y. Yasuno, “Noise-immune complex correlation for optical coherence angiography based on standard and Jones matrix optical coherence tomography,” Biomed. Opt. Express 7(4), 1525–1548 (2016).
[Crossref]

M. Sugita, A. Weatherbee, K. Bizheva, I. Popov, and A. Vitkin, “Analysis of scattering statistics and governing distribution functions in optical coherence tomography,” Biomed. Opt. Express 7(7), 2551–2564 (2016).
[Crossref]

T. Pfeiffer, W. Wieser, T. Klein, M. Petermann, J. P. Kolb, M. Eibl, and R. Huber, “Flexible A-scan rate MHz OCT: computational downscaling by coherent averaging,” Proc. SPIE 9697, 96970S (2016).
[Crossref]

A. C. Chan, K. Kurokawa, S. Makita, M. Miura, and Y. Yasuno, “Maximum a posteriori estimator for high-contrast image composition of optical coherence tomography,” Opt. Lett. 41(2), 321–324 (2016).
[Crossref]

C.-L. Chen, H. Ishikawa, G. Wollstein, R. A. Bilonick, L. Kagemann, and J. S. Schuman, “Virtual Averaging Making Nonframe-Averaged Optical Coherence Tomography Images Comparable to Frame-Averaged Images,” Trans. Vis. Sci. Tech. 5(1), 1 (2016).
[Crossref]

C. Blatter, E. F. J. Meijer, A. S. Nam, D. Jones, B. E. Bouma, T. P. Padera, and B. J. Vakoc, “In vivo label-free measurement of lymph flow velocity and volumetric flow rates using Doppler optical coherence tomography,” Sci. Rep. 6(1), 29035 (2016).
[Crossref]

S. Fialová, M. Augustin, M. Glösmann, T. Himmel, S. Rauscher, M. Gröger, M. Pircher, C. K. Hitzenberger, and B. Baumann, “Polarization properties of single layers in the posterior eyes of mice and rats investigated using high resolution polarization sensitive optical coherence tomography,” Biomed. Opt. Express 7(4), 1479–1495 (2016).
[Crossref]

2015 (3)

H. Zhang, Z. Li, X. Wang, and X. Zhang, “Speckle reduction in optical coherence tomography by two-step image registration,” J. Biomed. Opt. 20(3), 036013 (2015).
[Crossref]

A. Lozzi, A. Agrawal, A. Boretsky, C. G. Welle, and D. X. Hammer, “Image quality metrics for optical coherence angiography,” Biomed. Opt. Express 6(7), 2435–2447 (2015).
[Crossref]

S. Wang and K. V. Larin, “Optical coherence elastography for tissue characterization: a review,” J. Biophotonics 8(4), 279–302 (2015).
[Crossref]

2014 (4)

M. Sugita, S. Zotter, M. Pircher, T. Makihira, K. Saito, N. Tomatsu, M. Sato, P. Roberts, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Motion artifact and speckle noise reduction in polarization sensitive optical coherence tomography by retinal tracking,” Biomed. Opt. Express 5(1), 106–122 (2014).
[Crossref]

W. Drexler, M. Liu, A. Kumar, T. Kamali, A. Unterhuber, and R. A. Leitgeb, “Optical coherence tomography today: speed, contrast, and multimodality,” J. Biomed. Opt. 19(7), 071412 (2014).
[Crossref]

S. Makita, Y.-J. Hong, M. Miura, and Y. Yasuno, “Degree of polarization uniformity with high noise immunity using polarization-sensitive optical coherence tomography,” Opt. Lett. 39(24), 6783–6786 (2014).
[Crossref]

J. J. Liu, A. J. Witkin, M. Adhi, I. Grulkowski, M. F. Kraus, A.-H. Dhalla, C. D. Lu, J. Hornegger, J. S. Duker, and J. G. Fujimoto, “Enhanced Vitreous Imaging in Healthy Eyes Using Swept Source Optical Coherence Tomography,” PLoS One 9(7), e102950 (2014).
[Crossref]

