Multi-frequency inverse-phase fringe projection profilometry for nonlinear phase error compensation

Optics and Lasers in Engineering 66 (2015) 249–257

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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng

Multi-frequency inverse-phase fringe projection profilometry for nonlinear phase error compensation Zhenkun Lei a,n, Chunli Wang a, Canlin Zhou b a b

State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China College of Physics, Shandong University, Jinan 130012, China

art ic l e i nf o

a b s t r a c t

Article history: Received 26 November 2013 Received in revised form 22 September 2014 Accepted 25 September 2014

A new multi-frequency inverse-phase method was proposed to compensate for nonlinear phase errors in fringe projection profilometry and to measure the three-dimensional shape of discontinuous objects. After introducing a phase offset of π/4 into the multi-frequency four-step phase-shifting method the corresponding nonlinear phase error reversed its sign, which allowed the addition of unwrapped phases before and after the phase-offset operation to compensate for the error. For the four-step phase-shifting method, simulation analysis showed that the nonlinear phase error had quadrupled the fringe frequency. Moreover, experimental results verified the feasibility and applicability of the proposed method. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear phase error compensation Inverse-phase method Multi-frequency phase unwrapping Fringe projection profilometry

1. Introduction In recent decades three-dimensional (3D) shape measurement techniques have been applied in many fields, such as industrial inspection, reverse engineering, restoration of cultural relics, and human medicine [1–3]. Commonly used optical methods for 3D shape measurement include stereoscopic vision, laser fringe scanning, structured light, the projection grating method, and fringe projection profilometry (FPP). FPP has received extensive attention because of its many advantages, including automation, high resolution, and fast measurement. Recently, it has been established that object phase retrieval from a single-frame projection fringe pattern using TV-G-Shearlet image decomposition is more accurate than either the well-known Fourier transform method or the wavelet transform method [4]. In particular, both the measurement accuracy and repeatability of FPP have been greatly improved because of the introduction of the phase-shifting (PS) method [5–7]. White light is typically used in FPP, and a set of sinusoidal fringes are projected onto an object's surface by a projector. The projected fringes are modulated by the object's height, which leads to a change in the fringe phase. The wrapped phase for the distorted fringes is calculated using the PS method, and an unwrapped phase can be obtained from a phase-unwrapping (PU) algorithm. Finally, a 3D object shape can be reconstructed from the unwrapped phase and a geometric phase–height relationship.

n

Corresponding author. E-mail address: [email protected] (Z. Lei).

http://dx.doi.org/10.1016/j.optlaseng.2014.09.018 0143-8166/& 2014 Elsevier Ltd. All rights reserved.

When applying PS FPP to 3D shape measurement, a nonlinear phase error caused by the non-sinusoidal phase shift algorithm remains unavoidable because of errors resulting from nonlinear electronic noise in the projector and CCD camera [8], environmental vibration [9,10], and fringes with periodic inconsistency [11]; however the use of digital PS technology can circumvent the PS error produced in a traditional PS system, such as piezoelectric ceramic [12]. When an ideal sinusoidal fringe generated by a computer is projected onto the measured object by a projector and is imaged by a CCD camera, the captured image is a non-sinusoidal fringe pattern. Therefore, the relationship between the input gray of the fringe pattern and the output gray acquired by the CCD is nonlinear because of underexposure or solarization of the CCD image sensor array [13] and because of gamma error from the projector [14]. The nonlinear phase error generated by the non-sinusoidal fringe is the main error source in FPP and can be successfully analyzed and simulated by experiments and computers. The nonlinear phase error in the PS method can be described using exponential or polynomial models [15]. Huang et al. found that a second-order nonlinear error was produced by the camera and projector in the three-step PS method [16]. Similarly, second-order and third-order nonlinear errors appeared in the four-step PS method [17]. At present, to improve the measurement accuracy of FPP, it is possible to compensate for the nonlinear phase error. Coggrave's method can be used to eliminate the nonlinear phase error by increasing the number of PS steps [18]. This method requires additional images and a long computing time. The lookup table (LUT) method proposed by Zhang et al. utilizes a preestablished storage table between the input gray and the output

