Abstract

Optical quadrature microscopy (OQM) was invented in 1997 to reconstruct a full-field image of quantitative phase, and has been used to count the number of cells in live mouse embryos. Here we present a thorough SNR analysis that incorporates noise terms for fluctuations in the laser, aberrations within the individual paths of the Mach-Zehnder interferometer, and imperfections within the beamsplitters and CCD cameras to create a model for the resultant phase measurements. The current RMS error of the OQM phase images has been calculated to be 0.08 radians from substituting images from the instrumentation into the model.

© 2009 Optical Society of America

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2008 (4)

T. Yamauchi, H. Iwai, M. Miwa, and Y. Yamashita, "Low-coherent quantitative phase microscope for nanometer-scale measurement of living cells morphology," Opt. Express 16, 12227-12238 (2008).
[CrossRef] [PubMed]

W. C. WargerII, J. A. Newmark, C. M. Warner, and C. A. DiMarzio, "Phase subtraction cell counting method for live mouse embryos beyond the eight-cell stage," J. Biomed. Opt. 13, 034005 (2008).
[CrossRef] [PubMed]

W. C. WargerII and C. A. DiMarzio, "Modeling of optical quadrature microscopy for imaging mouse embryos," Proc. SPIE 6861, 68610T (2008).
[CrossRef]

C. L. Tsai, W. C. WargerII, G. S. Laevsky, and C. A. DiMarzio, "Alignment with sub-pixel accuracy for images of multi-modality microscopes using automatic calibration," J. Microsc.,  232, 164-176 (2008).
[CrossRef]

2007 (3)

W. C. WargerII, G. S. Laevsky, D. J. Townsend, M. Rajadhyaksha, and C. A. DiMarzio, "Multimodal optical microscope for detecting viability of mouse embryos in vitro," J. Biomed. Opt. 12, 044006 (2007).
[CrossRef] [PubMed]

J. A. Newmark, W. C. WargerII, C. C. Chang, G. E. Herrera, D. H. Brooks, C. A. DiMarzio, and C. M. Warner, "Determination of the Number of Cells in Preimplantation Embryos by Using Non-invasive Optical Quadrature Microscopy in Conjunction with Differential Interference Contrast Microscopy," Microsc. Microanal. 13, 118-127 (2007).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic Phase Microscopy," Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

2006 (5)

2005 (2)

2004 (4)

H. Iwai, C. Fang-Yen, G. Popescu, A. Wax, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Quantitative phase imaging using actively stabilized phase-shifting low-coherence interferometry," Opt. Lett. 29, 2399 (2004).
[CrossRef] [PubMed]

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

C. M. Warner, J. A. Newmark, M. Comiskey, S. R. De Fazio, D. M. O’Malley, M. Rajadhyaksha, D. J. Townsend, S. McKnight, B. Roysam, P. J. Dwyer, and C. A. DiMarzio, "Genetics and imaging to assess oocyte and preimplantation embryo health," Reprod. Fertil. Dev. 16, 729-741 (2004).
[CrossRef]

G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, "Fourier phase microscopy for investigation of biological structures and dynamics," Opt. Lett. 29, 2503-2505 (2004).
[CrossRef]

2003 (1)

D. J. Townsend, K. D. Quarles, A. L. Thomas, W. S. Rockward, C. M. Warner, J. A. Newmark, and C. A. DiMarzio, "Quantitative Phase Measurements Using a Quadrature Tomographic Microscope," Proc. SPIE 4964, 59-65 (2003).
[CrossRef]

2001 (2)

J. J. Stott, R. E. Bennett, C. M. Warner, and C. A. DiMarzio, "Three-dimensional imaging with a quadrature tomographic microscope," Proc. SPIE 4261, 24-32 (2001).
[CrossRef]

A. Dubois, L. Vabre, and A. C. Boccara, "Sinusoidally phase modulated interference microscope for high-speed high-resolution topographic imagery," Opt. Lett. 26, 1873-1875 (2001).
[CrossRef]

2000 (3)

C. Preza, "Rotational-diversity phase estimation from differential interference contrast microscopy images," J. Opt. Soc. Am. A 17, 415-424 (2000).
[CrossRef]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

1999 (2)

Y. Glina, G. A. Tsihrintzis, C. M. Warner, D. O. Hogenboom, and C. A. DiMarzio, "On the use of the optical quadrature method in tomographic microscopy," Proc. SPIE  3605, 101-106 (1999).
[CrossRef]

E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-293 (1999).
[CrossRef]

1998 (4)

1997 (2)

1930 (1)

A. Lebedeff, "Polarization interferometer and its applications," Rev. Opt., Theor. Instrum. 9, 385 (1930).

Arnison, M. R.

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
[CrossRef] [PubMed]

Badizadegan, K.

Barty, A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

Bennett, R. E.

J. J. Stott, R. E. Bennett, C. M. Warner, and C. A. DiMarzio, "Three-dimensional imaging with a quadrature tomographic microscope," Proc. SPIE 4261, 24-32 (2001).
[CrossRef]

Bevilacqua, F.

Boccara, A. C.

Brooks, D. H.

J. A. Newmark, W. C. WargerII, C. C. Chang, G. E. Herrera, D. H. Brooks, C. A. DiMarzio, and C. M. Warner, "Determination of the Number of Cells in Preimplantation Embryos by Using Non-invasive Optical Quadrature Microscopy in Conjunction with Differential Interference Contrast Microscopy," Microsc. Microanal. 13, 118-127 (2007).
[CrossRef] [PubMed]

Chang, C. C.

J. A. Newmark, W. C. WargerII, C. C. Chang, G. E. Herrera, D. H. Brooks, C. A. DiMarzio, and C. M. Warner, "Determination of the Number of Cells in Preimplantation Embryos by Using Non-invasive Optical Quadrature Microscopy in Conjunction with Differential Interference Contrast Microscopy," Microsc. Microanal. 13, 118-127 (2007).
[CrossRef] [PubMed]

Charrière, F.

Choi, W.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic Phase Microscopy," Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Cogswell, C. J.

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
[CrossRef] [PubMed]

Colomb, T.

