Abstract

Knowledge of the surface geometry of an imaging subject is important in many applications. This information can be obtained via a number of different techniques, including time of flight imaging, photogrammetry, and fringe projection profilometry. Existing systems may have restrictions on instrument geometry, require expensive optics, or require moving parts in order to image the full surface of the subject. An inexpensive generalised fringe projection profilometry system is proposed that can account for arbitrarily placed components and use mirrors to expand the field of view. It simultaneously acquires multiple views of an imaging subject, producing a cloud of points that lie on its surface, which can then be processed to form a three dimensional model. A prototype of this system was integrated into an existing Diffuse Optical Tomography and Bioluminescence Tomography small animal imaging system and used to image objects including a mouse-shaped plastic phantom, a mouse cadaver, and a coin. A surface mesh generated from surface capture data of the mouse-shaped plastic phantom was compared with ideal surface points provided by the phantom manufacturer, and 50% of points were found to lie within 0.1mm of the surface mesh, 82% of points were found to lie within 0.2mm of the surface mesh, and 96% of points were found to lie within 0.4mm of the surface mesh.

© 2013 OSA

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2011 (2)

J. Guggenheim, H. Dehghani, H. Basevi, I. Styles, and J. Frampton, “Development of a multi-view, multi-spectral bioluminescence tomography small animal imaging system,” Proc. SPIE8088, 80881K (2011).
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J. Geng, “Structured-light 3d surface imaging: a tutorial,” Adv. Opt. Photon.3, 128–160 (2011).
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2010 (2)

S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Laser. Eng.48, 133–140 (2010).
[CrossRef]

A. Cong, W. Cong, Y. Lu, P. Santago, A. Chatziioannou, and G. Wang, “Differential evolution approach for regularized bioluminescence tomography,” IEEE Trans. Biomed. Eng.57, 2229–2238 (2010).
[CrossRef] [PubMed]

2009 (3)

2008 (3)

T. Lasser, A. Soubret, J. Ripoll, and V. Ntziachristos, “Surface reconstruction for free-space 360° fluorescence molecular tomography and the effects of animal motion,” IEEE Trans. Med. Imag.27, 188–194 (2008).
[CrossRef]

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[CrossRef]

A. Kumar, S. Raymond, A. Dunn, B. Bacskai, and D. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imag.27, 1152–1163 (2008).
[CrossRef]

2007 (5)

M. Allard, D. Côté, L. Davidson, J. Dazai, and R. Henkelman, “Combined magnetic resonance and bioluminescence imaging of live mice,” J. Biomed. Opt.12, 034018 (2007).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

2006 (2)

G. Zavattini, S. Vecchi, G. Mitchell, U. Weisser, R. Leahy, B. Pichler, D. Smith, and S. Cherry, “A hyperspectral fluorescence system for 3d in vivo optical imaging,” Phys. Med. Biol.51, 2029–2043 (2006).
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[CrossRef] [PubMed]

2005 (3)

A. Chaudhari, F. Darvas, J. Bading, R. Moats, P. Conti, D. Smith, S. Cherry, and R. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

E. Li, X. Peng, J. Xi, J. Chicharo, J. Yao, and D. Zhang, “Multi-frequency and multiple phase-shift sinusoidal fringe projection for 3d profilometry,” Opt. Express13, 1561–1569 (2005).
[CrossRef] [PubMed]

A. Gibson, J. Hebden, and S. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005).
[CrossRef] [PubMed]

2004 (2)

J. Salvi, J. Pages, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn.37, 827–849 (2004).
[CrossRef]

R. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imag.23, 492–500 (2004).
[CrossRef]

2003 (3)

R. Schulz, J. Ripoll, and V. Ntziachristos, “Noncontact optical tomography of turbid media,” Opt. Lett.28, 1701–1703 (2003).
[CrossRef] [PubMed]

F. Berryman, P. Pynsent, and J. Cubillo, “A theoretical comparison of three fringe analysis methods for determining the three-dimensional shape of an object in the presence of noise,” Opt. Laser. Eng.39, 35–50 (2003).
[CrossRef]

B. Brooksby, H. Dehghani, B. Pogue, and K. Paulsen, “Near-infrared (nir) tomography breast image reconstruction with a priori structural information from mri: algorithm development for reconstructing heterogeneities,” IEEE J. Sel. Topics Quantum Electron.9, 199–209 (2003).
[CrossRef]

2001 (2)

R. Lange and P. Seitz, “Solid-state time-of-flight range camera,” IEEE J. Quantum Electron.37, 390–397 (2001).
[CrossRef]

X. Su and W. Chen, “Fourier transform profilometry:: a review,” Opt. Laser. Eng.35, 263–284 (2001).
[CrossRef]

1999 (1)

1997 (2)

1994 (1)

T. Judge and P. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Laser. Eng.21, 199–239 (1994).
[CrossRef]

1984 (1)

1983 (1)

1982 (1)

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A72, 156–160 (1982).
[CrossRef]

Ahn, S.

Allard, M.

M. Allard, D. Côté, L. Davidson, J. Dazai, and R. Henkelman, “Combined magnetic resonance and bioluminescence imaging of live mice,” J. Biomed. Opt.12, 034018 (2007).
[CrossRef] [PubMed]

Arridge, S.

