1. Introduction

Optical measurement [1] can be generally divided into interferometric methods and triangulation methods. Holographic interferometry, shearography, speckle method, moiré interferometry and photoelasticity belong to the former while techniques such as shadow moiré and fringe projection belong to the latter. In general, interferometric methods provide a much higher accuracy than triangulation methods and are suitable for micro- and nanoscale measurement. Triangulation methods, on the other hand, have a larger measurement range and can be flexibly applied for measurement ranging from microns to kilometers.

Among the interferometric techniques, shearography [2,3] owns the valuable property of not requiring a reference laser beam. This leads to a simpler optical setup and relaxes the stringent environmental stability requirement. Another feature of shearography is that it enables direct measurement of surface strain instead of displacement. These advantages render shearography a practical technique for nondestructive testing and evaluation. With recent developments, shearography has gained wide acceptance in rubber industry for routine tire testing and aerospace industry for structural integrity evaluation. Other applications of shearography include vibration characterization, residual stress evaluation, 3-D shape measurement and leakage detection.

A key step in applying shearography for nondestructive evaluation is phase retrieval. Phase, which represents the desired physical quantity such as displacement, strain and stress, is usually modulated into sinusoidal intensity variation which is called fringes. Phase retrieval from fringes pattern with abundant speckle noise is a lasting topic in optical measurement. Phase shifting and Fourier transform are methods commonly employed for quantitative phase measurement. Accurate as it is, the phase shifting method [4,5] requires the object under measurement remains static while capturing the phase-shifted fringe patterns. This renders a huge barrier for dynamic applications. Fourier Transform technique [6], on the other hand, requires additional equipments to generate a high frequency carrier fringe, which either makes the experimental setup complicated or sometimes is not achievable. Temporal phase analysis [7] using Wavelet transform has also been proposed for phase evaluation. However, a series of images need to be captured and a temporal carrier is also required to solve the phase ambiguity problem. Recently, there has been tremendous research effort on phase retrieval based on a single fringe pattern [8–14]. These methods try to solve the inherent ambiguity problem using smoothness and minimum-variation assumptions. Artful routing has to be implemented to avoid erroneous phase propagation, which makes the algorithm complex and less efficient. In the light of the aforementioned limitations, a convenient phase retrieval method without phase ambiguity problem would be desirable for dynamic shearographic phase measurement.

In our previous research [15,16], the authors had proposed a clustering method to solve the phase ambiguity problem and extract phase information from one single deformed speckle pattern. The clustering method works effectively for fringe patterns with low and medium fringe density, delivering accuracy comparable with phase shifting method. When dealing with high density fringe, however, the clustering method may generates noisy wrapped phase map with abundant phase residues due to environmental noise and the inherent information insufficiency with one single specklegram. This makes the subsequent phase unwrapping [17,18] a difficult or impossible task. In this paper, we propose a novel Fourier phase filtering method to analyze the fringe frequency and reconstruct a residue-free wrapped phase map which can be easily unwrapped. Together, the clustering method and the Fourier filtering b method constitute a complete solution for dynamic phase evaluation.

2. The clustering phase extraction method

2.1. Theoretical background

In holography and shearography, a general speckle pattern resulted from random interference of laser can be expressed as:

where $I(x,y)$is the intensity of the speckle, $a(x,y)$ and $b(x,y)$ are the background and fringe modulation terms respectively, and $\phi (x,y)$ is initial random speckle phase. For most applications in shearography, the deformation is in the range of microns and it is reasonable to assume that background and modulation are constant during the deformation. Thus the speckle intensity $I^{\prime }(x,y)$ after a small deformation can be expressed as:

where $\mathrm{\Delta }(x,y)$ is the phase change related to the deformation, which is the ultimate aim for any phase retrieval methods. From Eq. (2), the desired phase change Δ is readily determined as:

where a,and φ can be determined by four-step phase shifting technique before the object starts to deform, and n is an integer.

The phase term Δ in Eq. (3) still has two uncertainties to determine. One is the phase ambiguity problem associated with the arccosine function. The other is the phase unwrapping problem associated with the $2n\pi$ uncertainty. Phase unwrapping for low quality wrapped phase map with plenty of phase residues is very complicated and has attracted intensive research effort [17,18]. However, if the wrapped phase map is free of residues, the phase unwrapping process is straight forward. In this section, our main concern is to remove the sign ambiguity in Eq. (3). In next section, we will introduce a Fourier phase filtering method to remove possible phase residues to make phase unwrapping an easy task.

