Graphics processing unit accelerated non-uniform fast Fourier transform for ultrahigh-speed, real-time Fourier-domain OCT

Kang Zhang; Jin U. Kang

doi:10.1364/OE.18.023472

Graphics processing unit accelerated non-uniform fast Fourier transform for ultrahigh-speed, real-time Fourier-domain OCT

Kang Zhang and Jin U. Kang

Optics Express
Vol. 18,
Issue 22,
pp. 23472-23487
(2010)
•https://doi.org/10.1364/OE.18.023472

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Kang Zhang and Jin U. Kang, "Graphics processing unit accelerated non-uniform fast Fourier transform for ultrahigh-speed, real-time Fourier-domain OCT," Opt. Express 18, 23472-23487 (2010)

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Related Topics
Table of Contents Category
- Image Processing
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Digital image processing (100.2000)
- Optical coherence tomography (110.4500)
- Medical optics instrumentation (170.3890)

About this Article
History
- Original Manuscript: September 17, 2010
- Revised Manuscript: October 14, 2010
- Manuscript Accepted: October 15, 2010
- Published: October 22, 2010
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 6, Iss. 1

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Open Access

Abstract

We implemented fast Gaussian gridding (FGG)-based non-uniform fast Fourier transform (NUFFT) on the graphics processing unit (GPU) architecture for ultrahigh-speed, real-time Fourier-domain optical coherence tomography (FD-OCT). The Vandermonde matrix-based non-uniform discrete Fourier transform (NUDFT) as well as the linear/cubic interpolation with fast Fourier transform (InFFT) methods are also implemented on GPU to compare their performance in terms of image quality and processing speed. The GPU accelerated InFFT/NUDFT/NUFFT methods are applied to process both the standard half-range FD-OCT and complex full-range FD-OCT (C-FD-OCT). GPU-NUFFT provides an accurate approximation to GPU-NUDFT in terms of image quality, but offers >10 times higher processing speed. Compared with the GPU-InFFT methods, GPU-NUFFT has improved sensitivity roll-off, higher local signal-to-noise ratio and immunity to side-lobe artifacts caused by the interpolation error. Using a high speed CMOS line-scan camera, we demonstrated the real-time processing and display of GPU-NUFFT-based C-FD-OCT at a camera-limited rate of 122 k line/s (1024 pixel/A-scan).

