Abstract

We present a frequency domain analysis of the image resolution of optical tomography systems. The result of our analysis is a description of the spatially-variant resolution in optical tomographic image after reconstruction as a function of the properties of the imaging system geometry. We validate our model using optical projection tomography (OPT) measurements of fluorescent beads embedded in agarose gel. Our model correctly describes both the radial and tangential resolution of the measured images. In addition, we present a correction of the tomographic images for the spatially-varying resolution using a deconvolution algorithm. The resulting corrected tomographic reconstruction shows a homogeneous and isotropic pixel-limited resolution across the entire image. Our method is applied to OPT measurements of a zebrafish, showing improved resolution. Aside from allowing image correction and providing a resolution measure for OPT systems, our model provides a powerful tool for the design of optical tomographic systems.

© 2016 Optical Society of America

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References

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  1. P. J. Keller, A. D. Schmidt, J. Wittbrodt, and E. H. Stelzer, “Reconstruction of zebrafish early embryonic development by scanned light sheet microscopy,” Science 322, 1065–1069 (2008).
    [Crossref] [PubMed]
  2. J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
    [Crossref] [PubMed]
  3. J. Sharpe, “Optical projection tomography,” Annu. Rev. Biomed. Eng. 6, 209–228 (2004).
    [Crossref] [PubMed]
  4. L. Chen, N. Andrews, S. Kumar, P. Frankel, J. McGinty, and P. M. French, “Simultaneous angular multiplexing optical projection tomography at shifted focal planes,” Opt. Lett. 38, 851–853 (2013).
    [Crossref] [PubMed]
  5. Q. Miao, J. Hayenga, M. G. Meyer, T. Neumann, A. C. Nelson, and E. J. Seibel, “Resolution improvement in optical projection tomography by the focal scanning method,” Opt. Lett. 35, 3363–3365 (2010).
    [Crossref] [PubMed]
  6. W. Xia, R. M. Lewitt, and P. R. Edholm, “Fourier correction for spatially variant collimator blurring in SPECT,” IEEE Trans. Med. Imaging 14, 100–115 (1995).
    [Crossref] [PubMed]
  7. J. R. Walls, J. G. Sled, J. Sharpe, and R. M. Henkelman, “Resolution improvement in emission optical projection tomography,” Phys. Med. Biol. 52, 2775 (2007).
    [Crossref] [PubMed]
  8. A. Darrell, H. Meyer, K. Marias, M. Brady, and J. Ripoll, “Weighted filtered backprojection for quantitative fluorescence optical projection tomography,” Phys. Med. Biol. 53, 3863 (2008).
    [Crossref] [PubMed]
  9. L. Chen, J. McGinty, H. B. Taylor, L. Bugeon, J. R. Lamb, M. J. Dallman, and P. M. French, “Incorporation of an experimentally determined MTF for spatial frequency filtering and deconvolution during optical projection tomography reconstruction,” Opt. Express 20, 7323–7337 (2012).
    [Crossref] [PubMed]
  10. C. M. McErlean, E. Bräuer-Krisch, J. Adamovics, and S. J. Doran, “Assessment of optical CT as a future QA tool for synchrotron X-ray microbeam therapy,” Phys. Med. Biol. 61, 320 (2015).
    [Crossref] [PubMed]
  11. M. Gu, Advanced optical imaging theory, vol. 75 (Springer Science and Business Media, 2000).
    [Crossref]
  12. J. Hsieh, Computed Tomography: Principles, Design, Artifacts, and Recent Advances (SPIE, 2009), Chap. 3.
  13. H. Kogelnik and L. Tingye, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
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2015 (1)

C. M. McErlean, E. Bräuer-Krisch, J. Adamovics, and S. J. Doran, “Assessment of optical CT as a future QA tool for synchrotron X-ray microbeam therapy,” Phys. Med. Biol. 61, 320 (2015).
[Crossref] [PubMed]

2013 (1)

2012 (1)

2010 (1)

2008 (2)

P. J. Keller, A. D. Schmidt, J. Wittbrodt, and E. H. Stelzer, “Reconstruction of zebrafish early embryonic development by scanned light sheet microscopy,” Science 322, 1065–1069 (2008).
[Crossref] [PubMed]

A. Darrell, H. Meyer, K. Marias, M. Brady, and J. Ripoll, “Weighted filtered backprojection for quantitative fluorescence optical projection tomography,” Phys. Med. Biol. 53, 3863 (2008).
[Crossref] [PubMed]

2007 (1)

J. R. Walls, J. G. Sled, J. Sharpe, and R. M. Henkelman, “Resolution improvement in emission optical projection tomography,” Phys. Med. Biol. 52, 2775 (2007).
[Crossref] [PubMed]

2004 (1)

J. Sharpe, “Optical projection tomography,” Annu. Rev. Biomed. Eng. 6, 209–228 (2004).
[Crossref] [PubMed]

2002 (1)

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[Crossref] [PubMed]

1995 (1)

W. Xia, R. M. Lewitt, and P. R. Edholm, “Fourier correction for spatially variant collimator blurring in SPECT,” IEEE Trans. Med. Imaging 14, 100–115 (1995).
[Crossref] [PubMed]

1966 (1)

Adamovics, J.