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gray to compensate for the nonlinear phase error in the projection and acquirement system [19]. The LUT method has good accuracy, but it necessitates a cumbersome search procedure to preestablish the storage table among 256 grayscale images. Huang et al. [16] developed a double three-step algorithm to reduce the nonlinear phase error by introducing an initial phase offset of 601. Although the LUT method can reduce the nonlinear phase error, this method requires further improvement because of the timeconsuming process necessary for establishing the calibration table. In the present study, a new multi-frequency inverse-phase method (IPM) combined with the four-step PS method was proposed to compensate for nonlinear phase errors in FPP. After a brief introduction summarizing the FPP process a basic principle for reducing nonlinear phase errors is deduced, and an inversephase compensation method is proposed. The characteristics of the nonlinear phase error were analyzed by simulation and, for in conclusion, multi-frequency fringe experiments were performed to verify the feasibility and applicability of the proposed method.

on the phase. More details about FPP are given in the references [20,21].

3. Nonlinear phase error compensation 3.1. Nonlinearity In an ideal fringe projection system, when an ideal cosine fringe is projected onto an object's surface, the distorted fringe can be expressed as 
 M 2π n x¼ 1 þ cos þ δi þ φ ; ð1Þ 2 p where M is the maximum grayscale, δi is the phase shift, n is the pixel index in fringe period p, and φ is the object's true phase. Ideally the linear response f(x) between the camera and the projector can be written as f ðxÞ ¼ ε0 þ ε1 x; ;

2. Principle of FPP According to the schematic of the optical path shown in Fig. 1, the FPP measurement system consists of a digital projector, a digital camera, and a computer. The light axis (EO) of the LCD projector intersects the light axis (FO) of the CCD camera at Point O in reference plane R (along the x-axis), which is perpendicular to the z-axis of reference plane R that is oriented along the object's height. The distance between the two optical centers is d, and the distance between the camera and the reference plane is l. Point D is an arbitrary point on the object's surface with a height of h. Points A and C are the intersections of the light paths of the projector and the camera, respectively, with the reference axis. When a sinusoidal fringe pattern is projected onto the reference plane by the LCD projector the projected fringe remains parallel because EF is parallel to reference plane R, even if the projection direction is tilted. When an object is placed on the reference plane, the projected fringe is distorted by the object's height because of the distance between the object's surface and the reference plane. The distorted fringe is modulated by the object's surface and is related to the object's height. The phase corresponding to the distorted fringe can be obtained using the PS method. After the relationship between the phase and the object height is calibrated, the 3D object profile can be converted based

ð2Þ

which assumes that the object's surface can be totally reflected, regardless of the geometric effect of the camera and projector. Substituting Eq. (1) into Eq. (2), the ideal carrier fringe is given by

 ε1 M ε1 M 2π n þ cos þ δi þ φ : ð3Þ f ðxÞ ¼ ε0 þ 2 2 p Appling an N-step PS method (for N Z3) [specifically, δi ¼ 2π(i  1)/ N (i¼1, 2, …, N)], the ith phase-shifted image (Ii) can be written as I i ¼ σ 0 þ σ 1 cos ðφ þ δi Þ;

ð4Þ

where σ0 is the background intensity and σ1 is the fringe contrast. The corresponding N-step PS algorithm is expressed as tan φ ¼ 

∑N i ¼ 1 I i sin δi ∑N i ¼ 1 I i cos δi

ð5Þ

However, the ideal linear relationship given in Eq. (2) is not established because of the nonlinear response function between the camera and the projector in actual measurements. The nonlinear phase error contains both a second-order residual error and a third-order residual error in the four-step PS method. Assuming that the N  2 residues ranging from second-order residue ε2 to (N 1)-order residue εN  1 exist in both the camera and the projector, nonlinear response fN(x) between the camera and the projector can be rewritten from the linear case [Eq. (2)] as N1

f N ðxÞ ¼ ε0 þ ε1 x þ ∑ εj xj ;

Sinusoidal fringes

Projector

d

E

ð6Þ

j¼2

CCD

where εj (j ¼2, …, N  1) is the nonlinear residue. Substituting Eq. (1) into Eq. (6), the corresponding ith phase-shifted image Ii' (including nonlinear residues) simplifies to

F

N1   I i 0 ¼ σ 0 þ σ 1 cos ðφ þ δi Þ þ ∑ σ j cos j φ þ δi ; j¼2

z

l

where σj (j ¼2, …, N  1) is constant. The corresponding N-step PS algorithm from Eq. (5) is revised as tan φ0 ¼ 

D Reference plane R x

Fig. 1. Optical arrangement for FPP.