A. Gibson, J. Hebden, and S. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005).
[CrossRef] [PubMed]

S. Arridge and M. Schweiger, “Image reconstruction in optical tomography.” Phil. Trans. R. Soc. B352, 717–726 (1997).
[CrossRef] [PubMed]

Bacskai, B.

A. Kumar, S. Raymond, A. Dunn, B. Bacskai, and D. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imag.27, 1152–1163 (2008).
[CrossRef]

Bading, J.

A. Chaudhari, F. Darvas, J. Bading, R. Moats, P. Conti, D. Smith, S. Cherry, and R. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Basevi, H.

J. Guggenheim, H. Dehghani, H. Basevi, I. Styles, and J. Frampton, “Development of a multi-view, multi-spectral bioluminescence tomography small animal imaging system,” Proc. SPIE8088, 80881K (2011).
[CrossRef]

J. Guggenheim, H. Basevi, I. Styles, J. Frampton, and H. Dehghani, “Multi-view, multi-spectral bioluminescence tomography,” in Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BW4A.7.

Batlle, J.

J. Salvi, J. Pages, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn.37, 827–849 (2004).
[CrossRef]

Berryman, F.

F. Berryman, P. Pynsent, and J. Cubillo, “A theoretical comparison of three fringe analysis methods for determining the three-dimensional shape of an object in the presence of noise,” Opt. Laser. Eng.39, 35–50 (2003).
[CrossRef]

Bi, H.

Boas, D.

A. Kumar, S. Raymond, A. Dunn, B. Bacskai, and D. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imag.27, 1152–1163 (2008).
[CrossRef]

Brooksby, B.

B. Brooksby, H. Dehghani, B. Pogue, and K. Paulsen, “Near-infrared (nir) tomography breast image reconstruction with a priori structural information from mri: algorithm development for reconstructing heterogeneities,” IEEE J. Sel. Topics Quantum Electron.9, 199–209 (2003).
[CrossRef]

Bryanston-Cross, P.

T. Judge and P. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Laser. Eng.21, 199–239 (1994).
[CrossRef]

Busca, G.

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to fourier-transform profilometry,” Opt. Laser. Eng.46, 106–116 (2008).
[CrossRef]

Cable, M.

D. Nilson, M. Cable, B. Rice, and K. Kearney, “Structured light imaging apparatus,” U.S. Patent 7,298,415 (2007).

B. Rice, M. Cable, and K. Kearney, “3d in-vivo imaging and topography using structured light,” U.S. Patent 7,797,034 (2010).

D. Stearns, B. Rice, and M. Cable, “Method and apparatus for 3-d imaging of internal light sources,” U.S. Patent 7,860,549 (2010).

Cao, L.

X. Jiang, L. Cao, W. Semmler, and J. Peter, “A surface recognition approach for in vivo optical imaging applications using a micro-lens-array light detector,” in Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BTu3A.1.

Carnegie, D.

A. Dorrington, M. Cree, A. Payne, R. Conroy, and D. Carnegie, “Achieving sub-millimetre precision with a solid-state full-field heterodyning range imaging camera,” Meas. Sci. Technol.18, 2809–2816 (2007).
[CrossRef]

Carocci, M.

Carpenter, C.

H. Dehghani, M. Eames, P. Yalavarthy, S. Davis, S. Srinivasan, C. Carpenter, B. Pogue, and K. Paulsen, “Near infrared optical tomography using nirfast: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods En.25, 711–732 (2009).
[CrossRef]

Chatziioannou, A.

A. Cong, W. Cong, Y. Lu, P. Santago, A. Chatziioannou, and G. Wang, “Differential evolution approach for regularized bioluminescence tomography,” IEEE Trans. Biomed. Eng.57, 2229–2238 (2010).
[CrossRef] [PubMed]

Chaudhari, A.

A. Chaudhari, F. Darvas, J. Bading, R. Moats, P. Conti, D. Smith, S. Cherry, and R. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Chen, W.

X. Mao, W. Chen, and X. Su, “Improved fourier-transform profilometry,” Appl. Opt.46, 664–668 (2007).
[CrossRef] [PubMed]

X. Su and W. Chen, “Fourier transform profilometry:: a review,” Opt. Laser. Eng.35, 263–284 (2001).
[CrossRef]

Cherry, S.

C. Li, G. Mitchell, J. Dutta, S. Ahn, R. Leahy, and S. Cherry, “A three-dimensional multispectral fluorescence optical tomography imaging system for small animals based on a conical mirror design,” Opt. Express17, 7571–7585 (2009).
[CrossRef] [PubMed]

G. Zavattini, S. Vecchi, G. Mitchell, U. Weisser, R. Leahy, B. Pichler, D. Smith, and S. Cherry, “A hyperspectral fluorescence system for 3d in vivo optical imaging,” Phys. Med. Biol.51, 2029–2043 (2006).
[CrossRef] [PubMed]

A. Chaudhari, F. Darvas, J. Bading, R. Moats, P. Conti, D. Smith, S. Cherry, and R. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Chicharo, J.

Cong, A.

A. Cong, W. Cong, Y. Lu, P. Santago, A. Chatziioannou, and G. Wang, “Differential evolution approach for regularized bioluminescence tomography,” IEEE Trans. Biomed. Eng.57, 2229–2238 (2010).
[CrossRef] [PubMed]

Cong, W.

