1. Introduction

Electronic speckle pattern interferometry (ESPI) is a well-known technique for the measurement of deformation fields of the object surface. Generally, the phase-shifting method is one of the most successful and most widely used fringe analysis methods for ESPI [1-3]. However, owing to speckle decorrelation effects, the resultant phase fringe patterns are strongly affected by noise. This noise must somehow be dealt with in order to make the phase unwrapping process easier. There are some techniques to remove the noise of ESPI phase patterns, such as the phase-tracking method [4], the least-square method [5], the spin filtering [6], and a sine/cosine average filter [7]. However, for the fringes with high density, no matter which type of technique is used, filtering should be made along the fringe orientation. The traditional filtering methods along the fringe orientation need to establish the small filtering window along fringe orientation and perform a filtering method in the established filtering window, then move to next pixel and repeat above main steps a pixel by pixel, which is very inconvenient.

Partial differential equations (PDEs) image processing methods have been actively studied in the past few years. Again, the rapid development of mathematical models, solution tools, and high resolution numerical schemes has made PDEs-based methods be one of the major tools for image filtering and enhancement. The basic idea of PDEs-based methods is to deform a given image with a PDE, and obtain the desired result as the solution of this PDE with the image as initial conditions. Many different linear and nonlinear PDE models have been proposed for achieving image filtering and enhancement in past years. The original PDE filtering model is the linear heat equation [8] that diffuses in all directions and destroys edges. Some the second-order nonlinear PDE models have been proposed to correct this limitation from various points of view. For instance, the widely used Perona and Malik’s model is proposed from controlling the speed of the diffusion [9]. The degenerate diffusion PDE model is from controlling the direction of the diffusion, in which the diffusion is made only in the direction of the edge [10]. However, the second-order PDE models tend to cause the processed image to look “blocky” [11] and can’t remove impulse noise [12]. Thus some more complex models, such as the fourth-order PDE model [11] and the coupled nonlinear PDEs filtering model [12] have been proposed. In Ref. [12], Y. Chen et al. have reviewed some representative PDE filtering models. We have applied the PDEs image processing methods to process the ESPI fringe patterns [13-15], and also evaluated the performance of a few representative second-order PDE models for filtering in ESPI quantitatively [16]. Among these methods, the coupled nonlinear PDEs filtering models may be the better available filtering model for general image,

where u(x,y,t) is the evolving image, I denotes the initial image. α, β and b are the constant parameters, a(t) is a parameter which changes with time. The function g(|∇_v|) is a nonincreasing function of the gradient |∇_v|. However, when the fringe density is high, these PDE models blur the fringes. For solving this problem, we have proposed the second-order oriented PDE model based on the variational methods and controlling diffusion direction respectively in our former research [17],

Where θ is the angle between the fringe orientation with x coordinate. The function g(|∇_u|) on the right side of Eq. (2) controls the diffusion speed, and the remaining part (u_xx cos² θ + u_yy sin² θ + 2u_xy sinθ cosθ) makes the diffusion only along fringe orientation.

As far as we know, there are only our second-order oriented PDE models at present stage, which make the diffusion only along fringe orientation. Similar to the traditional second-order PDE filtering models, the performance of the second-order oriented PDE model should be improved. Therefore, the new oriented PDE filtering models with better performance, if possible, are very expected. In this paper, we derive the new oriented PDE filtering models, named the oriented-couple PDE models that have a better performance in the numerical stability and fidelity compared to our previous second-order oriented PDE model. In the filtering methods based on the oriented PDEs, filtering along fringe orientation for the entire image is simply realized through solving the oriented PDEs numerically, without having to laboriously establish the small filtering window along the fringe orientation and move this filtering window over each pixel in an image, which is a breakthrough for ESPI processing. We test the proposed models on two computer-simulated speckle phase fringe patterns and an experimentally obtained phase fringe pattern, respectively, in which the fringe density is variable, and compare our models with related PDE models, including the conventional coupled PDE models and our previous second-order oriented PDE model. Further, we quantitatively evaluate the performance of these PDE models with a comparative parameter, the image fidelity which quantifies how good image details are preserved after noise removal. In all cases, our results are much better than the ones obtained with the conventional coupled PDE models and our previous second-order oriented PDE model. Experimental results demonstrate that the filtering method based on the new oriented-couple PDE models can be able to preserve ESPI phase fringes perfectly and suppress the noise effectively, even in the area where the fringes are very dense.

