Abstract

The morphology of three-dimensional foams is of interest to physicists, engineers, and mathematicians. It is desired to image the 3-dimensional structure of the foam. Many different techniques have been used to image the foam, including magnetic resonance imaging, and short-focal length lenses. We use a camera and apply tomographic algorithms to accurately image a set of bubbles. We correct for the distortion of a curved plexiglas container using ray-tracing.

© 2000 Optical Society of America

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References

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  1. Denis Weaire, Stefan Hutzler, The Physics of Foams, (Oxford University, Oxford, 1999).
  2. D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple Light-Scattering Probes of Foam Structure and Dynamics,” Science 252686 (1991).
    [CrossRef] [PubMed]
  3. C. Monnereau, M. Vignes-Adler, “Optical Tomography of Real Three-Dimensional Foams,” Journal of Colloid and Interface Science 20245–53 (1998).
    [CrossRef]
  4. C. Monnereau, M. Vignes-Adler, “Dynamics of 3D Real Foam Coarsening,” Phys. Rev. Lett. 80 (23) 5228–5231 (1998).
    [CrossRef]
  5. C. P. Gonatas, J. S. Leigh, A. G. Yodh, J. A. Glazier, B. Prause, “Magnetic Resonance Images of Coarsening Inside a Foam,” Phys. Rev. Lett. 75 (3) 573–576 (1995).
    [CrossRef] [PubMed]
  6. H.P. Hiriyannaiah, “Computed Tomography for Medical Imaging,” IEEE Signal Processing Magazine, 42–59, (March 1997).
    [CrossRef]
  7. L.A. Feldkamp, L.C. Davis, J.W. Kress, “Practical Cone-beam Algorithm,” J. Opt. Soc. Am. A 1 (6) 612–619 (1984).
    [CrossRef]
  8. D. Marks, R.A. Stack, D.J. Brady, D.C. Munson “Visible Cone-Beam Tomography With a Lensless Interferometric Camera,” Science 2842164–2166 (1999).
    [CrossRef] [PubMed]
  9. D.L. Marks, R.A. Stack, D.J. Brady, D.C. Munson“Cone-beam Tomography with a digital camera,” Appl. Opt. (in review) 2000.
  10. VTK Toolkit, http://www.kitware.com/vtk.html
  11. H.K. Tuy, SIAM J. Appl. Math 43546 (1983).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics, (Cambridge University Press, Cambridge, 1980).
  13. P. Soille, Morphological Image Processing: Principles and Applications, (Springer, Heidelberg, 1999).
  14. S.A. Koehler, S. Hilgenfeldt, H.A. Stone, “A Generalized View of Foam Drainage: Experiment and Theory,” Langmuir (http://pubs.acs.org/journals/langd5) 16 (15) 6327–6341 (2000).
    [CrossRef]

2000 (1)

S.A. Koehler, S. Hilgenfeldt, H.A. Stone, “A Generalized View of Foam Drainage: Experiment and Theory,” Langmuir (http://pubs.acs.org/journals/langd5) 16 (15) 6327–6341 (2000).
[CrossRef]

1999 (1)

D. Marks, R.A. Stack, D.J. Brady, D.C. Munson “Visible Cone-Beam Tomography With a Lensless Interferometric Camera,” Science 2842164–2166 (1999).
[CrossRef] [PubMed]

1998 (2)

C. Monnereau, M. Vignes-Adler, “Optical Tomography of Real Three-Dimensional Foams,” Journal of Colloid and Interface Science 20245–53 (1998).
[CrossRef]

C. Monnereau, M. Vignes-Adler, “Dynamics of 3D Real Foam Coarsening,” Phys. Rev. Lett. 80 (23) 5228–5231 (1998).
[CrossRef]

1995 (1)

C. P. Gonatas, J. S. Leigh, A. G. Yodh, J. A. Glazier, B. Prause, “Magnetic Resonance Images of Coarsening Inside a Foam,” Phys. Rev. Lett. 75 (3) 573–576 (1995).
[CrossRef] [PubMed]

1991 (1)

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple Light-Scattering Probes of Foam Structure and Dynamics,” Science 252686 (1991).
[CrossRef] [PubMed]

1984 (1)

1983 (1)

H.K. Tuy, SIAM J. Appl. Math 43546 (1983).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, (Cambridge University Press, Cambridge, 1980).

Brady, D.J.