2013 (3)

2012 (2)

2011 (2)

J. W. Jacobs and S. J. Matcher, “Digital phase stabilization for improving sensitivity and degree of polarization accuracy in polarization sensitive optical coherence tomography,” Proc. SPIE 7889, 788938 (2011).
[Crossref]

Y. Lim, M. Yamanari, S. Fukuda, Y. Kaji, T. Kiuchi, M. Miura, T. Oshika, and Y. Yasuno, “Birefringence measurement of cornea and anterior segment by office-based polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2(8), 2392–2402 (2011).
[Crossref]

2009 (1)

2008 (3)

A. Sakamoto, M. Hangai, and N. Yoshimura, “Spectral-Domain Optical Coherence Tomography with Multiple B-Scan Averaging for Enhanced Imaging of Retinal Diseases,” Ophthalmology 115(6), 1071–1078.e7 (2008).
[Crossref]

X. Zhu, Y. Liang, Y. Mao, Y. Jia, Y. Liu, and G. Mu, “Analyses and calculations of noise in optical coherence tomography systems,” Front. Optoelectron. China 1(3-4), 247–257 (2008).
[Crossref]

S. Nadarajah, “A review of results on sums of random variables,” Acta Appl. Math. 103(2), 131–140 (2008).
[Crossref]

2007 (1)

P. H. Tomlins and R. K. Wang, “Digital phase stabilization to improve detection sensitivity for optical coherence tomography,” Meas. Sci. Technol. 18(11), 3365–3372 (2007).
[Crossref]

2006 (1)

D. M. Stein, H. Ishikawa, R. Hariprasad, G. Wollstein, R. J. Noecker, J. G. Fujimoto, and J. S. Schuman, “A new quality assessment parameter for optical coherence tomography,” Br. J. Ophthalmol. 90(2), 186–190 (2006).
[Crossref]

2003 (3)

2002 (1)

1991 (1)

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. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

1962 (1)

P. Beckmann, “Statistical distribution of the amplitude and phase of a multiply scattered field,” J. Res. Natl. Bur. Stand. (U. S.) 66D(3), 231–240 (1962).
[Crossref]

Adhi, M.

J. J. Liu, A. J. Witkin, M. Adhi, I. Grulkowski, M. F. Kraus, A.-H. Dhalla, C. D. Lu, J. Hornegger, J. S. Duker, and J. G. Fujimoto, “Enhanced Vitreous Imaging in Healthy Eyes Using Swept Source Optical Coherence Tomography,” PLoS One 9(7), e102950 (2014).
[Crossref]

Agrawal, A.

Almasian, M.

M. Almasian, T. G. van Leeuwen, and D. J. Faber, “OCT amplitude and speckle statistics of discrete random media,” Sci. Rep. 7(1), 14873 (2017).
[Crossref]

Augustin, M.

B. Baumann, M. Augustin, A. Lichtenegger, D. J. Harper, M. Muck, P. Eugui, A. Wartak, M. Pircher, and C. K. Hitzenberger, “Polarization-sensitive optical coherence tomography imaging of the anterior mouse eye,” J. Biomed. Opt. 23(08), 1 (2018).
[Crossref]

S. Fialová, M. Augustin, M. Glösmann, T. Himmel, S. Rauscher, M. Gröger, M. Pircher, C. K. Hitzenberger, and B. Baumann, “Polarization properties of single layers in the posterior eyes of mice and rats investigated using high resolution polarization sensitive optical coherence tomography,” Biomed. Opt. Express 7(4), 1479–1495 (2016).
[Crossref]

Baumann, B.

B. Baumann, M. Augustin, A. Lichtenegger, D. J. Harper, M. Muck, P. Eugui, A. Wartak, M. Pircher, and C. K. Hitzenberger, “Polarization-sensitive optical coherence tomography imaging of the anterior mouse eye,” J. Biomed. Opt. 23(08), 1 (2018).
[Crossref]

S. Fialová, M. Augustin, M. Glösmann, T. Himmel, S. Rauscher, M. Gröger, M. Pircher, C. K. Hitzenberger, and B. Baumann, “Polarization properties of single layers in the posterior eyes of mice and rats investigated using high resolution polarization sensitive optical coherence tomography,” Biomed. Opt. Express 7(4), 1479–1495 (2016).
[Crossref]

Beckmann, P.