0 ∑N i ¼ 1 I i sin δi ; 0 N ∑i ¼ 1 I i cos δi

ð8Þ

where φ0 is the unwrapped phase of the actual measured object. Nonlinear phase error Δφ (introduced by the nonlinear residues) is then expressed as

h C B A

ð7Þ

O

tan Δφ ¼

tan φ0  tan φ : 1 þ tan φ0 tan φ

ð9Þ

Z. Lei et al. / Optics and Lasers in Engineering 66 (2015) 249–257

Δφ produces

The above equation is further simplified as sin N φ and k ¼ σ 1 =σ N  1 : tan Δφ ¼  cos Nφ þ k

ð10Þ

It can be seen from Eq. (10) that the frequency of nonlinear phase error Δφ continues to use the product of N and the fringe frequency. The nonlinear phase error can be compensated for if nonlinear parameter k is known. For the four-step PS method (N ¼ 4) [specifically, δi ¼(i  1)π/2 (i¼ 1, 2, 3, 4)], the four phase-shifted images (Ii) without nonlinear residues can be written from Eq. (4) as 8 I 1 ¼ σ 0 þ σ 1 cos φ > > > > < I 2 ¼ σ 0 þ σ 1 cos ðφ þ π =2Þ : ð11Þ I ¼ σ 0 þ σ 1 cos ðφ þ π Þ > > > 3 > : I 4 ¼ σ 0 þ σ 1 cos ðφ þ 3π =2Þ Consequently, the wrapped phase can be expressed as
 I 4 I 2 wrapðφÞ ¼ arctan ; I 1 I 3

ð12Þ

where the function wrap() is a phase-wrapping operation. In actuality, the nonlinear error of the camera and the projector contains both a second-order residue (ε2) and a third-order residue (ε3) in the four-step PS method (N ¼4). Recorded light intensity distribution f4(x) is a function of theoretical intensity x: f 4 ðxÞ ¼ ε0 þ ε1 x þ ε2 x2 þ ε3 x3 :

ð13Þ 0

Substituting Eq. (1) into Eq. (13), recorded image Ii (N ¼4) is given by       I i 0 ¼ σ 0 þ σ 1 cos φ þ δi þ σ 2 cos 2 φ þ δi þ σ 3 cos 3 φ þ δi : ð14Þ Correspondingly, using the four-step PS method (N ¼4), the wrapped phase of the measured object is written as 8 0 0  I2 sin φ  σ 3 sin 3φ > < wrapðφ0 Þ ¼ arctan II40  ¼ arctan σσ11 cos 0 φ þ σ cos 3 φ  I 3 1 3 ; ð15Þ M2 15ε M3 ε3 M3 > 3 : σ1 ¼ M ε þ 2 ε þ and σ ¼ 1 2 3 2 2 2 4 4 2 where φ0 is the unwrapped phase of the actual measured object. The function wrap() is a phase-wrapping operation. Nonlinear phase error Δφ [introduced by the second-order and third-order residues in the four-step PS method (N ¼4)] is then expressed as tan Δφ ¼

tan φ0  tan φ σ 3 sin 4φ ¼ : σ 3 cos 4φ þ σ 1 1 þ tan φ0 tan φ

The above equation is further simplified as
 sin 4φ and k ¼ σ 1 =σ 3 : wrapðΔφÞ ¼ arctan cos 4φ þ k