A. Cong, W. Cong, Y. Lu, P. Santago, A. Chatziioannou, and G. Wang, “Differential evolution approach for regularized bioluminescence tomography,” IEEE Trans. Biomed. Eng.57, 2229–2238 (2010).
[CrossRef] [PubMed]

Conroy, R.

A. Dorrington, M. Cree, A. Payne, R. Conroy, and D. Carnegie, “Achieving sub-millimetre precision with a solid-state full-field heterodyning range imaging camera,” Meas. Sci. Technol.18, 2809–2816 (2007).
[CrossRef]

Conti, P.

A. Chaudhari, F. Darvas, J. Bading, R. Moats, P. Conti, D. Smith, S. Cherry, and R. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Coquoz, O.

C. Kuo, O. Coquoz, T. Troy, H. Xu, and B. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt.12, 024007 (2007).
[CrossRef] [PubMed]

Côté, D.

M. Allard, D. Côté, L. Davidson, J. Dazai, and R. Henkelman, “Combined magnetic resonance and bioluminescence imaging of live mice,” J. Biomed. Opt.12, 034018 (2007).
[CrossRef] [PubMed]

Cree, M.

A. Dorrington, M. Cree, A. Payne, R. Conroy, and D. Carnegie, “Achieving sub-millimetre precision with a solid-state full-field heterodyning range imaging camera,” Meas. Sci. Technol.18, 2809–2816 (2007).
[CrossRef]

Cubillo, J.

F. Berryman, P. Pynsent, and J. Cubillo, “A theoretical comparison of three fringe analysis methods for determining the three-dimensional shape of an object in the presence of noise,” Opt. Laser. Eng.39, 35–50 (2003).
[CrossRef]

Darvas, F.

A. Chaudhari, F. Darvas, J. Bading, R. Moats, P. Conti, D. Smith, S. Cherry, and R. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Davidson, L.

M. Allard, D. Côté, L. Davidson, J. Dazai, and R. Henkelman, “Combined magnetic resonance and bioluminescence imaging of live mice,” J. Biomed. Opt.12, 034018 (2007).
[CrossRef] [PubMed]

Davis, S.

H. Dehghani, M. Eames, P. Yalavarthy, S. Davis, S. Srinivasan, C. Carpenter, B. Pogue, and K. Paulsen, “Near infrared optical tomography using nirfast: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods En.25, 711–732 (2009).
[CrossRef]

Dazai, J.

M. Allard, D. Côté, L. Davidson, J. Dazai, and R. Henkelman, “Combined magnetic resonance and bioluminescence imaging of live mice,” J. Biomed. Opt.12, 034018 (2007).
[CrossRef] [PubMed]

Dehghani, H.

J. Guggenheim, H. Dehghani, H. Basevi, I. Styles, and J. Frampton, “Development of a multi-view, multi-spectral bioluminescence tomography small animal imaging system,” Proc. SPIE8088, 80881K (2011).
[CrossRef]

H. Dehghani, M. Eames, P. Yalavarthy, S. Davis, S. Srinivasan, C. Carpenter, B. Pogue, and K. Paulsen, “Near infrared optical tomography using nirfast: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods En.25, 711–732 (2009).
[CrossRef]

B. Brooksby, H. Dehghani, B. Pogue, and K. Paulsen, “Near-infrared (nir) tomography breast image reconstruction with a priori structural information from mri: algorithm development for reconstructing heterogeneities,” IEEE J. Sel. Topics Quantum Electron.9, 199–209 (2003).
[CrossRef]

J. Guggenheim, H. Basevi, I. Styles, J. Frampton, and H. Dehghani, “Multi-view, multi-spectral bioluminescence tomography,” in Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BW4A.7.

Dorrington, A.

A. Dorrington, M. Cree, A. Payne, R. Conroy, and D. Carnegie, “Achieving sub-millimetre precision with a solid-state full-field heterodyning range imaging camera,” Meas. Sci. Technol.18, 2809–2816 (2007).
[CrossRef]

Du, H.

Dunn, A.

A. Kumar, S. Raymond, A. Dunn, B. Bacskai, and D. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imag.27, 1152–1163 (2008).
[CrossRef]

Dutta, J.

Eames, M.

H. Dehghani, M. Eames, P. Yalavarthy, S. Davis, S. Srinivasan, C. Carpenter, B. Pogue, and K. Paulsen, “Near infrared optical tomography using nirfast: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods En.25, 711–732 (2009).
[CrossRef]

Economou, E.

Frampton, J.

J. Guggenheim, H. Dehghani, H. Basevi, I. Styles, and J. Frampton, “Development of a multi-view, multi-spectral bioluminescence tomography small animal imaging system,” Proc. SPIE8088, 80881K (2011).
[CrossRef]

J. Guggenheim, H. Basevi, I. Styles, J. Frampton, and H. Dehghani, “Multi-view, multi-spectral bioluminescence tomography,” in Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BW4A.7.

Garofalakis, A.

Geng, J.

Geng, Z.

Z. Geng, “Method and apparatus for omnidirectional three dimensional imaging,” U.S. Patent 6,744,569 (2004).

Z. Geng, “Diffuse optical tomography system and method of use,” U.S. Patent 7,242,997 (2007).

Gibson, A.

A. Gibson, J. Hebden, and S. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005).
[CrossRef] [PubMed]

Gorthi, S.

S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Laser. Eng.48, 133–140 (2010).
[CrossRef]

Guggenheim, J.