P. Beckmann, “Statistical distribution of the amplitude and phase of a multiply scattered field,” J. Res. Natl. Bur. Stand. (U. S.) 66D(3), 231–240 (1962).
[Crossref]

Bilonick, R. A.

C.-L. Chen, H. Ishikawa, G. Wollstein, R. A. Bilonick, L. Kagemann, and J. S. Schuman, “Virtual Averaging Making Nonframe-Averaged Optical Coherence Tomography Images Comparable to Frame-Averaged Images,” Trans. Vis. Sci. Tech. 5(1), 1 (2016).
[Crossref]

Bizheva, K.

Blackburn, B. J.

Blatter, C.

C. Blatter, E. F. J. Meijer, A. S. Nam, D. Jones, B. E. Bouma, T. P. Padera, and B. J. Vakoc, “In vivo label-free measurement of lymph flow velocity and volumetric flow rates using Doppler optical coherence tomography,” Sci. Rep. 6(1), 29035 (2016).
[Crossref]

Boretsky, A.

Bouma, B. E.

Braaf, B.

Brown, R. A.

M. Sugita, R. A. Brown, I. Popov, and A. Vitkin, “K-distribution three-dimensional mapping of biological tissues in optical coherence tomography,” J. Biophotonics 11(3), e201700055 (2018).
[Crossref]

Cense, B.

Chan, A. C.

Chang, W.

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. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref]

Chen, C.-L.

C.-L. Chen, H. Ishikawa, G. Wollstein, R. A. Bilonick, L. Kagemann, and J. S. Schuman, “Virtual Averaging Making Nonframe-Averaged Optical Coherence Tomography Images Comparable to Frame-Averaged Images,” Trans. Vis. Sci. Tech. 5(1), 1 (2016).
[Crossref]

Choma, M.

J. Izatt and M. Choma, “Theory of optical coherence tomography,” in Optical Coherence Tomography - Technology and Applications, (2008), pp. 47–72.

Choma, M. A.

Cole, E.

T. B. Dubose, D. Cunefare, E. Cole, P. Milanfar, J. A. Izatt, and S. Farsiu, “Statistical models of signal and noise and fundamental limits of segmentation accuracy in retinal optical coherence tomography,” IEEE Trans. Med. Imaging 37(9), 1978–1988 (2018).
[Crossref]

Cunefare, D.

T. B. Dubose, D. Cunefare, E. Cole, P. Milanfar, J. A. Izatt, and S. Farsiu, “Statistical models of signal and noise and fundamental limits of segmentation accuracy in retinal optical coherence tomography,” IEEE Trans. Med. Imaging 37(9), 1978–1988 (2018).
[Crossref]

Damodaran, K. V.

de Boer, J. F.

Dhalla, A.-H.

J. J. Liu, A. J. Witkin, M. Adhi, I. Grulkowski, M. F. Kraus, A.-H. Dhalla, C. D. Lu, J. Hornegger, J. S. Duker, and J. G. Fujimoto, “Enhanced Vitreous Imaging in Healthy Eyes Using Swept Source Optical Coherence Tomography,” PLoS One 9(7), e102950 (2014).
[Crossref]

Donner, S.

Drexler, W.

W. Drexler, M. Liu, A. Kumar, T. Kamali, A. Unterhuber, and R. A. Leitgeb, “Optical coherence tomography today: speed, contrast, and multimodality,” J. Biomed. Opt. 19(7), 071412 (2014).
[Crossref]

Duan, H.

W. Wu, O. Tan, R. R. Pappuru, H. Duan, and D. Huang, “Assessment of frame-averaging algorithms in OCT image analysis,” Ophthalmic Surg. Lasers Imag. Retin. 44(2), 168–175 (2013).
[Crossref]

Duan, L.

