# Comparative study on three-dimensional reconstruction of wheel surface by FTP and WTP methods.

1. IntroductionIt is a common way to achieve 3D measurement by inspecting phase displacement of fringe projection. In this method, optical fringe is projected on the surface of object, the variation of surface leads to a deformed fringe, which causes a displacement of fringe phase. Capturing and processing the deformed fringe image, extracting the phase displacement, then the surface distribution can be calculated.

Among 3D measurement methods (Chen et al., 2000) based on phase displacement extraction, Fourier Transform Profilometry (FTP) (Chen et al., 2005) has the characteristics of high accuracy and fast processing speed, and only a single frame of image needed to achieve 3D reconstruction. But for complex surface object, because the frequency filter window in FTP is fixed (Li et al., 2009), it can't get an accurate phase displacement extraction because of frequency spectrum aliasing (Bu et al., 2003).

An improved method called Window Fourier Transform (WFT) (Lei et al., 2010) is put up by Gbaor. It can solve the problem of spectrum aliasing to some extent, but the frequency filter window only has single resolution (Qian et al., 2008).

Wavelet transform is a time-frequency analyses method, which has multiple resolutions, and can detailed analysis signals with high resolution in time domain or space domain.

In recent years, wavelet transform (Zhong et al., 2004) is introduced into 3D Profilometry and it highly promotes technical development in this filed (Colonna et al., 1997).

To improve phase extraction accuracy on area of abrupt surface, complex Morlet wavelet transform Profilometry based on ridge detection theory is studied in this paper (Carmona et al., 1997). To quantitative assess reconstruction accuracy of FTP [Paolo et al., 2001] and WTP methods, peaks function is simulated and a train wheel is tested and analyzed, where the train wheel has flat surface as well as abrupt surface (Heng et al., 2004). By comparative analyzing the 3D surface reconstruction accuracy in the whole wheel and the detailed part of wheel tread, a better solution is recommended.

2. Principle of 3D reconstruction of phase extraction

2.1. Principle of FTP Method

In Figure 1, axis x is perpendicular to fringe, and axis y is parallel to fringe. Optical fringe is respectively projected on the reference plane and the object in the frequency of [f.sub.0]. Then light intensity distributions of fringe images captured by camera are shown in Eq. (1) and Eq. (2).

[g.sub.0] (x, y) = [alpha](x, y) + [r.sub.0](x, y) x cos(2[pi][f.sub.o] x + [[phi].sub.o] ( x, y)) (1)

g(x, y) = [alpha](x, y) + r(x, y) x cos(2[pi][f.sub.o] x + [phi](x, y)) (2)

Where [alpha](x,y) is light intensity of background, [r.sub.o](x,y) and r(x,y) are reflectivity of reference plane and the object, [[phi].sub.o](x, y) and [phi](x, y) are fringe phase of reference plane and the object.

Doing Fourier transform on each row of the fringe image to get spectrum distribution described in Eq. (3) and shown in Figure 2, where n is spectrum series.

G([f.sub.x],y) = A(x,y) + [[infinity].summation over (-[infinity])] [Q.sub.n]([f.sub.x] - n[f.sub.o], y) (3)

Band filtering on spectrum of deformed fringe image to get fundamental frequency component, then doing inverse Fourier transform to get light intensity distribution corresponding to fundamental frequency, which is described in Eq. (4).

g'(x, y) = r(x, y) x cos(2[pi][f.sub.o]x + [phi](x, y)) (4)

Light intensity distribution of reference fringe image corresponding to fundamental frequency can be got in the similar way, as described in Eq. (5).

[g.sub.o] (x, y) = [r.sub.o](x, y) x cos(2[pi][f.sub.o]x + [[phi].sub.o](x, y)) (5)

From Eq. (4) and Eq. (5), [phi](x,y) and [[phi].sub.o](x, y) is extracted by doing inverse trigonometric transform, and phase displacement between reference plane and the object is obtained by Eq. (6).

