Abstract

Optical coherence tomography of luminal structures, such as for intravascular or gastrointestinal imaging, is performed by using a fiber-optic catheter as a beam-delivery probe. The interrogating beam is scanned angularly by rotating the fiber around a fixed central axis. Because the beam is focused only at a fixed distance from the center of the fiber, only scatterers near this distance are resolved. We present a solution of the inverse scattering problem that provides an estimate of the susceptibility of the sample for an angularly scanned Gaussian beam focused at a fixed distance from the origin. This solution provides quantitatively meaningful reconstructions while also extending the volume of the sample that is resolvable by the instrument.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]

2006 (1)

2005 (2)

B. D. MacNeill, B. E. Bouma, H. Yabushita, I. K. Jang, and G. J. Tearney, "Intravascular optical coherence tomography: cellular imaging," J. Nucl. Cardiol. 12, 460-465 (2005).
[CrossRef] [PubMed]

I. K. Jang, G. J. Tearney, B. MacNeill, M. Takano, F. Moselewski, N. Iftima, M. Shishkov, S. Houser, H. T. Aretz, E. F. Halpern, and B. E. Bouma, "In vivo characterization of coronary atherosclerotic plaque by use of optical coherence tomography," Circulation 111, 1551-1555 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (3)

V. X. D. Yang, M. L. Gordon, S.-J. Tang, N. E. Marcon, G. Gardiner, B. Qi, B. Stuart, E. Seng-Yue, S. Lo, J. Pekar, B. C. Wilson, and V. I. Vitkin, "High speed, wide velocity dynamic range Doppler optical coherence tomography (Part iii): in vivo endoscopic imaging on blood flow in the rat and human gastrointestinal tracts," Opt. Express 11, 2416-2424 (2003).
[CrossRef] [PubMed]

B. E. Bouma, G. J. Tearney, H. Yabushita, M. Shishkov, C. R. Kauffman, D. DeJoseph Gauthier, B. D. MacNeill, S. L. Houser, H. T. Aretz, E. F. Halpern, and I. K. Jang, "Evaluation of intracoronary stenting by intravascular optical coherence tomography," Heart 89, 317-320 (2003).
[CrossRef] [PubMed]

L. M. H. Ulander, H. Hellsten, and G. Stenstrom, "Synthetic aperture radar processing using fast factorized back-projection," IEEE Trans. Aerosp. Electron. Syst. 39, 760-776 (2003).
[CrossRef]

2002 (3)

C. J. Nolan and M. Cheney, "Synthetic aperture inversion," Inverse Probl. 18, 221-235 (2002).
[CrossRef]

I. K. Jang, B. E. Bouma, D. H. Kang, S. J. Park, S. W. Park, K. B. Seung, K. B. Choi, M. Shishkov, K. Schlendorf, E. Pomerantsev, S. L. Houser, and H. T. Aretz, "Visualization of coronary atherosclerotic plaques in patients using optical coherence tomography: comparison with intravascular ultrasound," J. Am. Coll. Cardiol. 39, 604-609 (2002).
[CrossRef] [PubMed]

M. A. Haun, D. L. Jones, and W. D. O'Brien, Jr., "Efficient three-dimensional imaging from a small cylindrical aperture," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49, 861-870 (2002).
[CrossRef] [PubMed]

2001 (2)

D. Vray, C. Haas, T. Rastello, M. Krueger, E. Brusseau, K. Schroeder, G. Gimenez, and H. Ermert, "Synthetic aperture-based beam compression for intravascular ultrasound imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 189-201 (2001).
[CrossRef] [PubMed]

X. Li, T. H. Ko, and J. G. Fujimoto, "Intraluminal fiber-optic Doppler imaging catheter for structural and functional optical coherence tomography," Opt. Lett. 26, 1906-1908 (2001).
[CrossRef]

2000 (1)

S. Basu and Y. Bresler, "O(N2 log2N) filtered backprojection reconstruction algorithm for tomography," IEEE Trans. Image Process. 9, 1760-1773 (2000).
[CrossRef]

1999 (2)

1998 (2)

1997 (1)

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, and J. F. Southern, "In vivo endoscopic optical biopsy with optical coherence tomography," Science 276, 2037-2039 (1997).
[CrossRef] [PubMed]

1996 (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. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

1987 (2)

P. Edholm and G. T. Herman, "Linograms in image reconstruction from projections," IEEE Trans. Med. Imaging MI-6, 301-307 (1987).
[CrossRef]

H. Hellsten and L. E. Andersson, "An inverse method for the processing of synthetic aperture radar data," Inverse Probl. 3, 111-124 (1987).
[CrossRef]

1986 (1)

K. J. Langenberg, M. Berger, T. Kreutter, K. Mayer, and V. Schmitz, "Synthetic aperture focusing technique signal processing," NDT Int. 19, 177-189 (1986).
[CrossRef]

1985 (1)

J. A. Fawcett, "Inversion of N-dimensional spherical averages," SIAM J. Appl. Math. 45, 336-341 (1985).
[CrossRef]

1981 (1)

S. J. Norton and M. Linzer, "Ultrasonic reflectivity imaging in three dimensions: exact inverse scattering solutions for plane, cylindrical, and spherical apertures," IEEE Trans. Biomed. Eng. BME-28, 202-220 (1981).
[CrossRef]

Aguirre, A. D.

Andersson, L. E.

H. Hellsten and L. E. Andersson, "An inverse method for the processing of synthetic aperture radar data," Inverse Probl. 3, 111-124 (1987).
[CrossRef]

Aretz, H. T.

