Abstract

Optical projection tomography (OPT) is a noninvasive imaging technique that enables imaging of small specimens (<1cm), such as organs or animals in early developmental stages. In this paper, we present a set of computational methods that can be applied to the acquired data sets in order to correct for (a) unknown background or illumination intensity distributions over the field of view, (b) intensity spikes in single CCD pixels (so-called “hot pixels”), and (c) refractive index mismatch between the media in which the specimens are embedded and the environment. We have tested these correction methods using a variety of samples and present results obtained from Parhyale hawaiensis embedded in glycerol and in sea water. Successful reconstructions of fluorescence and absorption OPT images have been obtained for weakly scattering specimens embedded in media with nonmatched refractive index, thus advancing OPT toward routine in vivo imaging.

© 2011 Optical Society of America

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  1. J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, 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]
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    [CrossRef]
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2010 (2)

2009 (3)

2008 (4)

M. J. Boot, C. H. Westerberg, J. Sanz-Ezquerro, J. Cotterell, R. Schweitzer, M. Torres, and J. Sharpe, “In vitro whole-organ imaging: 4D quantification of growing mouse limb buds,” Nat. Method 5, 609–612 (2008).
[CrossRef]

A. Darrell, H. Meyer, K. Marias, M. Brady, and J. Ripoll, “Weighted filtered backprojection for quantitative fluorescence optical projection tomography,” Phys. Med. Biol. 53, 3863–3881 (2008).
[CrossRef] [PubMed]

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Method 5, 45–47 (2008).
[CrossRef]

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophotonics 1, 390–394 (2008).
[CrossRef]

2005 (2)

J. R. Walls, J. G. Sled, J. Sharpe, and R. M. Henkelman, “Correction of artefacts in optical projection tomography,” Phys. Med. Biol. 50, 4645–4665 (2005).
[CrossRef] [PubMed]

A. Pavlopoulos and M. Averof, “Establishing genetic transformation for comparative developmental studies in the crustacean Parhyale hawaiensis,” Proc. Natl. Acad. Sci. USA 102, 7888–7893 (2005).
[CrossRef] [PubMed]

2004 (1)

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

2003 (1)

2002 (1)

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, 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]

1993 (1)

1974 (1)

P. C. Lauterbur, “Magnetic resonance zeugmatography,” Pure Appl. Chem. 40, 149–157 (1974).
[CrossRef]

Ahlgren, U.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, 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]

Averof, M.

A. Pavlopoulos and M. Averof, “Establishing genetic transformation for comparative developmental studies in the crustacean Parhyale hawaiensis,” Proc. Natl. Acad. Sci. USA 102, 7888–7893 (2005).
[CrossRef] [PubMed]

Baldock, R.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, 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]

Birk, U.

Birk, U. J.

Boot, M. J.

M. J. Boot, C. H. Westerberg, J. Sanz-Ezquerro, J. Cotterell, R. Schweitzer, M. Torres, and J. Sharpe, “In vitro whole-organ imaging: 4D quantification of growing mouse limb buds,” Nat. Method 5, 609–612 (2008).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles in Optics (Wheaton, 1983).

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, 3863–3881 (2008).
[CrossRef] [PubMed]

A. Darrell, J. Swoger, L. Quintana, J. Sharpe, K. Marias, M. Brady, and J. Ripoll, “Improved fluorescence optical projection tomography reconstruction,” SPIE Newsroom1–4 (2008).
[CrossRef]

Carson, J. J. L.

Chapman, G. H.

Cotterell, J.

M. J. Boot, C. H. Westerberg, J. Sanz-Ezquerro, J. Cotterell, R. Schweitzer, M. Torres, and J. Sharpe, “In vitro whole-organ imaging: 4D quantification of growing mouse limb buds,” Nat. Method 5, 609–612 (2008).
[CrossRef]

Darrell, A.

