Abstract

A novel technique for lateral resolution improvement in optical coherence tomography (OCT) is presented. The proposed method is based on lateral oversampling of the image. The locations and weights of multiple high spatial resolution sub-volumes are calculated using a Capon estimator assuming each contributes a weighted portion to the detected signal. This technique is independent of the delivery optics and the depth of field. Experimental results demonstrate that it is possible to achieve ~4x lateral resolution improvement which can be diagnostically valuable, especially in cases where the delivery optics are constrained to low numerical aperture (NA).

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Improving lateral resolution and image quality of optical coherence tomography by the multi-frame superresolution technique for 3D tissue imaging

Kai Shen, Hui Lu, Sarfaraz Baig, and Michael R. Wang
Biomed. Opt. Express 8(11) 4887-4918 (2017)

Automatic estimation of point-spread-function for deconvoluting out-of-focus optical coherence tomographic images using information entropy-based approach

Guozhong Liu, Siavash Yousefi, Zhongwei Zhi, and Ruikang K. Wang
Opt. Express 19(19) 18135-18148 (2011)

Improving retinal image resolution with iterative weighted shift-and-add

Nizan Meitav and Erez N. Ribak
J. Opt. Soc. Am. A 28(7) 1395-1402 (2011)

References

  • View by:
  • |
  • |
  • |

  1. B. Hermann, E. J. Fernández, A. Unterhuber, H. Sattmann, A. F. Fercher, W. Drexler, P. M. Prieto, and P. Artal, “Adaptive-optics ultrahigh-resolution optical coherence tomography,” Opt. Lett. 29(18), 2142–2144 (2004).
    [Crossref] [PubMed]
  2. Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. 27(4), 243–245 (2002).
    [Crossref] [PubMed]
  3. Y. Wang, Y. Zhao, J. S. Nelson, Z. Chen, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography by broadband continuum generation from a photonic crystal fiber,” Opt. Lett. 28(3), 182–184 (2003).
    [Crossref] [PubMed]
  4. M. J. Cobb, X. Liu, and X. Li, “Continuous focus tracking for real-time optical coherence tomography,” Opt. Lett. 30(13), 1680–1682 (2005).
    [Crossref] [PubMed]
  5. T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A 23(5), 1027–1037 (2006).
    [Crossref] [PubMed]
  6. T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Inverse scattering for high-resolution interferometric microscopy,” Opt. Lett. 31(24), 3585–3587 (2006).
    [Crossref] [PubMed]
  7. T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
    [Crossref] [PubMed]
  8. L. Yu, B. Rao, J. Zhang, J. Su, Q. Wang, S. Guo, and Z. Chen, “Improved lateral resolution in optical coherence tomography by digital focusing using two-dimensional numerical diffraction method,” Opt. Express 15(12), 7634–7641 (2007).
    [Crossref] [PubMed]
  9. Y. Yasuno, J. Sugisaka, Y. Sando, Y. Nakamura, S. Makita, M. Itoh, and T. Yatagai, “Non-iterative numerical method for laterally superresolving Fourier domain optical coherence tomography,” Opt. Express 14(3), 1006–1020 (2006).
    [Crossref] [PubMed]
  10. T. S. Ralston, D. L. Marks, F. Kamalabadi, and S. A. Boppart, “Deconvolution methods for mitigation of transverse blurring in optical coherence tomography,” IEEE Trans. Image Process. 14(9), 1254–1264 (2005).
    [Crossref] [PubMed]
  11. Y. Liu, Y. Liang, G. Mu, and X. Zhu, “Deconvolution methods for image deblurring in optical coherence tomography,” J. Opt. Soc. Am. A 26(1), 72–77 (2009).
    [Crossref] [PubMed]
  12. G. Liu, S. Yousefi, Z. Zhi, and R. K. Wang, “Automatic estimation of point-spread-function for deconvoluting out-of-focus optical coherence tomographic images using information entropy-based approach,” Opt. Express 19(19), 18135–18148 (2011).
    [Crossref] [PubMed]
  13. E. Bousi and C. Pitris, “Lateral resolution improvement in Optical Coherence Tomography (OCT) images,” in 2012 IEEE 12th International Conference on Bioinformatics Bioengineering (BIBE), pp. 598–601 (2012).
    [Crossref]
  14. E. Bousi and C. Pitris, “Lateral resolution improvement in oversampled optical coherence tomography images assuming weighted oversampled multi-scatterer contributions,” Proc. SPIE 8213, 82132T (2012).
    [Crossref]
  15. T.-Y. Yu, G. Zhang, A. B. Chalamalasetti, R. J. Doviak, and D. Zrníc, “Resolution Enhancement Technique Using Range Oversampling,” J. Atmos. Ocean. Technol. 23(2), 228–240 (2006).
    [Crossref]
  16. G. Zhang, T.-Y. Yu, and R. J. Doviak, “Angular and range interferometry to refine weather radar resolution,” Radio Sci. 40(3), RS3013 (2005).
    [Crossref]
  17. J. Capon, “High-resolution frequency-wavenumber spectrum analysis,” Proc. IEEE 57(8), 1408–1418 (1969).
    [Crossref]
  18. R. T. Lacoss, “Data adaptive spectral analysis methods,” Geophysics 36(4), 661–675 (1971).
    [Crossref]
  19. P. J. Vaitkus, R. S. C. Cobbold, and K. W. Johnston, “A comparative study and assessment of Doppler ultrasound spectral estimation techniques. Part II: Methods and results,” Ultrasound Med. Biol. 14(8), 673–688 (1988).
    [Crossref] [PubMed]
  20. J. Li and P. Stoica, “An adaptive filtering approach to spectral estimation and SAR imaging,” Signal IEEE Trans. Sig. Proc. 44(6), 1469–1484 (1996).
    [Crossref]
  21. P. Stoica, A. Jakobsson, and J. Li, “Matched-Filter Bank Interpretation of Some Spectral Estimators,” Signal Process. 66(1), 45–59 (1998).
    [Crossref]
  22. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62(1), 55–59 (1972).
    [Crossref]

