Abstract

A model and a method providing a 3D reconstruction of a given translucent object from a series of image acquisitions performed with various focus tunings is proposed. The object is imaged by transmission; refraction, reflection and diffusion effects are neglected. It is modeled as a stack of translucent parallel slices and the acquisition process can be described by a set of linear equations. We propose an efficient inversion technique with O(n) complexity, allowing practical applications with a simple laptop computer in a very reasonable time. Examples of results obtained with a simulated 3D translucent object are presented and discussed.

© 2007 Optical Society of America

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References

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  1. Y. Y. Schechner, N. Kiryati, and R. Basri, “Separation of transparent layers using focus,” Proc. of IEEE 6th Int. Conf. On Computer Vision1061–1066 (1998).
  2. T. S. Choi and J. Yun, “Three-dimensional shape recovery from the focused-image surface,” Opt. Eng. 391321–1326 (2000).
    [Crossref]
  3. M. Noguchi and S. K. Nayar, “Microscopic shape from focus using a projected illumination pattern,” Math. Comput. Modelling 24 5/6 31–48 (1996).
    [Crossref]
  4. S. Nayar and Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Machine Intell. 16824–831 (1994).
    [Crossref]
  5. J. Ens and P. Lawrence, “An investigation of methods for determining depth from focus,” IEEE Trans. Pattern Anal. Machine Intell. 1597–108 (1993).
    [Crossref]
  6. J. Ens and P. Lawrence, “A matrix based method for determining depth from focus,” Proc. CVPR600–606 (1991).
  7. J. Vitria and J. Llacer, “Reconstructing 3D light microscopic images using the EM algorithm,” Pattern Recognition Letters 171491–1498 (1996).
    [Crossref]
  8. A. Pentland, T. Darell, M. Turk, and W. Huang, “A simple real-time range camera,” IEEE Comput. Soc. Conf. Comput. Vision Patt. Recogn. 256-261 (1989).
  9. A. Pentland, “A new sense for depth of field,” IEEE Trans. Pattern Anal. Mach. Intell. 9522–531 (1987).
    [Crossref]
  10. M. Asif and T. Choi, “Shape from focus using multilayer feedforward neural networks,” IEEE Trans. on Image Processing 101670–1675 (2001).
    [Crossref]
  11. N. Dey, A. Boucher, and M. Thonnat, “Image formation model a 3-d translucent object observed in light microscopy,” Proc. of IEEE ICIP (2002).
  12. W. Pratt, “Vector formulation of 2d signal processing operations,” Comp. Graph. and Image Proc. 41–24 (1975).
    [Crossref]
  13. G. Golub and C. V. Loan, Matrix Computations, Baltimore: John Hopkins University Press third ed. (1996).
  14. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. Royal Soc. London 231 A 91–103 (1955).
    [Crossref]
  15. F. Truchetet, “3D translucent object reconstruction from artificial vision,” Machine Vision Applications in Industrial Inspection XIV 6070 (2006).
  16. C. Preza, M. I. Miller, L. J. Thomas, and J. G. McNally, “Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A. 9 2 219–228 (1992).
    [Crossref] [PubMed]
  17. M. R. P. Homem, N. D.A. Mascarenhas, L. F. Costa, and C. Preza, “Biological image restoration in optical-sectioning microscopy using prototype image constraints,” Real-Time Imaging 8475–490 (2002).
    [Crossref]
  18. S. Joschi and M. Miller, “Maximum a posteriori estimation with good’s roughness for three-dimensional optical-sectioning microscopy,” J. Opt. Soc. Am. A. 10 5 1078–1085 (1993).
    [Crossref]
  19. The discrete convolution product is denoted as usual as h * s(l,c) =∑i∑jh(i,j)s(l-i,c - j)

2006 (1)

F. Truchetet, “3D translucent object reconstruction from artificial vision,” Machine Vision Applications in Industrial Inspection XIV 6070 (2006).

2002 (2)

N. Dey, A. Boucher, and M. Thonnat, “Image formation model a 3-d translucent object observed in light microscopy,” Proc. of IEEE ICIP (2002).

