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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  1. Y. Y. Schechner, N. Kiryati, and R. Basri, "Separation of transparent layers using focus," Proc. of IEEE 6th Int. Conf. On Computer Vision 1061-1066 (1998).
  2. T. S. Choi and J. Yun, "Three-dimensional shape recovery from the focused-image surface," Opt. Eng. 39, 1321- 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. 16, 824-831 (1994).
    [CrossRef]
  5. J. Ens and P. Lawrence, "An investigation of methods for determining depth from focus," IEEE Trans. Pattern Anal. Machine Intell. 15, 97-108 (1993).
    [CrossRef]
  6. J. Ens and P. Lawrence, "A matrix based method for determining depth from focus," Proc. CVPR 600-606 (1991).
  7. J. Vitria and J. Llacer, "Reconstructing 3D light microscopic images using the EM algorithm," Pattern Recognition Letters 17, 1491-1498 (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. 9, 522-531 (1987).
    [CrossRef]
  10. M. Asif and T. Choi, "Shape from focus using multilayer feedforward neural networks," IEEE Trans. on Image Processing 10, 1670-1675 (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. 4, 1-24 (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 14, 6070 (2006).
  16. C. Preza, M. I. Miller, L. J. Jr Thomas, J. G. McNally, "Regularized linear method for reconstruction of threedimensional microscopic objects from optical sections," J. Opt. Soc. Am. A. 9, 219-228 (1992).
    [CrossRef] [PubMed]
  17. M. R. P. Homem, N. D.A. Mascarenhas, L. F. Costa and C. Preza, "Biological image restoration in opticalsectioning microscopy using prototype image constraints," Real-Time Imaging 8, 475-490 (2002).
    [CrossRef]
  18. S. Joschi, M. Miller, "Maximum a posteriori estimation with good’s roughness for three-dimensional opticalsectioning microscopy," J. Opt. Soc. Am. A. 10, 1078-1085 (1993).
    [CrossRef]
  19. The discrete convolution product is denoted as usual as h s(l,c) =ΣiΣj h(i, j)s(l -i,c- j)

2006

F. Truchetet, "3D translucent object reconstruction from artificial vision," Machine Vision Applications in Industrial Inspection 14, 6070 (2006).

2002

M. R. P. Homem, N. D.A. Mascarenhas, L. F. Costa and C. Preza, "Biological image restoration in opticalsectioning microscopy using prototype image constraints," Real-Time Imaging 8, 475-490 (2002).
[CrossRef]

2001

M. Asif and T. Choi, "Shape from focus using multilayer feedforward neural networks," IEEE Trans. on Image Processing 10, 1670-1675 (2001).
[CrossRef]

2000

T. S. Choi and J. Yun, "Three-dimensional shape recovery from the focused-image surface," Opt. Eng. 39, 1321- 1326 (2000).
[CrossRef]

1996

J. Vitria and J. Llacer, "Reconstructing 3D light microscopic images using the EM algorithm," Pattern Recognition Letters 17, 1491-1498 (1996).
[CrossRef]

1994

S. Nayar and Y. Nakagawa, "Shape from focus," IEEE Trans. Pattern Anal. Machine Intell. 16, 824-831 (1994).
[CrossRef]

1993

J. Ens and P. Lawrence, "An investigation of methods for determining depth from focus," IEEE Trans. Pattern Anal. Machine Intell. 15, 97-108 (1993).
[CrossRef]

1987

A. Pentland, "A new sense for depth of field," IEEE Trans. Pattern Anal. Mach. Intell. 9, 522-531 (1987).
[CrossRef]

1975

W. Pratt, "Vector formulation of 2d signal processing operations," Comp. Graph. and Image Proc. 4, 1-24 (1975).
[CrossRef]

Asif, M.

M. Asif and T. Choi, "Shape from focus using multilayer feedforward neural networks," IEEE Trans. on Image Processing 10, 1670-1675 (2001).
[CrossRef]

Choi, T.

