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Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Editor: Joseph N. Mait
  • Vol. 48, Iss. 2 — Jan. 10, 2009
  • pp: 380–392

Experimental and theoretical inspection of the phase-to-height relation in Fourier transform profilometry

Agnès Maurel, Pablo Cobelli, Vincent Pagneux, and Philippe Petitjeans  »View Author Affiliations


Applied Optics, Vol. 48, Issue 2, pp. 380-392 (2009)
http://dx.doi.org/10.1364/AO.48.000380


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Abstract

The measurement of an object’s shape using projected fringe patterns needs a relation between the measured phase and the object’s height. Among various methods, the Fourier transform profilometry proposed by Takeda and Mutoh [Appl. Opt. 22, 3977–3982 (1983)] is widely used in the literature. Rajoub et al. have shown that the reference relation given by Takeda is erroneous [ J. Opt. A. Pure Appl. Opt. 9, 66–75 (2007)]. This paper follows from Rajoub’s study. Our results for the phase agree with Rajoub’s results for both parallel- and crossed-optical-axes geometries and for either collimated or noncollimated projection. Our two main results are: (i) we show experimental evidence of the error in Takeda’s formula and (ii) we explain the error in Takeda’s derivation and we show that Rajoub’s argument concerning Takeda’s error is not correct.

© 2009 Optical Society of America

OCIS Codes
(080.0080) Geometric optics : Geometric optics
(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

History
Original Manuscript: September 11, 2008
Revised Manuscript: October 28, 2008
Manuscript Accepted: October 29, 2008
Published: January 8, 2009

Citation
Agnès Maurel, Pablo Cobelli, Vincent Pagneux, and Philippe Petitjeans, "Experimental and theoretical inspection of the phase-to-height relation in Fourier transform profilometry," Appl. Opt. 48, 380-392 (2009)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-48-2-380


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References

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  18. Eq. (36) in is rather intricate. For the sake of clarity, it is worth mentioning that our expression for φ(Y) gives the same result as their Eq. (36) owing to the correspondences between Rajoub's notations and our notations: yfp=−D, zfp=Lp, yop=−D−fpsinθ, zop=Lp+fpcosθ, zfc=Lc, yoc=0, zoc=Lc+fc, Yiyoc=Y, zi=h, and ω′=ωp/cosθ.
  19. The ray bC′ has the direction of the vector (x′,y′,h−Lc) and the ray aC′ has the direction of the vector (x,y,h−Lc). The cross product of both vectors has to vanish and the first component of the cross product is xh+(x′−x)Lc. We get δxx′−x=−xh/L
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  21. P. Cobelli, A. Maurel, V. Pagneux, and P. Petitjeans, “Fast global measurement of water waves by Fourier transform profilometry,” submitted to Exp. in Fluids .
  22. We have used the unwrapping algorithm unwrap from MATLAB (The MathWorks, 2007).
  23. The useful relations are tan⁡(θ−β)=Ob′¯cosθ/L and tanβ=(D+y′)/(L−h), with y′/(L−h)=y/L. β is defined in Fig. 13.
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  25. F. Lilley, “An optical 3-D body surface measurement system to improve radiotherapy treatment of cancer,” Ph.D. thesis, Faculty of General Engineering (Liverpool John Moores University, 1999).
  26. L. C. Fang and L. Yang “A new approach to high precision 3-D measuring system,” Image Vis. Comput. 17, 80514(1999).

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