U. J. Birk, M. Rieckher, N. Konstantinides, A. Darrell, A. Sarasa-Renedo, H. Meyer, N. Tavernarakis, and J. Ripoll, “Correction for specimen movement and rotation errors for in-vivo optical projection tomography,” Biomed. Opt. Express 1, 87–96 (2010).
[CrossRef]

A. Darrell, H. Meyer, K. Marias, M. Brady, and J. Ripoll, “Weighted filtered backprojection for quantitative fluorescence optical projection tomography,” Phys. Med. Biol. 53, 3863–3881 (2008).
[CrossRef] [PubMed]

A. Darrell, J. Swoger, L. Quintana, J. Sharpe, K. Marias, M. Brady, and J. Ripoll, “Improved fluorescence optical projection tomography reconstruction,” SPIE Newsroom1–4 (2008).
[CrossRef]

Davidson, D.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, 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]

Dunsby, C.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophotonics 1, 390–394 (2008).
[CrossRef]

Favicchio, R.

Feruglio, P. F.

Fexon, L.

Fienup, J. R.

Figueiredo, J.

French, P. M. W.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophotonics 1, 390–394 (2008).
[CrossRef]

Hecksher-Sorensen, J.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, 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, “Correction of artefacts in optical projection tomography,” Phys. Med. Biol. 50, 4645–4665 (2005).
[CrossRef] [PubMed]

Hill, B.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, 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]

Jordan, K.

Kaminska, B.

Konstantinides, N.

Laine, R.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophotonics 1, 390–394 (2008).
[CrossRef]

Lauterbur, P. C.

P. C. Lauterbur, “Magnetic resonance zeugmatography,” Pure Appl. Chem. 40, 149–157 (1974).
[CrossRef]

Mamalaki, C.

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, 3863–3881 (2008).
[CrossRef] [PubMed]

A. Darrell, J. Swoger, L. Quintana, J. Sharpe, K. Marias, M. Brady, and J. Ripoll, “Improved fluorescence optical projection tomography reconstruction,” SPIE Newsroom1–4 (2008).
[CrossRef]

Marron, J. C.

McGinty, J.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophotonics 1, 390–394 (2008).
[CrossRef]

Meyer, H.

U. J. Birk, M. Rieckher, N. Konstantinides, A. Darrell, A. Sarasa-Renedo, H. Meyer, N. Tavernarakis, and J. Ripoll, “Correction for specimen movement and rotation errors for in-vivo optical projection tomography,” Biomed. Opt. Express 1, 87–96 (2010).
[CrossRef]

A. Darrell, H. Meyer, K. Marias, M. Brady, and J. Ripoll, “Weighted filtered backprojection for quantitative fluorescence optical projection tomography,” Phys. Med. Biol. 53, 3863–3881 (2008).
[CrossRef] [PubMed]

Nahrendorf, M.

Neil, M. A. A.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophotonics 1, 390–394 (2008).
[CrossRef]

Ng, E.

Ntziachristos, V.

Pavlopoulos, A.

A. Pavlopoulos and M. Averof, “Establishing genetic transformation for comparative developmental studies in the crustacean Parhyale hawaiensis,” Proc. Natl. Acad. Sci. USA 102, 7888–7893 (2005).
[CrossRef] [PubMed]

Perrimon, N.

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Method 5, 45–47 (2008).
[CrossRef]

Perry, P.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, 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]

Pitsouli, C.

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Method 5, 45–47 (2008).
[CrossRef]

Pivovarov, M.

Pozzo, A.

Quintana, L.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophotonics 1, 390–394 (2008).
[CrossRef]

A. Darrell, J. Swoger, L. Quintana, J. Sharpe, K. Marias, M. Brady, and J. Ripoll, “Improved fluorescence optical projection tomography reconstruction,” SPIE Newsroom1–4 (2008).
[CrossRef]

Razansky, D.

C. Vinegoni, D. Razansky, J. Figueiredo, M. Nahrendorf, V. Ntziachristos, and R. Weissleder, “Normalized Born ratio for fluorescence optical projection tomography,” Opt. Lett. 34, 319–321 (2009).
[CrossRef] [PubMed]

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Method 5, 45–47 (2008).
[CrossRef]

Rieckher, M.

Ripoll, J.

Ross, A.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, 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]

Sanz-Ezquerro, J.

M. J. Boot, C. H. Westerberg, J. Sanz-Ezquerro, J. Cotterell, R. Schweitzer, M. Torres, and J. Sharpe, “In vitro whole-organ imaging: 4D quantification of growing mouse limb buds,” Nat. Method 5, 609–612 (2008).
[CrossRef]

Sarasa-Renedo, A.

Sbarbati, A.

Schulz, R. B.

Schulz, T. J.

Schweitzer, R.

