Abstract

A mathematical description of the absolute out-of-plane height distribution in 3D shape measurement with an arbitrarily arranged fringe projection profilometry system is presented, and a corresponding algorithm is proposed to determine the parameters required for accurate 3D shape determination in practical applications. The proposed technique requires neither a specific and precise experimental setup nor a manual measurement of geometric parameters, and it yields high measurement accuracies while allowing the system components to be arbitrarily set and positioned. Computer simulations and a real experiment have been conducted to verify the validity of the technique.

© 2007 Optical Society of America

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References

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2007 (1)

P. Tavares and M. Vaz, Opt. Commun. 274, 307 (2007).
[CrossRef]

2006 (3)

2005 (1)

1983 (1)

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Figures (4)

Fig. 1
Fig. 1

Schematic illustration of a generalized FPP setup.

Fig. 2
Fig. 2

Simulation measurement of a partial sphere: (a) phase-shifted fringe patterns, (b) phase map, (c) 2D height map, (d) 3D shape map.

Fig. 3
Fig. 3

Comparison of techniques: (a) phase-shifted patterns, (b) 2D height map obtained by the conventional technique, (c) 2D height map obtained by the proposed technique, (d) height distribution along horizontal diameter, (e) height distribution along vertical diameter.

Fig. 4
Fig. 4

Experiment: (a) phase-shifted fringe patterns, (b) height distribution, (c) rendered 3D shape.

Equations (14)

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x P x A x B x A = y P y A y B y A = z P z A z B z A .
x P = [ ( z P z A ) x B + ( z B z P ) x A ] ( z B z A ) ,
y P = [ ( z P z A ) y B + ( z B z P ) y A ] ( z B z A ) .
x D = [ ( z D z C ) x P + ( z P z D ) x C ] ( z P z C ) ,
y D = [ ( z D z C ) y P + ( z P z D ) y C ] ( z P z C ) .
{ x B y B z B } T = { x O y O z O } T + R α , β , γ { x B y B z B } T ,
R α , β , γ = [ cos γ sin γ 0 sin γ cos γ 0 0 0 1 ] × [ cos β 0 sin β 0 1 0 sin β 0 cos β ] [ 1 0 0 0 cos α sin α 0 sin α cos α ] .
{ x D y D z D } T = { x O y O z O } T + R θ , ϕ , ψ { x D y D z D } T ,
x D = x O + cos ψ cos ϕ ( z D z C ) x P + ( z P z D ) x C ( z P z C ) + ( sin ψ cos θ + cos ψ sin ϕ sin θ ) ( z D z C ) y P + ( z P z D ) y C z P z C + ( sin ψ sin θ cos ψ sin ϕ cos θ ) z D .
z D = [ ( z C y P z P y C ) cos ϕ sin θ + ( z P x C z C x P ) sin ϕ z O z P + z O z C ] [ ( y P y C ) cos ϕ sin θ + ( z C z P ) cos ϕ cos θ + ( x C x p ) sin ϕ ] .
Φ B = Φ B ( x B , y B ) = Φ D ( x D , y D ) = Φ O + 2 π x D p .
z P = c 0 + c 1 Φ B + ( c 2 + c 3 Φ B ) x B + ( c 4 + c 5 Φ B ) y B d 0 + d 1 Φ B + ( d 2 + d 3 Φ B ) x B + ( d 4 + d 5 Φ B ) y B ,
z P = C 0 + C 1 Φ B + ( C 2 + C 3 Φ B ) I B + ( C 4 + C 5 Φ B ) J B D 0 + D 1 Φ B + ( D 2 + D 3 Φ B ) I B + ( D 4 + D 5 Φ B ) J B .
S = i = 1 m [ C 0 + C 1 Φ i + ( C 2 + C 3 Φ i ) I i + ( C 4 + C 5 Φ i ) J i D 0 + D 1 Φ i + ( D 2 + D 3 Φ i ) I i + ( D 4 + D 5 Φ i ) J i z i g ] 2 ,

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