D. Marks, R.A. Stack, D.J. Brady, D.C. Munson “Visible Cone-Beam Tomography With a Lensless Interferometric Camera,” Science 2842164–2166 (1999).
[CrossRef] [PubMed]

D.L. Marks, R.A. Stack, D.J. Brady, D.C. Munson“Cone-beam Tomography with a digital camera,” Appl. Opt. (in review) 2000.

Davis, L.C.

Durian, D. J.

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple Light-Scattering Probes of Foam Structure and Dynamics,” Science 252686 (1991).
[CrossRef] [PubMed]

Feldkamp, L.A.

Glazier, J. A.

C. P. Gonatas, J. S. Leigh, A. G. Yodh, J. A. Glazier, B. Prause, “Magnetic Resonance Images of Coarsening Inside a Foam,” Phys. Rev. Lett. 75 (3) 573–576 (1995).
[CrossRef] [PubMed]

Gonatas, C. P.

C. P. Gonatas, J. S. Leigh, A. G. Yodh, J. A. Glazier, B. Prause, “Magnetic Resonance Images of Coarsening Inside a Foam,” Phys. Rev. Lett. 75 (3) 573–576 (1995).
[CrossRef] [PubMed]

Hilgenfeldt, S.

S.A. Koehler, S. Hilgenfeldt, H.A. Stone, “A Generalized View of Foam Drainage: Experiment and Theory,” Langmuir (http://pubs.acs.org/journals/langd5) 16 (15) 6327–6341 (2000).
[CrossRef]

Hiriyannaiah, H.P.

H.P. Hiriyannaiah, “Computed Tomography for Medical Imaging,” IEEE Signal Processing Magazine, 42–59, (March 1997).
[CrossRef]

Hutzler, Stefan

Denis Weaire, Stefan Hutzler, The Physics of Foams, (Oxford University, Oxford, 1999).

Koehler, S.A.

S.A. Koehler, S. Hilgenfeldt, H.A. Stone, “A Generalized View of Foam Drainage: Experiment and Theory,” Langmuir (http://pubs.acs.org/journals/langd5) 16 (15) 6327–6341 (2000).
[CrossRef]

Kress, J.W.

Leigh, J. S.

C. P. Gonatas, J. S. Leigh, A. G. Yodh, J. A. Glazier, B. Prause, “Magnetic Resonance Images of Coarsening Inside a Foam,” Phys. Rev. Lett. 75 (3) 573–576 (1995).
[CrossRef] [PubMed]

Marks, D.

D. Marks, R.A. Stack, D.J. Brady, D.C. Munson “Visible Cone-Beam Tomography With a Lensless Interferometric Camera,” Science 2842164–2166 (1999).
[CrossRef] [PubMed]

Marks, D.L.

D.L. Marks, R.A. Stack, D.J. Brady, D.C. Munson“Cone-beam Tomography with a digital camera,” Appl. Opt. (in review) 2000.

Monnereau, C.

C. Monnereau, M. Vignes-Adler, “Dynamics of 3D Real Foam Coarsening,” Phys. Rev. Lett. 80 (23) 5228–5231 (1998).
[CrossRef]

C. Monnereau, M. Vignes-Adler, “Optical Tomography of Real Three-Dimensional Foams,” Journal of Colloid and Interface Science 20245–53 (1998).
[CrossRef]

Munson, D.C.

D. Marks, R.A. Stack, D.J. Brady, D.C. Munson “Visible Cone-Beam Tomography With a Lensless Interferometric Camera,” Science 2842164–2166 (1999).
[CrossRef] [PubMed]

D.L. Marks, R.A. Stack, D.J. Brady, D.C. Munson“Cone-beam Tomography with a digital camera,” Appl. Opt. (in review) 2000.

Pine, D. J.

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple Light-Scattering Probes of Foam Structure and Dynamics,” Science 252686 (1991).
[CrossRef] [PubMed]

Prause, B.

C. P. Gonatas, J. S. Leigh, A. G. Yodh, J. A. Glazier, B. Prause, “Magnetic Resonance Images of Coarsening Inside a Foam,” Phys. Rev. Lett. 75 (3) 573–576 (1995).
[CrossRef] [PubMed]

Soille, P.

P. Soille, Morphological Image Processing: Principles and Applications, (Springer, Heidelberg, 1999).

Stack, R.A.

D. Marks, R.A. Stack, D.J. Brady, D.C. Munson “Visible Cone-Beam Tomography With a Lensless Interferometric Camera,” Science 2842164–2166 (1999).
[CrossRef] [PubMed]

D.L. Marks, R.A. Stack, D.J. Brady, D.C. Munson“Cone-beam Tomography with a digital camera,” Appl. Opt. (in review) 2000.