J. Guggenheim, H. Dehghani, H. Basevi, I. Styles, and J. Frampton, “Development of a multi-view, multi-spectral bioluminescence tomography small animal imaging system,” Proc. SPIE8088, 80881K (2011).
[CrossRef]

J. Guggenheim, H. Basevi, I. Styles, J. Frampton, and H. Dehghani, “Multi-view, multi-spectral bioluminescence tomography,” in Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BW4A.7.

Halioua, M.

Hebden, J.

A. Gibson, J. Hebden, and S. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005).
[CrossRef] [PubMed]

Henkelman, R.

M. Allard, D. Côté, L. Davidson, J. Dazai, and R. Henkelman, “Combined magnetic resonance and bioluminescence imaging of live mice,” J. Biomed. Opt.12, 034018 (2007).
[CrossRef] [PubMed]

Huntley, J.

Ina, H.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A72, 156–160 (1982).
[CrossRef]

Jiang, X.

X. Jiang, L. Cao, W. Semmler, and J. Peter, “A surface recognition approach for in vivo optical imaging applications using a micro-lens-array light detector,” in Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BTu3A.1.

Judge, T.

T. Judge and P. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Laser. Eng.21, 199–239 (1994).
[CrossRef]

Kearney, K.

D. Nilson, M. Cable, B. Rice, and K. Kearney, “Structured light imaging apparatus,” U.S. Patent 7,298,415 (2007).

B. Rice, M. Cable, and K. Kearney, “3d in-vivo imaging and topography using structured light,” U.S. Patent 7,797,034 (2010).

Kioussis, D.

Kobayashi, S.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A72, 156–160 (1982).
[CrossRef]

Kraus, K.

K. Kraus, Photogrammetry: Geometry from Images and Laser Scans (de Gruyter, 2007).
[CrossRef]

Kumar, A.

A. Kumar, S. Raymond, A. Dunn, B. Bacskai, and D. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imag.27, 1152–1163 (2008).
[CrossRef]

Kuo, C.

C. Kuo, O. Coquoz, T. Troy, H. Xu, and B. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt.12, 024007 (2007).
[CrossRef] [PubMed]

Lange, R.

R. Lange and P. Seitz, “Solid-state time-of-flight range camera,” IEEE J. Quantum Electron.37, 390–397 (2001).
[CrossRef]

Lasser, T.

T. Lasser, A. Soubret, J. Ripoll, and V. Ntziachristos, “Surface reconstruction for free-space 360° fluorescence molecular tomography and the effects of animal motion,” IEEE Trans. Med. Imag.27, 188–194 (2008).
[CrossRef]

Leahy, R.

C. Li, G. Mitchell, J. Dutta, S. Ahn, R. Leahy, and S. Cherry, “A three-dimensional multispectral fluorescence optical tomography imaging system for small animals based on a conical mirror design,” Opt. Express17, 7571–7585 (2009).
[CrossRef] [PubMed]

G. Zavattini, S. Vecchi, G. Mitchell, U. Weisser, R. Leahy, B. Pichler, D. Smith, and S. Cherry, “A hyperspectral fluorescence system for 3d in vivo optical imaging,” Phys. Med. Biol.51, 2029–2043 (2006).
[CrossRef] [PubMed]

A. Chaudhari, F. Darvas, J. Bading, R. Moats, P. Conti, D. Smith, S. Cherry, and R. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Li, C.

Li, E.

Liu, H.

Lu, Y.

A. Cong, W. Cong, Y. Lu, P. Santago, A. Chatziioannou, and G. Wang, “Differential evolution approach for regularized bioluminescence tomography,” IEEE Trans. Biomed. Eng.57, 2229–2238 (2010).
[CrossRef] [PubMed]

Mamalaki, C.

Mao, X.

Meyer, H.

Mitchell, G.

C. Li, G. Mitchell, J. Dutta, S. Ahn, R. Leahy, and S. Cherry, “A three-dimensional multispectral fluorescence optical tomography imaging system for small animals based on a conical mirror design,” Opt. Express17, 7571–7585 (2009).
[CrossRef] [PubMed]

G. Zavattini, S. Vecchi, G. Mitchell, U. Weisser, R. Leahy, B. Pichler, D. Smith, and S. Cherry, “A hyperspectral fluorescence system for 3d in vivo optical imaging,” Phys. Med. Biol.51, 2029–2043 (2006).
[CrossRef] [PubMed]

Moats, R.

A. Chaudhari, F. Darvas, J. Bading, R. Moats, P. Conti, D. Smith, S. Cherry, and R. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Mutoh, K.

Nilson, D.

D. Nilson, M. Cable, B. Rice, and K. Kearney, “Structured light imaging apparatus,” U.S. Patent 7,298,415 (2007).

Ntziachristos, V.

T. Lasser, A. Soubret, J. Ripoll, and V. Ntziachristos, “Surface reconstruction for free-space 360° fluorescence molecular tomography and the effects of animal motion,” IEEE Trans. Med. Imag.27, 188–194 (2008).
[CrossRef]

H. Meyer, A. Garofalakis, G. Zacharakis, S. Psycharakis, C. Mamalaki, D. Kioussis, E. Economou, V. Ntziachristos, and J. Ripoll, “Noncontact optical imaging in mice with full angular coverage and automatic surface extraction,” Appl. Opt.46, 3617–3627 (2007).
[CrossRef] [PubMed]

R. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imag.23, 492–500 (2004).
[CrossRef]

R. Schulz, J. Ripoll, and V. Ntziachristos, “Noncontact optical tomography of turbid media,” Opt. Lett.28, 1701–1703 (2003).
[CrossRef] [PubMed]

Pages, J.

