Multiplication method for sparse interferometric fringes

Cong Liu; Xingyi Zhang; Youhe Zhou

doi:10.1364/OE.24.007693

1. Introduction

As it is known, fringe patterns which often contains the information of strain, stress or deformation gradient, etc., can be obtained by using many kinds of optical interferometry such as electronic speckle pattern interferometry (ESPI) [1], photoelasticity [2], coherent gradient sensor (CGS) [3]. However, sparse fringe (a half or no more than two fringes) often exists in the measurements because of many experimental restrictions such as small size of specimen and small deformation. For example, when doing measurements of deformation on a small piece of hard-brittle material, we often encounter this kind of fringe patterns that have extremely low density of fringes which provide little visual information, and also it is difficult to process. To solve this problem, a fringe multiplication method which intends to obtain the fringe pattern with the number of contour maps increased as several times as the original fringe pattern has been used. The multiplied fringe image can have large fringe density so that it not only gives the visual details of changing gradient of phase but also is easy to process. Several works have been dedicated to this subject. Post [4, 5] presented a fringe multiplication method by adding optical devices, which has been verified by the moiré and photoelasticity. However, this method causes great complexity to the experimental process. In 1989, Yu [6] proposed a digital processing method for the sparse fringe, and obtained the doubling and tripling of multiplication fringe according to multiple-angle formula. Recently, Baek et al. [7] utilized the fringe sharping method and acquired the double fringe multiplication. it is still difficult to obtain arbitrary integral-multiple fringe multiplication directly by these methods, Han [13] used phase stepping method to get multiplication fringe image. Though phase steeping is a very reliable technique, in some cases it is not applicable because of the system configuration. Considering these factors, we develop a multiplication method in a way of digital image processing on a single interference fringe pattern; we find a new relationship, which is different from the multiple-angle formula that used in [6, 7], between the arbitrary integral-multiple fringe pattern and the original one. For ideal fringe pattern, the intensity just depends on the phase value, namely the dependence of intensity on phase value at a point strictly comply with the governing interference equation. However, in practice, the experimental fringe pattern recorded by the camera is non-ideal, the main factors are non-uniform illumination; nonlinearity of the camera or digital random noise, etc., in these situations, the intensity distribution depends on both phase value and position of the point, so there are three steps for obtaining fringe multiplication in the general processes. First, the low-pass filter technique or smooth filtering can often be used to reduce the high frequency noise components contained in the fringe pattern. Second, one of the fringe normalization processes [6, 8–10] has been employed to make the non-ideal fringe to be ideal fringe. Third, the processed fringe image can be directly multiplied by using the method mentioned above. But the presented fringe normalization processes [6, 8–10] can be only applicable for the fringe pattern with many interferometry fringes, but not for those with less than two fringes in the whole field. Therefore, a more effective and general way for sparse fringe normalization still deserves our efforts. In this paper, a parameterized fringe multiplication method has been successfully developed for the ideal fringe pattern, which cannot only obtain the arbitrary integer-multiple fringe, but also separate the main frequency from the DC component. An experimental normalization method is also presented for the sparse photoelasticity fringe. It is found that the stress results from the fringe multiplication under normalization keep good agreements with the theoretical results.

2. Algorithm

In an optical ideal interferometric fringe, the form is

I (r) = A [1 + \cos (ϕ (r))],

where

I (r), r, ϕ (r)

and

A

denotes the light intensity of the fringe pattern, coordinate vector, phase and amplitude of the fringe variation, respectively. The phase

ϕ (r)

contains the desired information, e.g. strain, stress or deformation gradient. The light intensity of the N-fold order multiplied fringe can be expressed as:

I_{N} (r) = A [1 + \cos (ϕ_{N} (r))],

where

ϕ_{N} (r)

is the phase distribution of the multiplied pattern. The phase

ϕ (r)

before multiplication and

ϕ_{N} (r)

after multiplication can be considered as two parameters in the algorithm, where the relationship between them gives:

ϕ_{N} (r) = N \cdot ϕ (r) = N \cdot arc \cos (\frac{I (r)}{A} - 1) .

