Extraction of phase field from a single contoured correlation fringe pattern of ESPI

Qifeng Yu; Sihua Fu; Xia Yang; Xiangyi Sun; Xiaolin Liu

doi:10.1364/OPEX.12.000075

1. Introduction

Electronic speckle pattern of interferometry (ESPI) is a quickly developing and widely used metrology technique due to its simplicity and adaptability in field-site. However the speckle fringe patterns of ESPI are inherently full of speckle noise, and their signal to noise ratio is near unity [1]. In a general term speckle noise information has a large bandwidth and overlaps fringe spatial frequencies in many practical cases. This feature makes the post-processing of the phase extraction difficult for the ESPI fringe patterns. Much effort has been made to suppress the speckle noise [1–3].

In order to extract the information from speckle patterns, subtraction between two original speckle patterns with a phase difference is the most common way to generate a speckle fringe pattern of subtraction, as shown in the upper left part of Fig. 1, which is full of high-spatial-frequency and high contrast speckle noise. Other calculation methods, such as addition or multiplication between the two original speckle patterns with a phase difference, can also be used to get fringe patterns that have similar high-spatial-frequency speckle noise.

Fig.1. Comparison of three kind resultant fringe patterns by subtraction, correlation with rectangular windows, contoured correlation, respectively.

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Fig. 2. The fringe contour windows.

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There is an alternative way to obtain fringe patterns from original speckle patterns: the direct correlation between two original speckle patterns with a phase difference is performed within rectangular windows and the correlation coefficients generate fringe patterns [4]. This direct correlation method has an excellent advantage that it automatically normalizes the fringes to a great extent. The resultant fringe patterns of the method have good uniformed contrast over the whole field, which is independent from the local average illumination intensity. This advantage stems from the fact that the correlation operation measures the quality of the correlation between the two original speckle patterns on the window and does not depend on the local absolute value of the intensities.

Though the direct correlation method with rectangular windows suppresses the high-spatial-frequency and high contrast speckle noise, the resultant fringe patterns have some new large size speckles with low contrast, as shown in [4] and in the upper right part of Fig. 1. The direct correlation fringe patterns will benefit the skeleton method and maybe the phase shift methods since they provide improved fringes with high contrast. However the new large size speckle noise on the fringe patterns spoils the fine phase distribution. The gray levels of the resultant fringes, therefore, cannot correspond directly the phase field. The main source of the large size speckle noise may come from the rectangular windows because the real phase is not constant inside this window. The rectangular windows give one kind of average phase and some uncertain error is induced due to the random speckle noise inside the window.

In this paper, we propose a new method to derive the fringe pattern with smooth and consistent fringes without speckle noise from the two original speckle patterns, as shown in the lower part of Fig. 1. The main difference between the direct correlation method and proposed method is that we first establish fringe contour windows coinciding with fringe contours of equal-phase from the corresponding subtraction speckle fringe pattern and then the correlation is performed only on these contour windows instead of rectangular windows. We name the proposed method as the contoured correlation fringe pattern method. The new fringe patterns have smooth and normalized fringes with the same phase field as that of subtraction speckle fringe patterns. The contoured correlation fringe pattern method, therefore, removes the speckle noise completely and the gray levels of the resultant fringe patterns correspond directly to the phase field. With the new method, the ESPI fringe patterns have better quality even than that of moiré and hologram, which is unimaginable with traditional ESPI methods.

Then we improve the single-image phase shifting method to derive the full phase field directly [5] from the new fringe pattern, which is not possible for common ESPI fringe pattern. Commonly used phase shifting methods require at least three speckle fringe patterns with certain phase differences. The single-phase-step method can use two fringe patterns with π/2 phase-shift under the condition of nearly uniformed background [6]. Without this condition a double-image phase shifting method can be applied after the two fringe patterns are transformed as noise-free normalized fringe patterns [5,7]. The proposed method in this paper is a breakthrough to obtain the full phase field from a single ESPI speckle fringe pattern because the method eliminates the requirement of multi-fringe-patterns with phase differences for ESPI.

2. The main principle of the method

The main processing steps of the proposed method are as follows.

1. The two original speckle patterns with a phase difference are subtracted from each other to form a subtraction speckle fringe pattern and then the speckle fringe orientation map (SFOM) is determined from the subtraction speckle fringe pattern [3,8].