I. K. Jang, G. J. Tearney, B. MacNeill, M. Takano, F. Moselewski, N. Iftima, M. Shishkov, S. Houser, H. T. Aretz, E. F. Halpern, and B. E. Bouma, "In vivo characterization of coronary atherosclerotic plaque by use of optical coherence tomography," Circulation 111, 1551-1555 (2005).
[CrossRef] [PubMed]

B. E. Bouma, G. J. Tearney, H. Yabushita, M. Shishkov, C. R. Kauffman, D. DeJoseph Gauthier, B. D. MacNeill, S. L. Houser, H. T. Aretz, E. F. Halpern, and I. K. Jang, "Evaluation of intracoronary stenting by intravascular optical coherence tomography," Heart 89, 317-320 (2003).
[CrossRef] [PubMed]

I. K. Jang, B. E. Bouma, D. H. Kang, S. J. Park, S. W. Park, K. B. Seung, K. B. Choi, M. Shishkov, K. Schlendorf, E. Pomerantsev, S. L. Houser, and H. T. Aretz, "Visualization of coronary atherosclerotic plaques in patients using optical coherence tomography: comparison with intravascular ultrasound," J. Am. Coll. Cardiol. 39, 604-609 (2002).
[CrossRef] [PubMed]

Basu, S.

S. Basu and Y. Bresler, "O(N2 log2N) filtered backprojection reconstruction algorithm for tomography," IEEE Trans. Image Process. 9, 1760-1773 (2000).
[CrossRef]

Berger, M.

K. J. Langenberg, M. Berger, T. Kreutter, K. Mayer, and V. Schmitz, "Synthetic aperture focusing technique signal processing," NDT Int. 19, 177-189 (1986).
[CrossRef]

Boppart, S. A.

Bouma, B. E.

B. D. MacNeill, B. E. Bouma, H. Yabushita, I. K. Jang, and G. J. Tearney, "Intravascular optical coherence tomography: cellular imaging," J. Nucl. Cardiol. 12, 460-465 (2005).
[CrossRef] [PubMed]

I. K. Jang, G. J. Tearney, B. MacNeill, M. Takano, F. Moselewski, N. Iftima, M. Shishkov, S. Houser, H. T. Aretz, E. F. Halpern, and B. E. Bouma, "In vivo characterization of coronary atherosclerotic plaque by use of optical coherence tomography," Circulation 111, 1551-1555 (2005).
[CrossRef] [PubMed]

B. E. Bouma, G. J. Tearney, H. Yabushita, M. Shishkov, C. R. Kauffman, D. DeJoseph Gauthier, B. D. MacNeill, S. L. Houser, H. T. Aretz, E. F. Halpern, and I. K. Jang, "Evaluation of intracoronary stenting by intravascular optical coherence tomography," Heart 89, 317-320 (2003).
[CrossRef] [PubMed]

I. K. Jang, B. E. Bouma, D. H. Kang, S. J. Park, S. W. Park, K. B. Seung, K. B. Choi, M. Shishkov, K. Schlendorf, E. Pomerantsev, S. L. Houser, and H. T. Aretz, "Visualization of coronary atherosclerotic plaques in patients using optical coherence tomography: comparison with intravascular ultrasound," J. Am. Coll. Cardiol. 39, 604-609 (2002).
[CrossRef] [PubMed]

B. E. Bouma and G. J. Tearney, "Power-efficient nonreciprocal interferometer and linear-scanning fiber-optic catheter for optical coherence tomography," Opt. Lett. 24, 531-533 (1999).
[CrossRef]

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, and J. F. Southern, "In vivo endoscopic optical biopsy with optical coherence tomography," Science 276, 2037-2039 (1997).
[CrossRef] [PubMed]

B. E. Bouma and G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, 2001).

Bresler, Y.

S. Basu and Y. Bresler, "O(N2 log2N) filtered backprojection reconstruction algorithm for tomography," IEEE Trans. Image Process. 9, 1760-1773 (2000).
[CrossRef]

Brezinski, M. E.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, and J. F. Southern, "In vivo endoscopic optical biopsy with optical coherence tomography," Science 276, 2037-2039 (1997).
[CrossRef] [PubMed]

G. J. Tearney, S. A. Boppart, B. E. Bunma, M. E. Brezinski, N. J. Weissman, J. F. Southern, and J. G. Fujimoto, "Scanning single-mode fiber-optic catheter endoscope for optical coherence tomography," Opt. Lett. 21, 543-545 (1996).
[CrossRef] [PubMed]

Brusseau, E.

D. Vray, C. Haas, T. Rastello, M. Krueger, E. Brusseau, K. Schroeder, G. Gimenez, and H. Ermert, "Synthetic aperture-based beam compression for intravascular ultrasound imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 189-201 (2001).
[CrossRef] [PubMed]

Bunma, B. E.

Carney, P. S.

Chak, A.

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. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Chen, Y.

Cheney, M.

C. J. Nolan and M. Cheney, "Synthetic aperture inversion," Inverse Probl. 18, 221-235 (2002).
[CrossRef]

Choi, K. B.

I. K. Jang, B. E. Bouma, D. H. Kang, S. J. Park, S. W. Park, K. B. Seung, K. B. Choi, M. Shishkov, K. Schlendorf, E. Pomerantsev, S. L. Houser, and H. T. Aretz, "Visualization of coronary atherosclerotic plaques in patients using optical coherence tomography: comparison with intravascular ultrasound," J. Am. Coll. Cardiol. 39, 604-609 (2002).
[CrossRef] [PubMed]

Chumakov, Y. P.

DeJoseph Gauthier, D.

B. E. Bouma, G. J. Tearney, H. Yabushita, M. Shishkov, C. R. Kauffman, D. DeJoseph Gauthier, B. D. MacNeill, S. L. Houser, H. T. Aretz, E. F. Halpern, and I. K. Jang, "Evaluation of intracoronary stenting by intravascular optical coherence tomography," Heart 89, 317-320 (2003).
[CrossRef] [PubMed]

Edholm, P.

