Abstract

We present a product-of-convolutions (POC) model for phase microscopy images. The model was designed to simulate phase images of thick heterogeneous transparent objects. The POC approach attempts to capture phase delays along the optical axis by modeling the imaged object as a stack of parallel slices. The product of two-dimensional convolutions between each slice and the appropriate slice of the point spread function is used to represent the object at the image plane. The total electric field at the image plane is calculated as the product of the object function and the incident field. Phase images from forward models based on the first Born and Rytov approximations are used for comparison. Computer simulations and measured images illustrate the ability of the POC model to represent phase images from thick heterogeneous objects accurately, cases where the first Born and Rytov models have well-known limitations. Finally, measured phase microscopy images of mouse embryos are compared to those produced by the Born, Rytov, and POC models. Our comparisons show that the POC model is capable of producing accurate representations of these more complex phase images.

© 2009 Optical Society of America

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2009

2008

2007

2006

2005

2004

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62-70 (2004).
[CrossRef] [PubMed]

J. L. Hollmann, A. K. Dunn, and C. A. DiMarzio, “Computational microscopy in embryo imaging,” Opt. Lett. 29, 2267-2269 (2004).
[CrossRef] [PubMed]

2002

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194-203 (2002).
[CrossRef] [PubMed]

2000

D. Yevick and D. J. Thomson, “Complex Padé approximants for wide-angle acoustic propagators,” J. Acoust. Soc. Am. 108, 2784-2790 (2000).
[CrossRef]

G. A. Tsihrintzis and A. J. Devaney, “Higher-order (nonlinear) diffraction tomography: reconstruction algorithms and computer simulation,” IEEE Trans. Image Process. 9, 1560-1572 (2000).
[CrossRef]

1999

1998

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]

B. Chen and J. J. Stamnes, “Validity of diffraction tomography based on the first Born and the first Rytov approximations,” Appl. Opt. 37, 783-785 (1998).
[CrossRef]

F. Kagawala and T. Kanade, “Computational model of image formation process in DIC microscopy,” Proc. SPIE 3261, 193-204 (1998).
[CrossRef]

1994

C. J. R. Sheppard and M. Gu, “Modeling of 3-D brightfield microscope systems,” Proc. SPIE 2302, 352-358 (1994).
[CrossRef]

1993

H. Gundlach, “Phase contrast and differential interference contrast instrumentation and application in cell, developmental, and marine biology,” Opt. Eng. 32, 3223-3228 (1993).
[CrossRef]

1992

F. C. Lin and M. A. Fiddy, “The Born-Rytov controversy. I. comparing analytical and approximate expressions for the one-dimensional deterministic case,” J. Opt. Soc. Am. A 9, 1102-1110 (1992).
[CrossRef]

C. J. Cogswell and C. J. R. Sheppard, “Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81-101 (1992).
[CrossRef]

1989

1988

1985

N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121-127 (1985).
[CrossRef]

G. Beylkin and M. L. Oristaglio, “Distorted-wave Born and distorted-wave Rytov approximations,” Opt. Commun. 53, 213-216 (1985).
[CrossRef]

1981

1969

W. Lang, “Nomarski differential interference contrast microscopy. II. formation of the interference image,” Zeiss Inform. 17, 12-16 (1969).

Allman, B. E.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62-70 (2004).
[CrossRef] [PubMed]

Badizadegan, K.

Barone-Nugent, E. D.

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194-203 (2002).
[CrossRef] [PubMed]

Barty, A.

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194-203 (2002).
[CrossRef] [PubMed]

Bellair, C. J.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62-70 (2004).
[CrossRef] [PubMed]

Bevilacqua, F.

Beylkin, G.

G. Beylkin and M. L. Oristaglio, “Distorted-wave Born and distorted-wave Rytov approximations,” Opt. Commun. 53, 213-216 (1985).
[CrossRef]

Bienstman, P.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2001).

Boucher, A.

N. Dey, A. Boucher, and M. Thonnat, “Image formation model a 3 translucent object observed in light microscopy,” in Proceedings of 2002 Conference on Image Processing (IEEE, 2002), Vol. 2, pp. 469-472

Brooks, D. H.

H. Sierra, C. A. DiMarzio, and D. H. Brooks, “Modeling images of phase information for three-dimensional objects,” Proc. SPIE 6861, 68610A.1-68610A.9 (2008).