This equation constructs a direct correlation between the Eq. (1) and Eq. (2). Substituting Eq. (3) into Eq. (2), one finds:

I_{N} (r) = A [1 + \cos (N \cdot arc \cos (\frac{I (r)}{A} - 1))] .

For the non-ideal fringe pattern, the intensity of which can be expressed as [8]

I (r) = I_{1} (r) (1 + \cos [ϕ (r)]) + I_{b} (r) + I_{r a n d o m},

where

I (r)

is the intensity of the recorded image,

I_{b} (r)

is the intensity of background,

I_{1} (r)

is the amplitude of the fringe variation, and

I_{r a n d o m}

is random noise caused by light source and /or digital recording system. According to the description in the second paragraph, the random item can be reduced by using the filter technique, thus the Eq. (5) becomes

I (r) = I_{1} (r) (1 + \cos [ϕ (r)]) + I_{b} (r) .

When the fringe pattern is normalized, the Eq. (6) can be rewritten by

I^{'} (r) = I_{1} \cdot [1 + \cos (ϕ (r))],

where

I_{1}

denotes the constant amplitude that independent from the point position. The form of the Eq. (7) is consistent with the Eq. (1), thus our method can also be applicable for the no-ideal fringe pattern.

3. Discussion

3.1 Skeleton line extraction

Fig. 1 (a) The simulated digital image; (b) the 7-fold multiplied fringe pattern; (c) and (d) show the skeleton lines of the (a) and (b), respectively; (e) the intensity of centerline (seen the red line in the Figs. 1(a) and 1(b)).

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3.2 Fourier transform and phase unwrapping

Phase determination by Fourier-transform method [11, 12] is an effective way to obtain the phase information in the whole field. This method is applicable for the multiplied fringe pattern, the Fourier transform of the Eq. (7) can be written:

I^{'} (r) = C_{0} + C (r) + C^{*} (r),

where

C_{0}

is Fourier transform of the constant

I_{1}

,

C (r)

is the Fourier transform of

\frac{1}{2} I_{1} e^{j ϕ (r)}

and

C^{*} (r)

is the conjugate of

C (r)

. So we can filter out the DC term

C_{0}

and either the components

C (r)

or

C^{*} (r)

in frequency domain, if we leave the term

C (r)

, then we can get the wrapped phase with values between

- π

and

π

:

ϕ (r) = arc \tan \frac{Im [C (r)]}{Re [C (r)]} .

After that, phase unwrapping can be proceeds iteratively in x and y-directions [12]:

\begin{array}{l} n (x_{1}, y_{1}) = 0 \\ n (x_{1}, y_{i}) = {\begin{cases} n (x_{1}, y_{i - 1}) if | ϕ (x_{1}, y_{i}) - ϕ (x_{1}, y_{i - 1}) | < π \\ n (x_{1}, y_{i - 1}) + 1 if ϕ (x_{1}, y_{i}) - ϕ (x_{1}, y_{i - 1}) \leq - π \\ n (x_{1}, y_{i - 1}) - 1 if ϕ (x_{1}, y_{i}) - ϕ (x_{1}, y_{i - 1}) \geq π \end{cases} \\ i = 2, 3, ..., \\ n (x_{j}, y_{i}) = {\begin{cases} n (x_{j - 1}, y_{i}) if | ϕ (x_{j}, y_{i}) - ϕ (x_{j - 1}, y_{i}) | < π \\ n (x_{j - 1}, y_{i}) + 1 if ϕ (x_{j}, y_{i}) - ϕ (x_{j - 1}, y_{i}) \leq - π \\ n (x_{j - 1}, y_{i}) - 1 if ϕ (x_{j}, y_{i}) - ϕ (x_{j - 1}, y_{i}) \geq π \end{cases} . \\ j = 2, 3, ..., \\ ϕ_{u n w r e p} (x_{j}, y_{i}) = ϕ (x_{j}, y_{i}) + 2 π n (x_{j}, y_{i}), i,j=1,2, ... \end{array}