P. Edholm and G. T. Herman, "Linograms in image reconstruction from projections," IEEE Trans. Med. Imaging MI-6, 301-307 (1987).
[CrossRef]

Ermert, H.

D. Vray, C. Haas, T. Rastello, M. Krueger, E. Brusseau, K. Schroeder, G. Gimenez, and H. Ermert, "Synthetic aperture-based beam compression for intravascular ultrasound imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 189-201 (2001).
[CrossRef] [PubMed]

Fawcett, J. A.

J. A. Fawcett, "Inversion of N-dimensional spherical averages," SIAM J. Appl. Math. 45, 336-341 (1985).
[CrossRef]

Feldchtein, F. I.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Frazier, C. H.

C. H. Frazier and W. D. O'Brien, Jr., "Synthetic aperture techniques with a virtual source element," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 196-207 (l998).
[CrossRef]

Fujimoto, J. G.

Gardiner, G.

Gelikonov, G. V.

Gelikonov, V. M.

Gimenez, G.

D. Vray, C. Haas, T. Rastello, M. Krueger, E. Brusseau, K. Schroeder, G. Gimenez, and H. Ermert, "Synthetic aperture-based beam compression for intravascular ultrasound imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 189-201 (2001).
[CrossRef] [PubMed]

Gladkova, N. D.

Goodnow, J.

Gordon, M. L.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Haas, C.

D. Vray, C. Haas, T. Rastello, M. Krueger, E. Brusseau, K. Schroeder, G. Gimenez, and H. Ermert, "Synthetic aperture-based beam compression for intravascular ultrasound imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 189-201 (2001).
[CrossRef] [PubMed]

Halpern, E. F.

I. K. Jang, G. J. Tearney, B. MacNeill, M. Takano, F. Moselewski, N. Iftima, M. Shishkov, S. Houser, H. T. Aretz, E. F. Halpern, and B. E. Bouma, "In vivo characterization of coronary atherosclerotic plaque by use of optical coherence tomography," Circulation 111, 1551-1555 (2005).
[CrossRef] [PubMed]

B. E. Bouma, G. J. Tearney, H. Yabushita, M. Shishkov, C. R. Kauffman, D. DeJoseph Gauthier, B. D. MacNeill, S. L. Houser, H. T. Aretz, E. F. Halpern, and I. K. Jang, "Evaluation of intracoronary stenting by intravascular optical coherence tomography," Heart 89, 317-320 (2003).
[CrossRef] [PubMed]

Haun, M. A.

M. A. Haun, D. L. Jones, and W. D. O'Brien, Jr., "Efficient three-dimensional imaging from a small cylindrical aperture," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49, 861-870 (2002).
[CrossRef] [PubMed]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Hellsten, H.

L. M. H. Ulander, H. Hellsten, and G. Stenstrom, "Synthetic aperture radar processing using fast factorized back-projection," IEEE Trans. Aerosp. Electron. Syst. 39, 760-776 (2003).
[CrossRef]

H. Hellsten and L. E. Andersson, "An inverse method for the processing of synthetic aperture radar data," Inverse Probl. 3, 111-124 (1987).
[CrossRef]

Herman, G. T.

P. Edholm and G. T. Herman, "Linograms in image reconstruction from projections," IEEE Trans. Med. Imaging MI-6, 301-307 (1987).
[CrossRef]

Herz, P. R.

Houser, S.

I. K. Jang, G. J. Tearney, B. MacNeill, M. Takano, F. Moselewski, N. Iftima, M. Shishkov, S. Houser, H. T. Aretz, E. F. Halpern, and B. E. Bouma, "In vivo characterization of coronary atherosclerotic plaque by use of optical coherence tomography," Circulation 111, 1551-1555 (2005).
[CrossRef] [PubMed]

Houser, S. L.

B. E. Bouma, G. J. Tearney, H. Yabushita, M. Shishkov, C. R. Kauffman, D. DeJoseph Gauthier, B. D. MacNeill, S. L. Houser, H. T. Aretz, E. F. Halpern, and I. K. Jang, "Evaluation of intracoronary stenting by intravascular optical coherence tomography," Heart 89, 317-320 (2003).
[CrossRef] [PubMed]

I. K. Jang, B. E. Bouma, D. H. Kang, S. J. Park, S. W. Park, K. B. Seung, K. B. Choi, M. Shishkov, K. Schlendorf, E. Pomerantsev, S. L. Houser, and H. T. Aretz, "Visualization of coronary atherosclerotic plaques in patients using optical coherence tomography: comparison with intravascular ultrasound," J. Am. Coll. Cardiol. 39, 604-609 (2002).
[CrossRef] [PubMed]

Hsuing, P.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Iftima, N.

I. K. Jang, G. J. Tearney, B. MacNeill, M. Takano, F. Moselewski, N. Iftima, M. Shishkov, S. Houser, H. T. Aretz, E. F. Halpern, and B. E. Bouma, "In vivo characterization of coronary atherosclerotic plaque by use of optical coherence tomography," Circulation 111, 1551-1555 (2005).
[CrossRef] [PubMed]

Izatt, J. A.

Jang, I. K.

