Applicability analysis of wavelet-transform profilometry

Zibang Zhang; Jingang Zhong

doi:10.1364/OE.21.018777

1. Introduction

Profilometry is a technique to determine the surface structure of an object. Carrier-fringe projection [1, 2] is a kind of optical profilometry providing a non-contact approach for object surface measurement. It has been widely used in diverse fields: measurement of surface deformation and roughness [3], object shape measurement [4–6], non-invasive 3-D imaging [7], 3-D face reconstruction [8, 9], etc.

In the last few decades, the carrier-fringe projection profilometry has developed tremendously. To demodulate the underlying phase distribution from deformed fringe patterns is the main process of the carrier-fringe projection profilometry. Various implementations of the carrier-fringe projection profilometry can be mainly divided into two groups: single-shot fringe pattern methods such as Fourier transform profilometry (FTP) [10–13], wavelet-transform profilometry (WTP) [14–17], and multiple-shot fringe pattern methods such as phase stepping profilometry (PSP) [18–20]. Both kinds of methods have their pros and cons. The multiple-shot fringe pattern methods offer more accurate measurement than the single-shot methods do, but the methods are confined to static measurements. For example, the PSP outperforms the transform-based methods in terms of accuracy, but it requires more than one fringe pattern. Thus the PSP is unfit for dynamic measurements. The single-shot fringe pattern methods can be applied in the dynamic measurements, but the methods are restricted by some other limitations. For instance, the FTP needs only one fringe pattern and requires little computation. However, the Fourier transform is a global operation with the restriction that the signal should be globally stationary. Otherwise, the desired first-order spectrum will possibly overlap with the zero-order spectrum or the higher-order spectra. Once the first-order spectrum can’t be accurately extracted, the resulting demodulation errors will propagate. Windowed Fourier transform [21–23] has been proposed to extract and reconstruct the first-order spectrum with a sliding Gaussian window. The Gaussian window acts as a filter in both the spatial and the frequency domains, thus windowed Fourier transform can deal with the spectrum overlapping problem in a certain degree and it is robust to noise. However, the adaptive selection of the window width remains challenging.

Wavelet transform (WT) is an excellent tool for the local analysis. The WTP requires only one deformed fringe pattern for the phase demodulation [24, 25]. By using of the analytic Morlet wavelet, the argument of the wavelet ridge yields the fringe signal phase directly [26]. However, such a simple approach is limited by its applicability condition. Pronounced errors will occur when the applicability condition is not met. In [27], W. Chen et al. discussed how the spectrum overlapping affects the demodulation accuracy in the WTP.

In this paper, we look into the applicability condition [28, 29] of the WTP. The so called applicability condition refers to what kinds of object surfaces are suitable for the demodulation carried out by the WTP. The demodulation by the WTP is accurate when the applicability condition is satisfied. By analyzing the nature of the WT, we point out that the existing condition is defective and we propose a more robust applicability condition. Both the existing and the newly proposed applicability conditions lead to the same conclusion that one might reduce the demodulation errors by increasing the carrier frequency of the fringe.

2. Wavelet-transform profilometry

Fringe projection profilometry is to measure the height distribution of an object surface based on fringe analysis. As Fig. 1 shows, a standard sinusoid fringe pattern projected onto the object surface will be deformed due to the height distribution of the object surface. The deformed fringe pattern encodes the height information. To decode the height information, the fringe analysis is employed.

Fig. 1 Optical geometry of fringe projection profilometry.

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Essentially, a fringe pattern is a 2-D phase-modulated signal. Assuming the standard fringes are vertical or horizontal, the 2-D signal can be expressed in the 1-D form:

I (x) = I_{0} (x) + A (x) \cos [ϕ (x)],

where

I_{0} (x)

is the DC term resulted from the background illumination,

A (x)

is the modulated amplitude of the signal

I (x)

, and

ϕ (x)

is the signal phase. The signal phase is the sum of the linear phase

2 π f_{0} x

introduced by the carrier frequency

f_{0}

, and the modulated phase

Δ ϕ (x)

introduced by the object surface height distribution:

ϕ (x) = 2 π f_{0} x + Δ ϕ (x) .

According to the optical geometry shown in Fig. 1, the relation between the modulated

Δ ϕ

and the object surface height distribution

h

can be mathematically expressed as below [1]:

Δ ϕ (x, y) = - \frac{2 π f_{0} \cdot h (x, y) \cdot d}{l_{0} - h (x, y)},

where

(x, y)

is the 2-D Cartesian coordinate,

l_{0}

is the distance between the projector and the object, and

d

is the distance between the projector and the camera. Furthermore, when

l_{0} ≫ h

, the modulated phase

Δ ϕ (x)

is approximately linear to the height

h

and Eq. (3), therefore, can be re-written as:

Δ ϕ (x, y) = - \frac{2 π f_{0} \cdot h (x, y) \cdot d}{l_{0}} .