Comiskey, M.

C. M. Warner, J. A. Newmark, M. Comiskey, S. R. De Fazio, D. M. O’Malley, M. Rajadhyaksha, D. J. Townsend, S. McKnight, B. Roysam, P. J. Dwyer, and C. A. DiMarzio, "Genetics and imaging to assess oocyte and preimplantation embryo health," Reprod. Fertil. Dev. 16, 729-741 (2004).
[CrossRef]

Cuche, E.

Cui, X.

Dasari, R. R.

De Fazio, S. R.

C. M. Warner, J. A. Newmark, M. Comiskey, S. R. De Fazio, D. M. O’Malley, M. Rajadhyaksha, D. J. Townsend, S. McKnight, B. Roysam, P. J. Dwyer, and C. A. DiMarzio, "Genetics and imaging to assess oocyte and preimplantation embryo health," Reprod. Fertil. Dev. 16, 729-741 (2004).
[CrossRef]

Deflores, L. P.

Depeursinge, C.

Devaney, A. J.

DiMarzio, C. A.

W. C. WargerII, J. A. Newmark, C. M. Warner, and C. A. DiMarzio, "Phase subtraction cell counting method for live mouse embryos beyond the eight-cell stage," J. Biomed. Opt. 13, 034005 (2008).
[CrossRef] [PubMed]

W. C. WargerII and C. A. DiMarzio, "Modeling of optical quadrature microscopy for imaging mouse embryos," Proc. SPIE 6861, 68610T (2008).
[CrossRef]

C. L. Tsai, W. C. WargerII, G. S. Laevsky, and C. A. DiMarzio, "Alignment with sub-pixel accuracy for images of multi-modality microscopes using automatic calibration," J. Microsc.,  232, 164-176 (2008).
[CrossRef]

W. C. WargerII, G. S. Laevsky, D. J. Townsend, M. Rajadhyaksha, and C. A. DiMarzio, "Multimodal optical microscope for detecting viability of mouse embryos in vitro," J. Biomed. Opt. 12, 044006 (2007).
[CrossRef] [PubMed]

J. A. Newmark, W. C. WargerII, C. C. Chang, G. E. Herrera, D. H. Brooks, C. A. DiMarzio, and C. M. Warner, "Determination of the Number of Cells in Preimplantation Embryos by Using Non-invasive Optical Quadrature Microscopy in Conjunction with Differential Interference Contrast Microscopy," Microsc. Microanal. 13, 118-127 (2007).
[CrossRef] [PubMed]

C. M. Warner, J. A. Newmark, M. Comiskey, S. R. De Fazio, D. M. O’Malley, M. Rajadhyaksha, D. J. Townsend, S. McKnight, B. Roysam, P. J. Dwyer, and C. A. DiMarzio, "Genetics and imaging to assess oocyte and preimplantation embryo health," Reprod. Fertil. Dev. 16, 729-741 (2004).
[CrossRef]

D. J. Townsend, K. D. Quarles, A. L. Thomas, W. S. Rockward, C. M. Warner, J. A. Newmark, and C. A. DiMarzio, "Quantitative Phase Measurements Using a Quadrature Tomographic Microscope," Proc. SPIE 4964, 59-65 (2003).
[CrossRef]

J. J. Stott, R. E. Bennett, C. M. Warner, and C. A. DiMarzio, "Three-dimensional imaging with a quadrature tomographic microscope," Proc. SPIE 4261, 24-32 (2001).
[CrossRef]

Y. Glina, G. A. Tsihrintzis, C. M. Warner, D. O. Hogenboom, and C. A. DiMarzio, "On the use of the optical quadrature method in tomographic microscopy," Proc. SPIE  3605, 101-106 (1999).
[CrossRef]

D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, "Three-dimensional images generated by quadrature interferometry," Opt. Lett. 23, 783-785 (1998).
[CrossRef]

D. O. Hogenboom and C. A. DiMarzio, "Quadrature detection of a Doppler signal," Appl. Opt. 37, 2569-2572 (1998).
[CrossRef]

Dubois, A.

Dwyer, P. J.

C. M. Warner, J. A. Newmark, M. Comiskey, S. R. De Fazio, D. M. O’Malley, M. Rajadhyaksha, D. J. Townsend, S. McKnight, B. Roysam, P. J. Dwyer, and C. A. DiMarzio, "Genetics and imaging to assess oocyte and preimplantation embryo health," Reprod. Fertil. Dev. 16, 729-741 (2004).
[CrossRef]

Fang-Yen, C.

Fekete, P. W.

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
[CrossRef] [PubMed]

Feld, M. S.

Gaudette, T. J.

Glina, Y.

Y. Glina, G. A. Tsihrintzis, C. M. Warner, D. O. Hogenboom, and C. A. DiMarzio, "On the use of the optical quadrature method in tomographic microscopy," Proc. SPIE  3605, 101-106 (1999).
[CrossRef]

Heise, B.

B. Heise, A. Sonnleitner, and E. P. Klement, "DIC image reconstruction on large cell scans," Microsc. Res. Tech. 66, 312-320 (2005).
[CrossRef] [PubMed]

Heng, X.

Herrera, G. E.

J. A. Newmark, W. C. WargerII, C. C. Chang, G. E. Herrera, D. H. Brooks, C. A. DiMarzio, and C. M. Warner, "Determination of the Number of Cells in Preimplantation Embryos by Using Non-invasive Optical Quadrature Microscopy in Conjunction with Differential Interference Contrast Microscopy," Microsc. Microanal. 13, 118-127 (2007).
[CrossRef] [PubMed]

Hogenboom, D. O.

Ikeda, T.

Inoue, S.

Ishiwata, H.

H. Ishiwata, M. Itoh, and T. Yatagai, "A new method of three dimensional measurement by differential interference contrast microscope," Opt. Commun. 260, 117-126 (2006).
[CrossRef]

Itoh, M.

H. Ishiwata, M. Itoh, and T. Yatagai, "A new method of three dimensional measurement by differential interference contrast microscope," Opt. Commun. 260, 117-126 (2006).
[CrossRef]

Iwai, H.