I. K. Jang, G. J. Tearney, B. MacNeill, M. Takano, F. Moselewski, N. Iftima, M. Shishkov, S. Houser, H. T. Aretz, E. F. Halpern, and B. E. Bouma, "In vivo characterization of coronary atherosclerotic plaque by use of optical coherence tomography," Circulation 111, 1551-1555 (2005).
[CrossRef] [PubMed]

B. D. MacNeill, B. E. Bouma, H. Yabushita, I. K. Jang, and G. J. Tearney, "Intravascular optical coherence tomography: cellular imaging," J. Nucl. Cardiol. 12, 460-465 (2005).
[CrossRef] [PubMed]

B. E. Bouma, G. J. Tearney, H. Yabushita, M. Shishkov, C. R. Kauffman, D. DeJoseph Gauthier, B. D. MacNeill, S. L. Houser, H. T. Aretz, E. F. Halpern, and I. K. Jang, "Evaluation of intracoronary stenting by intravascular optical coherence tomography," Heart 89, 317-320 (2003).
[CrossRef] [PubMed]

I. K. Jang, B. E. Bouma, D. H. Kang, S. J. Park, S. W. Park, K. B. Seung, K. B. Choi, M. Shishkov, K. Schlendorf, E. Pomerantsev, S. L. Houser, and H. T. Aretz, "Visualization of coronary atherosclerotic plaques in patients using optical coherence tomography: comparison with intravascular ultrasound," J. Am. Coll. Cardiol. 39, 604-609 (2002).
[CrossRef] [PubMed]

Jones, D. L.

M. A. Haun, D. L. Jones, and W. D. O'Brien, Jr., "Efficient three-dimensional imaging from a small cylindrical aperture," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49, 861-870 (2002).
[CrossRef] [PubMed]

Kang, D. H.

I. K. Jang, B. E. Bouma, D. H. Kang, S. J. Park, S. W. Park, K. B. Seung, K. B. Choi, M. Shishkov, K. Schlendorf, E. Pomerantsev, S. L. Houser, and H. T. Aretz, "Visualization of coronary atherosclerotic plaques in patients using optical coherence tomography: comparison with intravascular ultrasound," J. Am. Coll. Cardiol. 39, 604-609 (2002).
[CrossRef] [PubMed]

Kauffman, C. R.

B. E. Bouma, G. J. Tearney, H. Yabushita, M. Shishkov, C. R. Kauffman, D. DeJoseph Gauthier, B. D. MacNeill, S. L. Houser, H. T. Aretz, E. F. Halpern, and I. K. Jang, "Evaluation of intracoronary stenting by intravascular optical coherence tomography," Heart 89, 317-320 (2003).
[CrossRef] [PubMed]

Ko, T. H.

Kobayashi, K.

Kreutter, T.

K. J. Langenberg, M. Berger, T. Kreutter, K. Mayer, and V. Schmitz, "Synthetic aperture focusing technique signal processing," NDT Int. 19, 177-189 (1986).
[CrossRef]

Krueger, M.

D. Vray, C. Haas, T. Rastello, M. Krueger, E. Brusseau, K. Schroeder, G. Gimenez, and H. Ermert, "Synthetic aperture-based beam compression for intravascular ultrasound imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 189-201 (2001).
[CrossRef] [PubMed]

Kuranov, R. V.

Kuznetova, I. A.

Langenberg, K. J.

K. J. Langenberg, M. Berger, T. Kreutter, K. Mayer, and V. Schmitz, "Synthetic aperture focusing technique signal processing," NDT Int. 19, 177-189 (1986).
[CrossRef]

Li, X.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Linzer, M.

S. J. Norton and M. Linzer, "Ultrasonic reflectivity imaging in three dimensions: exact inverse scattering solutions for plane, cylindrical, and spherical apertures," IEEE Trans. Biomed. Eng. BME-28, 202-220 (1981).
[CrossRef]

Lo, S.

MacNeill, B.

I. K. Jang, G. J. Tearney, B. MacNeill, M. Takano, F. Moselewski, N. Iftima, M. Shishkov, S. Houser, H. T. Aretz, E. F. Halpern, and B. E. Bouma, "In vivo characterization of coronary atherosclerotic plaque by use of optical coherence tomography," Circulation 111, 1551-1555 (2005).
[CrossRef] [PubMed]

MacNeill, B. D.

B. D. MacNeill, B. E. Bouma, H. Yabushita, I. K. Jang, and G. J. Tearney, "Intravascular optical coherence tomography: cellular imaging," J. Nucl. Cardiol. 12, 460-465 (2005).
[CrossRef] [PubMed]

B. E. Bouma, G. J. Tearney, H. Yabushita, M. Shishkov, C. R. Kauffman, D. DeJoseph Gauthier, B. D. MacNeill, S. L. Houser, H. T. Aretz, E. F. Halpern, and I. K. Jang, "Evaluation of intracoronary stenting by intravascular optical coherence tomography," Heart 89, 317-320 (2003).
[CrossRef] [PubMed]

Madden, K.

Marcon, N. E.

Marks, D. L.

Mayer, K.

K. J. Langenberg, M. Berger, T. Kreutter, K. Mayer, and V. Schmitz, "Synthetic aperture focusing technique signal processing," NDT Int. 19, 177-189 (1986).
[CrossRef]

Moselewski, F.

I. K. Jang, G. J. Tearney, B. MacNeill, M. Takano, F. Moselewski, N. Iftima, M. Shishkov, S. Houser, H. T. Aretz, E. F. Halpern, and B. E. Bouma, "In vivo characterization of coronary atherosclerotic plaque by use of optical coherence tomography," Circulation 111, 1551-1555 (2005).
[CrossRef] [PubMed]

Nolan, C. J.

C. J. Nolan and M. Cheney, "Synthetic aperture inversion," Inverse Probl. 18, 221-235 (2002).
[CrossRef]

Norton, S. J.

S. J. Norton and M. Linzer, "Ultrasonic reflectivity imaging in three dimensions: exact inverse scattering solutions for plane, cylindrical, and spherical apertures," IEEE Trans. Biomed. Eng. BME-28, 202-220 (1981).
[CrossRef]

O'Brien, W. D.