Dubose, T. B.

T. B. Dubose, D. Cunefare, E. Cole, P. Milanfar, J. A. Izatt, and S. Farsiu, “Statistical models of signal and noise and fundamental limits of segmentation accuracy in retinal optical coherence tomography,” IEEE Trans. Med. Imaging 37(9), 1978–1988 (2018).
[Crossref]

Duker, J. S.

J. J. Liu, A. J. Witkin, M. Adhi, I. Grulkowski, M. F. Kraus, A.-H. Dhalla, C. D. Lu, J. Hornegger, J. S. Duker, and J. G. Fujimoto, “Enhanced Vitreous Imaging in Healthy Eyes Using Swept Source Optical Coherence Tomography,” PLoS One 9(7), e102950 (2014).
[Crossref]

Eibl, M.

T. Pfeiffer, W. Wieser, T. Klein, M. Petermann, J. P. Kolb, M. Eibl, and R. Huber, “Flexible A-scan rate MHz OCT: computational downscaling by coherent averaging,” Proc. SPIE 9697, 96970S (2016).
[Crossref]

Eugui, P.

B. Baumann, M. Augustin, A. Lichtenegger, D. J. Harper, M. Muck, P. Eugui, A. Wartak, M. Pircher, and C. K. Hitzenberger, “Polarization-sensitive optical coherence tomography imaging of the anterior mouse eye,” J. Biomed. Opt. 23(08), 1 (2018).
[Crossref]

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Figures (9)