M. R. P. Homem, N. D.A. Mascarenhas, L. F. Costa, and C. Preza, “Biological image restoration in optical-sectioning microscopy using prototype image constraints,” Real-Time Imaging 8475–490 (2002).
[Crossref]

2001 (1)

M. Asif and T. Choi, “Shape from focus using multilayer feedforward neural networks,” IEEE Trans. on Image Processing 101670–1675 (2001).
[Crossref]

2000 (1)

T. S. Choi and J. Yun, “Three-dimensional shape recovery from the focused-image surface,” Opt. Eng. 391321–1326 (2000).
[Crossref]

1998 (1)

Y. Y. Schechner, N. Kiryati, and R. Basri, “Separation of transparent layers using focus,” Proc. of IEEE 6th Int. Conf. On Computer Vision1061–1066 (1998).

1996 (2)

J. Vitria and J. Llacer, “Reconstructing 3D light microscopic images using the EM algorithm,” Pattern Recognition Letters 171491–1498 (1996).
[Crossref]

M. Noguchi and S. K. Nayar, “Microscopic shape from focus using a projected illumination pattern,” Math. Comput. Modelling 24 5/6 31–48 (1996).
[Crossref]

1994 (1)

S. Nayar and Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Machine Intell. 16824–831 (1994).
[Crossref]

1993 (2)

J. Ens and P. Lawrence, “An investigation of methods for determining depth from focus,” IEEE Trans. Pattern Anal. Machine Intell. 1597–108 (1993).
[Crossref]

S. Joschi and M. Miller, “Maximum a posteriori estimation with good’s roughness for three-dimensional optical-sectioning microscopy,” J. Opt. Soc. Am. A. 10 5 1078–1085 (1993).
[Crossref]

1992 (1)

C. Preza, M. I. Miller, L. J. Thomas, and J. G. McNally, “Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A. 9 2 219–228 (1992).
[Crossref] [PubMed]

1991 (1)

J. Ens and P. Lawrence, “A matrix based method for determining depth from focus,” Proc. CVPR600–606 (1991).

1989 (1)

A. Pentland, T. Darell, M. Turk, and W. Huang, “A simple real-time range camera,” IEEE Comput. Soc. Conf. Comput. Vision Patt. Recogn. 256-261 (1989).

1987 (1)

A. Pentland, “A new sense for depth of field,” IEEE Trans. Pattern Anal. Mach. Intell. 9522–531 (1987).
[Crossref]

1975 (1)

W. Pratt, “Vector formulation of 2d signal processing operations,” Comp. Graph. and Image Proc. 41–24 (1975).
[Crossref]

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. Royal Soc. London 231 A 91–103 (1955).
[Crossref]

Asif, M.

M. Asif and T. Choi, “Shape from focus using multilayer feedforward neural networks,” IEEE Trans. on Image Processing 101670–1675 (2001).
[Crossref]

Basri, R.

Y. Y. Schechner, N. Kiryati, and R. Basri, “Separation of transparent layers using focus,” Proc. of IEEE 6th Int. Conf. On Computer Vision1061–1066 (1998).

Boucher, A.

N. Dey, A. Boucher, and M. Thonnat, “Image formation model a 3-d translucent object observed in light microscopy,” Proc. of IEEE ICIP (2002).

Choi, T.

M. Asif and T. Choi, “Shape from focus using multilayer feedforward neural networks,” IEEE Trans. on Image Processing 101670–1675 (2001).
[Crossref]

Choi, T. S.

T. S. Choi and J. Yun, “Three-dimensional shape recovery from the focused-image surface,” Opt. Eng. 391321–1326 (2000).
[Crossref]

Costa, L. F.

M. R. P. Homem, N. D.A. Mascarenhas, L. F. Costa, and C. Preza, “Biological image restoration in optical-sectioning microscopy using prototype image constraints,” Real-Time Imaging 8475–490 (2002).
[Crossref]

Darell, T.

A. Pentland, T. Darell, M. Turk, and W. Huang, “A simple real-time range camera,” IEEE Comput. Soc. Conf. Comput. Vision Patt. Recogn. 256-261 (1989).

Dey, N.