From Eq. (4), we can find that the object surface shares the same shape with the curve of the modulated phase.

The phase demodulation is the main process in the fringe analysis. With the signal phase map, one can obtain the height distribution easily. Therefore, the accuracy of the phase demodulation plays an extremely important role in the fringe projection profilometry. Since the wavelet transform has excellent localization in the spatial domain, it has been a powerful tool for the fringe analysis.

The wavelet transform (WT) is a kind of linear transforms. It computes the inner product between the signal $I$ and a series of wavelets in the spatial domain:

W (a, b) = \int_{- \infty}^{\infty} I (x) \frac{1}{a} ψ^{*} (\frac{x - b}{a}) d x,

where

ψ^{*}

denotes the complex conjugate of the ‘mother wavelet’

ψ

,

a

is the scale factor and

b

is the translation factor. Note that the coefficient

1 / a

in Eq. (5) has replaced

1 / \sqrt{a}

in the wavelet ridge detection. This substitution makes the WT coefficients independent on the scale factor, ensuring the accurate wavelet ridge detection. The amplitude of the wavelet coefficients is defined as:

A_{w} (a, b) = \sqrt{{real [W (a, b)]}^{2} + {imag [W (a, b)]}^{2}},

where

real [W (a, b)]

and

imag [W (a, b)]

represent the real part and the imaginary part of coefficients, respectively. The amplitude

A_{w}

represents the similarity between the signal and the scaled wavelet at the point

x = b

. The maximum represents that the signal has the largest similarity to the wavelet scaled with

a

, at the point

x = b

. Such a path consisted of the maximum amplitudes is called the wavelet ridge, which can be expressed as follows:

R (a_{r} (b), b) = \max [A_{w} (a, b)],

where

a_{r} (b)

is the corresponding scale on the ridge at the point

x = b

, and

\max [\cdot]

represents the maximum operator. The phase of the ridge is defined as the argument of the WT coefficient on the ridge:

ϕ_{w r} (a_{r} (b), b) = \arg [W (a_{r} (b), b)],

The analytic Morlet wavelet is one of the most used wavelets in the WTP. The 1-D analytic Morlet wavelet can be represented as:

ψ (x) = \frac{1}{\sqrt[4]{π}} \exp (j 2 π x) \exp (- \frac{x^{2}}{2}) .

From Eq. (9), we can see that the wavelet is a period of oscillation weighted by a Gaussian function. It has been demonstrated that if the modulated amplitude

A (x)

and the phase

ϕ (x)

given in Eq. (1) satisfy the following equations [25, 26]:

η_{A} (x) = \frac{{(2 π)}^{2}}{| ϕ^{'} (x) |} \frac{| A^{″} (x) |}{| A (x) |} ≪ 1

and

η_{ϕ} (x) = {(2 π)}^{2} \frac{| ϕ^{″} (x) |}{{| ϕ^{'} (x) |}^{2}} ≪ 1,

the phase

ϕ (x)

approximates to the phase of the wavelet ridge at the point

x = b

, that is,

ϕ (b) ≃ ϕ_{w r} (a_{r} (b), b) = \arg [W (a_{r} (b), b)] .

Thus, Eqs. (10) and (11) are called the applicability condition of the WTP, being an evaluation of the demodulation error. Since Eq. (10) is much easier to be satisfied than Eq. (11), the condition given in Eq. (10) is less concerned about in practice. Concentrating on Eq. (11), the first and the second derivatives of the signal phase are

ϕ^{'} (x) = 2 π f_{0} + Δ ϕ^{'} (x)

and

ϕ^{″} (x) = Δ ϕ^{″} (x),

respectively. Inserting Eq. (4) into Eqs. (13) and (14) yields

ϕ^{'} (x) = 2 π f_{0} - \frac{2 π f_{0} d}{l_{0}} h^{'} (x)

and

ϕ^{″} (x) = - \frac{2 π f_{0} d}{l_{0}} h^{″} (x) .