J. Salvi, J. Pages, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn.37, 827–849 (2004).
[CrossRef]

Park, S.

Paulsen, K.

H. Dehghani, M. Eames, P. Yalavarthy, S. Davis, S. Srinivasan, C. Carpenter, B. Pogue, and K. Paulsen, “Near infrared optical tomography using nirfast: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods En.25, 711–732 (2009).
[CrossRef]

B. Brooksby, H. Dehghani, B. Pogue, and K. Paulsen, “Near-infrared (nir) tomography breast image reconstruction with a priori structural information from mri: algorithm development for reconstructing heterogeneities,” IEEE J. Sel. Topics Quantum Electron.9, 199–209 (2003).
[CrossRef]

Payne, A.

A. Dorrington, M. Cree, A. Payne, R. Conroy, and D. Carnegie, “Achieving sub-millimetre precision with a solid-state full-field heterodyning range imaging camera,” Meas. Sci. Technol.18, 2809–2816 (2007).
[CrossRef]

Peng, X.

Peter, J.

X. Jiang, L. Cao, W. Semmler, and J. Peter, “A surface recognition approach for in vivo optical imaging applications using a micro-lens-array light detector,” in Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BTu3A.1.

Pichler, B.

G. Zavattini, S. Vecchi, G. Mitchell, U. Weisser, R. Leahy, B. Pichler, D. Smith, and S. Cherry, “A hyperspectral fluorescence system for 3d in vivo optical imaging,” Phys. Med. Biol.51, 2029–2043 (2006).
[CrossRef] [PubMed]

Pogue, B.

H. Dehghani, M. Eames, P. Yalavarthy, S. Davis, S. Srinivasan, C. Carpenter, B. Pogue, and K. Paulsen, “Near infrared optical tomography using nirfast: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods En.25, 711–732 (2009).
[CrossRef]

B. Brooksby, H. Dehghani, B. Pogue, and K. Paulsen, “Near-infrared (nir) tomography breast image reconstruction with a priori structural information from mri: algorithm development for reconstructing heterogeneities,” IEEE J. Sel. Topics Quantum Electron.9, 199–209 (2003).
[CrossRef]

Psycharakis, S.

Pynsent, P.

F. Berryman, P. Pynsent, and J. Cubillo, “A theoretical comparison of three fringe analysis methods for determining the three-dimensional shape of an object in the presence of noise,” Opt. Laser. Eng.39, 35–50 (2003).
[CrossRef]

Rastogi, P.

S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Laser. Eng.48, 133–140 (2010).
[CrossRef]

Raymond, S.

A. Kumar, S. Raymond, A. Dunn, B. Bacskai, and D. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imag.27, 1152–1163 (2008).
[CrossRef]

Rice, B.

C. Kuo, O. Coquoz, T. Troy, H. Xu, and B. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt.12, 024007 (2007).
[CrossRef] [PubMed]

D. Nilson, M. Cable, B. Rice, and K. Kearney, “Structured light imaging apparatus,” U.S. Patent 7,298,415 (2007).

B. Rice, M. Cable, and K. Kearney, “3d in-vivo imaging and topography using structured light,” U.S. Patent 7,797,034 (2010).

D. Stearns, B. Rice, and M. Cable, “Method and apparatus for 3-d imaging of internal light sources,” U.S. Patent 7,860,549 (2010).

Ripoll, J.

T. Lasser, A. Soubret, J. Ripoll, and V. Ntziachristos, “Surface reconstruction for free-space 360° fluorescence molecular tomography and the effects of animal motion,” IEEE Trans. Med. Imag.27, 188–194 (2008).
[CrossRef]

H. Meyer, A. Garofalakis, G. Zacharakis, S. Psycharakis, C. Mamalaki, D. Kioussis, E. Economou, V. Ntziachristos, and J. Ripoll, “Noncontact optical imaging in mice with full angular coverage and automatic surface extraction,” Appl. Opt.46, 3617–3627 (2007).
[CrossRef] [PubMed]

R. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imag.23, 492–500 (2004).
[CrossRef]

R. Schulz, J. Ripoll, and V. Ntziachristos, “Noncontact optical tomography of turbid media,” Opt. Lett.28, 1701–1703 (2003).
[CrossRef] [PubMed]

Rodella, R.

Saldner, H.

Salvi, J.

J. Salvi, J. Pages, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn.37, 827–849 (2004).
[CrossRef]

Sansoni, G.

Santago, P.

A. Cong, W. Cong, Y. Lu, P. Santago, A. Chatziioannou, and G. Wang, “Differential evolution approach for regularized bioluminescence tomography,” IEEE Trans. Biomed. Eng.57, 2229–2238 (2010).
[CrossRef] [PubMed]

Schulz, R.

R. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imag.23, 492–500 (2004).
[CrossRef]

R. Schulz, J. Ripoll, and V. Ntziachristos, “Noncontact optical tomography of turbid media,” Opt. Lett.28, 1701–1703 (2003).
[CrossRef] [PubMed]

Schweiger, M.