Klement, E. P.

B. Heise, A. Sonnleitner, and E. P. Klement, "DIC image reconstruction on large cell scans," Microsc. Res. Tech. 66, 312-320 (2005).
[CrossRef] [PubMed]

Kuehn, J.

Laevsky, G. S.

C. L. Tsai, W. C. WargerII, G. S. Laevsky, and C. A. DiMarzio, "Alignment with sub-pixel accuracy for images of multi-modality microscopes using automatic calibration," J. Microsc.,  232, 164-176 (2008).
[CrossRef]

W. C. WargerII, G. S. Laevsky, D. J. Townsend, M. Rajadhyaksha, and C. A. DiMarzio, "Multimodal optical microscope for detecting viability of mouse embryos in vitro," J. Biomed. Opt. 12, 044006 (2007).
[CrossRef] [PubMed]

Larkin, K. G.

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
[CrossRef] [PubMed]

Lebedeff, A.

A. Lebedeff, "Polarization interferometer and its applications," Rev. Opt., Theor. Instrum. 9, 385 (1930).

Lindberg, S. C.

Lue, N.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic Phase Microscopy," Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Marian, A.

Marx, D. S.

McKnight, S.

C. M. Warner, J. A. Newmark, M. Comiskey, S. R. De Fazio, D. M. O’Malley, M. Rajadhyaksha, D. J. Townsend, S. McKnight, B. Roysam, P. J. Dwyer, and C. A. DiMarzio, "Genetics and imaging to assess oocyte and preimplantation embryo health," Reprod. Fertil. Dev. 16, 729-741 (2004).
[CrossRef]

Miwa, M.

Montfort, F.

Newmark, J. A.

W. C. WargerII, J. A. Newmark, C. M. Warner, and C. A. DiMarzio, "Phase subtraction cell counting method for live mouse embryos beyond the eight-cell stage," J. Biomed. Opt. 13, 034005 (2008).
[CrossRef] [PubMed]

J. A. Newmark, W. C. WargerII, C. C. Chang, G. E. Herrera, D. H. Brooks, C. A. DiMarzio, and C. M. Warner, "Determination of the Number of Cells in Preimplantation Embryos by Using Non-invasive Optical Quadrature Microscopy in Conjunction with Differential Interference Contrast Microscopy," Microsc. Microanal. 13, 118-127 (2007).
[CrossRef] [PubMed]

C. M. Warner, J. A. Newmark, M. Comiskey, S. R. De Fazio, D. M. O’Malley, M. Rajadhyaksha, D. J. Townsend, S. McKnight, B. Roysam, P. J. Dwyer, and C. A. DiMarzio, "Genetics and imaging to assess oocyte and preimplantation embryo health," Reprod. Fertil. Dev. 16, 729-741 (2004).
[CrossRef]

D. J. Townsend, K. D. Quarles, A. L. Thomas, W. S. Rockward, C. M. Warner, J. A. Newmark, and C. A. DiMarzio, "Quantitative Phase Measurements Using a Quadrature Tomographic Microscope," Proc. SPIE 4964, 59-65 (2003).
[CrossRef]

Nugent, K. A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

D. Paganin and K. A. Nugent, "Noninterferometric Phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

O’Malley, D. M.

C. M. Warner, J. A. Newmark, M. Comiskey, S. R. De Fazio, D. M. O’Malley, M. Rajadhyaksha, D. J. Townsend, S. McKnight, B. Roysam, P. J. Dwyer, and C. A. DiMarzio, "Genetics and imaging to assess oocyte and preimplantation embryo health," Reprod. Fertil. Dev. 16, 729-741 (2004).
[CrossRef]

Oh, S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic Phase Microscopy," Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Paganin, D.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

D. Paganin and K. A. Nugent, "Noninterferometric Phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

Popescu, G.

Preza, C.

Psaltis, D.

Quarles, K. D.

D. J. Townsend, K. D. Quarles, A. L. Thomas, W. S. Rockward, C. M. Warner, J. A. Newmark, and C. A. DiMarzio, "Quantitative Phase Measurements Using a Quadrature Tomographic Microscope," Proc. SPIE 4964, 59-65 (2003).
[CrossRef]

Rajadhyaksha, M.

W. C. WargerII, G. S. Laevsky, D. J. Townsend, M. Rajadhyaksha, and C. A. DiMarzio, "Multimodal optical microscope for detecting viability of mouse embryos in vitro," J. Biomed. Opt. 12, 044006 (2007).
[CrossRef] [PubMed]

C. M. Warner, J. A. Newmark, M. Comiskey, S. R. De Fazio, D. M. O’Malley, M. Rajadhyaksha, D. J. Townsend, S. McKnight, B. Roysam, P. J. Dwyer, and C. A. DiMarzio, "Genetics and imaging to assess oocyte and preimplantation embryo health," Reprod. Fertil. Dev. 16, 729-741 (2004).
[CrossRef]

Roberts, A.

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

Rockward, W. S.

D. J. Townsend, K. D. Quarles, A. L. Thomas, W. S. Rockward, C. M. Warner, J. A. Newmark, and C. A. DiMarzio, "Quantitative Phase Measurements Using a Quadrature Tomographic Microscope," Proc. SPIE 4964, 59-65 (2003).
[CrossRef]

Roysam, B.

C. M. Warner, J. A. Newmark, M. Comiskey, S. R. De Fazio, D. M. O’Malley, M. Rajadhyaksha, D. J. Townsend, S. McKnight, B. Roysam, P. J. Dwyer, and C. A. DiMarzio, "Genetics and imaging to assess oocyte and preimplantation embryo health," Reprod. Fertil. Dev. 16, 729-741 (2004).
[CrossRef]

Sheppard, C. J.

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

Shribak, M.

Smith, N. I.

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
[CrossRef] [PubMed]

Sonnleitner, A.

B. Heise, A. Sonnleitner, and E. P. Klement, "DIC image reconstruction on large cell scans," Microsc. Res. Tech. 66, 312-320 (2005).
[CrossRef] [PubMed]

Stott, J. J.