When the analytic Morlet wavelet is used, the phase of the wavelet ridge can be expressed at the point

x = b

as:

\begin{matrix} ϕ (b) ≃ \arg [W (a_{r} (b), b)] \\ = \arg {\frac{1}{a_{r} (b) \cdot \sqrt[4]{π}} \int_{- \infty}^{+ \infty} I (x) \exp [j 2 π (\frac{x - b}{a_{r} (b)})] \exp [- \frac{1}{2} {(\frac{x - b}{a_{r} (b)})}^{2}] d x} . \end{matrix}

From Eq. (17), we can see that the phase

ϕ (b)

is essentially obtained by computing an integral of the fringe signal in the spatial domain. Figure 2 shows the distributions of the fringe signal amplitude, the real part of scaled analytic Morlet wavelets and the modulated phase. In numerical computation, the infinite integral interval in Eq. (17) is replaced by the finite interval

[L B, U B]

. The finite interval is determined by the support of the wavelet. In MATLAB, for example, the support of the analytic Morlet wavelet is [-8, 8], equal to the full width at

1 / e^{32}

maximum of its Gaussian envelope, as Fig. 2(b) shows. Therefore, the lower bound

L B = b - 8 a_{r} (b)

and the upper bound

U B = b + 8 a_{r} (b)

. With the replacement of the integral interval, Eq. (17) can be re-written as:

\begin{matrix} ϕ (b) ≃ \arg [W (a_{r} (b), b)] \\ = \arg {\frac{1}{a_{r} (b) \cdot \sqrt[4]{π}} \int_{L B}^{U B} I (x) \exp [j 2 π (\frac{x - b}{a_{r} (b)})] \exp [- \frac{1}{2} {(\frac{x - b}{a_{r} (b)})}^{2}] d x} . \end{matrix}

Note that the phase

ϕ (b)

obtained by Eq. (18) is determined by the integral over the interval of the fringe signal

I (x)

instead of the result which is only contributed by the signal at the point

x = b

. The value of the obtained phase is a weighted mean by the Gaussian envelope of the analytic Morlet wavelet within the integral interval.

Fig. 2 Illustration of the obtained modulated phase. Each point of the modulated phase (c) is obtained by computing an integral of the signal (a) and the analytic Morlet wavelet (b) within the interval that is the support of the scaled wavelet $[L B, U B]$ .

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For the applicability condition given in Eqs. (10) and (11), we find that it is defective. The reasons are listed below:

a) The applicability condition given in the differential form becomes unavailable at some points where the signal phase hasn’t the first or the second derivatives. According to Eqs. (15) and (16), the inexistence of the signal phase’s first or second derivatives is due to the height distribution of the object surface. In practice, the object surfaces are various. Points on the surfaces without the first derivative and the second derivatives are quite common. For instance, a V-shaped surface will result in the inexistence of the first derivative of $h^{'} (x)$ at the joint point. Even an S-shaped surface, a smooth-looking surface, will result in the inexistence of the second derivative at the inflection point. Therefore, the existing applicability condition in the differential form is not robust.
b) The evaluation by the existing applicability condition at every single point might not reflect the demodulation errors accurately. The existing applicability condition is only able to evaluate the errors at every single point while the modulated phase, according to Eq. (18), is substantially obtained by computing the integral whose interval is determined by the support of the scaled wavelet. That means the phase demodulation is contributed by the signal distribution in a local area instead of at a certain point. Therefore, the existing applicability condition is unable to reflect the influence from the abrupt changes in the neighbor.
c) The condition, $η_{ϕ} ≪ 1$ , is loose in terms of quantity, which cannot evaluate the errors effectively.

3. A new robust applicability condition

Since the existing applicability condition is defective, we propose a new robust condition. As has been discussed in [24–26], the error of demodulation is small when the second- and the higher-order terms in of Taylor series expansion of the phase can be negligible. From another point of view, it is beneficial to the error reduction that the phase is locally linear. In other words, the degree of local linearity of the phase is related to the accuracy of demodulation.

We define the nonlinearity $Δ S$ to evaluate the degree of linearity of the curve $ϕ (x)$ in the interval $[L B, U B]$ , which is expressed as below:

\begin{matrix} Δ S = | \int_{L B}^{U B} [ϕ (x) - L (x)] d x |, \\ w i t h L (x) = [\frac{ϕ (U B) - ϕ (L B)}{U B - L B}] (x - U B) + ϕ (U B), \end{matrix}

where

Δ S (x)

is the defined degree of nonlinearity,

L (x)

represents the straight-line equations which is the linear approximation to

ϕ (x)

in the integral interval

[L B, U B]

,

U B

represents the upper bound of the interval,

L B

represents the lower bound. In terms of its geometric meaning,

Δ S

represents the algebraic addition of area, corresponding to the shaded area in Fig. 3(c). Smaller