2. The main principle of our method

2.1 The derivation of the new oriented-couple PDE models

In this section, we restrict ourselves to derive the new oriented-couple PDE filtering models based on variational methods [12]. Let u(x, y) be a digital image, and functional E(u) (called the energy function) measure the oscillations in an image. A general formulation of the noise removal problem is to solve the minimization of E(u) in the image support Ω. The functional E(u) provides a degree of smoothing to the image u and may take many different forms. Here we propose the new functional:

Where ρ denotes the fringe orientation. A study of this functional would suggest the following: Firstly, the term |∂u/∂ρ|² provides a degree of the smoothness to the image u along fringe orientation as measured by |∂u/∂ρ|. Secondly, the coefficient of the first term of Eq. (3), namely, g(|∇_v|) serves the purpose of controlling the diffusion speed for different regions [9]. Thirdly, like the conventional coupled nonlinear PDE models, the term (u – I)² enforces the fidelity of the smoothed image u to the original image I.

In this case, the equivalent Euler equation based on variational method is

where

We can easily derive that

Note that although the fringe orientation angle θ of each pixel has various values, it has to be evaluated in advance. So the fringe orientation angle θ isn’t a variable in the above partial derivative calculation. Meanwhile, we fix v, since g(v) is not really a constant.

Inserting (6-8) into (4), the Euler–Lagrange optimality equation is

Consequently, the corresponding evolution equation is

Subsequently, we discuss the choice for v in Eq. (10). The choice of v plays an important role in the results [12].

We rearrange Eq. (1-b) as

The part of the first term on the right side of Eq. (11) namely, |∇_v| div(∇_v|∇_v|) is the degenerate diffusion PDE filtering model, which makes the diffusion only in the direction of the edge (see Ref. [10]). The second term on the right side of Eq. (11) b(v−u) enforces v not too far removed from u. Here we want to make the diffusion only along fringe orientation, so we replace the part |∇_v|div(∇_v|∇_v|) by (v_xx cos² θ + v_yy sin² θ + 2v_xy sinθ cosθ) (see Ref. [17]).

We take the following form for v:

Our oriented-couple PDE models are formed by Eq. (10) and (12) with initial conditions

The numerical solutions of the oriented-couple PDEs give the filtered image. For computing numerically Eqs. (10) and (12), it is needed to discretize them. It is easy to derive their discrete schemes (see Refs. [14, 17]).

where uⁿ_i,j is the numerical solution, the subscripts i, j denote the pixel position in a discrete two-dimensional grid, the superscript n denotes iteration time, then the discrete time t_n = nΔt, Δt is time step. In all mentioned models, gⁿ_i,j is calculated by

where k is a constant parameter, and the gradient of v is approximated by the upwind scheme [12],

where Δ^- _i, Δ⁺ _i represents the forward and backward difference operators, respectively. All the spatial derivatives are approximated using central differences.

2.2 Calculation for the orientation of fringes

The angle θ of fringe orientation of each pixel has to be evaluated before the implementation of our models. In Ref. [17], we described a method to obtain the smoothed orientation field of ESPI fringes based on the least mean square algorithm [18]. What follow is the main steps of this method.

We can find that the gradient operators used to calculate the gradients u_x and u_y in Eq. (16) must be chosen based on the width of ESPI fringes. In Ref. [17], we used the gradients of the Gaussian lowpass filter as the gradient operators to calculate the gradients u_x and u_y in Eq. (16) for dense and thin ESPI fringes, which is a very effective. However, the gradient operators aren’t suitable for wide ESPI fringes. Through many experiments, we find the plane-fit method [6] can be used to calculate the gradients u_x and u_y in Eq. (16) for wide fringes. What follows is a brief synopsis of this method used in this study. A small window (for instance, 11×11 pixels) is considered. Suppose the gray level of each pixel within this window is approximated by a plane polynomial,

Where x and y are coordinates and a, b, c are the plain coefficients.

Obviously, the gradients u_x and u_y are equal to the coefficients b and c respectively. Using the least-square fitting, one can get the coefficients b and c, i.e., the gradients u_x, u_y,

and

3. Experiments and discussion

In this section, we test our oriented-couple PDE models on two computer-simulated speckle phase fringe patterns and an experimentally obtained phase fringe pattern, respectively, in which the fringe density is variable, and compare our models with related PDE models, including the conventional coupled PDE models and our previous second-order oriented PDE model. Further, we calculate the image fidelity in order to quantitatively evaluate the performance of these PDE models. We also compare the computational time of our method with that of a traditional filtering method along the fringe orientation.