M. A. Haun, D. L. Jones, and W. D. O'Brien, Jr., "Efficient three-dimensional imaging from a small cylindrical aperture," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49, 861-870 (2002).
[CrossRef] [PubMed]

C. H. Frazier and W. D. O'Brien, Jr., "Synthetic aperture techniques with a virtual source element," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 196-207 (l998).
[CrossRef]

Park, S. J.

I. K. Jang, B. E. Bouma, D. H. Kang, S. J. Park, S. W. Park, K. B. Seung, K. B. Choi, M. Shishkov, K. Schlendorf, E. Pomerantsev, S. L. Houser, and H. T. Aretz, "Visualization of coronary atherosclerotic plaques in patients using optical coherence tomography: comparison with intravascular ultrasound," J. Am. Coll. Cardiol. 39, 604-609 (2002).
[CrossRef] [PubMed]

Park, S. W.

I. K. Jang, B. E. Bouma, D. H. Kang, S. J. Park, S. W. Park, K. B. Seung, K. B. Choi, M. Shishkov, K. Schlendorf, E. Pomerantsev, S. L. Houser, and H. T. Aretz, "Visualization of coronary atherosclerotic plaques in patients using optical coherence tomography: comparison with intravascular ultrasound," J. Am. Coll. Cardiol. 39, 604-609 (2002).
[CrossRef] [PubMed]

Pekar, J.

Petersen, C.

Pitris, C.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, and J. F. Southern, "In vivo endoscopic optical biopsy with optical coherence tomography," Science 276, 2037-2039 (1997).
[CrossRef] [PubMed]

Pomerantsev, E.

I. K. Jang, B. E. Bouma, D. H. Kang, S. J. Park, S. W. Park, K. B. Seung, K. B. Choi, M. Shishkov, K. Schlendorf, E. Pomerantsev, S. L. Houser, and H. T. Aretz, "Visualization of coronary atherosclerotic plaques in patients using optical coherence tomography: comparison with intravascular ultrasound," J. Am. Coll. Cardiol. 39, 604-609 (2002).
[CrossRef] [PubMed]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Qi, B.

Ralston, T. S.

Rastello, T.

D. Vray, C. Haas, T. Rastello, M. Krueger, E. Brusseau, K. Schroeder, G. Gimenez, and H. Ermert, "Synthetic aperture-based beam compression for intravascular ultrasound imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 189-201 (2001).
[CrossRef] [PubMed]

Rollins, A. M.

Schlendorf, K.

I. K. Jang, B. E. Bouma, D. H. Kang, S. J. Park, S. W. Park, K. B. Seung, K. B. Choi, M. Shishkov, K. Schlendorf, E. Pomerantsev, S. L. Houser, and H. T. Aretz, "Visualization of coronary atherosclerotic plaques in patients using optical coherence tomography: comparison with intravascular ultrasound," J. Am. Coll. Cardiol. 39, 604-609 (2002).
[CrossRef] [PubMed]

Schmitt, J.

Schmitz, V.

K. J. Langenberg, M. Berger, T. Kreutter, K. Mayer, and V. Schmitz, "Synthetic aperture focusing technique signal processing," NDT Int. 19, 177-189 (1986).
[CrossRef]

Schneider, K.

Schroeder, K.

D. Vray, C. Haas, T. Rastello, M. Krueger, E. Brusseau, K. Schroeder, G. Gimenez, and H. Ermert, "Synthetic aperture-based beam compression for intravascular ultrasound imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 189-201 (2001).
[CrossRef] [PubMed]

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Seng-Yue, E.

Sergeev, A. M.

Seung, K. B.

I. K. Jang, B. E. Bouma, D. H. Kang, S. J. Park, S. W. Park, K. B. Seung, K. B. Choi, M. Shishkov, K. Schlendorf, E. Pomerantsev, S. L. Houser, and H. T. Aretz, "Visualization of coronary atherosclerotic plaques in patients using optical coherence tomography: comparison with intravascular ultrasound," J. Am. Coll. Cardiol. 39, 604-609 (2002).
[CrossRef] [PubMed]

Shakhov, A. V.

Shakhova, N. M.

Shishkov, M.

I. K. Jang, G. J. Tearney, B. MacNeill, M. Takano, F. Moselewski, N. Iftima, M. Shishkov, S. Houser, H. T. Aretz, E. F. Halpern, and B. E. Bouma, "In vivo characterization of coronary atherosclerotic plaque by use of optical coherence tomography," Circulation 111, 1551-1555 (2005).
[CrossRef] [PubMed]

B. E. Bouma, G. J. Tearney, H. Yabushita, M. Shishkov, C. R. Kauffman, D. DeJoseph Gauthier, B. D. MacNeill, S. L. Houser, H. T. Aretz, E. F. Halpern, and I. K. Jang, "Evaluation of intracoronary stenting by intravascular optical coherence tomography," Heart 89, 317-320 (2003).
[CrossRef] [PubMed]

I. K. Jang, B. E. Bouma, D. H. Kang, S. J. Park, S. W. Park, K. B. Seung, K. B. Choi, M. Shishkov, K. Schlendorf, E. Pomerantsev, S. L. Houser, and H. T. Aretz, "Visualization of coronary atherosclerotic plaques in patients using optical coherence tomography: comparison with intravascular ultrasound," J. Am. Coll. Cardiol. 39, 604-609 (2002).
[CrossRef] [PubMed]

Sivak, M. V.

Snopova, L. B.

Southern, J. F.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, and J. F. Southern, "In vivo endoscopic optical biopsy with optical coherence tomography," Science 276, 2037-2039 (1997).
[CrossRef] [PubMed]

G. J. Tearney, S. A. Boppart, B. E. Bunma, M. E. Brezinski, N. J. Weissman, J. F. Southern, and J. G. Fujimoto, "Scanning single-mode fiber-optic catheter endoscope for optical coherence tomography," Opt. Lett. 21, 543-545 (1996).
[CrossRef] [PubMed]

Stenstrom, G.

