## 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

Full Article | PDF Article**OSA Recommended Articles**

A. H. Tewfik and H. Garnaoui

J. Opt. Soc. Am. A **8**(7) 1026-1037 (1991)

Amikam Borkowski, Zeev Zalevsky, and Bahram Javidi

J. Opt. Soc. Am. A **26**(3) 589-601 (2009)

Liji Cao and Jörg Peter

Opt. Express **19**(13) 11932-11943 (2011)

### References

- View by:
- Article Order
- |
- Year
- |
- Author
- |
- Publication

- 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).

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

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

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

[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).

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

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

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

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

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

- 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] - 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] - 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] - 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.

[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, “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.

[Crossref]

[Crossref]
[PubMed]

#### Schechner, Y. Y.

#### Thomas, L. J.

[Crossref]
[PubMed]

#### Thonnat, M.

#### Truchetet, F.

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

#### Turk, M.

#### 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)

#### IEEE Trans. on Image Processing (1)

[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)

[Crossref]

[Crossref]

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

[Crossref]
[PubMed]

[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)

[Crossref]

#### Opt. Eng. (1)

[Crossref]

#### Pattern Recognition Letters (1)

[Crossref]

#### Proc. CVPR (1)

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

#### Proc. of IEEE ICIP (1)

#### 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)

[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)

### Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

### Figures (11)

**Fig. 1.**

Stack of parallel transparent slices model.

**Fig. 2.**

Inversion computing time versus number of data.

**Fig. 3.**

Simulation device.

**Fig. 4.**

Sensibility to the gain setting.

**Fig. 5.**

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

**Fig. 6.**

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

**Fig. 7.**

Robustness against PSF width estimate.

**Fig. 8.**

Sensibility to focusing error.

**Fig. 9.**

Example of reconstructed images (rmse=0.052).

**Fig. 10.**

Example of error images: |

**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.