2012 (1)

E. Bousi and C. Pitris, “Lateral resolution improvement in oversampled optical coherence tomography images assuming weighted oversampled multi-scatterer contributions,” Proc. SPIE 8213, 82132T (2012).
[Crossref]

2011 (1)

2009 (1)

2007 (2)

2006 (4)

2005 (3)

G. Zhang, T.-Y. Yu, and R. J. Doviak, “Angular and range interferometry to refine weather radar resolution,” Radio Sci. 40(3), RS3013 (2005).
[Crossref]

T. S. Ralston, D. L. Marks, F. Kamalabadi, and S. A. Boppart, “Deconvolution methods for mitigation of transverse blurring in optical coherence tomography,” IEEE Trans. Image Process. 14(9), 1254–1264 (2005).
[Crossref] [PubMed]

M. J. Cobb, X. Liu, and X. Li, “Continuous focus tracking for real-time optical coherence tomography,” Opt. Lett. 30(13), 1680–1682 (2005).
[Crossref] [PubMed]

2004 (1)

2003 (1)

2002 (1)

1998 (1)

P. Stoica, A. Jakobsson, and J. Li, “Matched-Filter Bank Interpretation of Some Spectral Estimators,” Signal Process. 66(1), 45–59 (1998).
[Crossref]

1996 (1)

J. Li and P. Stoica, “An adaptive filtering approach to spectral estimation and SAR imaging,” Signal IEEE Trans. Sig. Proc. 44(6), 1469–1484 (1996).
[Crossref]

1988 (1)

P. J. Vaitkus, R. S. C. Cobbold, and K. W. Johnston, “A comparative study and assessment of Doppler ultrasound spectral estimation techniques. Part II: Methods and results,” Ultrasound Med. Biol. 14(8), 673–688 (1988).
[Crossref] [PubMed]

1972 (1)

1971 (1)

R. T. Lacoss, “Data adaptive spectral analysis methods,” Geophysics 36(4), 661–675 (1971).
[Crossref]

1969 (1)

J. Capon, “High-resolution frequency-wavenumber spectrum analysis,” Proc. IEEE 57(8), 1408–1418 (1969).
[Crossref]

Artal, P.