Charrière, F.

Chen, B.

B. Chen and J. J. Stamnes, “Validity of diffraction tomography based on the first Born and the first Rytov approximations,” Appl. Opt. 37, 783-785 (1998).
[CrossRef]

Choi, W.

Cogswell, C.

Cogswell, C. J.

C. J. Cogswell and C. J. R. Sheppard, “Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81-101 (1992).
[CrossRef]

Colomb, T.

Conchello, J. A.

C. Preza, D. L. Snyder, and J. A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential-interference contrast microscopy,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 16, 2185-2199 (1999).
[CrossRef] [PubMed]

Cuche, E.

Curl, C. L.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62-70 (2004).
[CrossRef] [PubMed]

Dasari, R. R.

Delbridge, L. M. D.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62-70 (2004).
[CrossRef] [PubMed]

Depeursinge, C.

Devaney, A. J.

G. A. Tsihrintzis and A. J. Devaney, “Higher-order (nonlinear) diffraction tomography: reconstruction algorithms and computer simulation,” IEEE Trans. Image Process. 9, 1560-1572 (2000).
[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]

Dey, N.

N. Dey, A. Boucher, and M. Thonnat, “Image formation model a 3 translucent object observed in light microscopy,” in Proceedings of 2002 Conference on Image Processing (IEEE, 2002), Vol. 2, pp. 469-472

DiMarzio, C. A.

H. Sierra, C. A. DiMarzio, and D. H. Brooks, “Modeling images of phase information for three-dimensional objects,” Proc. SPIE 6861, 68610A.1-68610A.9 (2008).

W. C. Warger II, 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. L. Hollmann, A. K. Dunn, and C. A. DiMarzio, “Computational microscopy in embryo imaging,” Opt. Lett. 29, 2267-2269 (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]

Drezek, R.

Dunn, A.

Dunn, A. K.

Emery, Y.

Fang-Yen, C.

Feld, M. S.

Fiddy, M. A.

Gaudette, T. J.

Ghiglia, D.

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

Gibson, S. F.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1999).

Gu, M.

C. J. R. Sheppard and M. Gu, “Modeling of 3-D brightfield microscope systems,” Proc. SPIE 2302, 352-358 (1994).
[CrossRef]

Gundlach, H.

H. Gundlach, “Phase contrast and differential interference contrast instrumentation and application in cell, developmental, and marine biology,” Opt. Eng. 32, 3223-3228 (1993).
[CrossRef]

Harris, P. J.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62-70 (2004).
[CrossRef] [PubMed]

Heger, T. J.

Hogenboom, D. O.

Hollmann, J. L.

Ikeda, T.

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]

Jenkins, F. A.

F. A. Jenkins and H. White, Fundamentals of Optics (McGraw-Hill, 1976).

Kagawala, F.

F. Kagawala and T. Kanade, “Computational model of image formation process in DIC microscopy,” Proc. SPIE 3261, 193-204 (1998).
[CrossRef]

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

Kanade, T.

F. Kagawala and T. Kanade, “Computational model of image formation process in DIC microscopy,” Proc. SPIE 3261, 193-204 (1998).
[CrossRef]

King, S.

Laevsky, G. S.

W. C. Warger II, 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]

Lagasse, P. E.

Laligant, O.

Lang, W.

W. Lang, “Nomarski differential interference contrast microscopy. II. formation of the interference image,” Zeiss Inform. 17, 12-16 (1969).

Lanni, F.

Le, K. Q.

Libertun, A.

Lin, F. C.

Lindberg, S. C.

Lue, N.

Magistretti, P. J.

Marks, D. L.

Marquet, P.

Mitchell, E. A. D.

Murphy, D. B.

D. B. Murphy, Fundamentals of Light Microscopy and Electronic Imaging (Wiley-Liss, 2001).

Nemoto, I.

Nugent, K. A.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62-70 (2004).
[CrossRef] [PubMed]

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194-203 (2002).
[CrossRef] [PubMed]

Oh, S.

Oristaglio, M. L.

G. Beylkin and M. L. Oristaglio, “Distorted-wave Born and distorted-wave Rytov approximations,” Opt. Commun. 53, 213-216 (1985).
[CrossRef]

Park, Y.

Pavani, S. R. P.

Pavillon, N.

Popescu, G.