251

ð16Þ

φ ¼ φ0  Δφ ¼ φ0 þ K: Similarly, shifted phase written as

ð19Þ

δi (with an extra phase offset of π/4) is

π π δ i ¼ ði  1Þ þ : 2

ð20Þ

4

Substituting Eq. (20) into Eq. (14) and selecting k⪢1, nonlinear phase error Δφ0 for the PS fringes with the π/4 phase offset is obtained using the four-step PS method: 
 sin 4φ ¼ K: ð21Þ Δφ0 ¼ unwrap arctan k Clearly, compared with Eq. (18), after introducing the π/4 phase offset into the four-step PS method, the corresponding nonlinear phase error in Eq. (21) has the opposite sign; specifically Δφ0 ¼  Δφ. Consequently, true phase φ equals measured phase φ″ after subtracting nonlinear phase error Δφ0 and the π/4 phase offset; specifically,

φ ¼ φ″  Δφ0  π =4 ¼ φ″  K  π =4:

ð22Þ

In summary, the basic principle governing inverse-phase compensation in FPP is described as follows. For two unwrapped phases (φ0 and φ″) with the identical fringe frequency (where φ″ has a π/4 phase offset), a simple arithmetic sum operation of φ0 þ φ″ can compensate for the nonlinear phase error and yield twice the true phase (φ). A simple subtraction operation of φ″  φ0 can yield twice the nonlinear phase error (2K þ π/4): ( φ″ þ φ0 ¼ 2φ þ π =4 : ð23Þ φ″  φ0 ¼ 2K þ π =4 In particular, when the projection fringe frequency is 1, it is unnecessary to unwrap the wrapped phase because this phase is equal to the unwrapping phase. With this feature two sets of phase-shifted fringes with a fringe frequency of 1 are generated, and the second set of phase-shifted fringes has a π/4 phase offset relative to the first set of fringes. Two unwrapped phases (φ0 and φ″ from before and after, respectively, implementing the π/4 phase offset) can be obtained using the four-step PS method. Applying the subtraction operation in Eq. (23), the overall distribution of nonlinear error k is given by    cos 2 φ″ þ φ0 sin 4φ ¼ : ð24Þ k¼ tan ððφ″  φ0 =2Þ  ðπ =8ÞÞ tan K

ð17Þ

As Eq. (17) demonstrates, third-order nonlinear error k for the four-step PS method contains second-order and third-order residues [17]. Similarly, a second-order nonlinear error appears when the three-step PS method is applied [16]. These special examples satisfy the general case of Eq. (10).

4. Simulations Using the four-step PS method for FPP, the projector's gamma (γ) nonlinear effect can be simulated by an exponential model as [15]

3.2. Inverse-phase compensation

zðx; yÞ ¼ ½I i ðx; yÞγ ;

Because of the condition k⪢1, after the PU operation is applied to the four-step PS method, Eq. (17) is simplified as 
 sin 4φ ¼  K; ð18Þ Δφ ¼  unwrap arctan k

There is no effect on fringe images Ii when γ is 1; consequently, the object's phase, given by φ ¼ φγ ¼ 1, is obtained using the four-step PS method and the PU procedure. When γ is 1 and the phase offset π =4 is π/4, the object's phase is φγ ¼ 1 ¼ φ þ π/4. When γ is 3, the object's phase (including the nonlinear error) is φγ ¼ 3 ¼ φ  K. When γ is 3 and the phase offset is π/4, the object's phase (including the nonlinear error and the phase offset) is

where the function unwrap() is a PU operation. Because true phase

φ equals measured phase φ0 , subtracting nonlinear phase error

i ¼ 1; 2; 3; 4:

ð25Þ

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π =4

φγ ¼ 3 ¼ φ þ K þ π/4. These relationships are summarized as 8 A : φγ ¼ 3  φγ ¼ 1 ¼  K > > > > π =4 π =4 > > < B : φγ ¼ 3  φγ ¼ 1 ¼ K

8 pixels. Consequently the nonlinear phase error has about four times the fringe frequency, in accordance with Eq. (18).