Boppart, S. A.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A 23(5), 1027–1037 (2006).
[Crossref] [PubMed]

T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Inverse scattering for high-resolution interferometric microscopy,” Opt. Lett. 31(24), 3585–3587 (2006).
[Crossref] [PubMed]

T. S. Ralston, D. L. Marks, F. Kamalabadi, and S. A. Boppart, “Deconvolution methods for mitigation of transverse blurring in optical coherence tomography,” IEEE Trans. Image Process. 14(9), 1254–1264 (2005).
[Crossref] [PubMed]

Bousi, E.

E. Bousi and C. Pitris, “Lateral resolution improvement in oversampled optical coherence tomography images assuming weighted oversampled multi-scatterer contributions,” Proc. SPIE 8213, 82132T (2012).
[Crossref]

Capon, J.

J. Capon, “High-resolution frequency-wavenumber spectrum analysis,” Proc. IEEE 57(8), 1408–1418 (1969).
[Crossref]

Carney, P. S.

Chalamalasetti, A. B.

T.-Y. Yu, G. Zhang, A. B. Chalamalasetti, R. J. Doviak, and D. Zrníc, “Resolution Enhancement Technique Using Range Oversampling,” J. Atmos. Ocean. Technol. 23(2), 228–240 (2006).
[Crossref]

Chen, Z.

Cobb, M. J.

Cobbold, R. S. C.

P. J. Vaitkus, R. S. C. Cobbold, and K. W. Johnston, “A comparative study and assessment of Doppler ultrasound spectral estimation techniques. Part II: Methods and results,” Ultrasound Med. Biol. 14(8), 673–688 (1988).
[Crossref] [PubMed]

Ding, Z.

Doviak, R. J.

T.-Y. Yu, G. Zhang, A. B. Chalamalasetti, R. J. Doviak, and D. Zrníc, “Resolution Enhancement Technique Using Range Oversampling,” J. Atmos. Ocean. Technol. 23(2), 228–240 (2006).
[Crossref]

G. Zhang, T.-Y. Yu, and R. J. Doviak, “Angular and range interferometry to refine weather radar resolution,” Radio Sci. 40(3), RS3013 (2005).
[Crossref]

Drexler, W.

Fercher, A. F.

Fernández, E. J.

Guo, S.

Hermann, B.

Itoh, M.

Jakobsson, A.

P. Stoica, A. Jakobsson, and J. Li, “Matched-Filter Bank Interpretation of Some Spectral Estimators,” Signal Process. 66(1), 45–59 (1998).
[Crossref]

Johnston, K. W.

P. J. Vaitkus, R. S. C. Cobbold, and K. W. Johnston, “A comparative study and assessment of Doppler ultrasound spectral estimation techniques. Part II: Methods and results,” Ultrasound Med. Biol. 14(8), 673–688 (1988).
[Crossref] [PubMed]

Kamalabadi, F.

T. S. Ralston, D. L. Marks, F. Kamalabadi, and S. A. Boppart, “Deconvolution methods for mitigation of transverse blurring in optical coherence tomography,” IEEE Trans. Image Process. 14(9), 1254–1264 (2005).
[Crossref] [PubMed]

Lacoss, R. T.

R. T. Lacoss, “Data adaptive spectral analysis methods,” Geophysics 36(4), 661–675 (1971).
[Crossref]

Li, J.

P. Stoica, A. Jakobsson, and J. Li, “Matched-Filter Bank Interpretation of Some Spectral Estimators,” Signal Process. 66(1), 45–59 (1998).
[Crossref]

J. Li and P. Stoica, “An adaptive filtering approach to spectral estimation and SAR imaging,” Signal IEEE Trans. Sig. Proc. 44(6), 1469–1484 (1996).
[Crossref]

Li, X.

Liang, Y.

Liu, G.

Liu, X.

Liu, Y.

Makita, S.