J. J. Stott, R. E. Bennett, C. M. Warner, and C. A. DiMarzio, "Three-dimensional imaging with a quadrature tomographic microscope," Proc. SPIE 4261, 24-32 (2001).
[CrossRef]

Thomas, A. L.

D. J. Townsend, K. D. Quarles, A. L. Thomas, W. S. Rockward, C. M. Warner, J. A. Newmark, and C. A. DiMarzio, "Quantitative Phase Measurements Using a Quadrature Tomographic Microscope," Proc. SPIE 4964, 59-65 (2003).
[CrossRef]

Townsend, D. J.

W. C. WargerII, G. S. Laevsky, D. J. Townsend, M. Rajadhyaksha, and C. A. DiMarzio, "Multimodal optical microscope for detecting viability of mouse embryos in vitro," J. Biomed. Opt. 12, 044006 (2007).
[CrossRef] [PubMed]

C. M. Warner, J. A. Newmark, M. Comiskey, S. R. De Fazio, D. M. O’Malley, M. Rajadhyaksha, D. J. Townsend, S. McKnight, B. Roysam, P. J. Dwyer, and C. A. DiMarzio, "Genetics and imaging to assess oocyte and preimplantation embryo health," Reprod. Fertil. Dev. 16, 729-741 (2004).
[CrossRef]

D. J. Townsend, K. D. Quarles, A. L. Thomas, W. S. Rockward, C. M. Warner, J. A. Newmark, and C. A. DiMarzio, "Quantitative Phase Measurements Using a Quadrature Tomographic Microscope," Proc. SPIE 4964, 59-65 (2003).
[CrossRef]

Tsai, C. L.

C. L. Tsai, W. C. WargerII, G. S. Laevsky, and C. A. DiMarzio, "Alignment with sub-pixel accuracy for images of multi-modality microscopes using automatic calibration," J. Microsc.,  232, 164-176 (2008).
[CrossRef]

Tsihrintzis, G. A.

Y. Glina, G. A. Tsihrintzis, C. M. Warner, D. O. Hogenboom, and C. A. DiMarzio, "On the use of the optical quadrature method in tomographic microscopy," Proc. SPIE  3605, 101-106 (1999).
[CrossRef]

Vabre, L.

Vaughan, J. C.

Warger, W. C.

C. L. Tsai, W. C. WargerII, G. S. Laevsky, and C. A. DiMarzio, "Alignment with sub-pixel accuracy for images of multi-modality microscopes using automatic calibration," J. Microsc.,  232, 164-176 (2008).
[CrossRef]

W. C. WargerII, J. A. Newmark, C. M. Warner, and C. A. DiMarzio, "Phase subtraction cell counting method for live mouse embryos beyond the eight-cell stage," J. Biomed. Opt. 13, 034005 (2008).
[CrossRef] [PubMed]

W. C. WargerII and C. A. DiMarzio, "Modeling of optical quadrature microscopy for imaging mouse embryos," Proc. SPIE 6861, 68610T (2008).
[CrossRef]

W. C. WargerII, G. S. Laevsky, D. J. Townsend, M. Rajadhyaksha, and C. A. DiMarzio, "Multimodal optical microscope for detecting viability of mouse embryos in vitro," J. Biomed. Opt. 12, 044006 (2007).
[CrossRef] [PubMed]

J. A. Newmark, W. C. WargerII, C. C. Chang, G. E. Herrera, D. H. Brooks, C. A. DiMarzio, and C. M. Warner, "Determination of the Number of Cells in Preimplantation Embryos by Using Non-invasive Optical Quadrature Microscopy in Conjunction with Differential Interference Contrast Microscopy," Microsc. Microanal. 13, 118-127 (2007).
[CrossRef] [PubMed]

Warner, C. M.

W. C. WargerII, J. A. Newmark, C. M. Warner, and C. A. DiMarzio, "Phase subtraction cell counting method for live mouse embryos beyond the eight-cell stage," J. Biomed. Opt. 13, 034005 (2008).
[CrossRef] [PubMed]

J. A. Newmark, W. C. WargerII, C. C. Chang, G. E. Herrera, D. H. Brooks, C. A. DiMarzio, and C. M. Warner, "Determination of the Number of Cells in Preimplantation Embryos by Using Non-invasive Optical Quadrature Microscopy in Conjunction with Differential Interference Contrast Microscopy," Microsc. Microanal. 13, 118-127 (2007).
[CrossRef] [PubMed]

C. M. Warner, J. A. Newmark, M. Comiskey, S. R. De Fazio, D. M. O’Malley, M. Rajadhyaksha, D. J. Townsend, S. McKnight, B. Roysam, P. J. Dwyer, and C. A. DiMarzio, "Genetics and imaging to assess oocyte and preimplantation embryo health," Reprod. Fertil. Dev. 16, 729-741 (2004).
[CrossRef]

D. J. Townsend, K. D. Quarles, A. L. Thomas, W. S. Rockward, C. M. Warner, J. A. Newmark, and C. A. DiMarzio, "Quantitative Phase Measurements Using a Quadrature Tomographic Microscope," Proc. SPIE 4964, 59-65 (2003).
[CrossRef]

J. J. Stott, R. E. Bennett, C. M. Warner, and C. A. DiMarzio, "Three-dimensional imaging with a quadrature tomographic microscope," Proc. SPIE 4261, 24-32 (2001).
[CrossRef]

Y. Glina, G. A. Tsihrintzis, C. M. Warner, D. O. Hogenboom, and C. A. DiMarzio, "On the use of the optical quadrature method in tomographic microscopy," Proc. SPIE  3605, 101-106 (1999).
[CrossRef]

Wax, A.

Wu, J.

Yamaguchi, I.

Yamashita, Y.

Yamauchi, T.

Yang, C.

Yaqoob, Z.

Yatagai, T.

H. Ishiwata, M. Itoh, and T. Yatagai, "A new method of three dimensional measurement by differential interference contrast microscope," Opt. Commun. 260, 117-126 (2006).
[CrossRef]

Zhang, T.