Fig. 1.
Fig. 1. Complex phasor representation of noise and signals in OCT and their probability density functions. (a) Cartoon of Beckmann distribution of noise phasors around the origin of the complex plane. A representative phasor $S_{noise}$ with real part $r_{noise}$ and imaginary part $i_{noise}$ is shown in green. (b) Complex OCT signals of 100 repeated noise measurements in the same pixel from real-world OCT data. (c) Histogram of noise amplitudes (1000 repeats, gray line) and Rayleigh PDF (red line) computed from the standard deviation $\sigma$ of the Beckmann distribution in (b) by Eq. (3). (d) Histogram of the noise intensity (gray line) and PDF (red line) computed from $\sigma$ in (b) by Eq. (9). (e) Cartoon of an OCT signal affected by noise. The green arrow represents the signal phasor. (f) Complex OCT signals of 100 repeated measurements of a weak reflection in the same pixel. (g) Histogram of signal amplitudes (1000 repeats, gray line) and Rice distribution (blue line) computed from the mean signal amplitude and $\sigma$ using Eq. (6). (h) Histogram of the signal intensity (gray line) and PDF (blue line) computed from the mean intensity and $\sigma$ by Eq. (10). The $+$ in (b) and (f) indicates the origin of the complex plane. The PDFs in (c,d,g,h) were scaled to match the count levels of the respective histograms.
Fig. 2.
Fig. 2. Intensity signal and noise background in a schematic OCT depth profile. The noise floor is characterized by its average intensity $\overline {I_{noise}}$ and its variance $\sigma _{I_{noise}}^2$. The measured OCT intensity signal $\overline {I}$ consists of the pure signal intensity $I$ biased by the average noise level $\overline {I_{noise}}$.
Fig. 3.
Fig. 3. Relative SNR improvement by signal averaging for strong input signals (without noise bias correction). (a) The ratios SNR$_{CPX}$/SNR$_1$ and SNR$_{MAG}$/SNR$_1$ are shown for $N$ from 1 to 100. SNR$_{MAG}$/SNR$_1$ is plotted as a dash-dotted line, whereas the SNR$_1$-dependent ratio SNR$_{CPX}$/SNR$_1$ is plotted in rainbow colors for several SNR$_1$ values between 5 dB and 50 dB. Note that SNR$_{CPX}$/SNR$_1$ converges to an $N$-fold improvement. (b) The ratio $\textrm {SNR}_{CPX}/\textrm {SNR}_{MAG} =\left (N-(N-1)/\textrm {SNR}_1 \right )/\sqrt []{N}$ is plotted for the spectrum of SNR$_1$ values used in (a). Note that in particular for strong input signals with high SNR$_1$, complex averaging outperforms magnitude averaging and converges to a $\sqrt []{N}$-fold better SNR performance. As the SNR profiles converge for large SNRs, the curves in (a) and (b) start to overlap for values greater than 15 dB.
Fig. 4.
Fig. 4. Influence of input signal level and number of averaged signals on the SNR (without noise bias correction). (a) SNR$_{MAG}$ and SNR$_{CPX}$ after magnitude and complex averaging of $N$ signals plotted for relative signal levels of $I/2\sigma ^2=1$ (left), $I/2\sigma ^2=0.5$ (middle), and $I/2\sigma ^2=0.1$ (right), respectively. (b) SNR$_{MAG}$ and SNR$_{CPX}$ after magnitude and complex averaging of signals $I/2\sigma ^2$ ranging from 0.01 through 10, plotted for averages of $N=2$ (left), $N=10$ (middle), and $N=100$ signals (right), respectively. Green arrows in (a) and (b) indicate the intercepts of the SNR profiles, i.e. the borderline SNR where magnitude and complex averaging perform equally. (c) Borderline plots of $I/2\sigma ^2 = 1/\sqrt []{N}$ as well as SNR$_1$ as described in Eq. (27).
Fig. 5.
Fig. 5. Relative SNR performance for magnitude and complex averaging (without noise bias correction). The heat map displays the ratio SNR$_{CPX}/$SNR$_{MAG}$ in decibels for relative input signals $I/2\sigma ^2$ ranging from -20 dB to +40 dB and up to a number of averaged signals $N=100$. For small input signals below the noise level, magnitude averaging yields a better SNR improvement (red range), while complex averaging performs better for greater $N$ and stronger input signals (blue range). The borderline SNR where SNR$_{MAG}=$ SNR$_{CPX}$ separates these two domains (white plot).