Δ S

implies better linearity. To better descript the local linearity and for the consistency to the Morlet-wavelet-based phase demodulation given in Eq. (18), we propose the local nonlinearity

Δ S_{w} (b)

at the point

x = b

:

\begin{matrix} Δ S_{w} (b) = | \int_{L B}^{U B} g (\frac{x - b}{a_{r} (b)}) \cdot [ϕ (x) - L (x)] d x |, \\ with g (x) = \exp (- \frac{x^{2}}{2}), \end{matrix}

where

g (x)

is the same Gaussian function as the one of the analytic Morlet wavelet. In terms of its geometric meaning,

Δ S_{w}

represents the algebraic addition of area weighted by the Gaussian function. The integral interval

[L B, U B]

in Eq. (20) is determined by the full width at

1 / e^{8}

maximum of the Gaussian function, i.e.,

L B = b - 4 a_{r} (b)

and

U B = b + 4 a_{r} (b)

. With the defined local nonlinearity, we propose the new applicability condition that the phase demodulation is accurate when the degree of local nonlinearity tends to zero, which can be expressed as:

Fig. 3 Illustration of local linearity. The height distribution of object surface (d), resulting phase (b), resulting modulated phase (c) and the Gaussian function (a).

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Δ S_{w} (b) \to 0.

The proposed condition, in the integral form, reflects the linearity within a local area instead of a certain point. The local area refers to the integral interval which corresponds to the support of the scaled wavelet. The integral doesn’t require that the phase can be written as Taylor series. Thus the proposed condition is suitable for the cases that the phase doesn’t have the first or the second derivatives. Smaller $Δ S$ implies that the local phase is of higher linearity. When the local phase is of high linearity, the second-order and the higher-order terms, if they exist, in its Taylor series expansion are negligible. The high local linearity of the phase leads to accurate demodulation result. Therefore, the proposed condition and the existing condition are consistent. In addition, inserting Eq. (2) into Eq. (20) yields:

\begin{matrix} Δ S_{w} (b) = | \int_{L B}^{U B} g (\frac{x - b}{a_{r} (b)}) \cdot [Δ ϕ (x) - L_{Δ} (x)] d x |, \\ w i t h L_{Δ} (x) = [\frac{Δ ϕ (U B) - Δ ϕ (L B)}{U B - L B}] (x - U B) + Δ ϕ (U B) . \end{matrix}

We find that the signal phase

ϕ

and the modulated phase

Δ ϕ

share the linearity in the same local area. What’s more, by inserting Eq. (4) into Eq. (22), we can further obtain

Δ S_{w} (b) = | - 2 π f_{0} d \int_{L B}^{U B} g (\frac{x - b}{a_{r} (b)}) {\begin{cases} \frac{h (x)}{l_{0} - h (x)} - \\ {[- \frac{h (U B)}{l_{0} - h (U B)} + \frac{h (L B)}{l_{0} - h (L B)}] (\frac{x - U B}{U B - L B}) - \frac{h (U B)}{l_{0} - h (U B)}} \end{cases}} d x | .

When

l_{0} ≫ h (x)

, Eq. (23) can be simplified as:

\begin{matrix} Δ S_{w} (b) = | - \frac{2 π f_{0} d}{l_{0}} \int_{L B}^{U B} g (\frac{x - b}{a_{r} (b)}) \cdot [h (x) - L_{h} (x)] d x |, \\ w i t h L_{h} (x) = [\frac{h (U B) - h (L B)}{U B - L B}] (x - U B) + h (U B) . \end{matrix}

The height

h

also shares the local linearity with the modulated phase

Δ ϕ

. In other words, the local linearity of the modulated phase can be directly reflected from the object surface (see Fig. 3). Since the signal phase

ϕ (x)

is linear to the carrier frequency

f_{0}

, the normalization

ϕ (x) / f_{0}

is carrier-frequency independent. We re-write the applicability condition given in Eq. (24) in the normalized form which only depends on the height distribution

h (x)

and the parameters of the experimental setup:

\begin{matrix} Δ S_{h} (b) = \frac{Δ S_{w} (b)}{f_{0}} \\ = | - \frac{2 π d}{l_{0}} {\int_{L B}^{U B} g (\frac{x - b}{a_{r} (b)}) \cdot [h (x) - L_{h} (x)] d x} | \end{matrix}

and

Δ S_{h} (b) \to 0.

Since the accurate demodulation results from the high local linearity of the height distribution

h (x)

, we can draw a conclusion that a simple surface, of high linearity, leads to the accurate demodulation result while a complex surface, of low linearity, leads to the relatively inaccurate demodulation result. Comparing to the existing applicability condition, the proposed condition has a clear physical meaning.