[DELTA][phi](x, y) = [phi](x,y) - [[phi].sub.o](x,y) = arccos [g'(x,y)/r(x,y)] - arccos [ [g'.sub.o](x, y)/[r.sub.o](x,y)] (6)

But abrupt surface or noise will lead to a spectrum expanding and aliasing, which will reduce phase extraction accuracy.

Since the phase displacement is truncated to [-[pi], [pi]), as shown in Figure 4. Phase unwrapping must be done to get the continual phase displacement.

It is known from Figure.1 that [DELTA][phi](x,y) = 2[pi][f.sub.o] x [bar.BC], and h(x,y) = (L x [bar.BC])/(d + [bar.BC]) is deduced from similar triangles of AABC and [??]A[O.sub.p][O.sub.c], so the height distribution of object comparing to the reference plane is deduced as Eq. (7), then the 3D surface of the object can be reconstructed from h(x, y) (Ricardo et al., 2004).

h(x,y) = L x [DELTA][phi](x-y)/2[pi][f.sub.o]d + [DELTA][phi](x, y) (7)

2.2. Principle of WTP Method

Wavelet is a new signal analysis method in time domain, which is constructed by scaling and shifting from mother wavelet in Eq. (8), where a is scaling factor, and b is shifting factor.

[[psi].sub.a,b](t) = 1/[square root of (a)] [psi](t - b/a) a,b [member of] R; (a > o) (8)

Apply one-dimensional Continuous Wavelet Transform (1D-CWT) on function f (t) by Eq. (9), where [W.sub.f] (a,b) is called wavelet transformation coefficient, which shows the similarity between wavelet and f (t) in the local part. When the frequency of wavelet is closed to the signal, the amplitude of [W.sub.f] (a,b) is larger.

[W.sub.f] (a,b) = -1/[square root of (a)] [[integral].sub.R] f(t) [[psi].sup.*.sub.a,b] (t)dt (9)

Wavelet analysis has the advantages of multi-resolution. It can also be applied to complex spatial signals analysis. WTP is a 3D reconstruction method by applying wavelet analysis on fringe image to improve phase extraction accuracy.

Apply one-dimensional Continuous Wavelet Transform (1D-CWT) along x axis on one row of fringe image by Eq. (10). The width of wavelet transform window will change according to scaling factor [alpha].

[W.sub.g] (a,b) = -1/[square root of (a)] [[integral].sub.R] g(t) [[psi].sup.*.sub.a,b] (t)dx (10)

The amplitude and phase of wavelet transformation coefficient is calculated by Eq. (11) and Eq. (12).

A(a,b) = [square root of ([[Im([W.sub.f] (a,b)].sup.2] + [[Re([W.sub.f](a,b)].sup.2])] (11)

[phi](a,b) = arctan[Im([W.sub.f] (a,b) / Re([W.sub.f](a,b)] (12)

When the frequency of wavelet is closed to the local frequency of the image analysised, the amplitude of A(a,b) is larger. The location where A(a,b) reaches to the maximum value along the scaling direction is defined as ridge of CWT, as shown in Eq. (13).

ridge(b) = max[ A([a.sub.i], b)] (13)

The value of the scaling parameter a contributing to the ridge is the fittest window width, which is named as [a.sub.ridge]. Put [a.sub.ridge] and b into Eq. (13), then phase of ridge is obtained by Eq. (14).

[phi][(a,b).sub.ridge] = arctan [Im([W.sub.f]([a.sub.ridge], b))/Re([W.sub.f]([a.sub.ridge], b))] (14)

Repeat the above process on the whole fringe image, detecting the ridge and extracting the phase of ridge, then the height distribution of object is obtained.

2.3. Research Method

In this paper, FTP method and WTP method are comparative studied to evaluate their adaption on 3D reconstruction of complex surface object. FTP method and WTP method are applied on the same deformed fringe image, which is formed by a complex surface reflection of a sinusoidal fringe projection. The process flow is shown in Figure 6. In FTP process, the orthogonal ellipse filter is used to get fundamental frequency and phase is extracted by Eq. (6). In WTP process, selecting complex Morlet wavelet as a mother wavelet, by finding the maxim value to detect ridge of wavelet transform, and extracting phase by Eq. (14). Those two methods use the same regional relative reliability-guided phase unwrapping algorithm.