The computer-simulated original speckle phase patterns are generated by means of the phase-shifting method. The intensities of the (i, j) pixel of the four phase-shifted original speckle patterns are then given by

where I _0,i,j and I_r,i,j are the intensities of the object and the reference beams, respectively. φ_i,j is the random interferometric phase of the speckle field, and ψ_i,j is the phase change that is due to deformation of the surface of the tested object. n _0,i,j is the random noise. The deformation phase ψ_i,j is calculated from

Here we let φ, I_o, I_r and n ₀ in Eqs. (25-a)-(25-d) randomly distribute over the intervals [-π,π], [−I_m,I_m], [−ρI_m,ρI_m], [−I_n,I_n], respectively, where I_m, ρ and I_n are the constant parameters.

It is well-known that applying directly filtering methods to original phase fringe patterns often smears out the 2π discontinuities and can’t produce the so-called sawtooth jumps. Therefore, here we apply the sine/cosine average filter method [7]. We firstly calculate the sine and cosine of the wrapped phase fringe pattern, which leads to continuous fringe patterns. Then these sine and cosine fringe patterns are filtered individually by our oriented-couple PDE models. The phase fringe pattern is finally obtained by the inverse tangent of the filtered sine and cosine fringe pattern.

The filtering results of PDE models are relative to the parameters in models, and discrete time step Δt and iteration time n. As far as we know in literatures, there isn’t a method to determine these parameters, so these parameters are chosen based on the better performance by trial. In our all implementation, the chosen parameters in the conventional couple PDE models are α = 0.27, β = 0.0005, k = 0.0001, b = 0.02, Δt = 0.8, the chosen parameters in the previous second-order oriented PDE model are k =0.0001, Δt = 0.5, and the chosen parameters in the new oriented couple PDE models are α = 0.40, k = 0.0001, Δt = 0.8 .

Figure 1(a) is a computer-simulated original speckle phase pattern with image size of 300×300 pixels based on the Eq. (26). Here we choose I_m = 180, ρ = 0.2, I_n = 20 .The phase ψ_i,j is calculated from

Fig. 1(b), Fig. 1(c) and Fig. 1(d) show the filtered images of the second-order oriented PDE model, the conventional coupled PDE models and our oriented-couple PDE models, respectively. Here we use 30 iterations in these models.

Figure 2(a) is another computer-simulated original speckle phase pattern with image size of 380×380 pixels. Here we choose I_m =150, ρ = 0.4, I_n = 20. The phase ψ_i,j is calculated from

where ψ _i,j1 and ψ _i,j2 are the exponential phase and the polynomial phase, respectively,

Figures 2(b), Fig. 2(c) and Fig. 2(d) give the filtered images of the second-order oriented PDE model, the conventional coupled PDE models and our oriented-couple PDE models, respectively. The iterative number in these models is chosen as 35.

 

Fig. 1. A computer-simulated phase fringe pattern and its filtered images. (a) Initial image.(b) The second-order oriented PDE model. (c) The conventional coupled PDE models.(d) Our oriented-couple PDE models.

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Figure 3(a) shows an experimentally obtained original speckle phase pattern, which depicts the derivative of the out-of-plane displacement of a square plate. The plate is rigidly clamped at its boundary and is subjected to a central load. Similar results for Fig. 3(a) as Fig. 1(a) are given in Figs. 3(b)-3(d). The noise level for this test image in Fig. 3(a) is very high, so we take 180 iterations in these models.

The fringes shown in Fig. 1(a) and Fig. 2(a) are dense and thin, so we use the gradients of the Gaussian lowpass filter as the gradient operators to calculate the gradients u_x and u_y in Eq. (16), whereas the fringes shown in Fig. 3(a) are wide, we use the plane-fit method to calculate the gradients u_x and u_y in Eq. (16) based on Eqs. (23) and (24).

Further, we quantitatively evaluate the performance of these PDE models with a comparative parameter, the image fidelity f [16]. The image fidelity f, which is a parameter that quantifies how good image details are preserved after noise removal, is defined as

where I ₀ and I are the noiseless image and the estimated image, respectively. A high fidelity value will indicate that the processed image is very similar to the noiseless one, i.e. has good fidelity. Since the noiseless image I ₀ for the experimentally obtained speckle phase pattern shown in Fig. 3(a) is unknown, here the image fidelity f is calculated for the filtered images of computer-simulated phase patterns (shown in Fig. 1(b)-Fig. 1(d) and Fig. 2(b)-Fig. 2(d)). The results are given in Table. 1.