Stone, H.A.

S.A. Koehler, S. Hilgenfeldt, H.A. Stone, “A Generalized View of Foam Drainage: Experiment and Theory,” Langmuir (http://pubs.acs.org/journals/langd5) 16 (15) 6327–6341 (2000).
[CrossRef]

Tuy, H.K.

H.K. Tuy, SIAM J. Appl. Math 43546 (1983).
[CrossRef]

Vignes-Adler, M.

C. Monnereau, M. Vignes-Adler, “Optical Tomography of Real Three-Dimensional Foams,” Journal of Colloid and Interface Science 20245–53 (1998).
[CrossRef]

C. Monnereau, M. Vignes-Adler, “Dynamics of 3D Real Foam Coarsening,” Phys. Rev. Lett. 80 (23) 5228–5231 (1998).
[CrossRef]

Weaire, Denis

Denis Weaire, Stefan Hutzler, The Physics of Foams, (Oxford University, Oxford, 1999).

Weitz, D. A.

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple Light-Scattering Probes of Foam Structure and Dynamics,” Science 252686 (1991).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, (Cambridge University Press, Cambridge, 1980).

Yodh, A. G.

C. P. Gonatas, J. S. Leigh, A. G. Yodh, J. A. Glazier, B. Prause, “Magnetic Resonance Images of Coarsening Inside a Foam,” Phys. Rev. Lett. 75 (3) 573–576 (1995).
[CrossRef] [PubMed]

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

Journal of Colloid and Interface Science (1)

C. Monnereau, M. Vignes-Adler, “Optical Tomography of Real Three-Dimensional Foams,” Journal of Colloid and Interface Science 20245–53 (1998).
[CrossRef]

Langmuir (1)

S.A. Koehler, S. Hilgenfeldt, H.A. Stone, “A Generalized View of Foam Drainage: Experiment and Theory,” Langmuir (http://pubs.acs.org/journals/langd5) 16 (15) 6327–6341 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

C. Monnereau, M. Vignes-Adler, “Dynamics of 3D Real Foam Coarsening,” Phys. Rev. Lett. 80 (23) 5228–5231 (1998).
[CrossRef]

C. P. Gonatas, J. S. Leigh, A. G. Yodh, J. A. Glazier, B. Prause, “Magnetic Resonance Images of Coarsening Inside a Foam,” Phys. Rev. Lett. 75 (3) 573–576 (1995).
[CrossRef] [PubMed]

Science (2)

D. J. Durian, D. A. Weitz, D. J. Pine, “Multiple Light-Scattering Probes of Foam Structure and Dynamics,” Science 252686 (1991).
[CrossRef] [PubMed]

D. Marks, R.A. Stack, D.J. Brady, D.C. Munson “Visible Cone-Beam Tomography With a Lensless Interferometric Camera,” Science 2842164–2166 (1999).
[CrossRef] [PubMed]

SIAM J. Appl. Math (1)

H.K. Tuy, SIAM J. Appl. Math 43546 (1983).
[CrossRef]

Other (6)

M. Born, E. Wolf, Principles of Optics, (Cambridge University Press, Cambridge, 1980).

P. Soille, Morphological Image Processing: Principles and Applications, (Springer, Heidelberg, 1999).

D.L. Marks, R.A. Stack, D.J. Brady, D.C. Munson“Cone-beam Tomography with a digital camera,” Appl. Opt. (in review) 2000.

VTK Toolkit, http://www.kitware.com/vtk.html

Denis Weaire, Stefan Hutzler, The Physics of Foams, (Oxford University, Oxford, 1999).

H.P. Hiriyannaiah, “Computed Tomography for Medical Imaging,” IEEE Signal Processing Magazine, 42–59, (March 1997).
[CrossRef]

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Figures (11)

Fig. 1.
Fig. 1.

A CCD video image of polyhedral aqueous foam showing the network of vertices and edges. The camera is set at a large depth of field to reveal the interior features.

Fig. 2.
Fig. 2.

Top. Experimental setup. The bubbles are in a cylindrical plexiglas container, that is placed on a rotation stage. The plexiglas container is held in a mount such that it is centered on the center of the rotation stage. A lightbox (a flat box with fluorescent lights that provides a uniform white light) is placed behind the cylinder, so that the camera sees the silhouetted image. The computer controls the rotation stage, and the computer also acquires images from the digital camera. Bottom. As shown above, the image rotates around on a stage, and the camera remains stationary. However, it is equivalent to view the object as stationary, while the camera rotates. The positions that the camera acquires an image at are referred to as the vertex points. The x, y, and z axes travel with the camera.