Marks, D. L.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A 23(5), 1027–1037 (2006).
[Crossref] [PubMed]

T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Inverse scattering for high-resolution interferometric microscopy,” Opt. Lett. 31(24), 3585–3587 (2006).
[Crossref] [PubMed]

T. S. Ralston, D. L. Marks, F. Kamalabadi, and S. A. Boppart, “Deconvolution methods for mitigation of transverse blurring in optical coherence tomography,” IEEE Trans. Image Process. 14(9), 1254–1264 (2005).
[Crossref] [PubMed]

Mu, G.

Nakamura, Y.

Nelson, J. S.

Pitris, C.

E. Bousi and C. Pitris, “Lateral resolution improvement in oversampled optical coherence tomography images assuming weighted oversampled multi-scatterer contributions,” Proc. SPIE 8213, 82132T (2012).
[Crossref]

Prieto, P. M.

Ralston, T. S.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Inverse scattering for high-resolution interferometric microscopy,” Opt. Lett. 31(24), 3585–3587 (2006).
[Crossref] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A 23(5), 1027–1037 (2006).
[Crossref] [PubMed]

T. S. Ralston, D. L. Marks, F. Kamalabadi, and S. A. Boppart, “Deconvolution methods for mitigation of transverse blurring in optical coherence tomography,” IEEE Trans. Image Process. 14(9), 1254–1264 (2005).
[Crossref] [PubMed]

Rao, B.

Ren, H.

Richardson, W. H.

Sando, Y.

Sattmann, H.

Stoica, P.

P. Stoica, A. Jakobsson, and J. Li, “Matched-Filter Bank Interpretation of Some Spectral Estimators,” Signal Process. 66(1), 45–59 (1998).
[Crossref]

J. Li and P. Stoica, “An adaptive filtering approach to spectral estimation and SAR imaging,” Signal IEEE Trans. Sig. Proc. 44(6), 1469–1484 (1996).
[Crossref]

Su, J.

Sugisaka, J.

Unterhuber, A.

Vaitkus, P. J.

P. J. Vaitkus, R. S. C. Cobbold, and K. W. Johnston, “A comparative study and assessment of Doppler ultrasound spectral estimation techniques. Part II: Methods and results,” Ultrasound Med. Biol. 14(8), 673–688 (1988).
[Crossref] [PubMed]

Wang, Q.

Wang, R. K.

Wang, Y.

Windeler, R. S.

Yasuno, Y.

Yatagai, T.

Yousefi, S.

Yu, L.

Yu, T.-Y.

T.-Y. Yu, G. Zhang, A. B. Chalamalasetti, R. J. Doviak, and D. Zrníc, “Resolution Enhancement Technique Using Range Oversampling,” J. Atmos. Ocean. Technol. 23(2), 228–240 (2006).
[Crossref]

G. Zhang, T.-Y. Yu, and R. J. Doviak, “Angular and range interferometry to refine weather radar resolution,” Radio Sci. 40(3), RS3013 (2005).
[Crossref]

Zhang, G.

T.-Y. Yu, G. Zhang, A. B. Chalamalasetti, R. J. Doviak, and D. Zrníc, “Resolution Enhancement Technique Using Range Oversampling,” J. Atmos. Ocean. Technol. 23(2), 228–240 (2006).
[Crossref]

G. Zhang, T.-Y. Yu, and R. J. Doviak, “Angular and range interferometry to refine weather radar resolution,” Radio Sci. 40(3), RS3013 (2005).
[Crossref]

Zhang, J.

Zhao, Y.

Zhi, Z.

Zhu, X.

Zrníc, D.

T.-Y. Yu, G. Zhang, A. B. Chalamalasetti, R. J. Doviak, and D. Zrníc, “Resolution Enhancement Technique Using Range Oversampling,” J. Atmos. Ocean. Technol. 23(2), 228–240 (2006).
[Crossref]

Geophysics (1)

R. T. Lacoss, “Data adaptive spectral analysis methods,” Geophysics 36(4), 661–675 (1971).
[Crossref]

IEEE Trans. Image Process. (1)

T. S. Ralston, D. L. Marks, F. Kamalabadi, and S. A. Boppart, “Deconvolution methods for mitigation of transverse blurring in optical coherence tomography,” IEEE Trans. Image Process. 14(9), 1254–1264 (2005).
[Crossref] [PubMed]