M. J. Boot, C. H. Westerberg, J. Sanz-Ezquerro, J. Cotterell, R. Schweitzer, M. Torres, and J. Sharpe, “In vitro whole-organ imaging: 4D quantification of growing mouse limb buds,” Nat. Method 5, 609–612 (2008).
[CrossRef]

Seldin, J. H.

Sharpe, J.

M. J. Boot, C. H. Westerberg, J. Sanz-Ezquerro, J. Cotterell, R. Schweitzer, M. Torres, and J. Sharpe, “In vitro whole-organ imaging: 4D quantification of growing mouse limb buds,” Nat. Method 5, 609–612 (2008).
[CrossRef]

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophotonics 1, 390–394 (2008).
[CrossRef]

J. R. Walls, J. G. Sled, J. Sharpe, and R. M. Henkelman, “Correction of artefacts in optical projection tomography,” Phys. Med. Biol. 50, 4645–4665 (2005).
[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-Sorensen, 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]

A. Darrell, J. Swoger, L. Quintana, J. Sharpe, K. Marias, M. Brady, and J. Ripoll, “Improved fluorescence optical projection tomography reconstruction,” SPIE Newsroom1–4 (2008).
[CrossRef]

Sled, J. G.

J. R. Walls, J. G. Sled, J. Sharpe, and R. M. Henkelman, “Correction of artefacts in optical projection tomography,” Phys. Med. Biol. 50, 4645–4665 (2005).
[CrossRef] [PubMed]

Swoger, J.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophotonics 1, 390–394 (2008).
[CrossRef]

A. Darrell, J. Swoger, L. Quintana, J. Sharpe, K. Marias, M. Brady, and J. Ripoll, “Improved fluorescence optical projection tomography reconstruction,” SPIE Newsroom1–4 (2008).
[CrossRef]

Tahir, K. B.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophotonics 1, 390–394 (2008).
[CrossRef]

Talbot, C. B.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophotonics 1, 390–394 (2008).
[CrossRef]

Tavernarakis, N.

Torres, M.

M. J. Boot, C. H. Westerberg, J. Sanz-Ezquerro, J. Cotterell, R. Schweitzer, M. Torres, and J. Sharpe, “In vitro whole-organ imaging: 4D quantification of growing mouse limb buds,” Nat. Method 5, 609–612 (2008).
[CrossRef]

Vasefi, F.

Vinegoni, C.

Walls, J. R.

J. R. Walls, J. G. Sled, J. Sharpe, and R. M. Henkelman, “Correction of artefacts in optical projection tomography,” Phys. Med. Biol. 50, 4645–4665 (2005).
[CrossRef] [PubMed]

Weissleder, R.

Westerberg, C. H.

M. J. Boot, C. H. Westerberg, J. Sanz-Ezquerro, J. Cotterell, R. Schweitzer, M. Torres, and J. Sharpe, “In vitro whole-organ imaging: 4D quantification of growing mouse limb buds,” Nat. Method 5, 609–612 (2008).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles in Optics (Wheaton, 1983).

Zacharakis, G.

Annu. Rev. Biomed. Eng. (1)

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

Appl. Opt. (2)

Biomed. Opt. Express (1)

J. Biophotonics (1)

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophotonics 1, 390–394 (2008).
[CrossRef]

Nat. Method (2)

M. J. Boot, C. H. Westerberg, J. Sanz-Ezquerro, J. Cotterell, R. Schweitzer, M. Torres, and J. Sharpe, “In vitro whole-organ imaging: 4D quantification of growing mouse limb buds,” Nat. Method 5, 609–612 (2008).
[CrossRef]

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Method 5, 45–47 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Med. Biol. (2)

A. Darrell, H. Meyer, K. Marias, M. Brady, and J. Ripoll, “Weighted filtered backprojection for quantitative fluorescence optical projection tomography,” Phys. Med. Biol. 53, 3863–3881 (2008).
[CrossRef] [PubMed]

J. R. Walls, J. G. Sled, J. Sharpe, and R. M. Henkelman, “Correction of artefacts in optical projection tomography,” Phys. Med. Biol. 50, 4645–4665 (2005).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

A. Pavlopoulos and M. Averof, “Establishing genetic transformation for comparative developmental studies in the crustacean Parhyale hawaiensis,” Proc. Natl. Acad. Sci. USA 102, 7888–7893 (2005).
[CrossRef] [PubMed]