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References

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K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation system for microsurgery guidance and intervention based on common-path optical coherence tomography,” IEEE Trans. Biomed. Eng. 56(9), 2318–2321 (2009).
[Crossref] [PubMed]
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[Crossref]
U. Sharma, N. M. Fried, and J. U. Kang, “All-fiber common-path optical coherence tomography: sensitivity optimization and system analysis,” IEEE J. Sel. Top. Quantum Electron. 11(4), 799–805 (2005).
[Crossref]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
B. Baumann, M. Pircher, E. Götzinger, and C. K. Hitzenberger, “Full range complex spectral domain optical coherence tomography without additional phase shifters,” Opt. Express 15(20), 13375–13387 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-20-13375 .
[Crossref] [PubMed]
L. An and R. K. Wang, “Use of a scanner to modulate spatial interferograms for in vivo full-range Fourier-domain optical coherence tomography,” Opt. Lett. 32(23), 3423–3425 (2007).
[Crossref] [PubMed]
S. Vergnole, G. Lamouche, and M. L. Dufour, “Artifact removal in Fourier-domain optical coherence tomography with a piezoelectric fiber stretcher,” Opt. Lett. 33(7), 732–734 (2008).
[Crossref] [PubMed]
H. M. Subhash, L. An, and R. K. Wang, “Ultra-high speed full range complex spectral domain optical coherence tomography for volumetric imaging at 140,000 A scans per second,” Proc. SPIE 7554, 75540K (2010).
[Crossref]
T. E. Ustun, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Real-time processing for Fourier domain optical coherence tomography using a field programmable gate array,” Rev. Sci. Instrum. 79(11), 114301 (2008).
[Crossref] [PubMed]
A. E. Desjardins, B. J. Vakoc, M. J. Suter, S. H. Yun, G. J. Tearney, and B. E. Bouma, “Real-time FPGA processing for high-speed optical frequency domain imaging,” IEEE Trans. Med. Imaging 28(9), 1468–1472 (2009).
[Crossref] [PubMed]
G. Liu, J. Zhang, L. Yu, T. Xie, and Z. Chen, “Real-time polarization-sensitive optical coherence tomography data processing with parallel computing,” Appl. Opt. 48(32), 6365–6370 (2009).
[Crossref] [PubMed]
J. Probst, D. Hillmann, E. Lankenau, C. Winter, S. Oelckers, P. Koch, and G. Hüttmann, “Optical coherence tomography with online visualization of more than seven rendered volumes per second,” J. Biomed. Opt. 15(2), 026014 (2010).
[Crossref] [PubMed]
Y. Watanabe and T. Itagaki, “Real-time display on Fourier domain optical coherence tomography system using a graphics processing unit,” J. Biomed. Opt. 14(6), 060506 (2009).
[Crossref]
K. Zhang and J. U. Kang, “Real-time 4D signal processing and visualization using graphics processing unit on a regular nonlinear-k Fourier-domain OCT system,” Opt. Express 18(11), 11772–11784 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-11-11772 .
[Crossref] [PubMed]
S. Van der Jeught, A. Bradu, and A. G. Podoleanu, “Real-time resampling in Fourier domain optical coherence tomography using a graphics processing unit,” J. Biomed. Opt. 15(3), 030511 (2010).
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[Crossref] [PubMed]
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[Crossref] [PubMed]
C. M. Eigenwillig, B. R. Biedermann, G. Palte, and R. Huber, “K-space linear Fourier domain mode locked laser and applications for optical coherence tomography,” Opt. Express 16(12), 8916–8937 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-8916 .
[Crossref] [PubMed]
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[Crossref]
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[PubMed]
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[Crossref] [PubMed]
S. Vergnole, D. Lévesque, 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), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-10-10446 .
[Crossref] [PubMed]
D. Hillmann, G. Huttmann, and P. Koch, “Using nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE 7372, 73720R (2009).
[Crossref]
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L. Greengard and J. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM Rev. 46(3), 443–454 (2004).
[Crossref]
C. Dorrer, N. Belabas, J. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17(10), 1795–1802 (2000).
[Crossref]

2010 (9)

J. U. Kang, J.-H. Han, X. Liu, K. Zhang, C. G. Song, and P. Gehlbach, ““Endoscopic functional Fourier domain common path optical coherence tomography for microsurgery,” IEEE J. Sel. Top. Quantum Electron. 16(4), 781–792 (2010).
[Crossref]

W.-Y. Oh, B. J. Vakoc, M. Shishkov, G. J. Tearney, and B. E. Bouma, “>400 kHz repetition rate wavelength-swept laser and application to high-speed optical frequency domain imaging,” Opt. Lett. 35(17), 2919–2921 (2010).
[Crossref] [PubMed]

W. Wieser, B. R. Biedermann, T. Klein, C. M. 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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14685 .
[Crossref] [PubMed]

H. M. Subhash, L. An, and R. K. Wang, “Ultra-high speed full range complex spectral domain optical coherence tomography for volumetric imaging at 140,000 A scans per second,” Proc. SPIE 7554, 75540K (2010).
[Crossref]

J. Probst, D. Hillmann, E. Lankenau, C. Winter, S. Oelckers, P. Koch, and G. Hüttmann, “Optical coherence tomography with online visualization of more than seven rendered volumes per second,” J. Biomed. Opt. 15(2), 026014 (2010).
[Crossref] [PubMed]

K. Zhang and J. U. Kang, “Real-time 4D signal processing and visualization using graphics processing unit on a regular nonlinear-k Fourier-domain OCT system,” Opt. Express 18(11), 11772–11784 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-11-11772 .
[Crossref] [PubMed]