2. Construction of fringe contour windows that coincide the fringe contours of the speckle fringe patterns, which is similar but different to the curved surface windows in [3].

3. Performing the correlation operation between the two original speckle patterns only on the contour windows. The correlation coefficients generate the smooth and normalized fringe pattern with the same phase field as that of subtraction speckle fringe patterns.

4. Performing a 90-degree digital phase-shifting transform to derive a new normalized fringe pattern with sine form instead of cosine form [5].

5. Making the arctangent transformation between the sine and cosine fringe patterns to derive the saw-tooth phase field (the principle value phase field).

6. Unwrapping the saw-tooth phase field by the unwrapping algorithms that are the same as those for common phase shifting methods.

The key step for the method is that the correlation is performed only within the fringe contour windows for each point.

3. Correlation between two original speckle patterns

In ESPI, two original speckle patterns with a phase difference Δ θ (x, y) are first acquired by a digital imaging system. The intensities of the two speckle patterns can be represented as follows:

f_{1} (x, y) = I_{1} (x, y) + I_{2} (x, y) + 2 \sqrt{I_{1} I_{2}} \cos [Ψ_{1} (x, y) - Ψ_{2} (x, y)]

f_{2} (x, y) = I_{1} (x, y) + I_{2} (x, y) + 2 \sqrt{I_{1} I_{2}} \cos [Ψ_{1} (x, y) - Ψ_{2} (x, y) + Δ θ (x, y)]

where I₁ (x, y) and I₂ (x, y) are the intensities of the two coherent lights or one object’s scattered light and one reference beam, respectively, depending on the measurement of in-plane or off-plane displacement. Ψ₁(x, y) and Ψ₂(x, y) are the phases of random speckles, and Δθ (x, y) is the phase difference between the two speckle patterns resulting from some displacement. The common correlation formula is:

C (x, y) = \frac{{< (f_{1} - {< f_{1} >}_{m \times n}) (f_{2} - {< f_{2} >}_{m \times n}) >}_{m \times n}}{{[{< {(f_{1} - {< f_{1} >}_{m \times n})}^{2} >}_{m \times n}]}^{\frac{1}{2}} {[{< {(f_{2} - {< f_{2} >}_{m \times n})}^{2} >}_{m \times n}]}^{\frac{1}{2}}}

where <f₁>,<f₂ > are the average values of f₁, f₂ within the windows, respectively. When the value of Δθ (x, y) equals to 2 _nπ, then f₁ (x, y)=f₂ (x, y) at the point (x, y), so the correlation coefficients, C(x, y), equal close to 1, and bright fringes are obtained. When the Δθ (x, y) equals to (2n+1)π that means no correlation, C(x, y) has the minimum value and forms dark fringes. When the Δθ (x, y) equals to other values, the C(x, y) values are between the bright and dark fringes. The value of Eq. (3) theoretically ranges from -1 to 1 but for our special case of speckle patterns it ranges from 0 to 1. These descriptions can be proofed further with the equation deductions as follows [8]:

We denote β=Ψ₁-Ψ₂. According to the speckle statistic theory, it is assumed that

{< I^{2} >}_{m \times n} = 2 {< I >}_{m \times n}^{2}

where I is I₁ (x, y) or I₂ (x, y) in Eq.(1) and (2) and

{< \cos β >}_{m \times n} = {< \cos (β + Δ θ) >}_{m \times n} = 0

Since the I₁, I₂ and β are independent from each other, their product’s average-value equals to their average-value’s product.

Let <I ₁>_m×n=<I ₂>_m×n=<I>_m×n, and under the condition that the phase term of Δθ (x, y) is a constant over the whole operating window of (m x n), the following relations can be yielded from Eq. (1) and Eq. (2), respectively:

< f_{1} > = 2 {< I >}_{m \times n}, < f_{2} > = 2 {< I >}_{m \times n}

{< f_{1} f_{2} >}_{m \times n} = < {(I_{1} + I_{2})}^{2} + 2 (I_{1} + I_{2}) \sqrt{I_{1} I_{2}} \cos β

< (f_{1} - < f_{1} >_{m \times n}) (f_{2} - < f_{2} >_{m \times n}) >_{m \times n} = < f_{1} f_{2} >_{m \times n} - < f_{1} >_{m \times n} < f_{2} >_{m \times n}