4. Improvement of applicability

As high local linearity of the modulated phase is beneficial to the demodulation accuracy, in this section we discuss how to improve the local linearity. Let’s look into the proposed applicability condition given in Eq. (20). The only one tunable parameter is the integral interval, $[L B, U B]$ , which is determined by the scale factor on the wavelet ridge, that is, $a_{r}$ . A smaller $a_{r}$ leads to a narrower integral interval in which the phase is more approximately linear. The smaller interval makes the proposed condition $Δ S_{w} \to 0$ better satisfied. There exists a simple relation between the scale factor on the ridge $a_{r}$ and the instantaneous frequency $f_{inst}$ as below:

a_{r} = \frac{1}{f_{inst}} .

In terms of its definition, the instantaneous frequency

f_{inst}

of the fringe signal can be obtained by:

f_{inst} (x) = \frac{1}{2 π} ϕ^{'} (x) .

Thus, detecting the ridge of the WT is an approach for estimating the instantaneous frequency [28, 30]. The smaller

a_{r}

can be obtained by increasing the instantaneous frequency of the fringe

f_{inst}

. Substituting Eq. (2) into Eq. (28) yields

f_{inst} (x) = f_{0} + \frac{1}{2 π} Δ ϕ^{'} (x) .

According to Eq. (29), a greater

f_{0}

results in a greater

f_{inst}

, since the instantaneous frequency is mainly contributed the carrier frequency

f_{0}

. In conclusion, increasing the carrier frequency is beneficial for improving the local linearity and, ultimately, reducing the errors. Figure 4 illustrates the relationship between the carrier frequency and the local nonlinearity.

Fig. 4 Chart of relationship between carrier frequency $f_{0}$ and local nonlinearity $Δ S_{h}$ .

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On the other hand, the conclusion, increasing the carrier frequency is beneficial for reducing the errors, can also be drawn from the existing condition given in Eqs. (10) and (11). For $η_{A}$ , its value depends on the variation of the amplitude $A$ which is contributed by the reflectivity of the object surface. In most cases, the reflectivity has small relative variations. The second derivative of amplitude $A^{″}$ is negligible, resulting in $η_{A}$ tending to zero. Thus Eq. (10) is easy to be satisfied. For $η_{ϕ}$ , its value depends on the variation of the phase $ϕ$ . The phase $ϕ$ , given in Eq. (2), is the sum of the standard fringe phase and the modulated phase introduced by the height of the object surface. When a certain object surface is given, the distributions of $h$ , $h^{'}$ and $h^{″}$ are fixed. According to Eqs. (15) and (16), we obtain

{| ϕ^{'} (x) |}^{2} = {| 2 π f_{0} [1 - \frac{h^{'} (x) \cdot d}{l_{0}}] |}^{2} \propto f_{0}^{2}

and

| ϕ^{″} (x) | = | 2 π f_{0} \frac{h^{″} (x) \cdot d}{l_{0}} | \propto f_{0} .

According to Eqs. (11), (30) and (31), we derive

η_{ϕ} \propto \frac{1}{f_{0}} .

Since

η_{ϕ}

is linear to

1 / f_{0}

, a higher carrier frequency

f_{0}

leads to a smaller

η_{ϕ}

. In conclusion, the higher carrier frequency

f_{0}

leads to smaller errors.

Both the differential-form and the integral-form conditions lead to the same conclusion that the errors can be improved by increasing the carrier frequency. In practice, we have observed the existence of the highest carrier frequency. The highest carrier frequency results from two factors: 1) the resolution of the digital projector and 2) the resolution of the imaging system. In most cases, the former dominates.

In the case that the carrier frequency is limited by the imaging system, the highest carrier frequency can be estimated by the fringe signal bandwidth. The first-order spectrum of the deformed fringe signal carriers the height information. The Fourier transform of the standard fringe signal is a pulse whose spatial frequency corresponds to the carrier frequency. The complexity of the object surface results in the pulse spreading. Thus, the bandwidth of fringe signal, i.e., the spectral range of the first-order spectrum, indicates the complexity of the object surface. In actual, the bandwidth is linear to the carrier frequency.

The Fourier spectrums (with the DC term removed) of the deformed fringes with three different carrier frequencies are plotted in Fig. 5, where $f_{1} = f$ , $f_{2} = 3 f$ , $f_{3} = 9 f$ , $B_{f_{i}}$ $(i = 1, 2, 3)$ represents the bandwidth of the first-order spectrum, $f_{0}$ is the carrier frequency of fringe and the normalized frequency is the actual frequency $f_{0}$ to the imaging system sampling frequency $f_{sample}$ ratio. Thus, to avoid under-sampled fringes, one should ensure

f_{0} < f_{sample} - \frac{B}{2},

where

B

is the bandwidth of the fringe signal.