As we know, shearographic phase represents continuous deformation without abrupt jumps, thus it would be reasonable to assume that the phase values within a small area are close to each other. In Eq. (3), there are two candidate phase values for each pixel. To pick up the correct phase value, an area of 3x3 pixels surrounding the specified pixel is chosen and the corresponding 18 candidate phase values determined from Eq. (3) are queued as $x_0,x_2\cdots x_{17}$in ascending sequence within the range of $-\pi$ to π (if some phase values are out of this range, adjust n in Eq. (3) to make them in range) as shown in Fig. 1 . Within these 18 phase values (represented by hollow dots), it can be observed that 9 correct phase data cluster together near the arrow position, while the other 9 wrong phase values distribute randomly. This observation inspires an algorithm to determine the clustering centre to represent the correct wrapped phase value for the specific pixel.

 

Fig. 1 Demonstration of clustering effect of wrapped phase values within an area of 3x3 pixels.

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For this purpose, a wrapped phase distance function $d(x_m,x_n)$ is first defined as:

The physical meaning of this distance function is well illustrated in Fig. 1 where the circle from$-\pi$ to π indicates the periodic feature of sinusoidal functions and the phase distances are measured as the shortest arc length. In this specific definition, the distance between $x_1=-0.85\pi$ and $x_{16}=0.75\pi$ (see Fig. (1)) is not $1.6\pi$ but $0.4\pi$. This is reasonable since $x_1$ is quite close to $x_{16}$ if the $2\pi$ jump which has not yet been settled by a phase unwrapping algorithm, is ignored. By using this wrapped phase distance function, a parameter called$SumDist(x_k)$, which represents the accumulated distance of a point $x_k$ with the surrounding 8 points (4 to the left and 4 to the right), is calculated for each point as:

In this calculation, if index $k+i$ exceeds 17 or is less than 0, it is subtracted or added by 18 to round it within 0 to 17. It is evident that if point $x_k$ resides within the cluster, the accumulated distance $SumDist(x_k)$ will be much smaller than points outside the cluster. Thus by looking for a smallest accumulated distance, the clustering centre $x_c$ is found as:

The clustering centre $x_c$ determined by Eq. (6) is an approximation of the wrapped phase in the center of the 3x3 pixels. It should be noted that the wrapped phase thus determined presents no phase ambiguity problem since the clustering algorithm has choose the correct phase value from Eq. (3). However, it should also be noted that a predetermined initial random phase φ without ambiguity is a necessity for the success of the clustering method. If φ is subject to phase ambiguity problem (i.e. φ is determined from arccosine function instead of phase shifting method), then the candidate phase values determined from Eq. (3) will have two clustering centers, making the resultant wrapped phase subject to phase ambiguity.

2.2. Comparison with phase shifting method

Shearographic experiment data have been used to verify the effectiveness of the proposed clustering phase extraction method. Figure 2(a) shows a typical shearographic fringe pattern which depicts the out-of-plane displacement derivative of a fully clamped rectangular plate subjected to a central loading. Figure 2(b) shows the wrapped phase map determined using the proposed clustering approach. It can be seen that the wrapped phase map agree well with fringe pattern without any phase ambiguity.

 

Fig. 2 Typical shearographic fringe pattern of a central loaded rectangular plate (a) and the corresponding wrapped phase obtained by the clustering approach (b)

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Quantitative comparison between the proposed clustering method and standard four-step phase shifting method has also been conducted. Both wrapped phase maps from clustering method and phase shifting method are unwrapped and the horizontal middle sections are plotted together as shown in Fig. 3 . It can be seen that the two cross section profiles agree pretty well with each other. Quantitative comparison shows that the maximum discrepancy is less than 0.8%. Thus it is confirmed that the proposed clustering method can effectively retrieve the phase map with accuracy similar to phase shifting method. While the proposed clustering method obtains similar wrapped phase map as standard four-step phase-shifting, a distinct advantage is that only one specklegram is required at each deformed stage. This advantage renders the proposed clustering method an ideal solution for continuous deformation measurement.

 

Fig. 3 Comparison of unwrapped phase maps of the clustering method and four-step phase shifting at the horizontal mid-section.