Preza, C.

C. Preza, D. L. Snyder, and J. A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential-interference contrast microscopy,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 16, 2185-2199 (1999).
[CrossRef] [PubMed]

Pritt, M.

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

Rajadhyaksha, M.

W. C. Warger II, 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]

Rappaz, B.

Richards-Kortum, R.

Roberts, A.

C. J. Bellair, C. L. Curl, B. E. Allman, P. J. Harris, A. Roberts, L. M. D. Delbridge, and K. A. Nugent, “Quantitative phase-amplitude microscopy IV: imaging thick specimens,” J. Microsc. 214, 62-70 (2004).
[CrossRef] [PubMed]

Seung, H. S.

Sheppard, C. J. R.

C. J. R. Sheppard and M. Gu, “Modeling of 3-D brightfield microscope systems,” Proc. SPIE 2302, 352-358 (1994).
[CrossRef]

C. J. Cogswell and C. J. R. Sheppard, “Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81-101 (1992).
[CrossRef]

Sierra, H.

H. Sierra, C. A. DiMarzio, and D. H. Brooks, “Modeling images of phase information for three-dimensional objects,” Proc. SPIE 6861, 68610A.1-68610A.9 (2008).

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

Snyder, D. L.

C. Preza, D. L. Snyder, and J. A. Conchello, “Theoretical development and experimental evaluation of imaging models for differential-interference contrast microscopy,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 16, 2185-2199 (1999).
[CrossRef] [PubMed]

Song, S.

Stamnes, J. J.

B. Chen and J. J. Stamnes, “Validity of diffraction tomography based on the first Born and the first Rytov approximations,” Appl. Opt. 37, 783-785 (1998).
[CrossRef]

Streibl, N.

Thomson, D. J.

D. Yevick and D. J. Thomson, “Complex Padé approximants for wide-angle acoustic propagators,” J. Acoust. Soc. Am. 108, 2784-2790 (2000).
[CrossRef]

Thonnat, M.

N. Dey, A. Boucher, and M. Thonnat, “Image formation model a 3 translucent object observed in light microscopy,” in Proceedings of 2002 Conference on Image Processing (IEEE, 2002), Vol. 2, pp. 469-472

Townsend, D. J.

W. C. Warger II, 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]

Truchetet, F.

Tsihrintzis, G. A.

G. A. Tsihrintzis and A. J. Devaney, “Higher-order (nonlinear) diffraction tomography: reconstruction algorithms and computer simulation,” IEEE Trans. Image Process. 9, 1560-1572 (2000).
[CrossRef]

van der Donk, J.

Van Roey, J.

Warger, W. C.

W. C. Warger II, 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]

White, H.

F. A. Jenkins and H. White, Fundamentals of Optics (McGraw-Hill, 1976).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2001).

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]

Yevick, D.

D. Yevick and D. J. Thomson, “Complex Padé approximants for wide-angle acoustic propagators,” J. Acoust. Soc. Am. 108, 2784-2790 (2000).
[CrossRef]

Appl. Opt.

IEEE Trans. Image Process.

G. A. Tsihrintzis and A. J. Devaney, “Higher-order (nonlinear) diffraction tomography: reconstruction algorithms and computer simulation,” IEEE Trans. Image Process. 9, 1560-1572 (2000).
[CrossRef]

J. Acoust. Soc. Am.

D. Yevick and D. J. Thomson, “Complex Padé approximants for wide-angle acoustic propagators,” J. Acoust. Soc. Am. 108, 2784-2790 (2000).
[CrossRef]

J. Biomed. Opt.

W. C. Warger II, 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.

C. J. Cogswell and C. J. R. Sheppard, “Confocal differential interference contrast (DIC) microscopy: including a theoretical analysis of conventional and confocal DIC imaging,” J. Microsc. 165, 81-101 (1992).
[CrossRef]

E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194-203 (2002).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Cartoon of 3D object description. The object is discretized along the optical axis and the light propagates perpendicularly through each slice. For illustration, a section of seven interior slices, each of thickness Δ z , is shown expanded in the bottom part of the figure.

Fig. 2
Fig. 2

Optical transmission system illustration. The light is propagated to the image plane using a PSF model defined in terms of the numerical aperture of the objective lens. Coherent illumination is assumed.