π =4 > C : φγ ¼ 3  φγ ¼ 1 ¼ K þ π =4 > > > > > : D : φγ ¼ 3  φπ =4 ¼  K  π =4 γ¼1

4.2. Identical

8 π =4 E : φγ ¼ 1  φγ ¼ 1 ¼ π =4 > > > < π =4 and F : φγ ¼ 3  φγ ¼ 3 ¼ 2K þ π =4 : > > > π =4 : G : φπ =4 þ φ γ ¼ 3 ¼ φγ ¼ 1 þ φγ ¼ 1 ¼ 2φ þ π =4 γ¼3

Next the nonlinear phase errors are simulated under conditions of fixed gamma (γ ¼ 3) but different fringe periods of p ¼20 and 40 pixels, as shown in Fig. 5. Clearly, when the fringe period is p ¼ 20 pixels, the period of the nonlinear phase error is about 5 pixels, which is four times the fringe frequency. A similar characteristic occurs for the fringe period of p ¼40 pixels. Therefore, for a fixed valued of γ and different fringe periods, the frequency of the nonlinear phase error is always four times the fringe frequency.

The above relationships can be verified using the following simulations. Two sets of four-step phase-shifted fringes with a fringe period (p) of 30 pixels are generated by the computer when γ is 1 and 3. Meanwhile, two sets of four-step phase-shifted fringes with identical fringe period p and a π/4 phase offset are also simulated when γ is 1 and 3. The phases of these sets of fringes are obtained using the four-step PS method and the PU procedure. When γ is 1 the nonlinear phase error does not occur, and the object's phase equals true phase φ without any nonlinear phase error. The relationships for the object's phase and the nonlinear phase error in Eq. (26) can be extracted and compared, as shown in Fig. 2. As Fig. 2 demonstrates, curve B represents nonlinear phase error K, and curve A can be inverted to equal curve B; in particular, curve A is given by  K. Curves C and D are the sum and difference, respectively, between nonlinear phase error K and the π/4 phase offset. Simultaneously, curve E approaches the π/4 phase offset and curve F is the sum of double the nonlinear phase error (2K) and the π/4 phase offset. The simulations in Fig. 2 verify the correctness of Eq. (26). As predicted based on the theoretical results of Eq. (26) curves A and B are reflections of each other, and adding curve A and curve B can compensate for the nonlinear phase error, as shown in Fig. 3. Theoretically, the IPM method can completely compensate for the nonlinear phase error. With a magnitude of 10  2 rad, the residual error in Fig. 3 is caused by the truncated error in numerical simulations based on the exponential model [Eq. (25)]. Experimentally, the IPM method can eliminate the majority of the nonlinear phase error in actual measurement. Nonlinear phase errors with different gamma values (γ ¼3, 5, and 7) and the same fringe period (p ¼30 pixels) are compared in Fig. 4. This figure shows that the nonlinear phase errors are distributed in the shape of a sine curve with the same period of

Residual error (x10-3 rad.)

γ 10 5 0 -5 -10 0

10

20

30

40

50

60

Position (pixel) Fig. 3. Residual error (γ ¼3) after compensation using the inverse-phase method.

γ=3

0.4

Phase error (rad.)

4.1. Identical fringe period p but different

ð26Þ

γ but different p

γ=5

γ=7

0.2 0.0 -0.2 -0.4

0

10

20

30

40

50

60

Position (pixel) Fig. 4. Nonlinear phase errors when γ is 3, 5, and 7.

Fig. 2. Distributions of the simulated nonlinear phase error in Eq. (26) when γ is 3.

0.08

0.08

0.06

0.06

0.04

0.04

Phase error (rad.)

Phase error (rad.)

Z. Lei et al. / Optics and Lasers in Engineering 66 (2015) 249–257

0.02 0.00 -0.02 -0.04

253

0.02 0.00 -0.02 -0.04 -0.06

-0.06 0

10

20

30

40

50

60

-0.08

0

10

Position (pixel)

20

30

40

50

60

Position (pixel)

Fig. 5. Nonlinear phase errors (γ ¼3) for fringe periods p of (a) 20 and (b) 40 pixels.

Phase offset of π/4

f1 φ1'w

φ1'

2φ1=φ1'+φ1''-π/4

φ1''w

φ1''

=

PS

PU

PS

f4 φ4'w PS

φ4'

2φ4=φ4'+φ4''-π/4

MFU

PS

f16 φ16'w PS

f16

φ16''w

φ16' 2φ16=φ16'+φ16''-π/4 φ16'' MFU

f4

φ4''w

φ4''

MFU

f1

MFU

PS

Fig. 6. Combination flowchart for the MFU algorithm and the inverse-phase method.

