Least-squares fitting of carrier phase distribution by using a rational ...

December 15, 2006 / Vol. 31, No. 24 / OPTICS LETTERS 3589
through the angles , , and (which are basic system
parameters for determining the orientation of
the camera in 3D space), respectively, the coordinates
of E in OXYZ can be calculated with
x E
y
E
cos sin 0
1
cos 0 − sin
E = − sin cos 0 0 1 0
z 0 0 sin 0 cos
1 0 0 E
0 cos sin y E
2
0 − sin cos x 0.
The line CE crosses R at D. By utilizing the equation
of CE, i.e., x−x C /x E −x C =y−y C /y E −y C =z
−z C /z E −z C , the equation of R, i.e., z=0, and Eq. (2),
the coordinates of D, x D ,y D ,z D , are calculated with
x D
y
D x Cz E − x E z C /z E − z C
D = y C z E − y E z C /z E − z C
z 0
1 x E + b 2 y E /1+a 1 x E + a 2 y E
=b
b 3 x E + b 4 y E /1+a 1 x E + a 2 y E , 3
0
where a 1 =−sin /z C , a 2 =sin cos /z C , b 1 =cos
cos −x C sin /z C , b 2 =cos sin +sin sin cos
+x C sin cos /z C , b 3 =−cos sin −y C sin /z C , and
b 4 = cos cos − sin sin sin + y C sin cos /z C .
Because D and E produce their images at the same
pixel, the carrier phase at i,j equals the reference
phase at D. That is,
i,j = D .
Figure 1(b) shows the relationship between the
projector and R, where P is the center of the projector
lens. The fictitious plane G passes through O and is
perpendicular to the optical axis of the projector lens.
Therefore the fringes projected on G are kept parallel
to each other and are equally spaced with pitch p.
The XOY plane of the system OXYZ is superposed
on G, and the Y axis is parallel to the fringes.
Assuming that OXYZ can be transformed to
OXYZ by rotations around the X, Y, and Z axes
in sequence, through the angles , , and , respectively,
and that the coordinates of P in OXYZ are
x P ,y P ,z P , the coordinates of D in OXYZ can be
calculated with
x D
= 1 0 0
cos 0 sin
y D 0 cos − sin 0 1 0
z D
0 sin cos − sin 0 cos
cos − sin 0
1 x 0
D
sin cos 0 y D .
5
0 0
4
PD crosses G at F. By using the equation for PD, i.e.,
x−x P / x D −x P =y−y P /y D −y P = z−z P /z D −z P ,
the equation for G, i.e., z=0, and Eq. (5), the coordinates
of F are obtained with
x F P z D − x D z P /z D − z P
y F y P z D − y D z P /z D − z P
z F =x
0
= d 1x D + d 2 y D /1+c 1 x D + c 2 y D
d 3 x D + d 4 y D /1+c 1 x D + c 2 y D , 6
0
where c 1 = cos sin cos − sin sin / z P , c 2
=−cos sin sin +sin cos /z P , d 1 =cos cos +x P
cos sin cos −sin sin /z P , d 2 =−cos sin −x P
cos sin sin +sin cos /z P , d 3 =sin sin cos
+cos sin +y P cos sin cos −sin sin /z P , and
d 4 = −sin sin sin + cos cos − y P cos sin sin
+sin cos /z P . Noting that DF is an equal-phase
line, we have
D = F = O +2x F /p,
7
where O is the phase at O. Utilizing Eqs. (1), (3), (4),
(6), and (7) yields
i,j = r + si + tj/1+ui + vj,
where u=a 1 + c 1 b 1 + c 2 b 3 /1 − a 1 + c 1 b 1 + c 2 b 3 i o
− a 2 +c 1 b 2 + c 2 b 4 j o , v = a 2 + c 1 b 2 + c 2 b 4 /1
−a 1 + c 1 b 1 + c 2 b 3 i o − a 2 + c 1 b 2 + c 2 b 4 j o , r = 0
−2 d 1 b 1 + d 2 b 3 i o +d 2 b 4 +d 1 b 2 j o /p1−a 1 +c 1 b 1
+c 2 b 3 i o −a 2 +c 1 b 2 +c 2 b 4 j o , s= 0 a 1 +c 1 b 1 +c 2 b 3
+2d 1 b 1 + d 2 b 3 /p/ 1 − a 1 + c 1 b 1 + c 2 b 3 i o − a 2
+ c 1 b 2 + c 2 b 4 j o t = 0 a 2 + c 1 b 2 +c 2 b 4 +2 d 2 b 4
+ d 1 b 2 /p/ 1− a 1 + c 1 b 1 +c 2 b 3 i o −a 2 +c 1 b 2 +c 2 b 4
j o . Differing from the existing methods, 5–8 Eq. (8)
can exactly model the carrier phase map for an arbitrary
system scheme.
Derivation of Eq. (8) makes it possible in principle
to determine the carrier phase distribution from the
system parameters. However, the difficulty remains
that these parameters are hard to know exactly. As
an alternative approach, the coefficients of Eq. (8)
can be estimated from the measured phase distribution
of a standard plane (i.e., reference plane). For
simplifying the computation, we recast Eq. (8) in the
following linear form,
r + si + tj − ui,ji − vi,jj = i,j, 9
8
where the reference phase i,j has been measured.
When i,j varies, a linear system based on Eq. (9) is
established, from which r, s, t, u, and v can be solved.
Using Eq. (9) involves a biased estimation, by which
the expectations of the estimated coefficients in the
presence of noise slightly deviate from their true values.
Because the deviations are very small, the results
are still satisfactory. If a higher accuracy is required,
the nonlinear system based on Eq. (8) must
be solved, and an iterative procedure has to be implemented.