Appl. Opt. (3)

J. Biomed. Opt. (2)

W. C. WargerII, J. A. Newmark, C. M. Warner, and C. A. DiMarzio, "Phase subtraction cell counting method for live mouse embryos beyond the eight-cell stage," J. Biomed. Opt. 13, 034005 (2008).
[CrossRef] [PubMed]

W. C. WargerII, G. S. Laevsky, D. J. Townsend, M. Rajadhyaksha, and C. A. DiMarzio, "Multimodal optical microscope for detecting viability of mouse embryos in vitro," J. Biomed. Opt. 12, 044006 (2007).
[CrossRef] [PubMed]

J. Microsc. (3)

C. L. Tsai, W. C. WargerII, G. S. Laevsky, and C. A. DiMarzio, "Alignment with sub-pixel accuracy for images of multi-modality microscopes using automatic calibration," J. Microsc.,  232, 164-176 (2008).
[CrossRef]

M. R. Arnison, C. J. Cogswell, N. I. Smith, P. W. Fekete, and K. G. Larkin, "Using the Hilbert transform for 3D visualization of differential interference contrast microscope images," J. Microsc. 199, 79-84 (2000).
[CrossRef] [PubMed]

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, "Linear phase imaging using differential interference contrast microscopy," J. Microsc. 214, 7-12 (2004).
[CrossRef] [PubMed]

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

Microsc. Microanal. (1)

J. A. Newmark, W. C. WargerII, C. C. Chang, G. E. Herrera, D. H. Brooks, C. A. DiMarzio, and C. M. Warner, "Determination of the Number of Cells in Preimplantation Embryos by Using Non-invasive Optical Quadrature Microscopy in Conjunction with Differential Interference Contrast Microscopy," Microsc. Microanal. 13, 118-127 (2007).
[CrossRef] [PubMed]

Microsc. Res. Tech. (1)

B. Heise, A. Sonnleitner, and E. P. Klement, "DIC image reconstruction on large cell scans," Microsc. Res. Tech. 66, 312-320 (2005).
[CrossRef] [PubMed]

Nat. Methods (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, "Tomographic Phase Microscopy," Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Opt. Commun. (2)

H. Ishiwata, M. Itoh, and T. Yatagai, "A new method of three dimensional measurement by differential interference contrast microscope," Opt. Commun. 260, 117-126 (2006).
[CrossRef]

A. Barty, K. A. Nugent, A. Roberts, and D. Paganin, "Quantitative phase tomography," Opt. Commun. 175, 329-336 (2000).
[CrossRef]

Opt. Express (2)

Opt. Lett. (10)

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998).
[CrossRef]

F. Charrière, A. Marian, F. Montfort, J. Kuehn, and T. Colomb, "Cell refractive index tomography by digital holographic microscopy," Opt. Lett. 31, 178-180 (2006).
[CrossRef] [PubMed]

E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett. 24, 291-293 (1999).
[CrossRef]

G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, "Fourier phase microscopy for investigation of biological structures and dynamics," Opt. Lett. 29, 2503-2505 (2004).
[CrossRef]

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett. 30, 1165-1167 (2005).
[CrossRef] [PubMed]

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, "Diffraction phase microscopy for quantifying cell structure and dynamics," Opt. Lett. 31, 775-777 (2006).
[CrossRef] [PubMed]

I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997).
[CrossRef] [PubMed]

A. Dubois, L. Vabre, and A. C. Boccara, "Sinusoidally phase modulated interference microscope for high-speed high-resolution topographic imagery," Opt. Lett. 26, 1873-1875 (2001).
[CrossRef]

H. Iwai, C. Fang-Yen, G. Popescu, A. Wax, K. Badizadegan, R. R. Dasari, and M. S. Feld, "Quantitative phase imaging using actively stabilized phase-shifting low-coherence interferometry," Opt. Lett. 29, 2399 (2004).
[CrossRef] [PubMed]

D. O. Hogenboom, C. A. DiMarzio, T. J. Gaudette, A. J. Devaney, and S. C. Lindberg, "Three-dimensional images generated by quadrature interferometry," Opt. Lett. 23, 783-785 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

D. Paganin and K. A. Nugent, "Noninterferometric Phase imaging with partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998).
[CrossRef]

Proc. SPIE (4)

W. C. WargerII and C. A. DiMarzio, "Modeling of optical quadrature microscopy for imaging mouse embryos," Proc. SPIE 6861, 68610T (2008).
[CrossRef]

Y. Glina, G. A. Tsihrintzis, C. M. Warner, D. O. Hogenboom, and C. A. DiMarzio, "On the use of the optical quadrature method in tomographic microscopy," Proc. SPIE  3605, 101-106 (1999).
[CrossRef]

J. J. Stott, R. E. Bennett, C. M. Warner, and C. A. DiMarzio, "Three-dimensional imaging with a quadrature tomographic microscope," Proc. SPIE 4261, 24-32 (2001).
[CrossRef]

D. J. Townsend, K. D. Quarles, A. L. Thomas, W. S. Rockward, C. M. Warner, J. A. Newmark, and C. A. DiMarzio, "Quantitative Phase Measurements Using a Quadrature Tomographic Microscope," Proc. SPIE 4964, 59-65 (2003).
[CrossRef]

Reprod. Fertil. Dev. (1)

C. M. Warner, J. A. Newmark, M. Comiskey, S. R. De Fazio, D. M. O’Malley, M. Rajadhyaksha, D. J. Townsend, S. McKnight, B. Roysam, P. J. Dwyer, and C. A. DiMarzio, "Genetics and imaging to assess oocyte and preimplantation embryo health," Reprod. Fertil. Dev. 16, 729-741 (2004).
[CrossRef]

Rev. Opt., Theor. Instrum. (1)

A. Lebedeff, "Polarization interferometer and its applications," Rev. Opt., Theor. Instrum. 9, 385 (1930).

Other (8)

V. N. Bringi and V. Chandrasekar, Polarimetric Doppler Weather Radar: Principles and Applications (Cambridge University Press, 2001), pp. 37-38.

C. DiMarzio, "Optical quadrature interferometry utilizing polarization to obtain in-phase and quadrature information," U.S. Patent No. 5,883,717, Mar. 16, 1999.