5. Experiments A multi-frequency phase-unwrapping (MFU) algorithm has the ability to measure the 3D shape of discontinuous objects without applying the PU operation [22]. Several sets of phase-shifted sinusoidal fringe images are produced by the computer with pre-designed fringe frequencies for each set, such as a fringe frequency of 1 in set I and 4 in set II. These fringe images are then sequentially projected onto the object's surface. Acquired by CCD, the images are calculated using the PS method to obtain wrapped phase φw with the corresponding frequencies. The MFU algorithm is written as

φi ¼ φwi þINT






φj U ðf i =f j Þ  φwi 2π ; 2π

ð27Þ

where INT() represents a function to take the rounded integer of a decimal number. φi and φj are the unwrapped phases with fringe frequencies of fi and fj, respectively. φw i is the wrapped phase with a fringe frequency of fi. It is noted that unwrapped phase φ1 equals wrapped phase φw 1 ; in particular, the PU process is unnecessary when the fringe frequency is 1. Because the accuracy of unwrapped phase φ1 with a fringe frequency of 1 is insufficient, a higher fringe frequency is needed to increase the measurement accuracy. In this paper, it is demonstrated that a combination of the above IPM method and the existing MFU algorithm has the advantages of compensating for the nonlinear phase error and also measuring the 3D shape of discontinuous objects. A four-step PS method with different fringe frequencies of 1, 4, and 16 is used to explain the combination flowchart for the IPM method and the MFU algorithm, as shown in Fig. 6.

1. Step 1: A set of four-step phase-shifted fringe patterns with the fringe frequency of f1 ¼ 1 are projected onto the objects. Unwrapped phase φ10 can be obtained using the four-step PS method without applying the PU operation (φ10 ¼ φw 10 ). This process includes nonlinear phase error Δφ1 for the fringe frequency of 1. Similarly, the other set of four-step phase-shifted fringe patterns with the same fringe frequency (f1 ¼ 1) and an initial phase offset of π/4 are projected onto the objects. Wrapped phase φw 1″ can also be obtained using the four-step PS method. Corresponding unwrapped phase φ1” can be obtained using a conventional PU algorithm. This process includes nonlinear phase error  Δφ1. Finally, the IPM method can compensate for the nonlinear phase error when the fringe frequency is 1; specifically, an unwrapped phase of φ1 ¼(φ10 þ φ1″  π/4)/2 (without the nonlinear phase error) can be obtained. 2. Step 2: When the fringe frequency is f4 ¼4, two wrapped phases w of φw 40 and φ4″ (before and after, respectively, the π/4 phase offset) can be obtained using the four-step PS method. Combining unwrapped phase φ1 (without the nonlinear error) that has a fringe frequency of 1 and is obtained from Step 1 with the MFU algorithm [Eq. (27)], unwrapped phases φ40 and φ4″ (before and after, correspondingly, the π/4 phase offset) can be obtained. These unwrapped phases include nonlinear errors Δφ4 and  Δφ4, respectively, for the fringe frequency of 4. Finally, the unwrapped phase of φ4 ¼(φ40 þ φ4″  π/4)/2 (without the nonlinear error) can be obtained using the IPM method. 3. Step 3: When the fringe frequency is f16 ¼16 or a higher value, Step 2 is repeated until the phase measurement accuracy meets the experimental requirements. The following experiments were performed to verify the multifrequency IPM method for FPP. A 3LCD Projector (Toshiba

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f1

f4

f16

f64

Fig. 7. (a) Photograph of the sphere cap and block and (b) four sets of PS fringes with frequencies of 1, 4, 16, and 64 projected onto two separate objects and captured by the CCD.

f1

f4

f16

f64

f1

f4

f16

f64

Fig. 8. Four wrapped phases of objects with frequencies of 1, 4, 16, and 64 (a) before and (b) after implementing a π/4 phase offset.

