1. Introduction

Electronic speckle pattern interferometry (ESPI) is a well-known, whole-field technique for measurement. Accurate extraction of phase value is of fundamental importance because it is related with a physical quantity. The phase-shifting technique is one of the most commonly used methods for phase extraction because of its high accuracy [1,2]. However, owing to speckle decorrelation effects, the resultant phase fringe patterns are strongly affected by noise. Moreover, the easily blurred high density regions in fringe patterns increase the difficulty of removing the noise while preserving the features [3]. In order to make the phase unwrapping process easier, the noise presented in wrapped phase patterns must somehow be dealt with.

Filtering methods along the fringe orientation have been demonstrated to be a powerful tool for filtering in ESPI fringe patterns, especially for patterns with high fringe density. In the past few years some works have been developed for directional filtering such as the contoured window filter [4] and our tangent fitting filter [5], in which the small filtering window is designed along the fringe orientation, pixel by pixel, and a filtering method is performed in this directional filtering window. Recently, some efforts have been made toward the directional filtering methods without having to laboriously establish the small filtering window along the fringe orientation. In our former works [6,7], we have proposed the second-order oriented partial differential equations (PDEs) filtering models and the oriented-couple PDE models respectively, which make the diffusion only along fringe orientation. In the oriented PDEs-based filtering methods, filtering along fringe orientation for the entire image is simply realized through solving the oriented PDEs numerically. However, when the oriented PDEs-based filtering methods are applied to the phase patterns that are seriously noisy, they usually require many iterations [7]. Furthermore, single-direction diffusion is not sufficient in the regions with low fringe density [3]. For solving this problem, Wang et al. [3] improved the second-order oriented PDEs filtering models, and proposed an application of coherence-enhancing diffusion to fringe-pattern denoising. It smoothed a fringe pattern along directions both parallel and perpendicular to fringe orientation with suitable diffusion speeds, and could give the better results in both high and low fringe density. However, for implementing the diffusion, the distance map is needed, which seem to be inconvenient. J. Villa et al. [8] used a different and powerful mathematical tool for directional filtering. They constructed the oriented, regularized quadratic-cost function. The minimization of this cost function is equivalent to smoothing the image only along the fringe’s tangent direction, which is easily implemented using iterative methods.

In this paper, we derive the new directional filtering method with better performance, based on the spatial masks. It is well-known that the spatial masks have been widely used in smoothing and sharpening of an image [9], and the processing based on spatial masks may be straightforward. Unfortunately, not all previous spatial filter masks take into account fringe orientation, and they cannot carry out filtering along fringe orientation. Inspired by image sharpening using the Laplace operator, we establish the oriented derivative operator that only highlights noise without edges of an image. The noise in the image can be removed while still preserving the edges simply by subtracting the oriented operator image from original image, which can be implemented with one pass of the mask (we name the oriented spatial filter mask). Further, we make an improvement on the oriented spatial filter mask for enhancing the smoothness in the regions with low fringe density. We test the proposed method on the simulated and experimentally obtained ESPI phase fringe pattern, respectively, and comparison with the oriented-couple PDE method and the oriented, regularized quadratic-cost function method. In all cases, our method can give desired results. In the following sections, we will first review image sharpening based on the Laplace operator briefly and then describe in detail the derivation of our oriented spatial filter masks. Section 3 contains experiments and discussion. Section 4 provides the summary for this paper.

2. The derivation of our oriented spatial filter masks

2.1 A brief review of the Laplacian for sharpening

The principal objective of sharpening is to highlight fine details in an image or to enhance details that have been blurred. Sharpening can be accomplished by spatial differentiation [9]. The Laplacian is a simple, widely used isotropic derivative operator, which is implemented using the Laplace masks. Its use highlights gray-level discontinuities in an image and deemphasizes regions with slowly varying gray levels. Background features can be recovered while still preserving the sharpening effect of the Laplacian operation simply by adding the original and Laplacian image. There are several ways to define the Laplace masks. As noted, it is very important to keep in mind which mask is used. If the mask used has a negative center coefficient, then we subtract, rather than add, to the Laplacian image to obtain a sharpened result. Thus, the basic way in which the Laplacian is used for image enhancement is [9]:

2.2 The derivation of the oriented spatial filter mask

It is well-known that filtering and sharpening are two distinct problems. Therefore, it is logical to conclude that we should subtract Laplacian image from the original for filtering. If the mask used has a negative center coefficient, then we add, rather than subtract for filtering. Contrary to Eq. (1), the basic way in which the Laplacian is used for image filtering is:

In order to be useful for digital image processing, Eq. (6) needs to be expressed in discrete form. Here we use the following fourth-order finite differences, derived by standard collocated formulas [10],

2.3 The improvement of the oriented spatial filter mask

Using above oriented spatial filter mask, filtering can be made only along fringe orientation. As above-mentioned, the single-direction smoothing is not sufficient, and the level of smoothness may be low in the regions with low fringe density [3]. For improving the performance of the oriented spatial filter mask, we add a Gaussian filter $F_G(x,y)$to the right side of Eq. (4)

For implementing our filtering method, the orientation angle θ of each pixel has to be evaluated in advance. Recently, we proposed an approach for estimating orientation field of optical fringes with poor quality, which is based on the Fourier transform [11]. What follows is a brief synopsis of the method used in this study.

A window with its center point$(x,y)$is considered. Let$(k,l)$denote pixel position in the window. The orientation angle $\theta (x,y)$ of this point can be estimated by

3. Experiments and discussion

In this section, we test the improved, oriented spatial filter mask (IOSFM) on a simulated and a real ESPI phase image respectively, and compare with the oriented-couple PDE model (OCPDE) [7] and the oriented, regularized quadratic-cost function (ORQCF) [8]. As noted, applying directly a filter to original phase image often smears out the $2\pi$discontinuities. This problem can be solved by the sine/cosine filter method [7]. The sine and cosine of the wrapped phase image are firstly calculated, and then are filtered individually. The arctangent of the ratio of the filtered sine image to the filtered cosine image is the denoised phase image.

Figure 1(a) is a simulated noisy ESPI phase pattern. Figures 1(b)-1(d) show the filtered images by ORQCF, OCPDE and IOSFM, respectively. For comparing, we take the same iterations in the three methods. We use 30 iterations in this case. Further, to quantitatively evaluate the performance, we calculate the image fidelity f, which is defined as $f=1-{\displaystyle \sum {\left(I_0-I\right)}^2}/{\displaystyle \sum I_0{}^2},$ where $I_0$ and I are the noiseless wrapped phase image and the estimated, wrapped phase image, respectively. A high fidelity value will indicate that the processed image is very similar to the noiseless one. The value f in the respective figure (Figs. 1(b)–1(d)) is 0.8457, 0.8716, and 0.9048. Figure 2 shows a real ESPI phase pattern and its filtered images obtained by the three methods with 80 iterations.

 

Fig. 1 A simulated phase pattern and its filtered images. (a) Initial image. (b) ORQCF with $\lambda =0.01,\mu =12$. (c) OCPDE with time step $\Delta t=0.8$. (d) IOSFM with$\alpha =0.35,\beta =1/33$.

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Fig. 2 A real phase pattern and its filtered images. (a) Initial image. (b) ORQCF$\lambda =0.01,\mu =10$. (c) OCPDE with time step $\Delta t=0.8$.(d) IOSFM with$\alpha =0.3,\beta =1/10$.

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As we can see, the two original phase patterns are seriously noisy; especially for the real ESPI phase pattern, and the fringe density of the test images is variable. Obviously, our results are even better than the ones obtained with ORQCF and OCPDE. The consequence is accordant with the quantitative evaluation result. Moreover, our directional filter is easily implemented, in which the fringe orientation angle of each pixel is first estimated, and then the filtered image is obtained by applying Eq. (13).

Like OCPDE and ORQCF, there isn’t the explicit formula or a method to determine the parameters α and β in Eq. (10). They should be set according to the fringe density. Generally, the higher the fringe density is, the larger value α is and smaller value β is. According to our own experience in this method, the appropriate absolute value of αis somewhere between 0.25 and 0.36, and β is between 1/50 and 1/10.

4. Conclusion

We propose the oriented spatial filter masks for directional filtering of ESPI phase patterns. We test our IOSFM on a simulated and a real ESPI phase fringe pattern that contain variable fringe density, and heavy noise, and compare with the OCPDE and the ORQCF. Simulation and experimental results show that our IOSFM outperforms ORQCF and OCPDE.

This work is supported the by National Natural Science Foundation of China (NNSFC) (grant 60877001).We thank the anonymous reviewers for their constructive and helpful comments.