S. Arridge and M. Schweiger, “Image reconstruction in optical tomography.” Phil. Trans. R. Soc. B352, 717–726 (1997).
[CrossRef] [PubMed]

Seitz, P.

R. Lange and P. Seitz, “Solid-state time-of-flight range camera,” IEEE J. Quantum Electron.37, 390–397 (2001).
[CrossRef]

Semmler, W.

X. Jiang, L. Cao, W. Semmler, and J. Peter, “A surface recognition approach for in vivo optical imaging applications using a micro-lens-array light detector,” in Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BTu3A.1.

Smith, D.

G. Zavattini, S. Vecchi, G. Mitchell, U. Weisser, R. Leahy, B. Pichler, D. Smith, and S. Cherry, “A hyperspectral fluorescence system for 3d in vivo optical imaging,” Phys. Med. Biol.51, 2029–2043 (2006).
[CrossRef] [PubMed]

A. Chaudhari, F. Darvas, J. Bading, R. Moats, P. Conti, D. Smith, S. Cherry, and R. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

Soubret, A.

T. Lasser, A. Soubret, J. Ripoll, and V. Ntziachristos, “Surface reconstruction for free-space 360° fluorescence molecular tomography and the effects of animal motion,” IEEE Trans. Med. Imag.27, 188–194 (2008).
[CrossRef]

Srinivasan, S.

H. Dehghani, M. Eames, P. Yalavarthy, S. Davis, S. Srinivasan, C. Carpenter, B. Pogue, and K. Paulsen, “Near infrared optical tomography using nirfast: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods En.25, 711–732 (2009).
[CrossRef]

Srinivasan, V.

Stearns, D.

D. Stearns, B. Rice, and M. Cable, “Method and apparatus for 3-d imaging of internal light sources,” U.S. Patent 7,860,549 (2010).

Styles, I.

J. Guggenheim, H. Dehghani, H. Basevi, I. Styles, and J. Frampton, “Development of a multi-view, multi-spectral bioluminescence tomography small animal imaging system,” Proc. SPIE8088, 80881K (2011).
[CrossRef]

J. Guggenheim, H. Basevi, I. Styles, J. Frampton, and H. Dehghani, “Multi-view, multi-spectral bioluminescence tomography,” in Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BW4A.7.

Su, X.

X. Mao, W. Chen, and X. Su, “Improved fourier-transform profilometry,” Appl. Opt.46, 664–668 (2007).
[CrossRef] [PubMed]

X. Su and W. Chen, “Fourier transform profilometry:: a review,” Opt. Laser. Eng.35, 263–284 (2001).
[CrossRef]

Takeda, M.

M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-d object shape,” Appl. Opt.22, 3977–3982 (1983).
[CrossRef] [PubMed]

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A72, 156–160 (1982).
[CrossRef]

Troy, T.

C. Kuo, O. Coquoz, T. Troy, H. Xu, and B. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt.12, 024007 (2007).
[CrossRef] [PubMed]

Vecchi, S.

G. Zavattini, S. Vecchi, G. Mitchell, U. Weisser, R. Leahy, B. Pichler, D. Smith, and S. Cherry, “A hyperspectral fluorescence system for 3d in vivo optical imaging,” Phys. Med. Biol.51, 2029–2043 (2006).
[CrossRef] [PubMed]

Wang, G.

A. Cong, W. Cong, Y. Lu, P. Santago, A. Chatziioannou, and G. Wang, “Differential evolution approach for regularized bioluminescence tomography,” IEEE Trans. Biomed. Eng.57, 2229–2238 (2010).
[CrossRef] [PubMed]

Wang, Z.

Weisser, U.

G. Zavattini, S. Vecchi, G. Mitchell, U. Weisser, R. Leahy, B. Pichler, D. Smith, and S. Cherry, “A hyperspectral fluorescence system for 3d in vivo optical imaging,” Phys. Med. Biol.51, 2029–2043 (2006).
[CrossRef] [PubMed]

Xi, J.

Xie, H.

Xu, H.

C. Kuo, O. Coquoz, T. Troy, H. Xu, and B. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt.12, 024007 (2007).
[CrossRef] [PubMed]

Yalavarthy, P.

H. Dehghani, M. Eames, P. Yalavarthy, S. Davis, S. Srinivasan, C. Carpenter, B. Pogue, and K. Paulsen, “Near infrared optical tomography using nirfast: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods En.25, 711–732 (2009).
[CrossRef]

Yao, J.

Zacharakis, G.

Zappa, E.

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to fourier-transform profilometry,” Opt. Laser. Eng.46, 106–116 (2008).
[CrossRef]

Zavattini, G.

G. Zavattini, S. Vecchi, G. Mitchell, U. Weisser, R. Leahy, B. Pichler, D. Smith, and S. Cherry, “A hyperspectral fluorescence system for 3d in vivo optical imaging,” Phys. Med. Biol.51, 2029–2043 (2006).
[CrossRef] [PubMed]

Zhang, D.

