Abstract

We derive the new oriented-couple partial differential equation (PDE) models based on the variational methods for filtering in electronic speckle pattern interferometry phase fringe patterns. In the filtering methods based on the oriented PDE models, filtering along fringe orientation for the entire image is simply realized through solving the 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. 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. Further, we quantitatively evaluate the performance of these PDE models with a comparative parameter, named the image fidelity. We also compare the computational time of our method with that of a traditional filtering method along the fringe orientation. The experimental results demonstrate the performance of our new oriented PDE models.

© 2009 Optical Society of America

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,

{ut=αg(v)udiv(uu)+α(g(v))uβ(uI)uvt=a(t)div(vv)b(vu)

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],

ut=g(u)(uxxcos2θ+uyysin2θ+2uxysinθcosθ)

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 (uxx cos2 θ + uyy sin2 θ + 2uxy 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:

E(u)=Ω{12g(v)uρ2+12β(uI)2}dxdy
=Ω{12g(v)(uxcosθ+uysinθ)2+12β(uI)2}dxdy

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 (uI)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

fux(fux)y(fuy)=0

where

f=12g(v)(uxcosθ+uysinθ)2+12β(uI)2

We can easily derive that

fu=β(uI)
x(fux)=gcosθ(uxxcosθ+uxysinθ)
y(fuy)=gsinθ(uyysinθ+uxycosθ)

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

αg(v)(uxxcos2θ+uyysin2θ+2uxysinθcosθ)+β(uI)u=0

Consequently, the corresponding evolution equation is

ut=αg(v)(uxxcos2θ+uyysin2θ+2uxysinθcosθ)β(uI)u

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

vt=a(t)vvdiv(vv)b(vu)

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(vu) 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 (vxx cos2 θ + vyy sin2 θ + 2vxy sinθ cosθ) (see Ref. [17]).

We take the following form for v:

vt=a(t)v(vxxcos2θ+vyysin2θ+2vxysinθcosθ)b(vu)

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

uxy0=Ixy,vxy0=Ixy

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]).

{ui,jn+1=ui,jn+Δtgi,jn[(uxx)i,jncos2(θi,j)+(uyy)i,jnsin2(θi,j)+2(uxy)i,jncos(θi,j)sin(θi,j)]β(ui,jnI)(u)i,jnvi,jn+1=vi,jn+Δta(t)(v)i,jn[(uxx)i,jncos2(θi,j)+(uyy)i,jnsin2(θi,j)+2(uxy)i,jncos(θi,j)sin(θi,j)]b(vi,jnui,jn)

where uni,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 tn = nΔt, Δt is time step. In all mentioned models, gni,j is calculated by

gi,jn=1(1+k(vi,jn)2)

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

vi,jn=((max(Δivi,jn,0))2+(min(Δi+vi,jn,0))2+(max(Δj+vi,jn,0))2+(min(Δjvi,jn,0))2)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.

  • Divide fringe pattern u into blocks of size w 1 × w 1.
  • Calculate the x and y directional gradients ux and uy of each pixel within the block using a gradient operator.
  • Calculate the orientation angle θ of the (i, j) centered w 1 × w 1 sized block using the following equation

    θij=12tan1k,l2uxkluyklk,l(ux2kluy2kl)

    where k and l are the subscripts of the pixel point in this w 1 × w 1 window, i–(w 1 −1)/2 ≤ ki + (w 1 −1)/2 and j−(w 1 −1)/2 ≤ lj + (w 1−1)/2.

  • Convert the orientation image obtained by Eq. (16) into a continuous vector field. The x and y components of the continuous vector field are defined as Φx and Φy respectively:

    Φxij=cos(2θij)

    and

    Φyij=sin(2θij)

  • Smooth the continuous vector fields Φx and Φy by a low-pass filter,

    Φxij=k=w212w212l=w212w212FklΦxi+kj+l

    and

    Φyij=k=w212w212l=w212w212FklΦyi+kj+l

    where F is a two-dimensional low-pass filter, w 2 × w 2 is the size of the filter. Here F is chosen as a rotationally symmetric Gaussian lowpass filter.