S. Van der Jeught, A. Bradu, and A. G. Podoleanu, “Real-time resampling in Fourier domain optical coherence tomography using a graphics processing unit,” J. Biomed. Opt. 15(3), 030511 (2010).
[Crossref] [PubMed]

Y. Watanabe, S. Maeno, K. Aoshima, H. Hasegawa, and H. Koseki, “Real-time processing for full-range Fourier-domain optical-coherence tomography with zero-filling interpolation using multiple graphic processing units,” Appl. Opt. 49(25), 4756–4762 (2010).
[Crossref] [PubMed]

S. Vergnole, D. Lévesque, 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), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-10-10446 .
[Crossref] [PubMed]

2009 (10)

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

Y. Watanabe and T. Itagaki, “Real-time display on Fourier domain optical coherence tomography system using a graphics processing unit,” J. Biomed. Opt. 14(6), 060506 (2009).
[Crossref]

A. E. Desjardins, B. J. Vakoc, M. J. Suter, S. H. Yun, G. J. Tearney, and B. E. Bouma, “Real-time FPGA processing for high-speed optical frequency domain imaging,” IEEE Trans. Med. Imaging 28(9), 1468–1472 (2009).
[Crossref] [PubMed]

G. Liu, J. Zhang, L. Yu, T. Xie, and Z. Chen, “Real-time polarization-sensitive optical coherence tomography data processing with parallel computing,” Appl. Opt. 48(32), 6365–6370 (2009).
[Crossref] [PubMed]

M. Gargesha, M. W. Jenkins, D. L. Wilson, and A. M. Rollins, “High temporal resolution OCT using image-based retrospective gating,” Opt. Express 17(13), 10786–10799 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-13-10786 .
[Crossref] [PubMed]

I. Grulkowski, M. Gora, M. Szkulmowski, I. Gorczynska, D. Szlag, S. Marcos, A. Kowalczyk, and M. Wojtkowski, “Anterior segment imaging with Spectral OCT system using a high-speed CMOS camera,” Opt. Express 17(6), 4842–4858 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-6-4842 .
[Crossref] [PubMed]

M. Gora, K. Karnowski, M. Szkulmowski, B. J. Kaluzny, R. Huber, A. Kowalczyk, and M. Wojtkowski, “Ultra high-speed swept source OCT imaging of the anterior segment of human eye at 200 kHz with adjustable imaging range,” Opt. Express 17(17), 14880–14894 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-17-14880 .
[Crossref] [PubMed]

M. S. Jafri, R. Tang, and C. M. Tang, “Optical coherence tomography guided neurosurgical procedures in small rodents,” J. Neurosci. Methods 176(2), 85–95 (2009).
[Crossref]

K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation system for microsurgery guidance and intervention based on common-path optical coherence tomography,” IEEE Trans. Biomed. Eng. 56(9), 2318–2321 (2009).
[Crossref] [PubMed]

K. Wang, Z. Ding, T. Wu, C. Wang, J. Meng, M. Chen, and L. Xu, “Development of a non-uniform discrete Fourier transform based high speed spectral domain optical coherence tomography system,” Opt. Express 17(14), 12121–12131 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-12121 .
[Crossref] [PubMed]

2008 (5)

C. M. Eigenwillig, B. R. Biedermann, G. Palte, and R. Huber, “K-space linear Fourier domain mode locked laser and applications for optical coherence tomography,” Opt. Express 16(12), 8916–8937 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-8916 .
[Crossref] [PubMed]

M. Gargesha, M. W. Jenkins, A. M. Rollins, and D. L. Wilson, “Denoising and 4D visualization of OCT images,” Opt. Express 16(16), 12313–12333 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-16-12313 .
[Crossref] [PubMed]

B. Potsaid, I. Gorczynska, V. J. Srinivasan, Y. Chen, J. Jiang, A. Cable, and J. G. Fujimoto, “Ultrahigh speed spectral / Fourier domain OCT ophthalmic imaging at 70,000 to 312,500 axial scans per second,” Opt. Express 16(19), 15149–15169 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-16-19-15149 .
[Crossref] [PubMed]