Adv. Opt. Photon. (1)

Appl. Opt. (7)

Commun. Numer. Methods En. (1)

H. Dehghani, M. Eames, P. Yalavarthy, S. Davis, S. Srinivasan, C. Carpenter, B. Pogue, and K. Paulsen, “Near infrared optical tomography using nirfast: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods En.25, 711–732 (2009).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. Lange and P. Seitz, “Solid-state time-of-flight range camera,” IEEE J. Quantum Electron.37, 390–397 (2001).
[CrossRef]

IEEE J. Sel. Topics Quantum Electron. (1)

B. Brooksby, H. Dehghani, B. Pogue, and K. Paulsen, “Near-infrared (nir) tomography breast image reconstruction with a priori structural information from mri: algorithm development for reconstructing heterogeneities,” IEEE J. Sel. Topics Quantum Electron.9, 199–209 (2003).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

A. Cong, W. Cong, Y. Lu, P. Santago, A. Chatziioannou, and G. Wang, “Differential evolution approach for regularized bioluminescence tomography,” IEEE Trans. Biomed. Eng.57, 2229–2238 (2010).
[CrossRef] [PubMed]

IEEE Trans. Med. Imag. (3)

T. Lasser, A. Soubret, J. Ripoll, and V. Ntziachristos, “Surface reconstruction for free-space 360° fluorescence molecular tomography and the effects of animal motion,” IEEE Trans. Med. Imag.27, 188–194 (2008).
[CrossRef]

R. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imag.23, 492–500 (2004).
[CrossRef]

A. Kumar, S. Raymond, A. Dunn, B. Bacskai, and D. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imag.27, 1152–1163 (2008).
[CrossRef]

J. Biomed. Opt. (2)

M. Allard, D. Côté, L. Davidson, J. Dazai, and R. Henkelman, “Combined magnetic resonance and bioluminescence imaging of live mice,” J. Biomed. Opt.12, 034018 (2007).
[CrossRef] [PubMed]

C. Kuo, O. Coquoz, T. Troy, H. Xu, and B. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt.12, 024007 (2007).
[CrossRef] [PubMed]

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

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A72, 156–160 (1982).
[CrossRef]

Meas. Sci. Technol. (1)

A. Dorrington, M. Cree, A. Payne, R. Conroy, and D. Carnegie, “Achieving sub-millimetre precision with a solid-state full-field heterodyning range imaging camera,” Meas. Sci. Technol.18, 2809–2816 (2007).
[CrossRef]

Opt. Express (3)

Opt. Laser. Eng. (5)

S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Laser. Eng.48, 133–140 (2010).
[CrossRef]

X. Su and W. Chen, “Fourier transform profilometry:: a review,” Opt. Laser. Eng.35, 263–284 (2001).
[CrossRef]

T. Judge and P. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Laser. Eng.21, 199–239 (1994).
[CrossRef]

F. Berryman, P. Pynsent, and J. Cubillo, “A theoretical comparison of three fringe analysis methods for determining the three-dimensional shape of an object in the presence of noise,” Opt. Laser. Eng.39, 35–50 (2003).
[CrossRef]

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to fourier-transform profilometry,” Opt. Laser. Eng.46, 106–116 (2008).
[CrossRef]

Opt. Lett. (1)

Pattern Recogn. (1)

J. Salvi, J. Pages, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn.37, 827–849 (2004).
[CrossRef]

Phil. Trans. R. Soc. B (1)

S. Arridge and M. Schweiger, “Image reconstruction in optical tomography.” Phil. Trans. R. Soc. B352, 717–726 (1997).
[CrossRef] [PubMed]

Phys. Med. Biol. (3)

A. Gibson, J. Hebden, and S. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005).
[CrossRef] [PubMed]

A. Chaudhari, F. Darvas, J. Bading, R. Moats, P. Conti, D. Smith, S. Cherry, and R. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50, 5421–5441 (2005).
[CrossRef] [PubMed]

G. Zavattini, S. Vecchi, G. Mitchell, U. Weisser, R. Leahy, B. Pichler, D. Smith, and S. Cherry, “A hyperspectral fluorescence system for 3d in vivo optical imaging,” Phys. Med. Biol.51, 2029–2043 (2006).
[CrossRef] [PubMed]

Proc. SPIE (1)

J. Guggenheim, H. Dehghani, H. Basevi, I. Styles, and J. Frampton, “Development of a multi-view, multi-spectral bioluminescence tomography small animal imaging system,” Proc. SPIE8088, 80881K (2011).
[CrossRef]

Other (13)

J. Guggenheim, H. Basevi, I. Styles, J. Frampton, and H. Dehghani, “Multi-view, multi-spectral bioluminescence tomography,” in Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BW4A.7.

K. Kraus, Photogrammetry: Geometry from Images and Laser Scans (de Gruyter, 2007).
[CrossRef]

Z. Geng, “Method and apparatus for omnidirectional three dimensional imaging,” U.S. Patent 6,744,569 (2004).

Z. Geng, “Diffuse optical tomography system and method of use,” U.S. Patent 7,242,997 (2007).

D. Nilson, M. Cable, B. Rice, and K. Kearney, “Structured light imaging apparatus,” U.S. Patent 7,298,415 (2007).

X. Jiang, L. Cao, W. Semmler, and J. Peter, “A surface recognition approach for in vivo optical imaging applications using a micro-lens-array light detector,” in Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BTu3A.1.

B. Rice, M. Cable, and K. Kearney, “3d in-vivo imaging and topography using structured light,” U.S. Patent 7,797,034 (2010).

D. Stearns, B. Rice, and M. Cable, “Method and apparatus for 3-d imaging of internal light sources,” U.S. Patent 7,860,549 (2010).

PerkinElmer, “Ivis 200 series,” http://www.perkinelmer.com/Catalog/Product/ID/IVIS200 .