J. Atmos. Ocean. Technol. (1)

T.-Y. Yu, G. Zhang, A. B. Chalamalasetti, R. J. Doviak, and D. Zrníc, “Resolution Enhancement Technique Using Range Oversampling,” J. Atmos. Ocean. Technol. 23(2), 228–240 (2006).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nat. Phys. (1)

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Lett. (5)

Proc. IEEE (1)

J. Capon, “High-resolution frequency-wavenumber spectrum analysis,” Proc. IEEE 57(8), 1408–1418 (1969).
[Crossref]

Proc. SPIE (1)

E. Bousi and C. Pitris, “Lateral resolution improvement in oversampled optical coherence tomography images assuming weighted oversampled multi-scatterer contributions,” Proc. SPIE 8213, 82132T (2012).
[Crossref]

Radio Sci. (1)

G. Zhang, T.-Y. Yu, and R. J. Doviak, “Angular and range interferometry to refine weather radar resolution,” Radio Sci. 40(3), RS3013 (2005).
[Crossref]

Signal IEEE Trans. Sig. Proc. (1)

J. Li and P. Stoica, “An adaptive filtering approach to spectral estimation and SAR imaging,” Signal IEEE Trans. Sig. Proc. 44(6), 1469–1484 (1996).
[Crossref]

Signal Process. (1)

P. Stoica, A. Jakobsson, and J. Li, “Matched-Filter Bank Interpretation of Some Spectral Estimators,” Signal Process. 66(1), 45–59 (1998).
[Crossref]

Ultrasound Med. Biol. (1)

P. J. Vaitkus, R. S. C. Cobbold, and K. W. Johnston, “A comparative study and assessment of Doppler ultrasound spectral estimation techniques. Part II: Methods and results,” Ultrasound Med. Biol. 14(8), 673–688 (1988).
[Crossref] [PubMed]

Other (1)

E. Bousi and C. Pitris, “Lateral resolution improvement in Optical Coherence Tomography (OCT) images,” in 2012 IEEE 12th International Conference on Bioinformatics Bioengineering (BIBE), pp. 598–601 (2012).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1 A schematic diagram of L = 3 lateral oversampling. Each oversampled signal Vi consists of independent signals from L subvolumes (Si to SL + i-1) with a 2/3 overlap between volumes which share signals Si to SL + i-2 with the previous volume [15]. The red rectangle indicated the shared volume which can be isolated from appropriately combining the oversampled signals.
Fig. 2
Fig. 2 Schematic illustrating graphically the process of lateral resolution enhancement using oversampling and Capon’s method to reconstruct the high spatial resolution signal. In this example, five high resolution volumes (S1-S5) are reconstructed from three oversampled signals (V1-V3) using the weights (w11-w35) estimated using Capon’s approach.
Fig. 3
Fig. 3 (A) OCT image of microspheres embedded in acrylamide gel. (B) The PSFs estimated at 3 different depths below the surface of (A) using blind deconvolution. The red arrow indicates the focal plane (Objective: f = 30 mm, NA = 0.075, b = 2zR = 0.6 mm)
Fig. 4
Fig. 4 (A) Standard OCT image. (B) The OCT image deconvoluted with the estimated PSF. (C) Inverse solution image without filtering. (D) Inverse solution image with filtering. The scale bar is 200 μm. (E) Single line at the position indicated by the arrow (Red: Standard OCT. Green: Deconvoluted OCT. Blue: Inverse solution).
Fig. 5
Fig. 5 An example of the Capon weights of single sub-volume calculated for a 56-pixel length around the sub-volume of interest (green circle) and normalized to unit total power. Note how the values of some adjacent sub-volumes are suppressed (red circles).
Fig. 6
Fig. 6 (A) The Standard OCT image. (B) The OCT image deconvoluted with the estimated PSF. (C) The OCT image after applying the Capon weight method. The scale bar is 200 μm. (D) Single line at the position indicated by the arrow (Red: Standard OCT. Green: Deconvoluted OCT. Blue: Capon weight OCT).
Fig. 7
Fig. 7 (A) The Standard OCT image. (B) The OCT image deconvoluted with the estimated PSF. (C) The OCT image after applying the Capon weight method. The scale bar is 200 μm. (D) Single line at the position indicated by the arrow (Red: Standard OCT. Green: Deconvoluted OCT. Blue: Capon weight OCT).
Fig. 8
Fig. 8 (A) The Standard OCT image. (B) The OCT image deconvoluted with the estimated PSF. (C) The OCT image after applying the Capon weight method. The scale bar is 200 μm. (D) Single line at the position indicated by the arrow (Red: Standard OCT. Green: Deconvoluted OCT. Blue: Capon weight OCT).
Fig. 9
Fig. 9 Human gastrointestinal tissue imaged during surgical excision. Normal colon: (A) Standard OCT image, (B) the same image after applying the Capon-based method, (C) H&E stained histology image. Invasive adenocarcinoma of the colon: (D) Standard OCT image, (E) the same image after applying the Capon-based method, (F) H&E stained histology image. The insets zoom on a portion of glands and crypts where the resolution improvement is evident.