C. DiMarzio, "Optical quadrature interferometer," U.S. Patent No. 6,020,963, Feb. 1, 2000.

L. W. CouchII, Digital and Analog Communication Systems, 4th Ed. (Macmillan Publishing Company, 1993) 303 and 389.

M. Born and E. Wolf, Principles of Optics, 7th Ed. (Cambridge University Press, 1999) 523-525.

K. Creath, "Phase-Measurement Interferometry Techniques," Progress in Optics 26, E. Wolf (Elsevier Science Publishers, 1988), 349-393.
[CrossRef]

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley & Sons, 1998).

S. Braganza and M. Lesser, "An efficient implementation of a phase unwrapping kernel on reconfigurable hardware," Proc. Application-Specific Systems, Architectures, and Processors (IEEE, 2008) 138-143.

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Figures (7)

Fig. 1.
Fig. 1.

Simplified block diagram for quadrature detection in a Doppler radar receiver [4].

Fig. 2.
Fig. 2.

Optical layout for OQM. The x⃗ and y⃗ basis vectors are labeled along the optical path within the individual arms of the interferometer and after recombination. The unlabeled lenses are single element lenses.

Fig. 3.
Fig. 3.

(a) Experimental image of the wavefront mismatch between reference and signal paths. (b) Model of the wavefront mismatch created from a Zernike polynomial fit that included bias, tilt, and focus.

Fig. 4.
Fig. 4.

Images for the noise terms that are associated with the phase reconstructions. The real (Re) and imaginary (Im) parts of the complex noise terms are shown in separate images. All of the dark current noise terms have similar distributions to those shown in (g) and (h).

Fig. 5.
Fig. 5.

Comparison of the SNR model to experimental images of wrapped phase from balanced mixing in Eq. (19), balanced mixing and DC term subtraction in Eq. (20), and balanced mixing, DC term subtraction, and camera normalization in Eq. (21) when the sample images were divided by a blank image created from the same reconstruction. The plot for each phase reconstruction shows the unwrapped phase values through the center of both the model and experimental beads.

Fig. 6.
Fig. 6.

Comparison of the SNR model to experimental images of wrapped phase from balanced mixing in Eq. (19), balanced mixing and DC term subtraction in Eq. (20), and balanced mixing, DC term subtraction, and camera normalization in Eq. (21) when the sample images were not divided by a blank image. The wavefront mismatch between the reference and signal paths is clearly visible in the resultant images. The plot for each phase reconstruction shows the unwrapped phase values through the center of both the model and experimental beads.

Fig. 7.
Fig. 7.

Phase reconstruction error for balanced mixing from (a) Eq. (43) and (b) Eq. (51), balanced mixing and DC term subtraction from (c) Eq. (60) and (d) Eq. (64), and balanced mixing, DC term subtraction, and camera normalization from (e) Eq. (72) and (f) Eq. (76). The plots in the left column show the result of dividing the sample images by a blank image using the same reconstruction, while the plots in the right column do not divide the sample image by a blank. The plots in the right column also assume a constant wavefront mismatch across the field of view to calculate the error with a constant ideal phase.

Equations (108)