Figures 2(a) and 2(b) display the frequency domain of the images obtained by the Fourier transform for the original and processed images, respectively. The plots of amplitude of corresponding frequency along the centerline in the Figs. 2(a) and 2(b) are shown in Figs. 2(c) and 2(d), respectively. We can find that the main frequencies of the multiplied fringe image are separated from the DC component, which allow us to use band-pass filtering and phase unwrapping technique to obtain the phase values in the whole field. Figure 3(a) shows the wrapped phase of the 7-fold multiplied fringe. Based on the relationship between the $ϕ (r)$ and $ϕ_{N} (r)$ in the Eq. (3), the distribution of the retrieved relative phase is demonstrated in Fig. 3(b), where the reference phase point is located at the center of the image, and the corresponding phase can be set to zero. Figure 3(c) shows the information for phase error of the whole field, and also indicates that there is no more than 0.07 phase error besides the margin of the picture. In summary, the presented method for ideal fringe multiplication has clear mathematics algorithm, which cannot only provide a way to extract the skeleton lines, but also separate the main frequency form the DC component in the Fourier transform.

Fig. 2 (a) and (b) are the frequency domain of the original and the multiplied image respectively (To increase the contrast, the displayed intensity $I = \ln (1 + | I_{F} |)$ is used, where $| I_{F} |$ is the amplitude of frequency, however, the intensity in center of (b) appears lower than the two neighboring lobes, this optical illusion is mainly caused by picture zooming out in _{the text and the displaying function} $I = \ln (1 + | I_{F} |)$ _{we used.}); (c) and (d) are the frequency amplitude distribution along centerline of frequency map of the original and the multiplied image respectively.

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Fig. 3 (a) The wraped pase of multiplied fringes pattern; (b) The unwrapped phase $(ϕ_{N} / N)$ distribution of the simulated image; (c) The phase error distribution.

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4. Experimental verification

4.1 Intensity calibration

Furthermore, the presented method has been realized experimentally. As illustrated in Fig. 4, the typical photoelasticity experiment [2] has been arranged. In this study, the light source is replaced by a 17-inch LCD in order to provide nearly uniform illumination, and a photoelastic disk composed by the epoxy resin with diameter of 50 mm and thickness of 5.4 mm is used. When the normal loading of 68.6 N is applied, we obtained a sparse fringe pattern with zero and one order fringes. Figure 5 shows our experimental result.

Fig. 4 The arrangement of photoelasticity experiment.

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Fig. 5 The original photoelastic fringe obtained in this experiment.

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For this kind of sparse fringe pattern, the intensity of the recording fringe image can be expressed as

I (r) = I_{1} \cdot [1 + \cos (ϕ (r))] + I_{b} + I_{r a n d o m} .

We assume that

I_{1}

and

I_{b}

in Eq. (7) are uniform (independent of the coordinate), and they are only affected by the response of the CCD camera. Moreover, we reduce the item of random noise by smooth filtering method. In order to normalize the fringe pattern (as shown in Fig. 5), we removed two 1/4 wave plates and the specimen at the first, and rotated the analyzer with unvaried interval of 10 degree while recording the intensity of image, which can be written as

I = I (r) = I_{1} \cdot [1 + \cos (2 \cdot θ_{r o t} + π)] + I_{b},

where

θ_{r o t}

denotes the rotation angle of analyzer. The intensity dependent on rotation angle is displayed by the black curve in Fig. 6(a). According to the experimental curve, we can propose a theoretic equation with the constant background

I_{b} = I_{\min}

and constant amplitude

I_{1} = (I_{\max} - I_{\min}) / 2

. Where

I_{m a x}

and

I_{\min}

are on behalf of the maximum and minimum intensity of the recording image, respectively. Based on Eq. (12), results are shown as the red curve in Fig. 6(a). It is found that there are some or little differences between the experimental and theoretic results. For eliminating these differences, the experimental intensity can be calibrated by using a polynomial fitting function with the maximum order of two. The comparison between the fitted results and the experimental results is displayed in Fig. 6(b). One can see that the fitting polynomial function adequately capture the characteristics of the calibration results.

Fig. 6 The process of calibration (a) the experimental intensituy dependent on rotation degree of analyzer, (b) the fitting function of calibration.