< {(f_{1} - < f_{1} >_{m \times n})}^{2} >_{m \times n} = < f_{1}^{2} - 2 f_{1} < f_{1} >_{m \times n} + < f_{1} >_{m \times n}^{2} >_{m \times n} = 4 < I >_{m \times n}^{2}

< {(f_{2} - < f_{2} >_{m \times n})}^{2} >_{m \times n} = < f_{2}^{2} - 2 f_{2} < f_{2} >_{m \times n} + < f_{2} >_{m \times n}^{2} >_{m \times n} = 4 < I >_{m \times n}^{2}

Substituting above equations into Eq.(3) yields the relationship between the correlation coefficients and the phase field as follows▫

C = \frac{2 < I >_{m \times n}^{2} + 2 < I >_{m \times n}^{2} \cos Δ θ}{4 < I >_{m \times n}^{2}} = (\cos Δ θ + 1) ⁄ 2

This relation indicates that the correlation fringe pattern is a pure cosine fringe pattern that is the same as normal interferometric fringe patterns. Please notice that during the above deduction, the phase term of Δθ (x, y) is assumed as constant within the correlation windows. When the correlation is performed within rectangular windows, this assumption is no valid and certain error will be induced. The larger the phase variation within the window is, the larger the error is for the resultant fringe pattern. The larger the window size is, therefore, the larger the error is. This should be the main reason why the direct correlation fringe patterns with rectangular windows have some new low contrast large size speckle noise, as shown in the upper right part of Fig.1. In order to receive an accurate fringe pattern of Eq. (11), we propose here that the correlation must be performed only on the fringe contour windows on which the phase term is constant.

4. Determination of fringe contour windows

As described previously, for making the correlation of Eq. (3) accurately, operating windows with a constant phase are extremely important. The fringe contour windows of equal phase coincide the local contours of local intensity or phase of the fringe pattern, which is just the window with a constant phase. The principle and method for determination of the contour windows are similar to the curved surface windows proposed by the author [3,9]. But the name of fringe contour windows is more meaningful than that of the curved surface window.

First the speckle fringe orientation map (SFOM) is constructed from a common speckle fringe pattern with subtraction. The least square fit with a polynomial is performed within a sliding square window for the speckle fringe pattern. The direction with the zero direction-derivative of the fitted polynomial is defined as the fringe direction, which generates the SFOM. Based on the SFOM, we can acquire the fringe contour curve segment by tracking the contour pixel by pixel in both the local fringe directions from the current point, as shown in Fig. 2. The contour curve of equal phase is taken as the central curve of the contour window. Theoretically, the phase values only on this contour are constant. Practically, to suppress the random noise efficiently with relatively more data, we make the window with a width more than one pixel. To maintain good accuracy, the window width should be taken much less than the window’s length, which is different with the curved surface windows [3,9].

5. Performing the correlation on the contour windows

Then the correlation of Eq. (3) between the two original speckle patterns of Eq. (1) (2) is performed within the contour windows for every point. The correlation coefficients generate an excellent fringe pattern with the same phase field as that of speckle fringe patterns with subtraction, but in a reversed pattern with 180° shift.

The correlation coefficient measures the similarity of the two original speckle patterns in the window regardless of the speckle profile, the background and contrast of the patterns. The resultant correlation coefficients, therefore, result in smooth fringes that have consistent fringe amplitude and background, which is more attractive to us. The new fringe patterns have the same feature as that of noise-free normalized fringe patterns with the equation as follows [10]:

I (x, y) = I_{0 c} + I_{1 c} \cos (φ (x, y))

where I_0c, I_1c are the constant background and constant fringe amplitude, respectively, φ(x, y) is the same as Δθ (x, y).

6. Single-image phase shifting method

For above normalized fringe patterns the gray levels, the intensity, correspond directly to the phase of φ(x, y) in the principle values of cosine. It is applicable, therefore, to derive the phase field from Eq. (12) by the arc-cosine function directly with a priori or additional information of phase sign. But the cosine is an even function and the principle value of the arc-cosine ranges from 0-π instead of 0-2π for arc-tan function. In addition the unwrapping algorithm for arc-cosine patterns is quite different from common unwrapping algorithms for arc-tan patterns. In order to make a 2π jump saw-tooth phase pattern that can be unwrapped directly by a common unwrapping algorithm and program for common phase-shifting techniques, we developed and use the single-image shifting method that is briefly described as follows [5].