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E ref = 1 2 E R e j ( ωt + ϕ ) ( x + y )
E sig = 1 2 E S e j ( ωt + ϕ ) ( x + y ) ,
E ref = 1 2 E R e j ( ωt + ϕ ) ( x + j y ) .
E sig = 1 2 A E S e j ( ωt + ϕ + α ) ( x + y ) .
α = 2 π λ ( n s n 0 ) h ,
E sig 2 + E ref 2 = 1 2 ( E sig + E ref ) 2 + 1 2 ( E sig E ref ) 2 ,
M 0 = 1 2 ( E sig + E ref ) · x 2
M 1 = 1 2 ( E sig + E ref ) · y 2
M 2 = 1 2 ( E sig + E ref ) · x 2
M 3 = 1 2 ( E sig + E ref ) · y 2 ,
M 0 = 1 4 ( A 2 E S E S * + E R E R * + A E S E R * e + A E R E S * e )
M 1 = 1 4 ( A 2 E S E S * + E R E R * j A E S E R * e + j A E R E S * e )
M 2 = 1 4 ( A 2 E S E S * + E R E R * A E S E R * e A E R E S * e )
M 3 = 1 4 ( A 2 E S E S * + E R E R * + j A E S E R * e j A E R E S * e ) .
M 0 = 1 4 ( A 2 E S 2 + E R 2 + A E S E R 2 cos α )
M 1 = 1 4 ( A 2 E S 2 + E R 2 + A E S E R 2 sin α )
M 2 = 1 4 ( A 2 E S 2 + E R 2 + A E S E R 2 cos α )
M 3 = 1 4 ( A 2 E S 2 + E R 2 + A E S E R 2 sin α ) ,
E BM = k = 0 3 j k M k = A e E S E R ,
E BM , DC = k = 0 3 j k ( M k R k S k + D k ) = A e E S E R ,
E BM , DC , Norm = k = 0 3 j k M k R k S k + D k R k D k = A e E S E R E R .
E BM , DC , Norm , blank = E S E R E R .
E BM , DC , Norm E BM , DC , Norm , blank = A e .
E sig = 1 2 ( E S e j ( ωt + ϕ ) + E N e j ( ωt + ζ ) ) × ( x + y )
E ref = 1 2 ( E R e j ( ωt + ϕ ) + E N e j ( ωt + ζ ) ) × ( x + j y ) ,
E sig = 1 2 ( A E S e j ( ωt + ϕ + α ) + A E N e j ( ωt + ζ + α ) ) × ( x + y ) .
E ref = 1 2 ( E R e j ( ωt + ϕ ) + E N e j ( ωt + ζ ) ) X r e j χ r E C k R × ( x + j y )
E sig = 1 2 ( A E S e j ( ωt + ϕ + α ) + A E N e j ( ωt + ζ + α ) ) X s e j χ s E C k S × ( x + y ) ,
i k = η k q A pixel I k + i D , k ,
M k = 1 4 η k q A pixel [ ( A 2 E S E S + A 2 E S E N e j ( ϕ ζ ) ) E C k S E C k S X s X s + ( j ) k ( A E S E R e + A E S E N e j ( ϕ + α ζ ) ) E C k S E C k R X s X r e j ( χ s χ r ) +
( A 2 E N E S e j ( ϕ ζ ) + A 2 E N E N ) E C k S E C k S X s X s + ( j ) k ( A E N E R e j ( ϕ ζ α ) + A E N E N e ) E C k S E C k R X s X r e j ( χ s χ r ) +
j k ( A E R E S e + A E R E N e j ( ϕ ζ α ) ) E C k R E C k S X r X s e j ( χ s χ r ) + ( E R E R + E R E N e j ( ϕ ζ ) ) E C k R E C k R X r X r +
j k ( A E N E S e j ( ϕ + α ζ ) + A E N E N e ) E C k R E C k S X r X s e j ( χ s χ r ) + ( E N E R e j ( ϕ ζ ) + E N E N ) E C k R E C k R X r X r + i M , k
E BM = 1 4 A pixel q [ A e ( E S E R + E S E N e j ( ϕ ζ ) + E N E R e j ( ϕ ζ ) + E N 2 ) X s X r e j ( χ s χ r ) k = 0 3 η k E C k S E C k R +
A e ( E R E S + E R E N e j ( ϕ ζ ) + E N E S e j ( ϕ ζ ) + E N 2 ) X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k R E C k S +
A 2 ( E S 2 + E S E N ( e j ( ϕ ζ ) + e j ( ϕ ζ ) ) + E N 2 ) X s 2 k = 0 3 j k η k ( E C k S ) 2 +
( E R 2 + E R E N ( e j ( ϕ ζ ) + e j ( ϕ ζ ) ) + E N 2 ) X r 2 k = 0 3 j k η k ( E C k R ) 2 ] + k = 0 3 j k i M , k .
E BM = A pixel q A e E S E R ,
i M , k = 0
η k = 1 ,
E C k R = E C k S = 1 ,
X s e j χ s = X r e j χ r = 1 ,
E N e j ( ωt + ζ ) = 0 .
S = 1 4 A pixel q [ A e E S E R X s X r e j ( χ s χ r ) k = 0 3 η k E C k S E C k R + A e E R E S X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k S E C k R +
A 2 E S E S X s X s k = 0 3 j k η k E C k S E C k S + E R E R X r X r k = 0 3 j k η k E C k R E C k R ] + k = 0 3 j k i M , k
N S = 1 4 A pixel q [ A e ( E S E N e j ( ϕ ζ ) + E N E R e j ( ϕ ζ ) + E N E N ) X s X r e j ( χ s χ r ) k = 0 3 η k E C k S E C k R +
A e ( E R E N e j ( ϕ ζ ) + E N E S e j ( ϕ ζ ) + E N E N ) X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k S E C k R +
A 2 ( E S E N e j ( ϕ ζ ) + E N E S e j ( ϕ ζ ) + E N E N ) X s X s k = 0 3 ( j ) k η k E C k S E C k S +
( E R E N e j ( ϕ ζ ) + E N E R e j ( ϕ ζ ) + E N E N ) X r X r k = 0 3 ( j ) k η k E C k R E C k R ] ,
B = 1 4 A pixel q [ E S E R X s X r e j ( χ s χ r ) k = 0 3 η k E C k S E C k R + E R E S X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k S E C k R +
E S E S X s X s k = 0 3 j k η k E C k S E C k S + E R E R X r X r k = 0 3 ( j ) k η k E C k R E C k R ] + k = 0 3 j k i BM , k
N B = 1 4 A pixel q [ ( E S E B e j ( ϕ β ) + E B E R e j ( ϕ β ) + E B E B ) X s X r e j ( χ s χ r ) k = 0 3 η k E C k S E C k R + ]
( E R E B e j ( ϕ β ) + E B E S e j ( ϕ β ) + E B E B ) X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k S E C k R +
( E S E B e j ( ϕ β ) + E B E S e j ( ϕ β ) + E B E B ) X s X s k = 0 3 j k η k E C k S E C k S +
( E R E B e j ( ϕ β ) + E B E R e j ( ϕ β ) + E B E B ) X r X r k = 0 3 j k η k E C k R E C k R ] .
S + N S B + N B S B + N S B N B S B 2 S B
E BM = A e [ 1 Φ e j 2 ( χ s χ r ) ( Γ S + Γ R ) e j ( χ s χ r ) I B , M e j ( χ s χ r ) ] +
A e [ Φ e j 2 ( χ s χ r ) Φ ( Γ S + Γ R ) e j 3 ( χ s χ r ) Φ 2 e j 4 ( χ s χ r ) ] +
( A 2 Γ S + Γ R ) [ e j ( χ s χ r ) ( Γ S + Γ R ) e j 2 ( χ s χ r ) Φ e j 3 ( χ s χ r ) ] + I S , M e j ( χ s χ r )