C. M. McErlean, E. Bräuer-Krisch, J. Adamovics, and S. J. Doran, “Assessment of optical CT as a future QA tool for synchrotron X-ray microbeam therapy,” Phys. Med. Biol. 61, 320 (2015).
[Crossref] [PubMed]

Ahlgren, U.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[Crossref] [PubMed]

Andrews, N.

Baldock, R.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[Crossref] [PubMed]

Brady, M.

A. Darrell, H. Meyer, K. Marias, M. Brady, and J. Ripoll, “Weighted filtered backprojection for quantitative fluorescence optical projection tomography,” Phys. Med. Biol. 53, 3863 (2008).
[Crossref] [PubMed]

Bräuer-Krisch, E.

C. M. McErlean, E. Bräuer-Krisch, J. Adamovics, and S. J. Doran, “Assessment of optical CT as a future QA tool for synchrotron X-ray microbeam therapy,” Phys. Med. Biol. 61, 320 (2015).
[Crossref] [PubMed]

Bugeon, L.

Chen, L.

Dallman, M. J.

Darrell, A.

A. Darrell, H. Meyer, K. Marias, M. Brady, and J. Ripoll, “Weighted filtered backprojection for quantitative fluorescence optical projection tomography,” Phys. Med. Biol. 53, 3863 (2008).
[Crossref] [PubMed]

Davidson, D.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[Crossref] [PubMed]

Doran, S. J.

C. M. McErlean, E. Bräuer-Krisch, J. Adamovics, and S. J. Doran, “Assessment of optical CT as a future QA tool for synchrotron X-ray microbeam therapy,” Phys. Med. Biol. 61, 320 (2015).
[Crossref] [PubMed]

Edholm, P. R.

W. Xia, R. M. Lewitt, and P. R. Edholm, “Fourier correction for spatially variant collimator blurring in SPECT,” IEEE Trans. Med. Imaging 14, 100–115 (1995).
[Crossref] [PubMed]

Frankel, P.

French, P. M.

Gu, M.

M. Gu, Advanced optical imaging theory, vol. 75 (Springer Science and Business Media, 2000).
[Crossref]

Hayenga, J.

Hecksher-Sørensen, J.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[Crossref] [PubMed]

Henkelman, R. M.

J. R. Walls, J. G. Sled, J. Sharpe, and R. M. Henkelman, “Resolution improvement in emission optical projection tomography,” Phys. Med. Biol. 52, 2775 (2007).
[Crossref] [PubMed]

Hill, B.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[Crossref] [PubMed]

Hsieh, J.

J. Hsieh, Computed Tomography: Principles, Design, Artifacts, and Recent Advances (SPIE, 2009), Chap. 3.

Keller, P. J.

P. J. Keller, A. D. Schmidt, J. Wittbrodt, and E. H. Stelzer, “Reconstruction of zebrafish early embryonic development by scanned light sheet microscopy,” Science 322, 1065–1069 (2008).
[Crossref] [PubMed]

Kogelnik, H.

Kumar, S.

Lamb, J. R.

Lewitt, R. M.

W. Xia, R. M. Lewitt, and P. R. Edholm, “Fourier correction for spatially variant collimator blurring in SPECT,” IEEE Trans. Med. Imaging 14, 100–115 (1995).
[Crossref] [PubMed]

Marias, K.

A. Darrell, H. Meyer, K. Marias, M. Brady, and J. Ripoll, “Weighted filtered backprojection for quantitative fluorescence optical projection tomography,” Phys. Med. Biol. 53, 3863 (2008).
[Crossref] [PubMed]

McErlean, C. M.

C. M. McErlean, E. Bräuer-Krisch, J. Adamovics, and S. J. Doran, “Assessment of optical CT as a future QA tool for synchrotron X-ray microbeam therapy,” Phys. Med. Biol. 61, 320 (2015).
[Crossref] [PubMed]

McGinty, J.

Meyer, H.

A. Darrell, H. Meyer, K. Marias, M. Brady, and J. Ripoll, “Weighted filtered backprojection for quantitative fluorescence optical projection tomography,” Phys. Med. Biol. 53, 3863 (2008).
[Crossref] [PubMed]

Meyer, M. G.

Miao, Q.

Nelson, A. C.

Neumann, T.

Perry, P.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[Crossref] [PubMed]

Ripoll, J.