Fig. 3
Fig. 3

OPD profile comparison of a synthetic uniform square object with n δ = 0.035 taken at the middle of the object. (a) A transverse slice through either object at the center of the glass block, (b) comparison of simulated OPD as a function of position along the x axis at y = 0 μ m for the object with a thickness of 2 μ m , and (c) similar comparison of simulated OPD for an object with a thickness of 35 μ m .

Fig. 4
Fig. 4

OPD comparison from images of uniform square objects as a function of varying thicknesses. The plot shows for each thickness the OPD averaged over the five OPD values closest to the center of the object along a profile taken at the middle of the object.

Fig. 5
Fig. 5

Simulated and measured OPD images of a PMMA bead in oil. All images are shown with the same scale and contrast. (a) Cartoon of the cross section of the real part of the simulated PMMA bead in oil, (b) projection model, (c) POC, (d) Rytov, (e) Born, (f) OQM measured image of a real bead matching the model, and (g) plot of the profile of simulated and measured OPD images.

Fig. 6
Fig. 6

Measured and simulated OPD images of a one-cell mouse embryo. All images are shown with the same scale and contrast. (a) Slice of the real part of the synthetic object. Small bright particles represent the mitochondria while the small and large spheres represent the nucleus and cytoplasm, respectively, and the outer shell surrounding the large sphere simulates the zona pellucida. (b) Projection, (c) POC, (d) Rytov, (e) Born, (f) experimental measured OQM image of a typical embryo, (g) profiles taken at the middle of the simulated OPD images, and (h) OPD profile taken at the middle of the OQM image.

Equations (17)

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U ( x , y , z I ) = U 0 ( x , y , z I ) + U p ( x , y , z I ) .
U ( x , y , z m ) = U 0 ( x , y , z m ) + U p ( x , y , z m ) ,
U ( x , y , z m + Δ z ) = [ U 0 ( x , y , z m ) + U p ( x , y , z m ) ] exp [ j k n δ ( x , y , z m ) Δ z ] .
U ( x , y , z m + Δ z ) U 0 ( x , y , z m ) exp [ j k n δ ( x , y , z m ) Δ z ] .
U ( x , y , z m + Δ z ) U 0 ( x , y , z m ) + j k U 0 ( x , y , z m ) n δ ( x , y , z m ) Δ z ,
U B ( x , y , z I ) m = N m = N Δ z r , s U 0 ( x ̃ , y ̃ , z m ) [ j k n δ ( x ̃ , y ̃ , z m ) ] h ( x x ̃ , y y ̃ , z I z m ) d x ̃ d y ̃ ,
U 0 ( x , y , z I ) = r , s U 0 ( x ̃ , y ̃ , z 0 ) h ( x x ̃ , y y ̃ , z I z 0 ) d x ̃ d y ̃ ,
U ( x , y , z I ) = exp [ ϕ ( x , y , z I ) ] ,
ϕ p ( x , y , z I ) = U B ( x , y , z I ) U 0 ( x , y , z I ) ,
U ( x , y , z I ) = U 0 ( x , y , z I ) exp ( U B ( x , y , z I ) U 0 ( x , y , z I ) ) .
U ( x , y , z I ) U 0 ( x , y , z I ) m = N N ( 1 + Δ z r , s [ U 0 ( x ̃ , y ̃ , z m ) ( j k n δ ( x ̃ , y ̃ , z m ) ) h ( x x ̃ , y y ̃ , z I z m ) ] d x ̃ d y ̃ U 0 ( x , y , z I ) ) ,
g ( x , y , z m ; z I ) = r , s exp [ j k n δ ( x ̃ , y ̃ , z m ) Δ z ] h ( x x ̃ , y y ̃ , z I z m ) d x ̃ d y ̃ .
g ( x , y , z m ; z I ) = exp [ ϕ p ( x , y , z m ; z I ) ] .
ϕ p ( x , y , z I ) = m = N m = N ϕ p ( x , y , z m ; z I ) ,
g ( x , y , z I ) = m = N m = N r , s exp [ j k n δ ( x ̃ , y ̃ , z m ) Δ z ] h ( x x ̃ , y y ̃ , z I z m ) d x ̃ d y ̃ .
U ( x , y , z I ) = U 0 ( x , y , z I ) g ( x , y , z I ) ,
OPD ( x , y ) = 0 d ( x , y ) n δ ( x , y , z ) d z .

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