Φ = k = 0 3 ( 1 ) k R k B S k k = 0 3 R k B S k
Γ R = k = 0 3 j k R k k = 0 3 B S k R k
Γ S = k = 0 3 j k B S k k = 0 3 R k B S k
I S , M = k = 0 3 j k i M , k k = 0 3 R k B S k
I B , M = k = 0 3 j k i BM , k k = 0 3 R k B S k
B S k 1 4 A pixel q η k E C k S X s E S 2
R k 1 4 A pixel q η k E C k R X r E R 2 .
E BM , NB = [ A e + A e Φ e j 2 ( χ s χ r ) + ( A 2 Γ S + Γ R ) e j ( χ s χ r ) ] e j ( χ s χ r ) k = 0 3 R k B S k + I M
I M = k = 0 3 j k i M , k .
S k = 1 4 η k q A pixel ( A 2 E S 2 + A 2 E S E N ( e j ( ϕ ζ ) + e j ( ϕ ζ ) ) + A 2 E N 2 ) ( E C k S X s ) 2 + i S , k ,
R k = 1 4 η k q A pixel ( E R 2 + E R E N ( e j ( ϕ ζ ) + e j ( ϕ ζ ) ) + E N 2 ) ( E C k R X r ) 2 + i R , k ,
E BM , DC = 1 4 A pixel q [ A e ( E S E R + E S E N e j ( ϕ ζ ) + E N E R e j ( ϕ ζ ) + E N 2 ) X s X r e j ( χ s χ r ) k = 0 3 η k E C k S E C k R +
A e ( E R E S + E R E N e j ( ϕ ζ ) + E N E S e j ( ϕ ζ ) + E N 2 ) X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k R E C k S ] +
k = 0 3 j k ( i M , k i R , k i S , k + i D . , k ) ,
S = 1 4 A pixel q [ A e E S E R X s X r e j ( χ s χ r ) k = 0 3 η k E C k S E C k R +
A e E R E S X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k R E C k S ] + k = 0 3 j k ( i M , k i R , k i S , k + i D , k )
N S = 1 4 A pixel q [ A e ( E S E N e j ( ϕ ζ ) + E N E R e j ( ϕ ζ ) + E N 2 ) X s X r e j ( χ s χ r ) k = 0 3 η k E C k S E C k R +
A e ( E R E N e j ( ϕ ζ ) + E N E S e j ( ϕ ζ ) + E N 2 ) X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k R E C k S ] .
B = 1 4 A pixel q [ E S E R X s X r e j ( χ s χ r ) k = 0 3 η k E C k S E C k R + E R E S X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k R E C k S ] +
k = 0 3 j k ( i BM , k i BR , k i BS , k + i D , k )
N B = 1 4 A pixel q [ ( E S E B e j ( ϕ β ) + E B E R e j ( ϕ β ) + E B 2 ) X s X r e j ( χ s χ r ) k = 0 3 η k E C k S E C k R +
( E R E B e j ( ϕ β ) + E B E S e j ( ϕ β ) + E B 2 ) X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k R E C k S ] ,
E BM , DC = A e [ 1 Φ e j 2 ( χ s χ r ) I B e j ( χ s χ r ) ] +
A e [ Φ e j 2 ( χ s χ r ) Φ 2 e j 4 ( χ s χ r ) ] + I S e j ( χ s χ r ) ,
Φ = k = 0 3 ( 1 ) k R k B S k k = 0 3 R k B S k
I S = k = 0 3 j k ( i M , k i R , k i S , k + i D , k ) k = 0 3 R k B S k
I B = k = 0 3 j k ( i BM , k i BR , k i BS , k + i D , k ) k = 0 3 R k B S k .
E BM , DC , NB = [ A e + A e Φ e j 2 ( χ s χ r ) ] e j ( χ s χ r ) k = 0 3 R k B S k + I BM
I BM = k = 0 3 j k ( i M , k i R , k i S , k + i D , k ) .
R k i D , k 1 2 η k q A pixel E C k R X r E R + E N ,
E BM , DC , Norm = 1 2 E R + E N A pixel q [ A e ( E S E R + E S E N e j ( ϕ ζ ) + E N E R e j ( ϕ ζ ) + E N 2 ) X s X r e j ( χ s χ r ) k = 0 3 η k E C k S E C k R E C k R X r +
A e ( E R E S + E R E N e j ( ϕ ζ ) + E N E S e j ( ϕ ζ ) + E N 2 ) X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k S E C k R E C k R X r ] +
k = 0 3 j k i M , k i R , k i S , k + i D , k 1 2 η k q A pixel E R E C k R X r + E N E C k R X r ,
S = 1 2 A pixel q 1 E R + E N [ A e E S E R X s X r e j ( χ s χ r ) k = 0 3 η k E C k S E C k R E C k R X r +
A e E R E S X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k S E C k R E C k R X r + k = 0 3 j k i M , k i R , k i S , k + i D , k 1 4 A pixel q η k E C k R X r E R + E N
N S = 1 2 A pixel q 1 E R + E N [ A e ( E S E N e j ( ϕ ζ ) + E N E R e j ( ϕ ζ ) + E N 2 ) X s X r e j ( χ s χ r ) k = 0 3 η k E C k S E C k R E C k R X r +
A e ( E R E N e j ( ϕ ζ ) + E N E S e j ( ϕ ζ ) + E N 2 ) X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k S E C k R E C k R X r ] .
B = 1 2 A pixel q 1 E R + E B [ E S E R X s X r e j ( χ s r ) k = 0 3 η k E C k S E C k R E C k R X r +
E R E S X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k S E C k R E C k R X r + k = 0 3 j k i BM , k i BR , k i BS , k + i D , k 1 4 A pixel q η k E C k R X r E R + E B ]
N B = 1 2 A pixel q 1 E R + E B [ ( E S E B e j ( ϕ β ) + E B E R e j ( ϕ β ) + E B 2 ) X s X r e j ( χ s χ r ) k = 0 3 η k E C k S E C k R E C k R X r +
( E R E B e j ( ϕ β ) + E B E S e j ( ϕ β ) + E B 2 ) X r X s e j ( χ s χ r ) k = 0 3 ( 1 ) k η k E C k S E C k R E C k R X r ] .
E BM , DC , Norm = A e [ 1 Ψ e j 2 ( χ s χ r ) I B , Norm e j ( χ s χ r ) ] +
A e [ Ψ e j 2 ( χ s χ r ) Ψ 2 e j 4 ( χ s χ r ) ] + I S , Norm e j ( χ s χ r )
Ψ = k = 0 3 ( 1 ) k B S k k = 0 3 B S k
I S , Norm = k = 0 3 j k i M , k i R , k i S , k + i D , k R k k = 0 3 B S k
I B , Norm = k = 0 3 j k i BM , k i BR , k i BS , k + i D , k R k k = 0 3 B S k .
E BM , DC , Norm , NB = [ A e + A e Ψ e j 2 ( χ s χ r ) ] e j ( χ s χ r ) k = 0 3 B S k + I BM , Norm
I BM , Norm = k = 0 3 j k i M , k i R , k i S , k + i D , k R k .
E BM , DC , Norm , blank = [ 1 + Ψ e j 2 ( χ s χ r ) ] e j ( χ s χ r ) k = 0 3 B S k + I BM , Norm .

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