As we know, the signs of the phase for fringe patterns are lost for a single fringe pattern owing to the even function of the cosine and this phase sign can be retrieved only under the condition that prior or additional information about the phase situation is provided. Therefore, to derive an unambiguous phase field with the single-image phase shifting method, a phase-sign map should be analyzed and derived beforehand. Based on the contoured correlation fringe pattern of Eq. (12) and the phase sign map, we can make a π/2 digital phase-shifting transform on the pattern of Eq. (12) to derive the sine fringe pattern [5]. The corresponding sine fringe pattern is constructed from the fringe pattern of Eq. (12):

J (x, y) = I_{0 c} + \sqrt{I_{1 c}^{2} - {(I - I_{0 c})}^{2}} = I_{0 c} \times I_{1 c} \sin φ (x, y)

Then the sign ambiguity in Eq.(13) is removed according to the equation:

\frac{\partial \cos φ (x, y)}{\partial x} = - \sin φ (x, y) \frac{\partial φ (x, y)}{\partial x}

where the sign of $\frac{\partial \cos φ (x, y)}{\partial x}$ can be obtained from the derivative-sign binary image [10]. Thus the sine fringes shifted digitally with π/2 from the original cosine fringe pattern are obtained:

J (x, y) = I_{oc} + I_{1 c} \sin (φ (x, y))

Consequently, the saw-tooth phase pattern can be derived by the arctangent transform:

φ (x, y) = \arctan (\frac{J (x, y) - I_{0 c}}{I (x, y) - I_{0 c}}) = \arctan (\frac{\sin [φ (x, y)]}{\cos [φ (x, y)]})

This saw-tooth phase pattern is the same as that of phase-shifting methods that require multi fringe patterns. Therefore the wrapped phase pattern of Eq. (16) can be unwrapped by common unwrapping algorithms [11]. Until now, the full phase field is extracted from the single speckle fringe pattern containing two original speckle images of Eqs. (1) and (2).

7. Experimental results and discussion

7.1 Experimental Rectification of the method

The upper part of Fig.3 is a speckle fringe pattern with subtraction of a specimen being loaded near to the center. The resultant contoured correlation fringe pattern with a window size of 67×7 pixels is shown in the lower part of Fig. 3. We can see that the contoured correlation fringe pattern is smooth and has no speckle noise at all, neither the high-spatial-frequency speckle noise in subtraction fringe patterns, nor low contrast new large-size speckle noise in direct correlation speckle fringe patterns with rectangular windows. Fig. 4 is the intensity distribution of an intersection of the correlation fringe pattern of Fig. 3, which shows that the intensities and the amplitudes of the fringes are quite uniform.

Fig. 3. Experimental results. The upper part is a speckle fringe pattern with subtraction and the lower part is the contoured correlation fringe pattern with a window size of 67×7 pixels for the same original speckle patterns.

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Fig. 4. The intensity distribution of an intersection of the contoured correlation fringe pattern of Fig. 3.

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The upper part of Fig.5 is the resultant saw-tooth phase field from the lower part of Fig.3 by the single-image phase shifting method. The lower part of Fig.5 is the unwrapped phase field by the robust unwrapping algorithm [11]. Fig.6 and Fig.7 are the phase distributions of an intersection of the upper part and the lower part of Fig.5, respectively, which proves that the full phase field can be extract directly from a single speckle fringe pattern of Fig.3.

Fig.8 shows another results for a speckle fringe pattern with varying fringe density and open fringe contours. The upper left part, upper right part and lower parts of Fig.8 are the resultant fringe patterns by subtraction, direct correlation with rectangular windows of 17×17 pixels and the contoured correlation with window size of 67×5 pixels, respectively. This result indicates that the proposed method is suitable for the complicated applications of the fringe patterns with varying fringe density, open fringe contours and different fringe orientation.

Fig. 5. The phase field results. The upper part is the saw-tooth phase field from the lower part of Fig. 3 and the lower part is its unwrapped phase field result.

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Fig. 6. The saw-tooth phase distribution of an intersection of the upper part of Fig. 5.

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Fig. 7. The full phase field distribution of an intersection of the lower part of Fig. 5.

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Fig. 8. Experimental results. The upper left part, upper right part and lower parts are the resultant fringe patterns by subtraction, direct correlation with rectangular windows of 17×17 pixels and the contour correlation with window size of 67×5 pixels, respectively.