N. Dey, A. Boucher, and M. Thonnat, “Image formation model a 3-d translucent object observed in light microscopy,” Proc. of IEEE ICIP (2002).

Ens, J.

J. Ens and P. Lawrence, “An investigation of methods for determining depth from focus,” IEEE Trans. Pattern Anal. Machine Intell. 1597–108 (1993).
[Crossref]

J. Ens and P. Lawrence, “A matrix based method for determining depth from focus,” Proc. CVPR600–606 (1991).

Golub, G.

G. Golub and C. V. Loan, Matrix Computations, Baltimore: John Hopkins University Press third ed. (1996).

Homem, M. R. P.

M. R. P. Homem, N. D.A. Mascarenhas, L. F. Costa, and C. Preza, “Biological image restoration in optical-sectioning microscopy using prototype image constraints,” Real-Time Imaging 8475–490 (2002).
[Crossref]

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. Royal Soc. London 231 A 91–103 (1955).
[Crossref]

Huang, W.

A. Pentland, T. Darell, M. Turk, and W. Huang, “A simple real-time range camera,” IEEE Comput. Soc. Conf. Comput. Vision Patt. Recogn. 256-261 (1989).

Joschi, S.

S. Joschi and M. Miller, “Maximum a posteriori estimation with good’s roughness for three-dimensional optical-sectioning microscopy,” J. Opt. Soc. Am. A. 10 5 1078–1085 (1993).
[Crossref]

Kiryati, N.

Y. Y. Schechner, N. Kiryati, and R. Basri, “Separation of transparent layers using focus,” Proc. of IEEE 6th Int. Conf. On Computer Vision1061–1066 (1998).

Lawrence, P.

J. Ens and P. Lawrence, “An investigation of methods for determining depth from focus,” IEEE Trans. Pattern Anal. Machine Intell. 1597–108 (1993).
[Crossref]

J. Ens and P. Lawrence, “A matrix based method for determining depth from focus,” Proc. CVPR600–606 (1991).

Llacer, J.

J. Vitria and J. Llacer, “Reconstructing 3D light microscopic images using the EM algorithm,” Pattern Recognition Letters 171491–1498 (1996).
[Crossref]

Loan, C. V.

G. Golub and C. V. Loan, Matrix Computations, Baltimore: John Hopkins University Press third ed. (1996).

Mascarenhas, N. D.A.

M. R. P. Homem, N. D.A. Mascarenhas, L. F. Costa, and C. Preza, “Biological image restoration in optical-sectioning microscopy using prototype image constraints,” Real-Time Imaging 8475–490 (2002).
[Crossref]

McNally, J. G.

C. Preza, M. I. Miller, L. J. Thomas, and J. G. McNally, “Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A. 9 2 219–228 (1992).
[Crossref] [PubMed]

Miller, M.

S. Joschi and M. Miller, “Maximum a posteriori estimation with good’s roughness for three-dimensional optical-sectioning microscopy,” J. Opt. Soc. Am. A. 10 5 1078–1085 (1993).
[Crossref]

Miller, M. I.

C. Preza, M. I. Miller, L. J. Thomas, and J. G. McNally, “Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A. 9 2 219–228 (1992).
[Crossref] [PubMed]

Nakagawa, Y.

S. Nayar and Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Machine Intell. 16824–831 (1994).
[Crossref]

Nayar, S.

S. Nayar and Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Machine Intell. 16824–831 (1994).
[Crossref]

Nayar, S. K.

M. Noguchi and S. K. Nayar, “Microscopic shape from focus using a projected illumination pattern,” Math. Comput. Modelling 24 5/6 31–48 (1996).
[Crossref]

Noguchi, M.

M. Noguchi and S. K. Nayar, “Microscopic shape from focus using a projected illumination pattern,” Math. Comput. Modelling 24 5/6 31–48 (1996).
[Crossref]

Pentland, A.

A. Pentland, T. Darell, M. Turk, and W. Huang, “A simple real-time range camera,” IEEE Comput. Soc. Conf. Comput. Vision Patt. Recogn. 256-261 (1989).

A. Pentland, “A new sense for depth of field,” IEEE Trans. Pattern Anal. Mach. Intell. 9522–531 (1987).
[Crossref]

Pratt, W.

