Abstract

Hologram tomography is a two-step method for three-dimensional topometry of extended objects. The first step consists of the hologram recording with a single laser pulse of 35ns duration and storage in a photosensitive material. In the second step the hologram is optically reconstructed and digitized, which leads to a set of two-dimensional projections at different axial positions. A maximization of a focus measure has to be performed to extract the surface position out of the projections. Unlike with well-established methods, where the comparison of sharpness values is done parallel to the optical axis, we propose an iterative solution to perform the maximization along the direction of image formation, which is evaluated for each surface point individually. This leads to a better reproducibility of the surface in the off-axis regions.

© 2005 Optical Society of America

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References

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  1. D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–778 (1948).
    [CrossRef]
  2. A. D. Gara, R. F. Majkowski, T. T. Stapelton, “Holographic system for automatic surface mapping,” Appl. Opt. 12, 2172–2179 (1973).
    [CrossRef] [PubMed]
  3. J. Bongartz, D. Giel, P. Hering, “Living human face measurement using pulsed holography,” in Holography 2000, T. H. Jeong and W. K. Sobotka, eds., Proc. SPIE4149, 303–308 (2000).
  4. R. W. Meier, “Magnification and third-order aberrations in holography,” J. Opt. Soc. Am. 55, 987–992 (1965).
  5. J. Bongartz, “Hochaufloesende dreidimensionale Gesichtsprofilvermessung mit kurzgepulster Holographie,” Ph.D. thesis (Mathematisch Naturwissenschaftliche Fakultät der Heinrich-Heine-Universitaet Duesseldorf, Duesseldorf, Germany, 2002). http://deposit.ddb.de/cgi-bin/dokserv?idn=964966670.
  6. D. Giel, “Hologram tomography for surface topometry,” Ph.D. thesis (Mathematisch Naturwissenschaftliche Fakultät der Heinrich-Heine-Universitaet Duesseldorf, Duesseldorf, Germany, 2003). http://deposit.ddb.de/cgi-bin/dokserv?idn=968530842.
  7. S. K. Nayar, Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).
    [CrossRef]
  8. M. Watanabe, S. K. Nayar, “Telecentric optics for constant magnification imaging,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 1360–1365 (1995).
    [CrossRef]

1995 (1)

M. Watanabe, S. K. Nayar, “Telecentric optics for constant magnification imaging,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 1360–1365 (1995).
[CrossRef]

1994 (1)

S. K. Nayar, Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).
[CrossRef]

1973 (1)

1965 (1)

1948 (1)

D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–778 (1948).
[CrossRef]

Bongartz, J.

J. Bongartz, D. Giel, P. Hering, “Living human face measurement using pulsed holography,” in Holography 2000, T. H. Jeong and W. K. Sobotka, eds., Proc. SPIE4149, 303–308 (2000).

J. Bongartz, “Hochaufloesende dreidimensionale Gesichtsprofilvermessung mit kurzgepulster Holographie,” Ph.D. thesis (Mathematisch Naturwissenschaftliche Fakultät der Heinrich-Heine-Universitaet Duesseldorf, Duesseldorf, Germany, 2002). http://deposit.ddb.de/cgi-bin/dokserv?idn=964966670.

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–778 (1948).
[CrossRef]

Gara, A. D.

Giel, D.

J. Bongartz, D. Giel, P. Hering, “Living human face measurement using pulsed holography,” in Holography 2000, T. H. Jeong and W. K. Sobotka, eds., Proc. SPIE4149, 303–308 (2000).

D. Giel, “Hologram tomography for surface topometry,” Ph.D. thesis (Mathematisch Naturwissenschaftliche Fakultät der Heinrich-Heine-Universitaet Duesseldorf, Duesseldorf, Germany, 2003). http://deposit.ddb.de/cgi-bin/dokserv?idn=968530842.

Hering, P.

J. Bongartz, D. Giel, P. Hering, “Living human face measurement using pulsed holography,” in Holography 2000, T. H. Jeong and W. K. Sobotka, eds., Proc. SPIE4149, 303–308 (2000).

Majkowski, R. F.

Meier, R. W.

Nakagawa, Y.

S. K. Nayar, Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).
[CrossRef]

Nayar, S. K.

M. Watanabe, S. K. Nayar, “Telecentric optics for constant magnification imaging,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 1360–1365 (1995).
[CrossRef]

S. K. Nayar, Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).
[CrossRef]

Stapelton, T. T.