Fig. 6.
Fig. 6. Theoretical improvement of the signal-to-noise ratio SNR$'$ after noise bias correction plotted on (a) linear scales and (b) log scales. A $\sqrt {N}$- and $N$-fold improvement of the SNR$'$ of a single signal can be observed for magnitude and complex averaging, respectively. Note that, unlike for the noise-afflicted SNR calculations in Figs. 3 through 5, neither of the averaging approaches depend on the input signal strength.
Fig. 7.
Fig. 7. Simulation of the effect of averaging $N$ signals with relative strength $I/2\sigma ^2=10$ in (a) and $I/2\sigma ^2=0.1$ in (b). Shown are the averaged signal-to-noise ratios calculated from $N$ simulated phasors ($\bullet$) alongside the corresponding theoretical plots ($-$). The SNRs without and with noise bias correction are plotted in the left and right panels, respectively. Without noise bias correction, the SNR$_{CPX}$ shows a better performance for the strong signal in (a), whereas SNR$_{MAG}$ dominates for $N=1$ to 100 both for theoretical calculation and simulation. With noise bias correction (rightmost column), complex averaging similarly provides an $N$-fold improvement of the respective SNR$' = I/2\sigma ^2$ whereas an $\sqrt {N}$-fold improvement can be observed for magnitude averaging.
Fig. 8.
Fig. 8. Experimental verification of SNR improvement by the different averaging approaches in a layered retina phantom. (a) Single B-scan image of the phantom (no attenuation). (b) Single B-scan image after attenuating the sample beam by 30 dB. (c) B-scan image after averaging the magnitudes of 100 repeated frames (with 30 dB attenuation). (d) B-scan image after averaging the phasors of 100 repeated frames (with 30 dB attenuation). Note that all B-scan images in (a-d) are displayed with identical dynamic ranges of 40 dB where 0 dB refers to the maximum signal intensity in the frame. (e) Depth profiles of a single A-scan before and after attenuation, 100 magnitude averaged, and 100 complex averaged A-scans at the locations indicated by the dotted lines in (a-d). Due to a beam offset caused by the ND filter, the scattering profile of the unattenuated case has a slightly different structure. Dynamic range as in (a-d). (f) SNR improvement without noise bias correction for three pixels with weak (left), borderline (middle) and strong signal strength $I/2\sigma ^2$ (right), respectively. Pixel locations are indicated by orange boxes numbered with 1-3 in panel (d). SNR curves are shown for magnitude and complex averaging of 1-100 repeated M-scan signals for the experimental data ($\bullet$) alongside the corresponding theoretical plots ($-$). (g) SNR$'$ improvement after noise bias correction for the data sets shown in panel (f). Note that the experimental data ($\bullet$) slightly fluctuates around the theoretical profiles ($-$). SNR$'$ data fluctuating below SNR$' = 0$ is not shown. The axes are scaled as in the respective plots in (f) in order to enable a direct comparison between the two SNR analyses.
Fig. 9.
Fig. 9. Phasor rotation by axial motion and signal penalty after complex averaging of axially displaced signals. (a) Example of a signal phasor trajectory of 1000 repeated measurements of a weak reflector (an attenuated glass surface). Over 1/70 second, the phasor (blue dots) is markedly moving around the origin. Corresponding noise signals are shown as red dots. The main source of displacement in this measurement was air flow from a nearby air condition outlet. (b) Relative signal intensity decrease caused by complex averaging of $N$ signals with relative displacements $W$ of 0.5 (solid line), 0.05 (dashed line), and 0.005 (dotted line). For reference, $1/N$ is plotted as well. While relative displacements of $\lambda /200$ ($W$ = 0.005) between consecutive signals almost do not affect the intensity of the averaged signal, greater displacements such as $\lambda /20$ and $\lambda /2$ cause a periodic, significant signal reduction. In unlucky cases, $P(N,W)$ reaches zero such that the averaged signal is completely annihilated.