A. Darrell, H. Meyer, K. Marias, M. Brady, and J. Ripoll, “Weighted filtered backprojection for quantitative fluorescence optical projection tomography,” Phys. Med. Biol. 53, 3863 (2008).
[Crossref] [PubMed]

Ross, A.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[Crossref] [PubMed]

Schmidt, A. D.

P. J. Keller, A. D. Schmidt, J. Wittbrodt, and E. H. Stelzer, “Reconstruction of zebrafish early embryonic development by scanned light sheet microscopy,” Science 322, 1065–1069 (2008).
[Crossref] [PubMed]

Seibel, E. J.

Sharpe, J.

J. R. Walls, J. G. Sled, J. Sharpe, and R. M. Henkelman, “Resolution improvement in emission optical projection tomography,” Phys. Med. Biol. 52, 2775 (2007).
[Crossref] [PubMed]

J. Sharpe, “Optical projection tomography,” Annu. Rev. Biomed. Eng. 6, 209–228 (2004).
[Crossref] [PubMed]

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[Crossref] [PubMed]

Sled, J. G.

J. R. Walls, J. G. Sled, J. Sharpe, and R. M. Henkelman, “Resolution improvement in emission optical projection tomography,” Phys. Med. Biol. 52, 2775 (2007).
[Crossref] [PubMed]

Stelzer, E. H.

P. J. Keller, A. D. Schmidt, J. Wittbrodt, and E. H. Stelzer, “Reconstruction of zebrafish early embryonic development by scanned light sheet microscopy,” Science 322, 1065–1069 (2008).
[Crossref] [PubMed]

Taylor, H. B.

Tingye, L.

Walls, J. R.

J. R. Walls, J. G. Sled, J. Sharpe, and R. M. Henkelman, “Resolution improvement in emission optical projection tomography,” Phys. Med. Biol. 52, 2775 (2007).
[Crossref] [PubMed]

Wittbrodt, J.

P. J. Keller, A. D. Schmidt, J. Wittbrodt, and E. H. Stelzer, “Reconstruction of zebrafish early embryonic development by scanned light sheet microscopy,” Science 322, 1065–1069 (2008).
[Crossref] [PubMed]

Xia, W.

W. Xia, R. M. Lewitt, and P. R. Edholm, “Fourier correction for spatially variant collimator blurring in SPECT,” IEEE Trans. Med. Imaging 14, 100–115 (1995).
[Crossref] [PubMed]

Annu. Rev. Biomed. Eng. (1)

J. Sharpe, “Optical projection tomography,” Annu. Rev. Biomed. Eng. 6, 209–228 (2004).
[Crossref] [PubMed]

Appl. Opt. (1)

IEEE Trans. Med. Imaging (1)

W. Xia, R. M. Lewitt, and P. R. Edholm, “Fourier correction for spatially variant collimator blurring in SPECT,” IEEE Trans. Med. Imaging 14, 100–115 (1995).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (2)

Phys. Med. Biol. (3)

C. M. McErlean, E. Bräuer-Krisch, J. Adamovics, and S. J. Doran, “Assessment of optical CT as a future QA tool for synchrotron X-ray microbeam therapy,” Phys. Med. Biol. 61, 320 (2015).
[Crossref] [PubMed]

J. R. Walls, J. G. Sled, J. Sharpe, and R. M. Henkelman, “Resolution improvement in emission optical projection tomography,” Phys. Med. Biol. 52, 2775 (2007).
[Crossref] [PubMed]

A. Darrell, H. Meyer, K. Marias, M. Brady, and J. Ripoll, “Weighted filtered backprojection for quantitative fluorescence optical projection tomography,” Phys. Med. Biol. 53, 3863 (2008).
[Crossref] [PubMed]

Science (2)

P. J. Keller, A. D. Schmidt, J. Wittbrodt, and E. H. Stelzer, “Reconstruction of zebrafish early embryonic development by scanned light sheet microscopy,” Science 322, 1065–1069 (2008).
[Crossref] [PubMed]

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[Crossref] [PubMed]

Other (2)

M. Gu, Advanced optical imaging theory, vol. 75 (Springer Science and Business Media, 2000).
[Crossref]

J. Hsieh, Computed Tomography: Principles, Design, Artifacts, and Recent Advances (SPIE, 2009), Chap. 3.

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Figures (9)