  • Compute the smoothed orientation field by

    θij=12tan1ΦyijΦxij

We can find that the gradient operators used to calculate the gradients ux and uy 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 ux and uy 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 ux and uy 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,

uxy=a+bx+cy

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

Obviously, the gradients ux and uy 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 ux, uy,

b=ux=x,yuxyxx,yx2

and

c=uy=x,yuxyyx,yy2

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

I1,i,j=Io,i,j+Ir,i,j+2Io,i,jIr,i,jcosφi,j+n0,i,j
I2,i,j=Io,i,j+Ir,i,j+2Io,i,jIr,i,jcos(φi,j+π2)+n0,i,j
I3,i,j=Io,i,j+Ir,i,j+2Io,i,jIr,i,jcos(φi,j+ψi,j)+n0,i,j
I4,i,j=Io,i,j+Ir,i,j+2Io,i,jIr,i,jcos(φi,j+ψi,j+π2)+n0,i,j

where I 0,i,j and Ir,i,j are the intensities of the object and the reference beams, respectively. φi,j is the random interferometric phase of the speckle field, and ψi,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 ψi,j is calculated from

ψi,j=2tan1(I4,i,jI2,i,j+I3,i,jI1,i,jI4,i,j+I2,i,jI3,i,jI1,i,j)

Here we let φ, Io, Ir and n 0 in Eqs. (25-a)-(25-d) randomly distribute over the intervals [-π,π], [−Im,Im], [−ρIm,ρIm], [−In,In], respectively, where Im, ρ and In 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 Im = 180, ρ = 0.2, In = 20 .The phase ψi,j is calculated from

ψi,j=3π[(6×i150300)2+(6×j150300)2]

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 Im =150, ρ = 0.4, In = 20. The phase ψi,j is calculated from

ψi,j=ψi,j1+ψi,j2

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

ψi,j1=80[exp((185)2+(j85)23500)+exp((i295)2+(j295)23500)]
ψi,j2=100(i190222)210(i190222)(j190222)+40(j190222)2

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 ux and uy in Eq. (16), whereas the fringes shown in Fig. 3(a) are wide, we use the plane-fit method to calculate the gradients ux and uy 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

f=1(I0I)2I02

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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Tables Icon

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

Tables Icon

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

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.

Acknowledgments

We would like to thank Prof. Jinlong Chen in Tianjin University for his kind help of providing some images, and also thank the anonymous reviewer for his constructive and helpful comment. This work is supported the by National Natural Science Foundation of China (NNSFC) (grant 60877001).

References and links

1. S. Nakadate and H. Saito, “Fringe scanning speckle-pattern interferometry,” Appl. Opt. 24, 2172–2180 (1985). [CrossRef]   [PubMed]  

2. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985). [CrossRef]   [PubMed]  

3. D. W. Robinson and D. C. Williams, “Digital phase stepping speckle interferometry,” Opt. Commun. 57, 26–30 (1986). [CrossRef]  

4. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, and R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999). [CrossRef]  

5. C. K. Hong, H. S. Ryu, and H. C. Lim, “Least-squares fitting of the phase map obtained in phase-shifting electronic speckle pattern interferometry,” Opt. Lett. 20, 931–933 (1995). [CrossRef]   [PubMed]  

6. Qifeng Yu, X. Sun, and X. Liu, “Spin filtering with curve windows for interferometric fringe patterns,” Appl. Opt. 412650–2654 (2002). [CrossRef]   [PubMed]  

7. H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999). [CrossRef]  

8. A. P. Witkin, “Scale-space filtering,” in Proceedings of IJCAI, (Karlsruhe, 1983), pp. 1019–1021.

9. P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE TPAMI. 12, 629–639 (1990). [CrossRef]  

10. L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 845–866 (1992). [CrossRef]  

11. Y. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000). [CrossRef]  

12. Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, “Smoothing and edge detection by time-varying coupled nonlinear diffusion equations,” Computer Vision and Image Understanding, 82, 85–100 (2001). [CrossRef]  