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4.2 fringe pattern normalization and multiplication

In order to obtain the multiplied image for the sparse photoelastic fringe, following steps have been applied. First, the fringe pattern should be calibrated by using the fitting polynomial function. Second, the background intensity $I_{b}$ can be subtracted directly. Third, the processed fringe pattern is multiplied by the method in Sec. 2. The normalized fringe patterns without and with calibration are shown in Figs. 7(a) and 7(d), respectively. Figures 7(b) and 7(c) show the 15-fold and 23-fold multiplication results of normalized fringe pattern without calibration while the Figs. 7(f) and 7(g) depict the 15-fold and 23-fold multiplication results of the one with calibration. As shown in the Figs. 7(b) and 7(c), the fringes are diseased at the left and right sides of the multiplication image, which can be understood as the contribution of the slow change in the illumination intensity. Moreover, other possible reasons for the diseased fringes after multiplication can be considered as that the fringes at the top and bottom of all the multiplication results are not normalized. The dependence of phase on the resultant difference between two principal stresses for the stress-optical law has the form as

{(σ_{1} - σ_{2})}_{r} = \frac{ϕ (r) \cdot f_{σ}}{2 π h},

where

ϕ (r)

denotes the real phase distribution of the fringe pattern, namely

ϕ_{m u l t i p l i e d} (r) / N

,

f_{σ}

is the material fringe value (equals to

18.4 K N / m

in this work), and

h

is the thickness of the photoelastic slice. Figure 8 shows the calculated stresses of the 15-fold and 23-fold multiplied fringes of the normalization results with and without calibration (along the white line in Figs. 7(b), 7(c), 7(e) and 7(f)). The blue line is on behalf of the theoretical results. We can see that the calculated results based on the multiplication fringes under calibration (dark cyan line and red line) have a good agreement with the analytical results except the margin of the sample. In comparison, the calculated results based on the multiplication fringes without calibration (orange line and dark yellow line)are much bigger than theoretical results, which can cause many errors of analysis.

Fig. 7 The multiplication result of normalized fringe pattern with and without calibration, (a) the normalized fringe without calibration (only the area interested is shown with mask); (b) the 15-fold multiplication result of (a); (c) the 23-fold multiplication result of (a); (d) the normalized fringe with calibration; (e) the 15-fold multiplication result of (d); (f) the 23-fold multiplication result of (d).

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Fig. 8 Comparison the stress distributions along the white line in the Figs. 7(b), 7(c), 7(e) and 7(f) with theory result, in the Fig. the curve 15-MC stands for 15-fold multiplication of calibrated normalization, 15-MNC stands for 15-fold multiplication of non-calibrated normalization, so do the 23MC and 23-MNC.

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5. Conclusions

An effective parameterized method for sparse fringe multiplication has been proposed. Compared with the traditional methods, the presented method is simple and efficient. Arbitrary integral-multiple fringe can be obtained, and it is convenient to extract the skeleton lines and separate the main frequency from the DC component in the Fourier transform. The present method is also suitable for the non-ideal fringe pattern, when appropriate fringe normalization is conducted. For the sparse photoelastic fringe, the stress results from the fringe multiplication normalization with and without calibration have been compared and it turns out that our results have good agreements with the theoretic ones.

Acknowledgments

This work is supported by the Fund of Natural Science Foundation of China (No. 11372121), Innovative Research Group of the National Natural Science Foundation of China (No. 11421062), the National Key Project of Scientific Instrument and Equipment Development (No. 11327802), the National Program for Special Support of Top-Notch Young Professionals. This work is also supported by the National Key Project of Magneto-Constrained Fusion Energy Development Program (No. 2013GB110001, 2013GB110002), the Program for New Central Excellent Talents in University (NCET-12-0245).

References and links

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Multiplication method for sparse interferometric fringes

Abstract

1. Introduction

2. Algorithm

3. Discussion

3.1 Skeleton line extraction

3.2 Fourier transform and phase unwrapping

4. Experimental verification

4.1 Intensity calibration

4.2 fringe pattern normalization and multiplication

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Equations (13)

Optics Express