M. Asif and T. Choi, "Shape from focus using multilayer feedforward neural networks," IEEE Trans. on Image Processing 10, 1670-1675 (2001).
[CrossRef]

Choi, T. S.

T. S. Choi and J. Yun, "Three-dimensional shape recovery from the focused-image surface," Opt. Eng. 39, 1321- 1326 (2000).
[CrossRef]

Costa, L. F.

M. R. P. Homem, N. D.A. Mascarenhas, L. F. Costa and C. Preza, "Biological image restoration in opticalsectioning microscopy using prototype image constraints," Real-Time Imaging 8, 475-490 (2002).
[CrossRef]

Ens, J.

J. Ens and P. Lawrence, "An investigation of methods for determining depth from focus," IEEE Trans. Pattern Anal. Machine Intell. 15, 97-108 (1993).
[CrossRef]

J. Ens and P. Lawrence, "A matrix based method for determining depth from focus," Proc. CVPR 600-606 (1991).

Homem, M. R. P.

M. R. P. Homem, N. D.A. Mascarenhas, L. F. Costa and C. Preza, "Biological image restoration in opticalsectioning microscopy using prototype image constraints," Real-Time Imaging 8, 475-490 (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]

Joschi, S.

S. Joschi, M. Miller, "Maximum a posteriori estimation with good’s roughness for three-dimensional opticalsectioning microscopy," J. Opt. Soc. Am. A. 10, 1078-1085 (1993).
[CrossRef]

Jr Thomas, L. J.

C. Preza, M. I. Miller, L. J. Jr Thomas, J. G. McNally, "Regularized linear method for reconstruction of threedimensional microscopic objects from optical sections," J. Opt. Soc. Am. A. 9, 219-228 (1992).
[CrossRef] [PubMed]

Lawrence, P.

J. Ens and P. Lawrence, "An investigation of methods for determining depth from focus," IEEE Trans. Pattern Anal. Machine Intell. 15, 97-108 (1993).
[CrossRef]

J. Ens and P. Lawrence, "A matrix based method for determining depth from focus," Proc. CVPR 600-606 (1991).

Llacer, J.

J. Vitria and J. Llacer, "Reconstructing 3D light microscopic images using the EM algorithm," Pattern Recognition Letters 17, 1491-1498 (1996).
[CrossRef]

Mascarenhas, N. D.A.

M. R. P. Homem, N. D.A. Mascarenhas, L. F. Costa and C. Preza, "Biological image restoration in opticalsectioning microscopy using prototype image constraints," Real-Time Imaging 8, 475-490 (2002).
[CrossRef]

McNally, J. G.

C. Preza, M. I. Miller, L. J. Jr Thomas, J. G. McNally, "Regularized linear method for reconstruction of threedimensional microscopic objects from optical sections," J. Opt. Soc. Am. A. 9, 219-228 (1992).
[CrossRef] [PubMed]

Miller, M.

S. Joschi, M. Miller, "Maximum a posteriori estimation with good’s roughness for three-dimensional opticalsectioning microscopy," J. Opt. Soc. Am. A. 10, 1078-1085 (1993).
[CrossRef]

Miller, M. I.

C. Preza, M. I. Miller, L. J. Jr Thomas, J. G. McNally, "Regularized linear method for reconstruction of threedimensional microscopic objects from optical sections," J. Opt. Soc. Am. A. 9, 219-228 (1992).
[CrossRef] [PubMed]

Nakagawa, Y.

S. Nayar and Y. Nakagawa, "Shape from focus," IEEE Trans. Pattern Anal. Machine Intell. 16, 824-831 (1994).
[CrossRef]

Nayar, S.

S. Nayar and Y. Nakagawa, "Shape from focus," IEEE Trans. Pattern Anal. Machine Intell. 16, 824-831 (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, "A new sense for depth of field," IEEE Trans. Pattern Anal. Mach. Intell. 9, 522-531 (1987).
[CrossRef]

Pratt, W.