Pure Appl. Chem. (1)

P. C. Lauterbur, “Magnetic resonance zeugmatography,” Pure Appl. Chem. 40, 149–157 (1974).
[CrossRef]

Science (1)

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, 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)

A. Darrell, J. Swoger, L. Quintana, J. Sharpe, K. Marias, M. Brady, and J. Ripoll, “Improved fluorescence optical projection tomography reconstruction,” SPIE Newsroom1–4 (2008).
[CrossRef]

M. Born and E. Wolf, Principles in Optics (Wheaton, 1983).

Supplementary Material (1)

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Figures (6)

Fig. 1
Fig. 1

OPT setup for fluorescence imaging. Small specimens are mounted in a glass capillary, which is attached to a rotation stage. Excitation is performed in quasi-epifluorescence mode, with the light source placed in close proximity to the objective lens. Detection optics (objective lens, filters, iris diaphragm, tube lens, and CCD camera) are connected by an InfiniTube system mounted directly to the optical breadboard.

Fig. 2
Fig. 2

(a) 2D projections (raw image data) can be used to automatically determine the positions of the capillary walls (horizontal lines) by means of a forward radon transform. b) Automated object segmentation algorithm is implemented to separate the image area into the background (object number 0), specimens (2), area outside the capillary (1, including the glass walls), and sometimes additional fragments (3, e.g., residues from the culture media).

Fig. 3
Fig. 3

Comparison of background correction using measured flatfield data and background estimation from 2D projection data. Depicted are the deviations of the measured intensity I after subtraction of the background data I bkg , normalized to the measured intensities. (a) Measured flatfield data without the capillary has been used as background data. (b) Zernike fit has been applied to each of the 2D projection images after segmentation to determine the background intensities once the images were acquired. It can be seen that the capillaries also affect the illumination intensities: the flatfield-corrected data is brighter along the center of the capillary ( y = 250 ), i.e., at the pixels surrounding the object. Zernike decomposition can be used to overcome this problem. Arrows point at areas where the improvements are most pronounced.

Fig. 4
Fig. 4

So-called hot pixels of the EM-CCD camera, where the intensity in a single pixel is suddenly boosted because of electronic noise during the EM gain stage, appear as line artifacts (top right) in the reconstructed images. Nonlinear response of the individual CCD pixels appear as concentric rings in the reconstructions, as shown by Walls et al. [13].

Fig. 5
Fig. 5

(a) Standard filtered back-projection reconstruction along the parallel (green) lines in (b). (b) Cross section through the capillary with radius R = 137.8 pixels (cyan). As embedding medium, sea water with 4% methyl cellulose is used. Because of the lower index of refraction ( n = 1.344 ), the acquired projections are no longer parallel to the optical axis (apart from the central ray at x = 0 ). The geometrical focus points are shown for light rays focused inside the capillary ( y = 0 ) at positions x = 0 , 50 , 100 . The dotted bright green lines depict the apparent focus positions at 0.7 NA , whereas the true focal positions within the capillary are shown in black. (c) Result of the reconstruction along the true lines of projection [black lines in (b)]. Arrowheads indicate where reconstruction artifacts have been removed, and the arrow indicates that the overall magnification in the reconstruction has been corrected.

Fig. 6
Fig. 6

P. hawaiensis tail with muscle-specific labeling with dsRed, 40 mM sodium-azide for 1 h , no fixation, embedded in sea water with 4% methyl cellulose. Sagittal, coronal, and transverse cross sections obtained from the 3D reconstruction of the acquired 2D OPT projections. Bottom right: View of the 3D volume rendering (Media 1, which additionally shows the anatomy of the other half of the animal reconstructed from the white light data).

Tables (1)

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Table 1 Coefficients c n , m of the Zernike Polynomial Decomposition (First Three Orders) on the Cartesian Unit Interval [ 1 ; 1 ] Obtained from a Nonlinear Least Squares Fit to the Background Intensities of the 2D Projection Images after Segmentation a

Equations (2)

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t k , l = k cos ϑ + l sin ϑ + s k , l ( ϑ ) .
I k , l = ( t k , l floor ( t k , l ) ) d + + ( ceil ( t k , l ) t k , l ) d .

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