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The correlation between f₁(x, y) and f ₂(x, y) of Eqs.(1) (2) is perform within the fringe contour windows that have the equal-Δθ (x, y). Because β=Ψ₁(x, y)-Ψ₂(x, y) in Eqs. (1) (2) is changing randomly, the difference between cos β and cos(β+Δθ) is changing randomly too. Within the fringe contour windows, on one hand Δθ(x, y) is accumulated and enhanced and on the other hand the random components are suppressed and removed by the correlation averaging. Thus the smooth fringe pattern with phase Δθ(x, y) is constructed through the contoured correlation. Therefore, the fringe contour windows instead of rectangular windows are the key technique to derive the smooth correlation fringe patterns from speckle patterns.

7.2 Influence of window sizes

The size of the fringe contour windows is important during the practical processing. The window with too large size may blur the phase field and that with too small size may not give smooth enough fringes. The results of the effects of window sizes are shown in Figs. 9(a)–(d) where the upper and lower parts show the results by direct correlation and contoured correlation, respectively. The window sizes of Fig. 9(a) to (d) are 3×3 and 9×1, 7×7 and 15×3, 13×13 and 55×3, and 19×19 and 73×5 pixels, respectively, for the upper and lower parts of each pattern. The pixel numbers within both kind windows, respectively, are similar to each other for each image for Fig. 9(a) to (d). Figures 9(a) and (b) show that a small window size gives a noisy fringe pattern where speckle can not be suppressed efficiently. Figure 9(d) and the lower part of Fig. 8 give the best results of the smooth fringe pattern. Further increase of the window sizes will not increase the quality of the resultant fringe pattern significantly.

Fig. 9. The resultant patterns of the effects of window sizes. The upper and lower parts are the results by direct correlation and contoured correlation, respectively The window sizes are 3×3 and 9×1 (a), 7×7 and 15×3(b), 13×13 and 55×3(c), 19×19 and 73×5 pixels(d), respectively, for the upper and lower parts.

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Though increase of window sizes will cost more computation time, the main concerning factor is quality instead of computation since the modern computer hardware provides sufficient power for the calculation that costs only some seconds.

As we can see from the results, the increase of window sizes will not influence the fringe contrast considerably for our method. The increase of window size for rectangular windows, however, will decrease the fringe contrast significantly [4] even blur the fringes especially for high density fringes, as shown in Fig. 9(c). This demonstrates again that the contour windows have no so much average effect as that of rectangular windows and will not give significant distortion to the phase distribution.

The central area of the circular fringes may have some singularity for fringe directions. However, the average effect of the neighboring area for the center may give roughly correct answer to the final results.

As described previously, in order to keep the phase constant on the window, the width of the contour window should be small enough compare with the window length. Further research work will be done to find adaptive optimal window sizes for different fringe cases and integrate the fringe spaces into the window width.

8. Conclusions

To extract phase field from a single speckle fringe pattern of ESPI is still a difficult task and is an unsolved problem due to its feature of speckle noise, though the phase shifting methods are successful to extract the phase field from multi-speckle fringe patterns. We propose a novel method to extract the full phase field directly from a single speckle fringe pattern in this paper. With the method, the contoured correlation fringe pattern is established by performing correlation between two original speckle patterns within fringe contour windows. The new fringe pattern is smooth and speckle-noise-free. The most important for the fringe patterns is that the fringe amplitude and background are normalized automatically and the ideal fringe patterns whose gray levels correspond directly to the phase field are derived. Based on this fringe pattern and the additional sign information of the phase-field, we can extract the full phase field by the single-image phase shifting method. The proposed method enlarges considerably the techniques and applications of ESPI, especially for the dynamic tests that only a single speckle fringe pattern can be obtained.

Acknowledgments

This research was supported by Natural Science Foundation of China with contract number of 19872077. We thank the topical editor and a reviewer for their valuable suggestions for improvement of the paper.

References and links

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Extraction of phase field from a single contoured correlation fringe pattern of ESPI

Abstract

1. Introduction

2. The main principle of the method

3. Correlation between two original speckle patterns

4. Determination of fringe contour windows

5. Performing the correlation on the contour windows

6. Single-image phase shifting method

7. Experimental results and discussion

7.1 Experimental Rectification of the method

7.2 Influence of window sizes

8. Conclusions

Acknowledgments

References and links

Cited By

Figures (9)

Equations (21)

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