Tables (1)

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Table 1. Overview of variables and abbreviations

Equations (45)

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S O C T ( x , t ) = A ( x , t ) exp [ i ϕ ( x , t ) ]
p r i ( r n o i s e , i n o i s e ) = 1 2 π σ 2 exp [ r n o i s e 2 + i n o i s e 2 2 σ 2 ] .
p A ( A n o i s e ) = A n o i s e σ 2 exp [ A n o i s e 2 2 σ 2 ] .
A n o i s e ¯ = π 2 σ ,
σ n o i s e 2 = ( 2 π 2 ) σ 2 .
p A ( A , A n o i s e ) = A n o i s e σ 2 exp [ A n o i s e 2 + A 2 2 σ 2 ] B 0 [ A n o i s e A σ 2 ] .
A ¯ = π 2 σ L 1 / 2 [ A 2 2 σ 2 ] ,
σ A 2 = A 2 + 2 σ 2 + π 2 σ L 1 / 2 2 [ A 2 2 σ 2 ] ,
p I ( I n o i s e ) = 1 2 σ 2 exp [ I n o i s e 2 σ 2 ] ,
p I ( I , I n o i s e ) = 1 2 σ 2 exp [ I n o i s e + I 2 σ 2 ] B 0 [ I n o i s e I σ 2 ] ,
I n o i s e ¯ = 2 σ 2 ,
σ I n o i s e 2 = 4 σ 4 ,
I ¯ = I + 2 σ 2 ,
σ I 2 = 4 σ 2 ( I + σ 2 ) .
I M A G = 1 N j = 1 N I j .
I n o i s e M A G ¯ = 2 σ 2 = I n o i s e ¯ ,
σ I n o i s e 2 M A G = 1 N 4 σ 4 = 1 N σ I n o i s e 2 ,
I M A G ¯ = I + 2 σ 2 = I ¯ ,
σ I 2 M A G = 1 N 4 σ 2 ( I + σ 2 ) = 1 N σ I 2 .
I n o i s e C P X ¯ = 1 N 2 σ 2 = 1 N I n o i s e ¯ ,
σ I n o i s e 2 C P X = 1 N 2 4 σ 4 = 1 N 2 σ I n o i s e 2 .
I C P X ¯ = I + 1 N 2 σ 2 = I ¯ N 1 N 2 σ 2 ,
σ I 2 C P X = 1 N 4 σ 2 ( I + 1 N σ 2 ) = 1 N σ I 2 N 1 N 2 σ 2 .
SNR 1 = I ¯ σ I n o i s e = I + I n o i s e ¯ σ I n o i s e ,
SNR M A G = I M A G ¯ σ I n o i s e M A G = I + I n o i s e ¯ 1 N σ I n o i s e = N SNR 1 ,
SNR C P X = I C P X ¯ σ I n o i s e C P X = I + 1 N I n o i s e ¯ 1 N σ I n o i s e = N SNR 1 ( N 1 ) .
SNR 1 , b o r d e r l i n e = 1 + 1 N
SNR 1 ( I = 0 ) = I n o i s e ¯ σ I n o i s e = 1 ,
SNR M A G ( I = 0 ) = I n o i s e ¯ 1 N σ I n o i s e = N ,
SNR C P X ( I = 0 ) = 1 N I n o i s e ¯ 1 N σ I n o i s e = 1.
I ¯ = I ¯ I n o i s e ¯ = I + 2 σ 2 2 σ 2 = I ,
I M A G ¯ = I M A G ¯ I n o i s e M A G ¯ = I + 2 σ 2 2 σ 2 = I ,
I C P X ¯ = I C P X ¯ I n o i s e C P X ¯ = I + 1 N 2 σ 2 1 N 2 σ 2 = I .
S N R 1 = I ¯ σ I n o i s e = I 2 σ 2 ,
S N R M A G = I M A G ¯ σ I n o i s e M A G = N I 2 σ 2 = N S N R 1 ,
S N R C P X = I C P X ¯ σ I n o i s e C P X = N I 2 σ 2 = N S N R 1 .
P ( N ) = 1 N | j = 1 N e i 2 π W j |
P ( N , W ) = 1 N | j = 1 N e i 2 π W j | = 1 N | sin [ π W ( N + 1 ) ] sin [ π W ] e i π N W 1 | .
Δ ϕ b u l k ( m ) = arg [ z = z 0 z m a x ( A m j ( z ) e i ϕ m j ( z ) ) ( A m r e f ( z ) e i ϕ m r e f ( z ) ) ] = = arg [ z = z 0 z m a x A m j ( z ) A m r e f ( z ) e i [ ϕ m j ( z ) ϕ m r e f ( z ) ] ]
S O C T ( m o c o ) ( x , z ) = A m j ( x , z ) exp [ i ( ϕ m j ( x , z ) Δ ϕ b u l k ( m ) ) ] .
S O C T ( j , m o c o ) ( x , z ) = A j ( x , z ) A r e f ( x , z ) exp [ i ( ϕ j ( x , z ) ϕ r e f ( x , z ) ) ] .
Δ ϕ j ¯ ( x , z ) = arg [ x = x Δ x / 2 x + Δ x / 2 z = z Δ z / 2 z + Δ z / 2 A j ( x , z ) A r e f ( x , z ) exp [ i Δ ϕ j ( x , z ) ] ] .
p Δ ϕ ( Δ ϕ , A ) = 1 2 π exp [ A 2 4 σ Δ ϕ 2 ] + + A 4 π σ Δ ϕ cos Δ ϕ exp [ A 2 sin 2 Δ ϕ 4 σ Δ ϕ 2 ] erf [ A 2 σ Δ ϕ cos Δ ϕ ]
p Δ ϕ ( Δ ϕ , A = 0 ) = 1 2 π .
S O C T ( j , m o c o ¯ ) ( x , z ) = A j ( x , z ) A r e f ( x , z ) exp [ i ( ϕ j ( x , z ) Δ ϕ j ¯ ( x , z ) ) ] ,