T. E. Ustun, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Real-time processing for Fourier domain optical coherence tomography using a field programmable gate array,” Rev. Sci. Instrum. 79(11), 114301 (2008).
[Crossref] [PubMed]

S. Vergnole, G. Lamouche, and M. L. Dufour, “Artifact removal in Fourier-domain optical coherence tomography with a piezoelectric fiber stretcher,” Opt. Lett. 33(7), 732–734 (2008).
[Crossref] [PubMed]

2007 (4)

Z. Hu and A. M. Rollins, “Fourier domain optical coherence tomography with a linear-in-wavenumber spectrometer,” Opt. Lett. 32(24), 3525–3527 (2007).
[Crossref] [PubMed]

B. Baumann, M. Pircher, E. Götzinger, and C. K. Hitzenberger, “Full range complex spectral domain optical coherence tomography without additional phase shifters,” Opt. Express 15(20), 13375–13387 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-20-13375 .
[Crossref] [PubMed]

L. An and R. K. Wang, “Use of a scanner to modulate spatial interferograms for in vivo full-range Fourier-domain optical coherence tomography,” Opt. Lett. 32(23), 3423–3425 (2007).
[Crossref] [PubMed]

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1(12), 709–716 (2007).
[Crossref]

2006 (4)

Y. Yasuno, S. Makita, T. Endo, G. Aoki, M. Itoh, and T. Yatagai, “Simultaneous B-M-mode scanning method for real-time full-range Fourier domain optical coherence tomography,” Appl. Opt. 45(8), 1861–1865 (2006).
[Crossref] [PubMed]

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]

M. W. Jenkins, F. Rothenberg, D. Roy, V. P. Nikolski, Z. Hu, M. Watanabe, D. L. Wilson, I. R. Efimov, and A. M. Rollins, “4D embryonic cardiography using gated optical coherence tomography,” Opt. Express 14(2), 736–748 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-14-2-736 .
[Crossref] [PubMed]

A. F. Low, G. J. Tearney, B. E. Bouma, and I. K. Jang, “Technology Insight: optical coherence tomography--current status and future development,” Nat. Clin. Pract. Cardiovasc. Med. 3(3), 154–162, quiz 172 (2006).
[Crossref] [PubMed]

2005 (1)

U. Sharma, N. M. Fried, and J. U. Kang, “All-fiber common-path optical coherence tomography: sensitivity optimization and system analysis,” IEEE J. Sel. Top. Quantum Electron. 11(4), 799–805 (2005).
[Crossref]

2004 (1)

L. Greengard and J. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM Rev. 46(3), 443–454 (2004).
[Crossref]

2000 (1)

C. Dorrer, N. Belabas, J. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17(10), 1795–1802 (2000).
[Crossref]

Adler, D. C.

An, L.

Aoki, G.

Aoshima, K.

Baumann, B.

Belabas, N.

Biedermann, B. R.

Bouma, B. E.

Bradu, A.

Cable, A.

Chen, M.

Chen, Y.

Chen, Z.

Connolly, J.

Desjardins, A. E.

Ding, Z.

Dorrer, C.

Dufour, M. L.

Efimov, I. R.

Eigenwillig, C. M.

Endo, T.

Ferguson, R. D.

Fried, N. M.

Fujimoto, J. G.

Gargesha, M.

Gehlbach, P.

Gora, M.

Gorczynska, I.

Götzinger, E.

Greengard, L.

L. Greengard and J. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM Rev. 46(3), 443–454 (2004).
[Crossref]

Grulkowski, I.

Hammer, D. X.

Han, J.

Han, J.-H.

Hasegawa, H.

Hillmann, D.

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

Hitzenberger, C. K.

Hu, Z.

Z. Hu and A. M. Rollins, “Fourier domain optical coherence tomography with a linear-in-wavenumber spectrometer,” Opt. Lett. 32(24), 3525–3527 (2007).
[Crossref] [PubMed]

Huber, R.

Huttmann, G.

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

Hüttmann, G.

Iftimia, N. V.

Itagaki, T.