Subsequently, we compare the computational time of our method with that of a traditional filtering method along the fringe orientation. Here we take Fig. 1(a) as the test image for timing measurement. The sine and cosine of Fig. 1(a) are filtered by our oriented-couple PDE models and a traditional filtering method along the fringe orientation, respectively. In this traditional filtering method along the fringe orientation, the small filtering window of 1×3 pixels is established along the fringe orientation, and the mean filtering method is performed in the established filtering window. Then move to next pixel and repeat above processing a pixel by pixel. Table. 2 compares the computational time of our oriented-couple PDE filtering method and this traditional filtering method along the fringe orientation on different iterative number. All the tests are implemented on a personal Pentium 4 computer, at 1.7 GHz by use of MATLAB.

 

Fig. 2. A computer-simulated phase fringe pattern and its filtered images. (a) Initial image. (b) The second-order oriented PDE model. (c) The conventional coupled PDE models.(d) Our oriented-couple PDE models.

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Fig. 3. An experimentally obtained ESPI phase fringe pattern and its filtered images. (a) Initial image. (b) The second-order oriented PDE model. (c) The conventional coupled PDE models. (d) Our oriented-couple PDE models.

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Table 1. Performance evaluation results for the various PDE filtering models

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Table 2. Comparison the computational time of our method and the traditional method

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As we can see, the original phase patterns are obviously very noisy; especially for the experimentally obtained original speckle phase pattern, and the fringe density of all test images is variable. The challenge here is to de-noise and yet preserve all fringes perfectly and fine detail, even in the area where the fringes are very dense. One can find that the conventional coupled PDE models can suppress noise effectively in the area where the fringes aren’t dense, whereas blur images where the fringes are dense. Although the previous second-order oriented PDE models improve the filtering effect to a certain extent, they still blur images in the area where the fringes are densest. In addition, it is noted that the chosen time step in the coupled PDE models is bigger than that in the previous second-order oriented PDE model. The conclusion is that the coupled PDE models have a better numerical stability compared to the second-order oriented PDE model. One can also find that using our proposed models, all fringes are perfectly preserved, and the noise is very effectively suppressed. Obviously, our results shown in Fig. 1(d), Fig. 2(d) and Fig. 3(d) represent a clear improvement over the ones shown in Fig. 1(b), Fig. 2(b) and Fig. 3(b) (the previous second-order oriented PDE model), and Fig. 1(c), Fig. 2(c) and Fig. 3(c) (the conventional coupled PDE models). On the other hand, one can find from Table. 1 that the image fidelity f for the filtered images obtained by our method is the highest. The results indicate that the processed image by our oriented-couple PDE models is closest to the noiseless one, i.e. has good fidelity. One can also find from Table. 2 that the traditional filtering method along the fringe orientation requires more computational time compared to our oriented-couple PDE method. The results are expected because our method doesn’t need to establish and move small filtering window a pixel by pixel.

4. Conclusion

With all above-mentioned advantages, the filtering methods based on the oriented PDEs are a useful tool for filtering and enhancement in ESPI phase fringe patterns of varying fringe density (which represent the usual case). The construction of new oriented PDEs with a better performance is of fundamental importance for the filtering methods. In this paper we derive the new oriented-couple PDE models. We test the new models on two computer-simulated speckle phase fringe patterns and an experimentally obtained phase fringe pattern, respectively, and compare our models with related PDE models. Further, we calculate the fidelity of the filtered images in order to evaluate the performance of these PDE models quantitatively. We also compare the computational time of our method with that of a traditional filtering method along the fringe orientation. The results clearly indicate that the conventional coupled PDE models blur dense ESPI phase fringe patterns seriously, our previous second-order oriented PDE model improves the filtering effect partly, whereas the oriented-couple PDE models can give desired results in all cases. In addition, the oriented-couple PDE models have better numerical stability and fidelity as compared with our previous second-order oriented PDE model, and our filtering method has less computational time than traditional filtering method along the fringe orientation. Here we propose the new oriented PDE models with a better performance for filtering in ESPI phase fringe patterns, and the proposed method can be also applied to digital holography [19], which develops the PDEs-based image filtering methods significantly. Further, we believe the new oriented-couple PDE models will be very useful in many tasks such as skeletonization and segmentation. This work is currently underway.