L. M. H. Ulander, H. Hellsten, and G. Stenstrom, "Synthetic aperture radar processing using fast factorized back-projection," IEEE Trans. Aerosp. Electron. Syst. 39, 760-776 (2003).
[CrossRef]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Stuart, B.

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Takano, M.

I. K. Jang, G. J. Tearney, B. MacNeill, M. Takano, F. Moselewski, N. Iftima, M. Shishkov, S. Houser, H. T. Aretz, E. F. Halpern, and B. E. Bouma, "In vivo characterization of coronary atherosclerotic plaque by use of optical coherence tomography," Circulation 111, 1551-1555 (2005).
[CrossRef] [PubMed]

Tang, S.-J.

Tearney, G. J.

B. D. MacNeill, B. E. Bouma, H. Yabushita, I. K. Jang, and G. J. Tearney, "Intravascular optical coherence tomography: cellular imaging," J. Nucl. Cardiol. 12, 460-465 (2005).
[CrossRef] [PubMed]

I. K. Jang, G. J. Tearney, B. MacNeill, M. Takano, F. Moselewski, N. Iftima, M. Shishkov, S. Houser, H. T. Aretz, E. F. Halpern, and B. E. Bouma, "In vivo characterization of coronary atherosclerotic plaque by use of optical coherence tomography," Circulation 111, 1551-1555 (2005).
[CrossRef] [PubMed]

B. E. Bouma, G. J. Tearney, H. Yabushita, M. Shishkov, C. R. Kauffman, D. DeJoseph Gauthier, B. D. MacNeill, S. L. Houser, H. T. Aretz, E. F. Halpern, and I. K. Jang, "Evaluation of intracoronary stenting by intravascular optical coherence tomography," Heart 89, 317-320 (2003).
[CrossRef] [PubMed]

B. E. Bouma and G. J. Tearney, "Power-efficient nonreciprocal interferometer and linear-scanning fiber-optic catheter for optical coherence tomography," Opt. Lett. 24, 531-533 (1999).
[CrossRef]

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, and J. F. Southern, "In vivo endoscopic optical biopsy with optical coherence tomography," Science 276, 2037-2039 (1997).
[CrossRef] [PubMed]

G. J. Tearney, S. A. Boppart, B. E. Bunma, M. E. Brezinski, N. J. Weissman, J. F. Southern, and J. G. Fujimoto, "Scanning single-mode fiber-optic catheter endoscope for optical coherence tomography," Opt. Lett. 21, 543-545 (1996).
[CrossRef] [PubMed]

B. E. Bouma and G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, 2001).

Terent'eva, A. B.

Ulander, L. M. H.

L. M. H. Ulander, H. Hellsten, and G. Stenstrom, "Synthetic aperture radar processing using fast factorized back-projection," IEEE Trans. Aerosp. Electron. Syst. 39, 760-776 (2003).
[CrossRef]

Ung-arunyawee, R.

Vitkin, V. I.

Vray, D.

D. Vray, C. Haas, T. Rastello, M. Krueger, E. Brusseau, K. Schroeder, G. Gimenez, and H. Ermert, "Synthetic aperture-based beam compression for intravascular ultrasound imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 189-201 (2001).
[CrossRef] [PubMed]

Weissman, N. J.

Wilson, B. C.

Wong, R. C. K.

Yabushita, H.

B. D. MacNeill, B. E. Bouma, H. Yabushita, I. K. Jang, and G. J. Tearney, "Intravascular optical coherence tomography: cellular imaging," J. Nucl. Cardiol. 12, 460-465 (2005).
[CrossRef] [PubMed]

B. E. Bouma, G. J. Tearney, H. Yabushita, M. Shishkov, C. R. Kauffman, D. DeJoseph Gauthier, B. D. MacNeill, S. L. Houser, H. T. Aretz, E. F. Halpern, and I. K. Jang, "Evaluation of intracoronary stenting by intravascular optical coherence tomography," Heart 89, 317-320 (2003).
[CrossRef] [PubMed]

Yang, V. X. D.

Zagainova, E. V.

Circulation (1)

I. K. Jang, G. J. Tearney, B. MacNeill, M. Takano, F. Moselewski, N. Iftima, M. Shishkov, S. Houser, H. T. Aretz, E. F. Halpern, and B. E. Bouma, "In vivo characterization of coronary atherosclerotic plaque by use of optical coherence tomography," Circulation 111, 1551-1555 (2005).
[CrossRef] [PubMed]

Heart (1)

B. E. Bouma, G. J. Tearney, H. Yabushita, M. Shishkov, C. R. Kauffman, D. DeJoseph Gauthier, B. D. MacNeill, S. L. Houser, H. T. Aretz, E. F. Halpern, and I. K. Jang, "Evaluation of intracoronary stenting by intravascular optical coherence tomography," Heart 89, 317-320 (2003).
[CrossRef] [PubMed]

IEEE Trans. Aerosp. Electron. Syst. (1)

L. M. H. Ulander, H. Hellsten, and G. Stenstrom, "Synthetic aperture radar processing using fast factorized back-projection," IEEE Trans. Aerosp. Electron. Syst. 39, 760-776 (2003).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

S. J. Norton and M. Linzer, "Ultrasonic reflectivity imaging in three dimensions: exact inverse scattering solutions for plane, cylindrical, and spherical apertures," IEEE Trans. Biomed. Eng. BME-28, 202-220 (1981).
[CrossRef]

IEEE Trans. Image Process. (1)