Fig. 5 Fourier spectrums of deformed fringe patterns.

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5.Numerical simulation

In practice, the object surfaces are various. To compare the proposed and the existing applicability conditions, we perform the numerical simulation in three cases: Case I: surface with every point infinitely differentiable; Case II: surface with the inexistence of the second derivative at the inflection point; Case III: surface with the inexistence of the first derivative at the joint point.

Case I: Surface with every point infinitely differentiable

We use the build-in function of MATLAB, peaks(), to simulate an object surface. The object is assumed to be $1 \times 1 m^{2}$ in size, represented by a $512 \times 512$ matrix. The height $h$ can be represented as follows:

h (x, y) = \frac{1}{40} peaks (6 x - 3, 6 y - 3),

\begin{matrix} peaks (x, y) = 3 {(1 - x)}^{2} \exp [- x^{2} - {(y + 1)}^{2}] - 10 (\frac{x}{5} - x^{3} - y^{5}) \exp (- x^{2} - y^{2}) \\ - \frac{1}{3} \exp [- {(x + 1)}^{2} - y^{2}], \end{matrix}

where

(x, y)

is the 2-D Cartesian coordinate. The peaks() function is analytic with every point infinitely differentiable. The simulated object is shown in Fig. 6.

Fig. 6 Height distribution of object for simulation.

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The distance between the camera and the reference plane $l_{0} = 1 m$ and the distance between the camera and the projector $d = 0.15 m$ . Three carrier frequencies are specified: $f_{0} = 25$ , $35$ and $75 cycle/m$ . The resulting modulated phase $Δ ϕ$ is obtained according to Eq. (3). We take 256th row of the matrix for comparison. The deformed fringes of three different carrier frequencies are shown in Fig. 7.

Fig. 7 Deformed fringes patterns at 256th row.

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The phase $ϕ (x)$ of the deformed fringes is demodulated by using the WT. The resulting phase is wrapped into $[- π, + π]$ and we use the MATLAB build-in function, unwrap(), for the unwrapping phase process. The first and the second derivatives of the phase $ϕ^{'} (x)$ are given in Fig. 8. The distributions of $η_{ϕ}$ and $Δ S_{h} (x)$ are computed with Eqs. (11) and (25) respectively and are shown in Fig. 9. The distributions of the demodulated height are plotted in Fig. 10 with the error distributions plotted in Fig. 11.

Fig. 8 Distributions of $ϕ^{'} (x)$ (a) and of $ϕ^{″} (x)$ (b).

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Fig. 9 Distributions of $η_{ϕ}$ (a) and of $Δ S_{h} (x)$ (b).

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Fig. 10 Comparison of the demodulated height with different carrier frequencies.

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Fig. 11 Comparison of demodulated absolute height error.

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Comparing Fig. 11 with Fig. 9, it is demonstrated that the increment of carrier frequency makes the approximation condition $η_{ϕ} \to 0$ better satisfied and the increment improves the accuracy of height demodulation, especially where the height varies rapidly. The demodulated height is absolutely accurate at the points where $η_{ϕ} = 0$ or $Δ S_{h} (x) = 0$ , despite of the carrier frequency. In the case that the modulated phase is infinitely differentiable, $η_{ϕ}$ and $Δ S_{h} (x)$ have the same distribution. The errors of the demodulated height can be evaluated by the value of $η_{ϕ}$ and the value of $Δ S_{h} (x)$ .

Case II: Surface with the inexistence of the second derivative at the inflection point

We simulated another 1-D object that leads to the modulated phase without the second derivative at $x = 0.5$ . The height distribution of the simulated object surface can be expressed as:

h (x) = \frac{1}{3} {\exp [- {(6 x - 3)}^{2}] - 1} \exp [- {(6 x - 3)}^{2}] sgn (- x + 0.5),

where

sgn (x)

is the sign function and

x \in [0, 1]

. The simulated height is S-shape in the neighbor of

x = 0.5

. The S-shape distribution is due to the sign function flipping the left-hand side part of the peak upside down. The modulated phase with its first and second derivatives is given in Fig. 12. The second derivative is inexistent at the inflection point

x = 0.5

. The modulated phase has the positive second derivative near

x = {0.5}^{-}

and the negative second derivative near

x = {0.5}^{+}

. The absolute value of the second derivative near

x = 0.5

is large.

