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

3588 OPTICS LETTERS / Vol. 31, No. 24 / December 15, 2006
Least-squares fitting of carrier phase distribution
by using a rational function in fringe
projection profilometry
Hongwei Guo, Mingyi Chen, and Peng Zheng
Department of Precision Mechanical Engineering, Laboratory of Applied Optics and Metrology,
Shanghai University, Shanghai 200072, China
Received August 7, 2006; revised September 8, 2006; accepted September 22, 2006;
posted October 2, 2006 (Doc. ID 73835); published November 22, 2006
A least-squares fitting technique for the carrier phase component in fringe projection profilometry is presented.
The carrier phase distribution for an arbitrary measurement system can be perfectly described with
a rational function, whose coefficients can be estimated by least-squares fitting to the measured reference
phases, so that the restrictions and limitations in the existing techniques are eliminated. © 2006 Optical
Society of America
OCIS codes: 120.2830, 120.5050, 150.6910.
In fringe projection profilometry, the depth maps of
object surfaces are calculated from the phase distributions
from which the carrier phases have been removed.
Therefore exactly determining the carrier
phase component is important for guaranteeing measurement
accuracies. The difficulty is that the carrier
phase distribution generally contains nonlinearity in
its profile due to the restrictions of system schemes.
In the conventional techniques, 1–4 this nonlinearity
has been corrected on the bases of geometrical analyses
of measurement systems. With these techniques,
however, the system parameters must be known or
adjusted carefully to satisfy some stringent conditions.
The polynomial fitting methods 5–8 can overcome
this problem, but with them the accuracies cannot
be guaranteed even if high-degree polynomials
are used because of their poor asymptotic properties;
moreover, the error will increase when the data are
extrapolated outside the reference plane. Reference 7
suggests a rational function representation of the
carrier phases, but it assumes the optical axis of the
imaging lens to be perpendicular to the reference
plane, which is not always valid in measurement
practice. To the best of our knowledge, models that
can exactly describe the carrier phases for an arbitrary
measurement system are still not available.
In this Letter, we demonstrate that the carrier
phase distribution for an arbitrary system can be
perfectly represented with a 2D rational function.
Based on this fact, a least-squares fitting technique
for the carrier phase component is proposed. Because
it is a pixelwise operation to remove the carrier component
from the object phases, we shall represent the
carrier phase as a function of pixel coordinates. Figure
1(a) shows the position and orientation of the
camera relative to the reference plane, where C is the
center of the imaging lens, and R is the reference
plane. The coordinate system OXYZ is established,
with the XOY plane being superposed on R. The coordinates
of C are x C ,y C ,z C . The fictitious plane I
passes through the origin O and is perpendicular to
the optical axis of the imaging lens. In another system
OXYZ, the XOY plane is superposed on I,
and the X and Y axes are parallel to the image coordinate
axes, i.e., i and j axes, respectively. For a
general pixel i,j, which is the image of the point E
on I, the mapping between the coordinates x E ,y E ,z E
and i,j is linear, viz.,
x E y E z E = i − i o j − j o 0, 1
where i o ,j o are pixel coordinates of the image of O,
and is a scale factor. Assuming that the system
OXYZ can be transformed to OXYZ by rotations
around the X, Y, and Z axes in sequence,
Fig. 1. (Color online) Geometry for fringe projection profilometry.
Positions and orientations of (a) the camera and
(b) the projector relative to the reference plane,
respectively.
0146-9592/06/243588-3/$15.00 © 2006 Optical Society of America

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.

3590 OPTICS LETTERS / Vol. 31, No. 24 / December 15, 2006
In experiment, a standard plane is used as a reference,
on which the checkerboard pattern is printed
for implementing the lateral calibration simultaneously.
In the scheme, the optical axes of both projector
and camera incline to the reference plane, a
case that cannot be well handled by the existing
techniques. 1–8 The phase-shifting technique is used
for phase recovery, and the sinusoid fringe patterns
are projected with a DLP (digital light processing)
projector, first on the reference plane and then on the
tested object. The deformed patterns are captured by
a CCD camera. Figure 2(a) shows a pattern on the
reference plane, from which the oblique perspective
of the reference plane induced by the incline of the
camera is easily observed. Figure 2(b) shows the recovered
phases. In the conventional techniques (see
Ref. 9, for example) the carrier phases are usually removed
by subtracting the measured reference phases
from the object phases. However, such a method is inoperable
here, because the phase values are not
available for the pixels inside the black boxes or outside
the reference plane region. By fitting Fig. 2(b)
with the rational function, the carrier phase distribution
can be reconstructed as shown in Fig. 2(c), where
the lost data in Fig. 2(b) have been padded by interpolation
or extrapolation. [The maximum difference
between the phase maps reconstructed with Eqs. (8)
and (9) is 0.009 rad, which is neglectable]. Figure 2(d)
shows the residual of (b) subtracting (c), which consists
mainly of the noises and the local deformations
of the reference plane. With the proposed technique,
such error factors are effectively eliminated.
As indicated in Ref. 5, inaccurate carrier phases
will introduce additional bending (i.e., curvature errors)
in the measurement results. This fact allows us
to verify the validity of this technique by measuring a
cylindrical object and then checking the resulting radius.
For this purpose, a cookie jar is measured. In
single-view measurements, we first recover the object
phases from the captured fringe patterns [one of
which is shown in Fig. 2(e)] and then subtract the
carrier phases in Fig. 2(c) from the object phases. Finally,
the 3D shape of each view is calculated by using
the mapping relationship between the phases and
depths. 9 By the registration of six views, 10 the 360°
shape is obtained and shown in Fig. 2(f). Although
additional errors may arise during the registration
procedure, the final achieved accuracy depends
mainly on single-view measurements. In Fig. 2(f), the
least-squares radius of the cylinder is calculated as
95.335 mm. As a comparison, the object is also measured
by using a coordinate measuring machine, and
the least-squares radius is obtained as 95.294 mm.
From the results, the accuracy of this technique is
evident.
In conclusion, we have proposed a least-squares fitting
technique for determining the carrier phase
component. This technique offers some important advantages
over others. First, the model perfectly describes
the carrier phase distribution for an arbitrary
measurement system. Second, it allows for exactly
determining the carrier phase map in the absence of
system parameters. Third, the influence of noise and
local deformations of the reference plane on the measurement
can be effectively restrained.
This work was sponsored by the National Natural
Science Foundation of China (60678036), and in part
by Shanghai Leading Academic Discipline Project
(Y0102). H. Guo’s e-mail address is hwguo@yeah.net.
Fig. 2. (Color online) Experimental results. (a) Fringe pattern
on the reference plane. (b) Phase map (in radians) reconstructed
from (a). (c) Least-squares fitting result (in radians)
of (b) using the rational function. (d) Residual (in
radians) obtained by subtracting (c) from (b). (e) Deformed
fringe pattern on the object. (f) Reconstructed 3D shape.
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