Y. Watanabe and T. Itagaki, “Real-time display on Fourier domain optical coherence tomography system using a graphics processing unit,” J. Biomed. Opt. 14(6), 060506 (2009).
[Crossref]

Itoh, M.

Jafri, M. S.

M. S. Jafri, R. Tang, and C. M. Tang, “Optical coherence tomography guided neurosurgical procedures in small rodents,” J. Neurosci. Methods 176(2), 85–95 (2009).
[Crossref]

Jang, I. K.

Jenkins, M. W.

Jiang, J.

Joffre, M.

Kaluzny, B. J.

Kang, J. U.

Karnowski, K.

Klein, T.

Koch, P.

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

Koseki, H.

Kowalczyk, A.

Lamouche, G.

Lankenau, E.

Lee, J.

L. Greengard and J. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM Rev. 46(3), 443–454 (2004).
[Crossref]

Lévesque, D.

Likforman, J.

Liu, G.

Liu, X.

Low, A. F.

Maeno, S.

Makita, S.

Marcos, S.

Meng, J.

Nikolski, V. P.

Oelckers, S.

Oh, W.-Y.

Palte, G.

Pircher, M.

Podoleanu, A. G.

Potsaid, B.

Probst, J.

Rollins, A. M.

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Supplementary Material (9)

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Fig. 1

System configuration: CMOS, CMOS line scan camera; L, spectrometer lens; G, grating; C1, C2, C3, achromatic collimators; C, 50:50 broadband fiber coupler; CL, camera link cable; COMP, host computer; GPU, graphics processing unit; PCIE, PCI Express x16 interface; MON, Monitor; GV, galvanometer (only the first galvanometer is illustrated for simplicity); SL, scanning lens; DCL, dispersion compensation lens; M, reference mirror; PC, polarization controller; SP, Sample.

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Fig. 2

Processing flowchart for GPU-NUDFT based FD-OCT: CL, Camera Link; FG, frame grabber; HM, host memory; GM, graphics global memory; DC, DC removal; MT, matrix transpose; FFT-x, Fast Fourier transform in x direction; IFFT-x, inverse Fast Fourier transform in x direction; BPF-x, band pass filter in x direction; Log, logarithmical scaling. The solid arrows describe the main data stream and the hollow arrows indicate the internal data flow of the GPU. The blue dashed arrows indicate the direction of inter-thread triggering. The hollow dashed arrow denotes standard FD-OCT without the Hilbert transform in x direction. Blue blocks: memory for pre-stored data; Yellow blocks: memory for real-timely refreshed data.

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Fig. 3

Convolution with a Gaussian interpolation kernel on a uniform grid when R = 2, M_sp = 2.

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Fig. 4

Processing flowchart for GPU-NUFFT based FD-OCT: CL, Camera Link; FG, frame grabber; HM, host memory; GM, graphics global memory; DC, DC removal; MT, matrix transpose; CON; convolution with Gaussian kernel; FFT-x, Fast Fourier transform in x direction; IFFT-x, inverse Fast Fourier transform in x direction; BPF-x, band pass filter in x direction; FFT-k_r, FFT in k_r direction; TRUC, truncation of redundant data in k_r direction; DECON, deconvolution with Gaussian kernel; Log, logarithmical scaling. The blue dashed arrows indicate the direction of inter-thread triggering. The solid arrows describe the main data stream and the hollow arrows indicate the internal data flow of the GPU. The hollow dashed arrow denotes standard FD-OCT without the Hilbert transform in x direction.

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Fig. 5

Benchmark line rate test of different FD-OCT processing method. (a) 1024-pixel FD-OCT; (b) 2048-pxiel FD-OCT; (c) 1024-pixel NUFFT-C with different frame size; (d) 2048-pixel NUFFT-C with different frame size; Both the peak internal processing line rate and the reduced line rate considering the data transfer bandwidth of PCIE x16 interface are listed.

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Fig. 6

Point spread function and sensitivity roll-off of different processing methods: (a) LIFFT; (b) CIFFT; (c) NUDFT; (d) NUFFT; (e) Comparison of PSF at certain image depth using different processing; (f) Comparison of sensitivity roll-off using different processing methods; (g) A-scan FWHM with depth; (h) Performance of NUFFT with different M_sp values.