S. Basu and Y. Bresler, "O(N2 log2N) filtered backprojection reconstruction algorithm for tomography," IEEE Trans. Image Process. 9, 1760-1773 (2000).
[CrossRef]

IEEE Trans. Med. Imaging (1)

P. Edholm and G. T. Herman, "Linograms in image reconstruction from projections," IEEE Trans. Med. Imaging MI-6, 301-307 (1987).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (3)

D. Vray, C. Haas, T. Rastello, M. Krueger, E. Brusseau, K. Schroeder, G. Gimenez, and H. Ermert, "Synthetic aperture-based beam compression for intravascular ultrasound imaging," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 48, 189-201 (2001).
[CrossRef] [PubMed]

C. H. Frazier and W. D. O'Brien, Jr., "Synthetic aperture techniques with a virtual source element," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 196-207 (l998).
[CrossRef]

M. A. Haun, D. L. Jones, and W. D. O'Brien, Jr., "Efficient three-dimensional imaging from a small cylindrical aperture," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49, 861-870 (2002).
[CrossRef] [PubMed]

Inverse Probl. (2)

C. J. Nolan and M. Cheney, "Synthetic aperture inversion," Inverse Probl. 18, 221-235 (2002).
[CrossRef]

H. Hellsten and L. E. Andersson, "An inverse method for the processing of synthetic aperture radar data," Inverse Probl. 3, 111-124 (1987).
[CrossRef]

J. Am. Coll. Cardiol. (1)

I. K. Jang, B. E. Bouma, D. H. Kang, S. J. Park, S. W. Park, K. B. Seung, K. B. Choi, M. Shishkov, K. Schlendorf, E. Pomerantsev, S. L. Houser, and H. T. Aretz, "Visualization of coronary atherosclerotic plaques in patients using optical coherence tomography: comparison with intravascular ultrasound," J. Am. Coll. Cardiol. 39, 604-609 (2002).
[CrossRef] [PubMed]

J. Nucl. Cardiol. (1)

B. D. MacNeill, B. E. Bouma, H. Yabushita, I. K. Jang, and G. J. Tearney, "Intravascular optical coherence tomography: cellular imaging," J. Nucl. Cardiol. 12, 460-465 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

NDT Int. (1)

K. J. Langenberg, M. Berger, T. Kreutter, K. Mayer, and V. Schmitz, "Synthetic aperture focusing technique signal processing," NDT Int. 19, 177-189 (1986).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Science (2)

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, and J. F. Southern, "In vivo endoscopic optical biopsy with optical coherence tomography," Science 276, 2037-2039 (1997).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

SIAM J. Appl. Math. (1)

J. A. Fawcett, "Inversion of N-dimensional spherical averages," SIAM J. Appl. Math. 45, 336-341 (1985).
[CrossRef]

Other (1)

B. E. Bouma and G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, 2001).

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Figures (5)

Fig. 1
Fig. 1

Diagram of an OCT catheter. A single-mode fiber guides an optical signal, which is focused into a converging beam by the lens. The prism directs the beam at a right angle to the catheter. The beam is scanned azimuthally, rotating around the long axis of the catheter and is translated along the long axis of the catheter to collect cylindrical 3-D volumes of data.

Fig. 2
Fig. 2

Geometry of the derivation. A Gaussian beam originates at a position x = 0 , y = p , z = 0 , and travels parallel to the x z plane at an angle θ from the x axis. The Gaussian beam is focused so the beam waist is a distance z 0 from the y axis.

Fig. 3
Fig. 3

Simulated and reconstructed OCT catheter images. The simulated area is 135 by 135 λ, and the Gaussian beam has a 2.5 λ beam waist, focused 45 λ from the origin. (a) Simulated OCT data for randomly scattered point objects. The dotted circle coincides with the focus radius of the Gaussian beam, and the dark center represents the location of the OCT catheter. Points inside the circle have wavefronts that curve toward the origin, and points outside the circle have wavefronts that curve away from the origin. (b) Reconstruction of the point sources from the simulated data.

Fig. 4
Fig. 4

Resolution chart showing FWHM transverse point-spread-function (PSF) resolution of simulated point sources situated at difference distances from the catheter axis, as a function of focus radius and beam width. The abscissa is the distance from the origin from which the simulated point source is placed. In each part, the number next to each curve is the distance away from the origin from which the beam was focused. The beam waist for each part is (a) 1.5 λ, (b) 3 λ, (c) 4.5 λ, (d) 6 λ.

Fig. 5
Fig. 5

Resolution chart showing FWHM axial resolution of simulated point sources imaged with two different beam widths focused 25 λ from the catheter axis, for various fractional bandwidths of the source. The dotted curve corresponds to a 1.5 λ beam waist, and the solid curve corresponds to a 5 λ beam waist. PSF, point-spread function.

Equations (23)