Fig. 12 Distributions of modulated phase (a) with its first derivative (b) and second derivative (c).

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As Fig. 13 shows, the higher carrier frequency leads to the smaller demodulation error. In addition, the higher carrier frequency improves the boundary effect because it results in the scaled wavelets that have better spatial localization. Particularly, the locally maximum demodulation error becomes zero at $x = 0.5$ where the top of the peak becomes the inflection point. We consider that the positive and the negative second derivative near $x = 0.5$ are evened out by the integral of the WT. The height errors are improved in the neighbor of $x = 0.5$ . As Fig. 14 shows, the distribution of $η_{ϕ}$ is no longer able to reflect the errors in the neighbor of the point that has not the second derivative while $Δ S_{h} (x)$ still gives a correct evaluation of the errors.

Fig. 13 Comparison of demodulated heights (a) and absolute height errors (b).

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Fig. 14 Distributions of $η_{ϕ}$ (a) and $Δ S_{h} (x)$ (b).

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Case III: Surface with the inexistence of the first derivative at the joint point

We also simulated a 1-D object with the height distribution that results in the modulated phase without the first derivative at one point:

h (x) = | 0.36 (x - 0.5) |,

where

x \in [0, 1]

. As Fig. 15 shows, the V-shaped height distribution results in the phase touching at the join point at

x = 0.5

. In other words, the phase doesn’t have either the first derivative or the second derivative, which leads to the discontinuities in Fig. 15(b) and (c). The phase distributes almost linearly except for the point

x = 0.5

. The demodulated heights are accurate outside the neighbor of the point

x = 0.5

, as Fig. 16 shows. The higher carrier frequency leads to the smaller demodulation error. The boundary effect is also improved by the higher carrier frequency. In terms of error evaluation,

Δ S_{h} (x)

is still available in this case while the existing condition

η_{ϕ}

doesn’t give an effective evaluation at all.

Fig. 15 Distributions of modulated phase (a) with its first derivative (b) and second derivative (c).

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Fig. 16 Comparison of demodulated heights (a) and absolute height errors (b).

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Fig. 17 Distributions of $η_{ϕ}$ (a) and $Δ S_{h} (x)$ (b).

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In summary, the proposed condition is suitable for evaluating the demodulation errors in all the cases considered while the existing condition is only suitable for the case where the height distribution is infinitely differentiable. And, the high carrier frequency helps reduce the demodulation errors, especially for the areas of low local linearity.

6. Experiment

A plaster model of Ludwig van Beethoven was used as the object to measure. The object has rapid height variations from Beethoven’s brow to the bridge of the nose and from the end of the nose to the lip. The rapid height variations are considered as a challenge to the WTP. As the optical geometry shown in Fig. 1, the distance between the camera and the reference plane $l_{0} = 1.05 m$ and the distance between the camera and the projector $d = 0.22 m$ . We projected 3 sets of sinusoidal fringe in turn by using a Toshiba TDp-T90 digital projector. The carrier frequencies $f_{0}$ of the sets of fringe are 0.045 cycle/pixel, 0.06 cycle/pixel and 0.075 cycle/pixel, respectively. Note that the carrier frequency used in this section is the normalized frequency whose unit is cycle/pixel. Each set includes 4 fringe patterns which have carrier frequency in common. Every fringe pattern in the set has a phase step $δ = π / 2$ . An Olympus digital camera, C-770, was used to capture the deformed fringe patterns. The captured fringe patterns were transferred to a PC for a further analysis. A $1000 \times 1000$ -pixel area was cropped from each captured image. We chose one image in each set for the WTP, as Fig. 18 shows.

Fig. 18 Captured deformed fringe pattern for WTP. Carrier frequencies of the fringe pattern are 0.045 cycle/pixel (a), 0.06 cycle/pixel (b) and 0.075 cycle/pixel (c), respectively. Red line in (a) indicates the 518th column of the fringe pattern.

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The phase of the deformed fringes was demodulated by the WT. Since the accuracy of the PSP is independent on the height variations, we also demodulated the height with the four-step PSP for a reference. The set, shown in Fig. 19, with the carrier frequencies $f_{0} = 0.045 cycle/pixel$ was chosen for the demodulation by the PSP.

Fig. 19 Set of deformed fringe pattern for phase demodulation by four-step PSP.