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Fig. 7

Real-time image of multilayered phantom using different processing methods, where the bars represent 1mm in both dimensions for all images: (a) LIFFT (Media 1, 29.8 fps); (b) CIFFT (Media 2, 29.8 fps); (c) NUDFT (Media 3, 9.3 fps); (d) NUFFT (Media 4, 29.8 fps). All images are originally 4096 pixel (lateral) × 1024 pixel (axial) and rescaled to 1024 pixel (lateral) × 512 pixel (axial) for display on the monitor. (e)~(h): Magnified view corresponding to the blue-boxed area in (a)~(d). ZDL: zero delay line. The red arrows in (a) and (b) indicate the ghost image due to the presence of side-lobes of the reflective surface at a large image depth relative to ZDL. The red lines correspond to the A-scans extracted from the same lateral position of each image, shown collectively in (i). The side-lobes of LIFFT/CIFFT are indicated by the blue arrow in (i).

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Fig. 8

Real-time C-FD-OCT images using GPU-NUFFT, where the bars represent 1mm in both dimensions for all images: (a) (Media 5) Finger tip, (coronal). (b) (Media 6) Finger palm (coronal). (c)~(d) (Media 7) Finger nail fold (coronal); (e)~(f) (Media 8, Media 9) Finger nail (sagittal). SD, sweat duct; SC, stratum corneum; SS, stratum spinosum; NP, nail plate; NB, nail bed; NR, nail root; E, epidermis; D, dermis.

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Equations (16)

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A [z_{m}] = \sum_{i = 0}^{N - 1} I [k_{i}] \exp [- j \frac{2 π}{Δ k} (k_{i} - k_{0}) * m], m=0,1,2, ...,N-1,

A [z_{m}] = \sum_{i = 0}^{N - 1} I [k_{i}] \exp [- j \frac{2 π}{Δ k} (k_{i} - k_{0}) * m], m=0,1,2, ...,N/2-1
                     .

A [z_{m}] = \sum_{i = 0}^{N - 1} I [k_{i}] \exp [- j \frac{2 π}{Δ k} (k_{i} - k_{0}) * (m - \frac{N}{2})], m=0,1,2, ..., N-1
                        .

A_{half} = D_{half} I_{real}, (Standard half-range FD-OCT),

A_{half} = [\begin{matrix} A_{0} [z_{0}] & A_{1} [z_{0}] & \dots & A_{M-2} [z_{0}] & A_{M- 1} [z_{0}] \\ A_{0} [z_{1}] & A_{1} [z_{1}] & \dots & A_{M-2} [z_{1}] & A_{M- 1} [z_{1}] \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ A_{0} [z_{N / 2 - 2}] & A_{1} [z_{N / 2 - 2}] & \dots & A_{M-2} [z_{N / 2 - 2}] & A_{M- 1} [z_{N / 2 - 2}] \\ A_{0} [z_{N / 2 - 1}] & A_{1} [z_{N / 2 - 1}] & \dots & A_{M-2} [z_{N / 2 - 1}] & A_{M- 1} [z_{N / 2 - 1}] \end{matrix}],

I_{real} = [\begin{matrix} I_{0} [z_{0}] & I_{1} [z_{0}] & \dots & I_{M - 2} [z_{0}] & I_{M - 1} [z_{0}] \\ I_{0} [z_{1}] & I_{1} [z_{1}] & \dots & I_{M - 2} [z_{1}] & I_{M - 1} [z_{1}] \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ I_{0} [z_{N - 2}] & I_{1} [z_{N - 2}] & \dots & I_{M - 2} [z_{N - 2}] & I_{M - 1} [z_{N - 2}] \\ I_{0} [z_{N - 1}] & I_{1} [z_{N - 1}] & \dots & I_{M - 2} [z_{N - 1}] & I_{M - 1} [z_{N - 1}] \end{matrix}],