3. Experiments and analysis

3.1. Simulation and Validation

Based on peaks function by Matlab simulation, a virtual 3D object is constructed in 362 pixels plus 362 pixels, and the Gauss noise with 0.3 standard deviation is mixed on the surface. Sinusoidal fringe is projected on it and the deformed fringe is shown in Figure 7.

The extracted phase and unwrapped phase by FTP and by WTP method are shown in Figure 8 and Figure 9. Reconstruction error distributions of those two methods are shown in Figure 10 and Figure 11. The mean square error is 0.1982 for FTP method and 0.1830 for WTP method. So, either FTP method or WTP method can reconstruct the virtual 3D object in a high accuracy (Gasca-Hurtado et al., 2015).

3.2. Train Wheel 3D Reconstruction and Analysis

A new manufactured train wheel width in 135mm is used to compare the reconstruction accuracy of FTP and WTP method on real object. The experiment is done in Figure 12. The parameter L is 600 mm, D is 200 mm, and the stripe frequency [f.sub.o] is 1/16 (stripe/pixel). The wheel surface is complex characteristic in a flat surface in wheel tread and an abrupt surface in wheel flange, as shown in Figure.12. Besides that, there are two orthogonal artificial cracks in length 10mm, width 2mm, and depth 2mm on tread, and the No.1 defect is along circumferential direction while No.2 defect is along axial direction. The wheel is made by metal which cause a strong reflection. Fringe is skewed projected onto wheel surface in 45 degrees. Defects on wheel tread and the deformed fringe image is shown in Figure.13. The extracted phase and unwrapped phase by FTP and by WTP method are shown in Figure 14 and Figure 15. The phase distribution of wheel surface and details in defects such as depth and width by two methods are shown in Figure 16 and Figure 17.

Figure 18 and Figure 19 show phases of two selected rows of fringe image across the two defects by FTP and WTP method. Two curves have similar tendency in WTP method, but they are sensitive to noise, which results in curves fluctuation. For FTP method, two curves trend to different tendency, but curves are smooth comparing to WTP method. As for processing time, FTP is two times faster than WTP.

To qualitatively compare reconstruction accuracy of FTP and WTP, another method called S-transform Profilometry also based on ridge extraction is introduced.

Phase extracted by WTP and S-transform method have similar tendency, but differ from FTP method, as shown in Figure 20 and Figure 21. That means WTP method has a higher accuracy than FTP method for wheel surface reconstruction.

The reasons caused a large phase deviation of FTP method is the abrupt surface and high reflectivity on wheel flange area. Figure 22 shows extracted phase by FTP and WTP method with wheel flange image cut off, and the phase deviation of two methods largely decreased.

4. Conclusion

From experiment on train wheel, conclusions are drawn as following:

1. Either FTP or WTP method can well reconstruct the whole 3D surface of wheel, and can get wheel profile information include details of crack width and depth.

2. Compared to FTP method, WTP method has a higher accuracy for wheel surface reconstruction, but it is sensitive to noise, so the reconstructed surface is not smooth as FTP method, and it is a time-consuming method.

3. For wheel tread area with flat surface, FTP method is a good solution because of the performance of smooth reconstructed surface and the high efficiency of data processing. But for wheel flange area with abrupt surface, WTP method is better because of its multi-resolution to avoid a wrong reconstruction.

Recebido/Submission: 09/04/2016

Aceitacao/Acceptance: 29/07/2016

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Yu Zhang (1) *, Rui Chen (1), Jiayuan Hu (1), Haiqing Wang (1), Zhengyin Ding (2)

(1) School of Physical Science and Technology Southwest Jiaotong University Chengdu, Sichuan, China

(2) College of Energy Engineering, Zhejiang University Hangzhou, Zhejiang, China