Tables (4)

Tables Icon

Table 1 Measurements of PSFs for Standard, Deconvoluted and Inverse Solution OCT images, as well as SNR for Standard, Deconvoluted, and Inverse Solution OCT images.

Tables Icon

Table 2 Measurements of PSFs for Standard OCT, Deconvoluted OCT and Weighting method, as well as SNR for Standard OCT, Deconvoluted OCT, and Weighting method.

Tables Icon

Table 3 Measurements of PSFs for Standard OCT, Deconvoluted OCT and Weighting method, as well as SNR for Standard OCT, Deconvoluted OCT, and Weighting method.

Tables Icon

Table 4 Measurements of PSFs f r Standard OCT, Deconvoluted OCT and Weighting method, as well as SNR for Standard OCT, Deconvoluted OCT, and Weighting method.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

V(t)=AS(t)
V(t)= [ V 1 (t) V 2 (t) ... V L (t) ] T
S(t)= [ S 1 (t) S 2 (t) ... S 2L1 (t) ] T
A(t)=[ a 1 a 2 ... a 2L1 ]
[ V 1 (t) V 2 (t) V 3 (t) ]=[ 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 ] [ S 1 (t) S 2 (t) S 3 (t) S 4 (t) S 5 (t) ] T
[ V 1 (t) V 2 (t) V 3 (t) V L (t) ]=[ a 1 ... a L 0 0 0 0 0 0 a 1 ... a L 0 0 0 0 0 0 a 1 ... a L 0 0 0 0 0 0 0 0 a 1 ... a L ] [ S 1 (t) S 2 (t) S 2 (t) ... S 2L1 (t) ] T
S(t)= A 1 V(t)
g H =[ g 0 g 1 g M ]
y(n)= m=0 M g m x(nm)=[ g 0 g 1 g M ] [ x n x n1 x nM ] H
R=E{x(n) x H (n)}
E{ | y(n) | 2 }= g H Rg
H(ω)= m0 M g m e iωm = g H a(ω)
a(ω)= [ 1 e iω ... e iMω ] T
g=arg min g g H Rg
g H a(ω)=1
F= g H Rg+μ[1 g H a(ω)]
g= R 1 a(ω) a (ω) H R 1 a(ω)
V(t)=AS(t)+N
W=[ w 1 w 2 ... w 2L1 ]
S est =[ S est(1) S est(2) ... S est(2L1) ]
S est(i) = w i V= w i AS+ w i N= w i a 1 S 1 +.+ w i a i S i + + w i a 2L-1 S 2L-1 + i=1 2L-1 w i N i
Min( P )=min( w i H R v w i ) subject to w i a i =1
w i = R V 1 a i a i H R V 1 a i
R V =E{V V H }
R Vij = V i V j H
Ο n+1 (x,y)=[ I (P O n ) * P ¯ ](x,y) O n (x,y)
P n+1 (x,y)=[ I ( P n O n ) * O n ¯ ](x,y) P n (x,y)
SNR=20log( I max / σ b )

Metrics