TLP-X2000), a CCD camera (Guppy F-080B), and an objective lens (Computar M3514MP) were used in our experiments. Firstly, four sets of four-step phase-shifted sinusoidal fringe patterns with frequencies of 1, 4, 16, and 64 and another four sets of patterns with corresponding fringe frequencies and a π/ 4 phase offset were generated by the computer. The fringe patterns were then projected onto two separate objects on a glass plate using the projector, and the fringe patterns were then collected by the CCD camera. The sphere cap used in these experiments had a bottom circle diameter of 110 mm and a height of 60 mm. The other object was a block with a 20 mm width and a 90 mm length; this block had four stages, and each stage was 4 mm in height, as shown in Fig. 7(a). Fig. 7(b) shows the four sets of object images corresponding to the four-step phase-shifted sinusoidal fringe patterns with different frequencies. After applying the PS algorithm, two sets of four wrapped phases for the objects with different frequencies (before and after implementing a phase offset of π/4) were obtained, as shown in Fig. 8.

After using the IPM method and the MFU algorithm (IPM and MFU) to compensate for the nonlinear phase error, unwrapped phase φ64 of the objects with a fringe frequency of 64 is shown in Fig. 9(a). Fig. 9(b) gives the unwrapped phase for the fringe frequency of 64, which is measured using the conventional MFU algorithm. Fig. 9(c) and (d) shows the 3D display corresponding to Fig. 9(a) and (b), respectively. As the figures demonstrate the MFU algorithm could be used to measure two separate objects, and compared with the results of the conventional MPU algorithm in the absence of the IPM method [Fig. 9(b)] the IPM and MPU method [Fig. 9(a)] can significantly improve the object's smoothness. The phase distributions along the same line for each object are shown in Fig. 10. It is noted that the IPM and MPU method mostly compensated for the nonlinear phase error, and the measurement accuracy was clearly improved. The image saturation contributed to the error at the top of the sphere, which was larger than the error at the bottom. The influence of image noise was also unavoidable.

Z. Lei et al. / Optics and Lasers in Engineering 66 (2015) 249–257

255

30 mm

30 mm

Fig. 9. Unwrapped phases of two objects obtained by (a) the proposed multi-frequency IPM and (b) the conventional MFU algorithm; panels (c) and (d) are the 3D displays corresponding to panels (a) and (b), respectively.

2.5

8 7

2.0

Phase (rad.)

Phase (rad.)

6 5 4 3 2

1.5 1.0 0.5

MFU IPM&MFU

1 0 0

50

100

150

MFU IPM&MFU

0.0

200

250

0

50

100

150

200

Position (pixel)

Position (pixel)

Fig. 10. Phase distributions of (a) a spherical cap and (b) a stage obtained using different methods.

A head statue was also measured, as shown in Fig. 11(a). The measured results obtained by the conventional MPU algorithm and the proposed IPM and MPU methods are shown in Fig. 11 (a) and (b), respectively. The head status' shadow was artificially repaired. The positive effect of the proposed method was obvious, and the majority of the nonlinear error was eliminated.

6. Conclusions A new multi-frequency inverse-phase method was proposed to compensate for nonlinear phase errors in FPP and also to realize 3D shape measurement of discontinuous objects. After presenting the theoretical derivation and simulation analysis of the nonlinear

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Fig. 11. (a) Photograph of the head statue with shadow and 3D phase maps measured by (b) the MPU algorithm and (c) the IPM and MPU method.

phase error for the four-step PS method it was determined that the gamma value did not change the frequency of the nonlinear phase error, which was always four times the fringe frequency. On introducing a phase offset of π/4 into the four-step PS method the corresponding nonlinear phase error reversed its sign, which allowed an addition operation of the unwrapped phases before and after the π/4 phase offset to compensate for the errors. The feasibility of the proposed method was verified by experiments, and the proposed method had higher measurement accuracy than the conventional method.

Acknowledgments The authors thank the National Basic Research Program of China (No. 2014CB046506), the National Natural Science Foundation of China (Nos. 11172054 and 11472070), and the Fundamental Research Funds for the Central Universities (No. DUT14LK11). References [1] Chen F, Brown GM, Song M. Overview of three dimensional shape measurement using optical methods. Opt Eng 2000;39(1):10–22.

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