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3. The Fourier phase filtering method

The quality of wrapped phase maps depends on several factors such as environmental noise, signal to noise ratio and resolution of the imaging element (normally CCD cameras), redundancy of information available, and the fringe density. Normally the initial noisy wrapped phase maps determined from either clustering method or phase shifting method can be improved by a fringe averaging method, which first converts the wrapped phase to both sine and cosine fringe maps, then averages the fringe maps on a small area (i.e. 3x3 pixels), and finally reconstructs a filtered wrapped phase map using the averaged sine and cosine fringe maps. However, for areas with rapid phase change, the clustering effect is not distinct and the clustering method may result in incorrect phase values which lead to severe phase residues after successive fringe averaging. Figure 4(a) shows an initial wrapped phase map determined by the clustering method. This wrapped phase map is subject to large amount of noise and difficult to unwrap. Figure 4(b) shows a clean wrapped phase map after applying the fringe averaging method for noise removal. However, in the central area indicated by a square and highlighted in Fig. 4(c), it can be seen that there are plenty of phase residues remain. This poses much difficulty for the subsequent phase unwrapping process. Figure 4(d) shows the phase unwrapping result using Raster’s unwrapping algorithm [17]. It can be seen that erroneous phase values propagate and damage the whole unwrapped phase map.

 

Fig. 4 (a) Initial wrapped phase map determined from the clustering method; (b) Wrapped phase map after 50 times fringe averaging; (c) The central area of (b) showing plenty of phase residues; (d) unwrapped phase map of (b) using Raster unwrapping algorithm.

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The phase residues in Fig. 4(c) cannot be removed by either fringe averaging or fringe median filtering. In fact, successive averaging and median filtering will only make the situation worse by enlarging the erroneous area. Noted in Fig. 4(c) that the fringe frequency is much lower than the residue frequency, we propose a frequency based Fourier filtering method to remove the unwanted phase residues while conserving the desired fringe information in the wrapped phase.

The wrapped phase map in Fig. 4(a) is converted to fringe patterns using both cosine and sine operations. In this process, the high frequency $2\pi$ phase jumps in wrapped phase map are replaced by low frequency continuous fringes due to the periodic feature of cosine and sine functions. The resultant fringe patterns $f(x,y)$ and $g(x,y)$can be expressed as:

Where $\cos (\mathrm{\Delta }(x,y))$and $\sin (\mathrm{\Delta }(x,y))$ corresponding to a low frequency fringe containing the deformation information, and $r(x,y),r^{\prime }(x,y)$ corresponding to the high frequency phase residues.

A typical rewrapped fringe pattern is shown in Fig. 5(a) . When applying a two-dimensional Fourier transformation to the fringe patterns, we have:

 

Fig. 5 (a) Cosine fringe map obtained from the wrapped phase map determined from clustering method; (b) Fourier spectrum of (a); (c) low-pass filter determined by thresholding; (d) reconstructed cosine fringe map after Fourier filtering; (e) reconstructed wrapped phase map after Fourier phase filtering; (f) unwrapped phase map by Raster unwrapping algorithm.

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where$\delta (u,v)$is a two-dimensional Dirac delta function, ${\mathrm{\Delta }}_x,{\mathrm{\Delta }}_y$are the fringe frequency components in $x,y$directions respectively, and$R(u,v),R^{\prime }(u,v)$are the Fourier transformation of high frequency residues. For a typical rewrapped shearographic fringe pattern shown in Fig. 5(a), the fringe frequencies ${\mathrm{\Delta }}_x,{\mathrm{\Delta }}_y$are confined to a small range and can be separated from the high frequency residues. Figure 5(b) shows the Fourier spectrum (complex modulus of$F(u,v)$) of Fig. 5(a). The bright central ellipse corresponds to the low frequency fringe pattern while the residue spectrum distribute mostly at higher frequency area. Since the fringe pattern dominates in Fig. 5(a), the Fourier energy for fringe pattern is much larger than that for the residues. Thus the central bright part can be easily separated using feature extraction methods or thresholding method. In this study, a simple thresholding operation is applied to Fig. 5(b) with a default threshold value of 10/255 of the maximum power. Then the two binary images resulted from cosine and sine spectrums are combined. After that a dilation operation is applied and the resultant frequency filter is shown in Fig. 5(c). It can be seen that this low-pass filter contains the central bright spectrum, but seems a little bit larger than necessary and can be further improved. However, the filtering effect using this filter is quite satisfying already. By applying this filter and subsequently an inverse Fourier transform to Eq. (8), clean cosine and sine fringe patterns are reconstructed as follows:

where $ifft\{•\}$represent the inverse Fourier transform, ∗is pixel-wise product operation, and $flt(u,v)$ is the low-pass filter shown in Fig. 5(c). A typical reconstructed fringe pattern using Eq. (9) is shown in Fig. 5(d). It can be observed that most of the noise has been removed and the original noisy fringe pattern has been smoothed. Based on both the reconstructed cosine and sine fringe maps, a new wrapped phase map is reconstructed as follows:

where the pixel-wise function $\arctan 2(y,x)$retrieves a phase angle in the range of $[-\pi ,\pi )$from complex number $x+yi$. Figure 5(e) shows the reconstructed wrapped phase map after the whole Fourier filtering process. For clearer viewing, Fig. 5(e) is re-plotted in Fig. 6 . It can be seen that a noise-free wrapped phase map has been obtained. Any simple phase unwrapping algorithms can now be applied to easily obtain the unwrapped phase map. The result using Raster unwrapping algorithm is shown in Fig. 5(f).

 

Fig. 6 Reconstructed wrapped phase map by the proposed clustering method and Fourier phase filtering method.

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It should be noted that phase residues is not a specific problem for clustering method. For standard phase shifting method where abundant information are available, the resultant phase maps may also be subject to phase residues in the cases of high phase gradient or large amount of environmental noise. In such cases, the proposed Fourier phase filtering method is an ideal tool to remove phase residues in wrapped phase map, and making the subsequent phase unwrapping process simple. It should also be noted that this Fourier filtering method is quite tolerant to change in parameters. For the example in Fig. 5, thresholding values ranging from 8/255 to 15/255 of the maximum power all result in acceptable filtering effect. It is also possible to design a more precise adaptive filter using feature extraction method based on the sudden changing of spectrum power between the high-energy central ellipse and the low-energy surrounding area. This more precise adaptive filter would be desirable for wrapped phase map with worse quality.

Different from previous works where Fourier transform is applied on fringe patterns for preprocessing or wrapped phase maps for post-processing [19], or applied on a complex field reconstructed from phase shifted fringes [20], our filtering method is based on pure wrapped phase maps where the background and fringe modulation have been eliminated. We also applied image processing techniques for constructing an ideal low pass filter, thus the method is more robust against the influence of speckle noise. More powerful speckle noise filtering techniques employing scale-space filtering [21], windowed Fourier transform [22] and windowed Fourier frames [23] have also been proposed for local frequency analysis to deliver better quantitative phase measurement. These methods require much more computation and thus efficient Graphics Processing Unit has recently been employed [24] to improve the computation speed. However for normal dynamic applications, the proposed combination of clustering method and Fourier wrapped phase filtering technique would be adequate to deliver efficient and accurate measurement.

In our experimental study using shearographic setup on tripod, the noise level would be larger than interferometric systems placed on vibration isolated optical table. Under such situation, the maximum fringe density workable for our proposed combined method is about 12 pixels per fringe. When the deformation becomes larger and larger, the fringe would be too dense to accurately extract the phase. It would be thus desirable to capture intermediate speckle patterns to reduce the fringe density, and then combine the incremental phase change to deliver the final phase map. We will explain more details and give more results to handle such situations in a forthcoming paper.

4. Dynamic phase measurement experiment

The combination of the clustering method and the Fourier filtering method provide an ideal solution to dynamic phase measurement where the object starts non-repeatable continuous deformation and only one image can be captured at each deformed stage. Figure 7 shows the experiment results of shearographic nondestructive evaluation. A pressurized steel tube is subject to quick pressure variation and shearography is employed for quantitative crack localization and evaluation. Figure 7(a)-(c) show the shearographic fringe pattern at different instants, and Fig. 7(d)-(f) show the corresponding quantitative deformation obtained using clustering method and Fourier filtering method based on a single deformed specklegram at each stage. Noted that in our previous publication [15], similar results have been reported, but at that time only clustering method is employed and the resultant wrapped phase map is subject to abundant phase residues which make it difficult to unwrap for quantitative evaluation. In Fig. 7(d)-(f), the wrapped phase maps are free of residues and quantitative results can been easily obtained, which can be further used to determine the depth and length of the crack.

 

Fig. 7 Fringe pattern depicting a vertical crack of a steel tube subject to continuous pressurization at different time (a), (b), (c) and the corresponding wrapped phase map obtained from the clustering method and Fourier phase filtering method (d), (e), (f).

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5. Conclusions

A clustering method has been proposed for ambiguity-free phase extraction from one single specklegram. A Fourier phase filtering method is further proposed for efficient removal of phase residues to facilitate phase unwrapping process. The integrated method opens a door for simple and reliable dynamic phase measurement. With further refinement, the proposed methods will have great potential for practical applications, especially in aircraft and automobile industry for vibration characterization.