Fig. 3.
Fig. 3.

Top. The angular notation used in this paper. Consider a particular voxel and vertex point. We write α for the angle in the vertex plane, β for the angle normal to the vertex plane, and r for the vector that connects the center of the vertex path to the voxel. Bottom. Angular notation, continued. For a given image, recorded by a camera, Ψ y and Ψ z refer to the coordinates on the camera. It is necessary to find a mapping function between the camera coordinates Ψ y and Ψ z , and the points in the reconstructed voxel space, which are denoted by the angles α, β, and the vector r.

Fig. 4.
Fig. 4.

Left. A typical image from the experimental setup. In this case the object is a cylinder with needles inside. A transparent plexiglas cylinder encloses the needles. The cylinder cannot be seen in this picture except for the white line on the right. The object is backlit, so that this image is a silhouette. Right. An image of the bubbles. Note the polyhedral bubble structure.

Fig. 5.
Fig. 5.

Compensating for distortion. Top. This illustrates the case of a generalized (2-D) distortion. Assume that we have a known index profile n(x, y) that is completely contained within a circle C with radius Rc . Outside this circle, the index of refraction is n=1.0. Here, we show the case when n(x, y)=1. Then we can connect the vertex point V and a voxel P with a straight line, and the equations described in section 2 apply. Bottom. Consider the situation when n(x, y) is a known function. In this case, the blue circle could represent a region where n(x, y)=2; everywhere else, n(x, y)=1. Using Snell’s Law and numerical iteration, we find the ray that connects P to V. Starting at a voxel P, and given a direction vector, we find the intersection of the ray with the circle C at P , Then the angles (for Eq.4 and Eq.5) can be found from the points M and P .

Fig. 6.
Fig. 6.

This shows one step in the construction to find the ray that connects the voxel to the vertex point. Note that this is a two-dimensional calculation. In this step, we project a ray from the voxel towards the vertex point. Using Snell’s law, we find the angle of refraction that occurs when the ray intersects the inner radius of the cylinder. Not shown is the next step, in which we calculate the refraction of the ray at the outer air-cylinder interface.

Fig. 7.
Fig. 7.

Solving for the rays such that they intersect the vertex point. Top. Magnified view of cylinder. Bottom. This image shows the cylinder as well as the vertex point.

Fig. 8.
Fig. 8.

Left. Reconstructing the test object of the needles without applying the distortion algorithm. This three-dimensional image was generated by vtk[10]. The colors in this picture are an arbitrary colormap and have no significance. Right. The image is improved through application of the distortion algorithm. The red area in the back of this image is a piece of paper that was in the original object.

Fig. 9.
Fig. 9.

Left. This is a slice through the dataset of Fig. 8, which is a reconstruction of the needles test object without applying the distortion correction. The needles, which should appear as points, in this case appear as blobs. Some of the features appear as crosses. Right. The distortion correction algorithm is applied. The cross section of the needles now appear as points.

Fig. 10.
Fig. 10.

Left. Reconstructing the bubbles without applying the distortion algorithm. Only a slice of the image from Fig. 4 is reconstructed. Right. The image is improved through application of the distortion algorithm.

Fig. 11.
Fig. 11.

Top. This is a slice through the dataset of Fig. 10, which is a reconstruction of the bubbles without applying the distortion correction. There is some blurring in this image. Bottom Left. The distortion correction algorithm is applied, improving the bubbles images. Bottom Right. An erosion algorithm is applied to reduce each blob in the image at left (the corrected bubbles images) to a single point.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

g y ( Ψ y ) = ω y 0 ω y 0 d ω ω exp ( i ω Ψ y 2 ω ω y 0 )
P ¯ ϕ ( Ψ y , Ψ z ) = d Ψ y d Ψ z g y ( Ψ y Ψ y ) g z ( Ψ z Ψ z ) P ϕ ( Ψ y , Ψ y ) 1 1 + Ψ y 2 + Ψ z 2
f ( r ) = 1 4 π 3 V d ϕ d 2 ( d + r x ̂ ) 2 P ¯ ϕ ( tan α , tan β )
α = arctan r y ̂ d + r x ̂
β = arctan r z ̂ d + r x ̂

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