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

R ( θ ) = [ cos θ 0 sin θ 0 1 0 sin θ 0 cos θ ] .
S ̃ ( r , k ) = i ( 2 π ) 2 A ( k ) k 1 V d 3 r η ( r ) f 2 ( r r 0 ; k ) ,
f 2 ( r ; k ) = 1 ( 2 π ) 2 d 2 ξ exp ( i ξ r ) 1 2 ( α 2 k 2 + i ( z z 0 ) k ) 1 × exp ( ξ 2 α 2 4 k 2 ) exp [ i ( z z 0 ) ( 2 k ) 2 ξ 2 ] ,
S ̃ ( k , p , θ ) = i ( 2 π ) 2 A ( k ) k 1 V d 3 r η [ R ( θ ) r ] f 2 ( r p y ̂ ; k ) .
S ̃ ( k , p , θ ) = i 2 A ( k ) k 1 d 2 ξ exp [ i z 0 ( 2 k ) 2 ξ 2 ] exp ( ξ 2 α 2 4 k 2 ) d 3 r exp [ i ξ ( r p y ̂ ) ] η [ R ( θ ) r ] [ α 2 k 2 + i ( z z 0 ) k ] 1 exp [ i z ( 2 k ) 2 ξ 2 ] .
S ̃ ( k , p , θ ) = i 2 A ( k ) d 2 ξ exp [ i z 0 ( 2 k ) 2 ξ 2 ] exp ( ξ 2 α 2 4 k 2 ) × d 3 r exp [ i ξ ( r p y ̂ ) ] η [ R ( θ ) r ] [ α 2 k 0 + i ( ρ z 0 ) ] 1 exp [ i z ( 2 k ) 2 ξ 2 ] .
S ̃ ( k , p , θ ) = i 2 A ( k ) d 2 ξ exp [ i z 0 ( 2 k ) 2 ξ 2 ] exp ( ξ 2 α 2 4 k 2 ) exp ( i ξ y p ) d 3 r exp { i [ ξ z ̂ ( 2 k ) 2 ξ 2 ] R [ ( θ ) r ] } η ( r ) [ α 2 k 0 + i ( ρ z 0 ) ] 1 ,
S ̃ ( k , p , θ ) = i 2 A ( k ) d 2 ξ exp [ i z 0 ( 2 k ) 2 ξ 2 ] exp ( ξ 2 α 2 4 k 2 ) exp ( i ξ y p ) η ̃ { R ( θ ) [ ξ z ̂ ( 2 k ) 2 ξ 2 ] } ,
η ̃ ( β ) = d 3 r exp ( i r β ) η ( r ) [ α 2 k 0 + i ( ρ z 0 ) ] 1 .
S ̃ ( k , p , θ ) = i 2 A ( k ) d ξ y exp ( i ξ y p ) 0 π d ϕ { [ ( 2 k ) 2 ξ y 2 sin ϕ ] exp [ i z 0 ( 2 k ) 2 ξ 2 ] exp ( ξ 2 α 2 4 k 2 ) } η ̃ { R ( θ ) [ x ̂ cos ϕ ( 2 k ) 2 ξ y 2 + y ̂ ξ y z ̂ sin ϕ ( 2 k ) 2 ξ y 2 ] } .
K ( k , ξ y , ϕ ) = i 2 A ( k ) [ ( 2 k ) 2 ξ y 2 sin ϕ ] exp [ i z 0 ( 2 k ) 2 ξ 2 ] exp ( ξ 2 α 2 4 k 2 ) .
S ̃ ( k , p , θ ) = d ξ y exp ( i ξ y p ) 0 π d ϕ K ( k , ξ y , ϕ ) η ̃ { R ( θ ϕ ) [ x ̂ ( 2 k ) 2 ξ y 2 + y ̂ ξ y ] } .
S ̃ ̃ ( k , ξ p , n 0 ) = π π d p d θ exp ( i p ξ p ) exp ( i θ n θ ) S ̃ ( k , p , θ ) ,
K ̃ ( k , ξ y , n θ ) = 0 π d θ exp ( i θ n θ ) K ( k , ξ y , θ ) ,
η ̃ ̃ ( k , ξ y , n θ ) = π π d θ exp ( i θ n θ ) η ̃ { [ x ̂ cos θ ( 2 k ) 2 ξ y 2 + y ̂ ξ y + z ̂ sin θ ( 2 k ) 2 ξ y 2 ] } ,
S ̃ ̃ ( k , ξ p , n θ ) = K ̃ ( k , ξ p , n θ ) η ̃ ̃ ( k , ξ p , n θ ) .
[ K ̃ η ̃ ̃ ] ( k , ξ p , n θ ) = d k d ξ p n θ K ̃ ( k , ξ p , n θ ) δ ( k k ) δ ( ξ p ξ p ) δ n θ , n θ η ̃ ̃ ( k , ξ p , n θ ) .
S ̃ ( k , p , θ ) = 1 4 π 2 d ξ y exp ( i ξ y p ) n θ = exp ( i θ n θ ) K ( k , ξ y , n θ ) d 3 r η ( r ) [ α 2 k 0 + i ( ρ z 0 ) ] 1 π π d ϕ exp ( i ϕ n θ ) exp { i r R ( ϕ ) [ x ̂ 2 ( k ) 2 ξ y 2 + y ̂ ξ y ] } ,
K ̃ ( k , ξ p , n θ ) = i 2 A ( k ) 0 π d θ exp ( i θ n θ ) exp [ ( 2 k ) 2 cos 2 θ + ξ p 2 sin 2 θ 2 α 2 2 k 2 ] exp [ i z 0 sin θ ( 2 k ) 2 ξ p 2 ] ( 2 k ) 2 ξ p 2 sin θ .
S ( k , p , θ ) = κ η = d 3 r κ ( k , p , θ , r ) η ( r ) .
η + = arg min η S κ η 2 = arg min η 0 d k d p π π d θ S ( k , p , θ ) κ η ( r ) 2 .
η ̃ ̃ + = arg min η ̃ ̃ S ̃ ̃ K η ̃ ̃ 2 + λ η ̃ ̃ 2 = arg min η ̃ ̃ 0 d k d ξ p n θ = S ̃ ̃ ( k , ξ p , n θ ) K ( k , ξ p , n θ ) η ̃ ̃ ( k , ξ p , n θ ) 2 + λ η ̃ ̃ ( k , ξ p , n θ ) 2 .
η ̃ ̃ + ( k , ξ p , n θ ) = S ̃ ̃ ( k , ξ p , n θ ) K * ( k , ξ p , n θ ) K ( k , ξ p , n θ ) 2 + λ .

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