Berthold Technologies, “Nightowl lb 983 in vivo imaging system,” https://www.berthold.com/en/bio/in_vivo_imager_NightOWL_LB983 .

Biospace Lab, “Photonimager,” http://www.biospacelab.com/m-31-optical-imaging.html .

Biospace Lab, “4-view module,” http://www.biospacelab.com/m-89-4-view-module.html .

P. Cignoni, “Meshlab home page,” http://meshlab.sourceforge.net/ .

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Figures (9)

Fig. 1
Fig. 1

Instrumental configuration. A camera is located above the object to be imaged. Two projectors project structured illumination patterns upon the object. Mirrors placed at either side of the object expand the field of view of the camera to include edges and concave regions of the object. Mirrors can be placed in arbitrary positions, orientations and number to expand the field of view. The black dotted lines represent light leaving the projectors and scattering off of the object’s surface. The blue lines indicate the regions of the object that are illuminated by at least one projector and visible directly to the camera. The red dashed lines indicate the regions of the object that are illuminated by at least one projector and visible to the camera through a mirror. It can be seen that there are regions of the object that are illuminated and visible through the mirror, but not directly.

Fig. 2
Fig. 2

Schematic of system. c and p are the locations of the camera and projector pupils respectively. o is a point on a plane orthogonal to the axis of the projector on which the spatial frequency, and direction as encoded by , of the projected pattern is known. The direction of the pattern is the direction in the plane of maximal increasing pattern phase. is a ray originating at the camera pupil, corresponding to a camera pixel. The unknown point x lines somewhere along the line containing c and spanned by r̂. y is the intersection point of the plane containing o and the line between x and p (spanned by ). For simplicity, this schematic is given in two dimensions, but this representation is also valid in three dimensions.

Fig. 3
Fig. 3

An XPM-2 Phantom Mouse (Caliper Life Sciences, A PerkinElmer Company, Hopkinton, Massachusetts, United States of America) placed in the imaging system, utilising two mirrors, under structured illumination generated by one projector.

Fig. 4
Fig. 4

Three wrapped phase maps acquired of an XPM-2 Phantom Mouse (Caliper Life Sciences, A PerkinElmer Company, Hopkinton, Massachusetts, United States of America) using low, medium and high frequency spatial patterns, and used to unwrap a high frequency phase map. The low frequency phase map resulted from a projected pattern with a wavelength of 1024 pixels. The medium frequency phase map resulted from a projected pattern with a wavelength of 45.3 pixels. The high frequency phase map resulted from a projected pattern with a wavelength of 11.3 pixels. The horizontal resolution of the projector was 800 pixels. Masking of background noise was performed using thresholding of fully illuminated images and applied to all phase maps. The projector was placed to the right of the subject.

Fig. 5
Fig. 5

Surface capture of an XPM-2 Phantom Mouse. The phantom (see Fig. 5(a)) was imaged using the surface capture system, generating a point cloud which was then converted into a surface mesh using MeshLab [47] (see Fig. 5(b)). A collection of points lying on the XPM-2 Phantom was kindly provided by Caliper Life Sciences (see Fig. 5(c)) and used to estimate the measurement error in the surface capture acquisition (see Fig. 5(d)).

Fig. 6
Fig. 6

Surface capture coverage of an XPM-2 Phantom Mouse. The addition of mirrors increases the surface area imaged by 15% over the surface area imaged directly.

Fig. 7
Fig. 7

Surface capture of a two pence coin. The surface capture data was acquired using one projector and no mirrors. The colour maps in Figs. 7(b) and 7(d) indicate height in millimetres. Note that no texture was applied to the renderings in Figs. 7(b) and 7(d).

Fig. 8
Fig. 8

Surface capture imaging data acquired of a mouse cadaver.

Fig. 9
Fig. 9

Point cloud of a mouse cadaver (see Fig. 8). All views using projector 1 were acquired simultaneously, as were all views using projector 2. The mesh in Fig. 9(b) was created from a sub-sampled version of the point cloud in Fig. 9(a).

Equations (15)

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I ( x ) = r ( x ) 1 2 ( 1 + cos ( 2 π f x + ψ b ( x ) ) )
I ( x ) = r ( x ) 1 2 ( 1 + cos ( 2 π f x + ψ b ( x ) + ψ o ( x ) ) )
h ( x ) = l ψ o ( x ) ψ o ( x ) 2 π f d
p ( y , f ) = 1 + cos ( 2 π f ( y o ) v ^ ) 2
y = p | p o | 2 q ^ ( p o ) q ^
q ^ = x p | x p |
ψ = 2 π f ( y o ) v ^
x = c + α ( p c ) α r ^ r ^
α = ψ 2 π f p o | p o | + | p o | v ^
p n ( y , f ) = 1 + cos ( 2 π f ( y o ) v ^ + ϕ n ) 2
g n ( x , f ) = b ( x ) + a ( x ) 1 + cos ( ψ ( x , f ) + ϕ n ) 2
ψ ( x , f ) = 2 π f ( y o ) v ^
ψ ( x , f ) ( mod 2 π ) = arctan ( n = 1 N g n ( x , f ) sin ( 2 π n N ) n = 1 N g n ( x , f ) cos ( 2 π n N ) )
f n = n d
ψ ˜ ( x , f 2 ) = 2 ψ ( x , f 1 )

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