W. Pratt, “Vector formulation of 2d signal processing operations,” Comp. Graph. and Image Proc. 41–24 (1975).
[Crossref]

Preza, C.

M. R. P. Homem, N. D.A. Mascarenhas, L. F. Costa, and C. Preza, “Biological image restoration in optical-sectioning microscopy using prototype image constraints,” Real-Time Imaging 8475–490 (2002).
[Crossref]

C. Preza, M. I. Miller, L. J. Thomas, and J. G. McNally, “Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A. 9 2 219–228 (1992).
[Crossref] [PubMed]

Schechner, Y. Y.

Y. Y. Schechner, N. Kiryati, and R. Basri, “Separation of transparent layers using focus,” Proc. of IEEE 6th Int. Conf. On Computer Vision1061–1066 (1998).

Thomas, L. J.

C. Preza, M. I. Miller, L. J. Thomas, and J. G. McNally, “Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A. 9 2 219–228 (1992).
[Crossref] [PubMed]

Thonnat, M.

N. Dey, A. Boucher, and M. Thonnat, “Image formation model a 3-d translucent object observed in light microscopy,” Proc. of IEEE ICIP (2002).

Truchetet, F.

F. Truchetet, “3D translucent object reconstruction from artificial vision,” Machine Vision Applications in Industrial Inspection XIV 6070 (2006).

Turk, M.

A. Pentland, T. Darell, M. Turk, and W. Huang, “A simple real-time range camera,” IEEE Comput. Soc. Conf. Comput. Vision Patt. Recogn. 256-261 (1989).

Vitria, J.

J. Vitria and J. Llacer, “Reconstructing 3D light microscopic images using the EM algorithm,” Pattern Recognition Letters 171491–1498 (1996).
[Crossref]

Yun, J.

T. S. Choi and J. Yun, “Three-dimensional shape recovery from the focused-image surface,” Opt. Eng. 391321–1326 (2000).
[Crossref]

Comp. Graph. and Image Proc. (1)

W. Pratt, “Vector formulation of 2d signal processing operations,” Comp. Graph. and Image Proc. 41–24 (1975).
[Crossref]

IEEE Comput. Soc. Conf. Comput. Vision Patt. Recogn. (1)

A. Pentland, T. Darell, M. Turk, and W. Huang, “A simple real-time range camera,” IEEE Comput. Soc. Conf. Comput. Vision Patt. Recogn. 256-261 (1989).

IEEE Trans. on Image Processing (1)

M. Asif and T. Choi, “Shape from focus using multilayer feedforward neural networks,” IEEE Trans. on Image Processing 101670–1675 (2001).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

A. Pentland, “A new sense for depth of field,” IEEE Trans. Pattern Anal. Mach. Intell. 9522–531 (1987).
[Crossref]

IEEE Trans. Pattern Anal. Machine Intell. (2)

S. Nayar and Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Machine Intell. 16824–831 (1994).
[Crossref]

J. Ens and P. Lawrence, “An investigation of methods for determining depth from focus,” IEEE Trans. Pattern Anal. Machine Intell. 1597–108 (1993).
[Crossref]

J. Opt. Soc. Am. A. (2)

C. Preza, M. I. Miller, L. J. Thomas, and J. G. McNally, “Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A. 9 2 219–228 (1992).
[Crossref] [PubMed]

S. Joschi and M. Miller, “Maximum a posteriori estimation with good’s roughness for three-dimensional optical-sectioning microscopy,” J. Opt. Soc. Am. A. 10 5 1078–1085 (1993).
[Crossref]

Machine Vision Applications in Industrial Inspection XIV (1)

F. Truchetet, “3D translucent object reconstruction from artificial vision,” Machine Vision Applications in Industrial Inspection XIV 6070 (2006).