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Particularly, we selected the 518th column of the demodulated height distributions for comparison, where rapid height variations exist. For the WTP, the phase was demodulated by the wavelet ridge detection method. The intensity distribution of the 518th column is given in Fig. 20(a). Figure 20(b) shows the amplitude distribution of the WT coefficients corresponding to the 518th column of fringe ( $f_{0} = 0.075 cycle/pixel$ ) and Fig. 20(c) shows the phase map of the WT coefficients.

Fig. 20 Distributions of intensity (a), wavelet coefficients amplitudes (b) and wavelet coefficients phase map (c) corresponding to the 518th-column fringe with the carrier frequency $f_{0} = 0.075 cycle/pixel$ .

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We used the flood fill algorithm [31] to unwrap the phase and then obtained the profile of the object by converting the height distribution from the unwrapped phase map.

As Fig. 21 shows, the results of the WTP are comparable to the one of the PSP, except for the areas where the height varies rapidly. For a better comparison, let’s look into two areas, from Beethoven’s brow to the bridge of the nose (a peak/valley-shape distribution shown in Fig. 22, Local 1) and from the end of the nose to the lip (an S-shape distribution shown in Fig. 23, Local 2).

Fig. 21 Comparison of the demodulated height at the 518th column.

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Fig. 22 Comparison of demodulated height at the 518th column (corresponding to Local 1, $y \in [560, 610]$ in Fig. 21).

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Fig. 23 Comparison of demodulated height at the 518th column (corresponding to Local 2, $y \in [760, 830]$ in Fig. 21).

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For Local 1, the peak/valley-shape surface, the PSP is able to recover the steep peaks and deep valleys. Although the demodulations by the WTP appear gentler, such a problem can be improved by increasing the carrier frequency. In terms of noise immunity, the WTP outperforms the PSP, providing a much smoother curve. This is because the integral operation in the WT acts like a smoothing filter. Each point of the demodulation result is a weighted mean within the integral interval. By increasing the carrier frequency, the result demodulated by the WTP is remarkably improved, which represents that either the peaks or the valleys become steeper or deeper. For Local 2, the S-shape surface similar to Case II in the numerical simulation, there exists an inflection point at $x = 796$ . The errors are evened out in the neighbor of the inflection point. The higher carrier frequency results in the better demodulation. We summarize the errors of the demodulated height in Table 1.

Table 1. Comparison of Height Errors Demodulated by WTP^a

View Table

The 2-D wrapped phase map and the 3-D distribution of demodulated object surface from the fringe pattern with the carrier frequency $f_{0} = 0.075 cycle/pixel$ are shown in Fig. 24 and Fig. 25, respectively. The 3-D distribution was generated by using the surf() function of MATLAB, and the lighting effect was employed. The artifacts off the face are the failures of the phase unwrapping process due to the shadow and the under-sampled fringes.

Fig. 24 2-D distribution of the wrapped phase $Δ ϕ$ with the carrier frequency $f_{0} = 0.075 cycle/pixel$ .

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Fig. 25 3-D distribution of demodulated height with the carrier frequency $f_{0} = 0.075 cycle/pixel$ .

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7. Conclusion

We have analyzed the applicability of the WTP in this paper and found that the existing applicability condition is defective. We propose a new robust applicability condition in the integral form. The proposed condition defines the local linearity of the modulated phase and is capable of evaluating the demodulation errors. We point out that the WTP can achieve accurate demodulation for the object surface with high resulting local linearity of the modulated phase. By analyzing both the existing and the proposed conditions, we also point out that the demodulation errors can be reduced by increasing the carrier frequency. It has been demonstrated by the numerical simulation and the experiment that increasing the carrier frequency can help satisfy the applicability conditions well and improve the measurement accuracy.

Acknowledgments

This work was supported by National Natural Science Foundation of China under the Grant No. 61077003

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Carrier frequency (cycle/pixel)	Minimum (m)	Maximum (m)	Average (m)
0.045	1.718 × 10⁻⁷	1.509 × 10⁻³	0.032 × 10⁻³
0.06	1.829 × 10⁻⁸	1.311 × 10⁻³	0.025 × 10⁻³
0.075	3.826 × 10⁻⁸	0.869 × 10⁻³	0.018 × 10⁻³

Applicability analysis of wavelet-transform profilometry

Abstract

1. Introduction

2. Wavelet-transform profilometry

3. A new robust applicability condition

4. Improvement of applicability

5.Numerical simulation

Case I: Surface with every point infinitely differentiable

Case II: Surface with the inexistence of the second derivative at the inflection point

Case III: Surface with the inexistence of the first derivative at the joint point

6. Experiment

7. Conclusion

Acknowledgments

References and links

Cited By

Figures (25)

Tables (1)

Equations (37)

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