D_{half} = [\begin{matrix} 1 & 1 & \dots & 1 & 1 \\ p_{0}^{- 1} & p_{1}^{- 1} & \dots & p_{N - 2}^{- 1} & p_{N - 1}^{- 1} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ p_{0}^{- (N / 2 - 2)} & p_{1}^{- (N / 2 - 2)} & \dots & p_{N - 2}^{- (N / 2 - 2)} & p_{N - 1}^{- (N / 2 - 2)} \\ p_{0}^{- (N / 2 - 1)} & p_{1}^{- (N / 2 - 1)} & \dots & p_{N - 2}^{- (N / 2 - 1)} & p_{N - 1}^{- (N / 2 - 1)} \end{matrix}],

A_{full} = D_{full} I_{complex}, (Standard half-range FD-OCT),

A_{full} = [\begin{matrix} A_{0} [z_{0}] & A_{1} [z_{0}] & \dots & A_{M - 2} [z_{0}] & A_{M - 1} [z_{0}] \\ A_{0} [z_{1}] & A_{1} [z_{1}] & \dots & A_{M - 2} [z_{1}] & A_{M - 1} [z_{1}] \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ A_{0} [z_{N - 2}] & A_{1} [z_{N - 2}] & \dots & A_{M - 2} [z_{N - 2}] & A_{M - 1} [z_{N - 2}] \\ A_{0} [z_{N - 1}] & A_{1} [z_{N - 1}] & \dots & A_{M - 2} [z_{N - 1}] & A_{M - 1} [z_{N - 1}] \end{matrix}],

I_{complex} = [\begin{matrix} I_{0} [z_{0}] & I_{1} [z_{0}] & \dots & I_{M - 2} [z_{0}] & I_{M - 1} [z_{0}] \\ I_{0} [z_{1}] & I_{1} [z_{1}] & \dots & I_{M - 2} [z_{1}] & I_{M - 1} [z_{1}] \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ I_{0} [z_{N - 2}] & I_{1} [z_{N - 2}] & \dots & I_{M - 2} [z_{N - 2}] & I_{M - 1} [z_{N - 2}] \\ I_{0} [z_{N - 1}] & I_{1} [z_{N - 1}] & \dots & I_{M - 2} [z_{N - 1}] & I_{M - 1} [z_{N - 1}] \end{matrix}],

D_{full} = [\begin{matrix} p_{0}^{+ (N / 2)} & p_{1}^{+ (N / 2)} & \dots & p_{N - 2}^{+ (N / 2)} & p_{N - 1}^{+ (N / 2)} \\ p_{0}^{+ (N / 2 - 1)} & p_{1}^{+ (N / 2 - 1)} & \dots & p_{N - 2}^{+ (N / 2 - 1)} & p_{N - 2}^{+ (N / 2 - 1)} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ p_{0}^{- (N / 2 - 2)} & p_{1}^{- (N / 2 - 2)} & \dots & p_{N - 2}^{- (N / 2 - 2)} & p_{N - 1}^{- (N / 2 - 2)} \\ p_{0}^{- (N / 2 - 1)} & p_{1}^{- (N / 2 - 1)} & \dots & p_{N - 2}^{- (N / 2 - 1)} & p_{N - 1}^{- (N / 2 - 1)} \end{matrix}],

\begin{matrix} F_{x \to u} [I (k, x)] = {| E_{r} (k) |}^{2} δ (u) \\ + Γ_{u} {F_{x \to u} [E_{s} (k, x)]} \\ + F_{x \to u} [E_{r}^{*} (k, x) E_{r} (k)] \otimes δ (u + β) \\ + F_{x \to u} [E_{s} (k, x) E_{r}^{*} (k)] \otimes δ (u - β), \end{matrix}

I_{τ} [u] = \sum_{i} I [i] g_{τ} [k_{τ} [u] - k [i]], u=0,1,2,…,Mr-1,

g_{τ} [k] = \exp [\frac{- k^{2}}{4 τ}],

τ = \frac{1}{N^{2}} \frac{π}{R (R - 0.5)} M_{s p},

G_{τ} [n] = \exp [n^{2} τ],