W. Pratt, "Vector formulation of 2d signal processing operations," Comp. Graph. and Image Proc. 4, 1-24 (1975).
[CrossRef]

Preza, C.

M. R. P. Homem, N. D.A. Mascarenhas, L. F. Costa and C. Preza, "Biological image restoration in opticalsectioning microscopy using prototype image constraints," Real-Time Imaging 8, 475-490 (2002).
[CrossRef]

C. Preza, M. I. Miller, L. J. Jr Thomas, J. G. McNally, "Regularized linear method for reconstruction of threedimensional microscopic objects from optical sections," J. Opt. Soc. Am. A. 9, 219-228 (1992).
[CrossRef] [PubMed]

Truchetet, F.

F. Truchetet, "3D translucent object reconstruction from artificial vision," Machine Vision Applications in Industrial Inspection 14, 6070 (2006).

Vitria, J.

J. Vitria and J. Llacer, "Reconstructing 3D light microscopic images using the EM algorithm," Pattern Recognition Letters 17, 1491-1498 (1996).
[CrossRef]

Yun, J.

T. S. Choi and J. Yun, "Three-dimensional shape recovery from the focused-image surface," Opt. Eng. 39, 1321- 1326 (2000).
[CrossRef]

Comp. Graph. and Image Proc.

W. Pratt, "Vector formulation of 2d signal processing operations," Comp. Graph. and Image Proc. 4, 1-24 (1975).
[CrossRef]

IEEE Trans. on Image Processing

M. Asif and T. Choi, "Shape from focus using multilayer feedforward neural networks," IEEE Trans. on Image Processing 10, 1670-1675 (2001).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

A. Pentland, "A new sense for depth of field," IEEE Trans. Pattern Anal. Mach. Intell. 9, 522-531 (1987).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell.

S. Nayar and Y. Nakagawa, "Shape from focus," IEEE Trans. Pattern Anal. Machine Intell. 16, 824-831 (1994).
[CrossRef]

J. Ens and P. Lawrence, "An investigation of methods for determining depth from focus," IEEE Trans. Pattern Anal. Machine Intell. 15, 97-108 (1993).
[CrossRef]

Machine Vision Applications in Industrial Inspection

F. Truchetet, "3D translucent object reconstruction from artificial vision," Machine Vision Applications in Industrial Inspection 14, 6070 (2006).

Opt. Eng.

T. S. Choi and J. Yun, "Three-dimensional shape recovery from the focused-image surface," Opt. Eng. 39, 1321- 1326 (2000).
[CrossRef]

Pattern Recognition Letters

J. Vitria and J. Llacer, "Reconstructing 3D light microscopic images using the EM algorithm," Pattern Recognition Letters 17, 1491-1498 (1996).
[CrossRef]

Real-Time Imaging

M. R. P. Homem, N. D.A. Mascarenhas, L. F. Costa and C. Preza, "Biological image restoration in opticalsectioning microscopy using prototype image constraints," Real-Time Imaging 8, 475-490 (2002).
[CrossRef]

Other

S. Joschi, M. Miller, "Maximum a posteriori estimation with good’s roughness for three-dimensional opticalsectioning microscopy," J. Opt. Soc. Am. A. 10, 1078-1085 (1993).
[CrossRef]

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

C. Preza, M. I. Miller, L. J. Jr Thomas, J. G. McNally, "Regularized linear method for reconstruction of threedimensional microscopic objects from optical sections," J. Opt. Soc. Am. A. 9, 219-228 (1992).
[CrossRef] [PubMed]

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

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

H. H. Hopkins, "The frequency response of a defocused optical system," Proc. Royal Soc. London 231, A 91-103 (1955).
[CrossRef]

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).

Y. Y. Schechner, N. Kiryati, and R. Basri, "Separation of transparent layers using focus," Proc. of IEEE 6th Int. Conf. On Computer Vision 1061-1066 (1998).

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]

J. Ens and P. Lawrence, "A matrix based method for determining depth from focus," Proc. CVPR 600-606 (1991).

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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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