Math. Comput. Modelling (1)

M. Noguchi and S. K. Nayar, “Microscopic shape from focus using a projected illumination pattern,” Math. Comput. Modelling 24 5/6 31–48 (1996).
[Crossref]

Opt. Eng. (1)

T. S. Choi and J. Yun, “Three-dimensional shape recovery from the focused-image surface,” Opt. Eng. 391321–1326 (2000).
[Crossref]

Pattern Recognition Letters (1)

J. Vitria and J. Llacer, “Reconstructing 3D light microscopic images using the EM algorithm,” Pattern Recognition Letters 171491–1498 (1996).
[Crossref]

Proc. CVPR (1)

J. Ens and P. Lawrence, “A matrix based method for determining depth from focus,” Proc. CVPR600–606 (1991).

Proc. of IEEE 6th Int. Conf. On Computer Vision (1)

Y. Y. Schechner, N. Kiryati, and R. Basri, “Separation of transparent layers using focus,” Proc. of IEEE 6th Int. Conf. On Computer Vision1061–1066 (1998).

Proc. of IEEE ICIP (1)

N. Dey, A. Boucher, and M. Thonnat, “Image formation model a 3-d translucent object observed in light microscopy,” Proc. of IEEE ICIP (2002).

Proc. Royal Soc. London (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. Royal Soc. London 231 A 91–103 (1955).
[Crossref]

Real-Time Imaging (1)

M. R. P. Homem, N. D.A. Mascarenhas, L. F. Costa, and C. Preza, “Biological image restoration in optical-sectioning microscopy using prototype image constraints,” Real-Time Imaging 8475–490 (2002).
[Crossref]

Other (2)

G. Golub and C. V. Loan, Matrix Computations, Baltimore: John Hopkins University Press third ed. (1996).

The discrete convolution product is denoted as usual as h * s(l,c) =∑i∑jh(i,j)s(l-i,c - j)

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Figures (11)

Fig. 1.
Fig. 1.

Stack of parallel transparent slices model.

Fig. 2.
Fig. 2.

Inversion computing time versus number of data.

Fig. 3.
Fig. 3.

Simulation device.

Fig. 4.
Fig. 4.

Sensibility to the gain setting.

Fig. 5.
Fig. 5.

Error in the reconstructed image from noisy measurements (exact inverse).

Fig. 6.
Fig. 6.

Error in the reconstructed image from noisy measurements (pseudoinverse).

Fig. 7.
Fig. 7.

Robustness against PSF width estimate.

Fig. 8.
Fig. 8.

Sensibility to focusing error.

Fig. 9.
Fig. 9.

Example of reconstructed images (rmse=0.052).

Fig. 10.
Fig. 10.

Example of error images: |reconstructedmodel| (rmse=0.052).

Fig. 11.
Fig. 11.

Error in the reconstructed image from noisy measurements (the same amount of noise is added to each image in the measurement stage).

Equations (14)

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

T x ( l , c ) = 1 a x ( l , c ) .
a x i ( l , c ) = x i x i + 1 a x ( l , c ) dx .
T ( l , c ) = ( 1 a x 1 ) ( 1 a x 2 ) ( 1 a x 3 ) ,
T ( l , c ) = 1 i a x i ( l , c ) ,
s y ( l , c ) = 1 x h x y a x ( l , c ) dx .
s x i ( l , c ) = 1 j h x j x i a x j ( l , c ) .
S i = ( 1 s xi ( 1 , 1 ) , 1 s xi ( 1 , 2 ) , 1 s xi ( 1 , 3 ) , , 1 s xi ( 2 , 1 ) , ) . . . , 1 s xi ( L , C ) ) t ,
E j = ( a x j ( 1 , 1 ) , a x j ( 1 , 2 ) , a x j ( 1 , 3 ) , , a x j ( 2 , 1 ) , , a x j ( L , C ) ) t .
[ S 1 S 2 . S P ] = [ H 11 H 12 H 1 K H 21 H 22 H 2 K H P 1 H P 2 H PK ] [ E 1 E 2 . E K ] .
s i ( l , c ) = j = 1 K h j i a j ( l , c ) .
s i ̂ ( n , m ) = j = 1 K h ̂ ji ( n , m ) a ̂ j ( n , m ) ,
S ̂ ( n , m ) = H ̂ ( n , m ) A ̂ ( n , m ) .
h ji ( l , c ) = 1 2 π σ 2 exp ( l 2 + c 2 2 σ 2 )
d ji = fD x j f x i x j x i

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