Fig. 1
Fig. 1 Schematic representation of the lens imaging system for the detection of a single projection
Fig. 2
Fig. 2 (a) Schematic representation of the coordinate systems in the object and camera frames of reference. The point object is represented by coordinates (xo, yo), (b) Frequency domain representation of the reconstructed image for a point object. The frequency content is composed of the frequency content of the individual projections, as indicated for a single projection at angle θ. (c) Reconstructed image resulting from a point object
Fig. 3
Fig. 3 (a) Effect of the PSF on the tomographic image for point sources located at various positions (xo, yo), indicated at the top and right. (b) Theoretical FWHM resolution in the radial and tangential direction as function of the radial distance from the center of rotation.
Fig. 4
Fig. 4 Schematic representation of the experimental OPT setup. LS: Light source, NDF: neutral density filter, PH: pinhole, C: cuvette, TS: translation stage assembly, F: filter, L1 L2 and L3 Lenses, CA, camera, XY: manual x and y translation mount, TT: tip-tilt mount, θ: rotation stage, X: x translation stage, Y: y translation stage, Z: z translation stage.
Fig. 5
Fig. 5 (a) Reconstruced slice of the phantom showing single fluorescent beads. A ROI (b) around the bead is selected and the fluorescence emission is fitted (c).
Fig. 6
Fig. 6 (a) Radial (blue) and tangential (red) FWHM resolution as function of the radial distance from the center of rotation. Theoretical curves (solid lines) are compared to measured FWHM resolution in radial (open blue squares) and tangential (open red circles) directions. In most cases the markersize exeeds the errormargin. (b) Radial (open blue squares) and tangential (open red circles) FWHM resolution after deconvolution. Theoretical curves plotted for comparison.
Fig. 7
Fig. 7 (a) Standard reconstructed slice of the fluorescence phantom. The image of a typical bead (b) and (f) show significant sharpening after the deconvolution (c) and (g). Radial (blue) and tangential (red) cross-sections of the initial and deconvoluted images of the bead are shown in (d) and (h), and (e) and (i) respectively.
Fig. 8
Fig. 8 (a) Projection of a zebrafish with a reconstructed slice before (b) and after (c) deconvolution. The reconstruction corresponds to the vertical position indicated by the white line in (a). The inset compares the intensity profiles allong the indicated lines.
Fig. 9
Fig. 9 Tangential FWHM resolution as function of the beam waist, plotted for several distances from the rotation center.

Equations (22)

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p ( t , s , z ) = f ( t , s , z ) | h ( t , s , z ) | 2 .
p ( t , z ) = f ( t t , s s , z z ) | h ( t , s , z ) | 2 d t d s d z | s = 0 .
[ t s z ] = [ cos ( θ ) sin ( θ ) 0 sin ( θ ) cos ( θ ) 0 0 0 1 ] [ x y z ] .
p ( θ , t , z ) = | h ( t t o , s o , z ) | 2 .
p ( θ , t ) = Δ z δ z Δ z + δ z | h ( t t o , s o , z ) | 2 d z ,
P ( θ , f t ) = { p ( θ , t ) } .
s o = s o ( θ ) = r o sin ( θ o θ ) ,
θ = tan 1 ( f y f x ) ,
r o 2 = x o 2 + y o 2 ,
f t = f x 2 + f y 2 .
| h ( t , s , z ) | 2 = 2 π w 2 ( s ) exp ( 2 ( t 2 + z 2 ) w 2 ( s ) ) ,
w ( s ) = w 0 2 + λ 2 s 2 π 2 w 0 2 ,
p ( θ , t ) = 2 π A ( θ ) π w 2 ( s o ( θ ) ) exp ( 2 [ t t o ] 2 w 2 ( s o ( θ ) ) ) ,
A ( θ ) = 1 2 [ erf ( 2 w 2 ( s o ( θ ) ) ( Δ z + δ z ) ) erf ( 2 w 2 ( s o ( θ ) ) ( Δ z δ z ) ) ] .
P ( θ , f t ) = A ( θ ) exp ( π 2 w 2 ( s o ( θ ) ) f t 2 2 ) exp ( 2 π i t o f t ) .
OTF ( f x , f y ) = exp ( π 2 ( f x 2 + f y 2 ) 2 [ w 0 2 + λ 2 r o 2 sin 2 ( θ o θ ) π 2 w 0 2 ] ) exp ( 2 π i ( x o f x + y o f y ) )
[ u v ] = [ cos ( θ o ) sin ( θ o ) sin ( θ o ) cos ( θ o ) ] [ x x o y y o ] ,
OTF ( f u , f v ) = exp ( 1 2 π 2 ( f u 2 + f v 2 ) [ w 0 2 + λ 2 π 2 w 0 2 ( r o 2 sin 2 ( θ u ) ) ] ) ,
OTF ( f u , f v ) = exp ( 1 2 π 2 [ ( f u 2 + f v 2 ) w 0 2 + λ 2 π 2 w 0 2 ( r o 2 f v 2 ) ] ) ,
OTF ( f u , f v ) = 1 2 π w 0 2 exp ( π 2 ( f u 2 a u + f v 2 a v ) ) ,
PSF ( u , v ) = 1 π 2 a u a v exp ( [ u 2 a u + v 2 a v ] ) ,
w 0 , opt = λ r π .

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