Watanabe, M.

M. Watanabe, S. K. Nayar, “Telecentric optics for constant magnification imaging,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 1360–1365 (1995).
[CrossRef]

Appl. Opt. (1)

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

S. K. Nayar, Y. Nakagawa, “Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).
[CrossRef]

M. Watanabe, S. K. Nayar, “Telecentric optics for constant magnification imaging,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 1360–1365 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

Nature (London) (1)

D. Gabor, “A new microscopic principle,” Nature (London) 161, 777–778 (1948).
[CrossRef]

Other (3)

J. Bongartz, “Hochaufloesende dreidimensionale Gesichtsprofilvermessung mit kurzgepulster Holographie,” Ph.D. thesis (Mathematisch Naturwissenschaftliche Fakultät der Heinrich-Heine-Universitaet Duesseldorf, Duesseldorf, Germany, 2002). http://deposit.ddb.de/cgi-bin/dokserv?idn=964966670.

D. Giel, “Hologram tomography for surface topometry,” Ph.D. thesis (Mathematisch Naturwissenschaftliche Fakultät der Heinrich-Heine-Universitaet Duesseldorf, Duesseldorf, Germany, 2003). http://deposit.ddb.de/cgi-bin/dokserv?idn=968530842.

J. Bongartz, D. Giel, P. Hering, “Living human face measurement using pulsed holography,” in Holography 2000, T. H. Jeong and W. K. Sobotka, eds., Proc. SPIE4149, 303–308 (2000).

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Figures (9)

Fig. 1
Fig. 1

(a) Recording of a hologram. (b) Reconstruction with the complex-conjugate reference beam and formation of the real image. Unlike as indicated in this figure, the recording and reconstruction is done for all scattering points simultaneously.

Fig. 2
Fig. 2

Reconstruction of the hologram and digitalization of a set of 2D projections at different axial positions.

Fig. 3
Fig. 3

Section of 2D projection of the real image at a specific diffusor position.

Fig. 4
Fig. 4

Dependency of the focus measure F j ( x , y ) on the slice position j and a schematic view of the slices and the line along which the comparison of the focus measure F is performed.

Fig. 5
Fig. 5

Schematic view of the image formation of an off-axis point. The line of image formation (IFL) cannot be considered as being parallel to the optical axis.

Fig. 6
Fig. 6

Dependence of the focus measure F along the IFL and schematic view of the slices and the points that belong to the IFL.

Fig. 7
Fig. 7

(a) Computer model of a human face obtained with the method described in Section 4. (b) Model of the same person found with the iterative algorithm described in this paper. The improvement can be seen mainly in the outer regions of the face.

Fig. 8
Fig. 8

Profile of the autocorrelation pattern along the indicated line in the displayed pattern. The difference σ between the first local minimum and the following maximum is used as a quantitative measure for the recognizability of the periodic structure.

Fig. 9
Fig. 9

Reproducibility of the periodic structure quantified through the S value as a function of the number of iterations. After approximately ten iterations no further significant increase occurs.

Equations (14)

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I = o + r 2 = o 2 + r 2 + o * r + o r * ,
r * I = r * o 2 + r * r 2 + o * r 2 + o r * 2 .
H ( x , y ) = { z j max F j max ( x , y ) F j ( x , y ) , j = 1 N } .
F j ( x , y ) = ( ξ , η ) U ( x , y ) [ G j ( ξ , η ) G ¯ j ( x , y ) ] 2 ,
θ = arccos z ( x 2 + y 2 + z 2 ) 1 2 ,
ϕ = sgn ( x ) arctan y x .
r j = z j cos θ
= z j z ( x 2 + y 2 + z 2 ) 1 2 .
x j = [ r j cos ϕ sin θ ] ,
y j = [ r j sin ϕ sin θ ] ,
IFL ( x , y , z j max 0 ) = { x j , y j } j = 1 N ,
F j max i ( x j max i , y j max i ) = max { F j ( x j , y j ) ( x j , y j ) IFL ( x , y , z j max i 1 ) j j max i 1 r } .
AC ( δ x , δ y ) = x , y [ H ( x , y ) z ¯ ] [ H ( x + δ x , y + δ y ) z ¯ ] ,
S σ height map σ object ,

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