13. C. Tang, F. Zhang, and Z. Chen, “Contrast enhancement for electronic speckle pattern interferometry fringes by the differential equation enhancement method,” Appl. Opt. 45, 2287–2294, (2006). [CrossRef]   [PubMed]  

14. C. Tang, W. Lu, S. Chen, Z. Zhang, B. Li, W. Wang, and Lin Han, “Denoising by coupled partial differential equations and extracting phase by backpropagation neural networks for electronic speckle pattern interferometry,” Appl. Opt. 46, 7475–7484 (2007). [CrossRef]   [PubMed]  

15. C. Tang, W. Lu, Y. Cai, L. Han, and G. Wang, “Nearly preprocessing-free method for skeletonization of gray-scale electronic speckle pattern interferometry fringe patterns via partial differential equations,” Opt. Lett. 33,183–185 (2008). [CrossRef]   [PubMed]  

16. C. Tang, F. Zhang, B. Li, and H. Yan, “Performance evaluation of partial differential equation models in electronic speckle pattern interferometry and δ-mollification method of phase map,” Appl. Opt. 457392–7400 (2006). [CrossRef]   [PubMed]  

17. C. Tang, L. Han, H. Ren, D. Zhou, Y. Chang, X. Wang, and X. Cui, “Second-order oriented partial-differential equations for denoising in electronic-speckle-pattern interferometry fringes,” Opt. Lett. 332179–2181 (2008). [CrossRef]   [PubMed]  

18. L. Hong, Y. Wan, and A. Jain, “Fingerprint Image Enhancement: Algorithm and Performance Evaluation,” IEEE Transactions on pattern analysis and machine intelligence, 20, 777–789 (1998). [CrossRef]  

19. U. Schnars and W. P. O. Jueptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002) [CrossRef]  

References

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  1. S. Nakadate and H. Saito, “Fringe scanning speckle-pattern interferometry,” Appl. Opt. 24, 2172–2180 (1985).
    [Crossref] [PubMed]
  2. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).
    [Crossref] [PubMed]
  3. D. W. Robinson and D. C. Williams, “Digital phase stepping speckle interferometry,” Opt. Commun. 57, 26–30 (1986).
    [Crossref]
  4. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, and R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
    [Crossref]
  5. C. K. Hong, H. S. Ryu, and H. C. Lim, “Least-squares fitting of the phase map obtained in phase-shifting electronic speckle pattern interferometry,” Opt. Lett. 20, 931–933 (1995).
    [Crossref] [PubMed]
  6. Qifeng Yu, X. Sun, and X. Liu, “Spin filtering with curve windows for interferometric fringe patterns,” Appl. Opt. 412650–2654 (2002).
    [Crossref] [PubMed]
  7. H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
    [Crossref]
  8. A. P. Witkin, “Scale-space filtering,” in Proceedings of IJCAI, (Karlsruhe, 1983), pp. 1019–1021.
  9. P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE TPAMI. 12, 629–639 (1990).
    [Crossref]
  10. L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 845–866 (1992).
    [Crossref]
  11. Y. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
    [Crossref]
  12. Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, “Smoothing and edge detection by time-varying coupled nonlinear diffusion equations,” Computer Vision and Image Understanding, 82, 85–100 (2001).
    [Crossref]
  13. C. Tang, F. Zhang, and Z. Chen, “Contrast enhancement for electronic speckle pattern interferometry fringes by the differential equation enhancement method,” Appl. Opt. 45, 2287–2294, (2006).
    [Crossref] [PubMed]
  14. C. Tang, W. Lu, S. Chen, Z. Zhang, B. Li, W. Wang, and Lin Han, “Denoising by coupled partial differential equations and extracting phase by backpropagation neural networks for electronic speckle pattern interferometry,” Appl. Opt. 46, 7475–7484 (2007).
    [Crossref] [PubMed]
  15. C. Tang, W. Lu, Y. Cai, L. Han, and G. Wang, “Nearly preprocessing-free method for skeletonization of gray-scale electronic speckle pattern interferometry fringe patterns via partial differential equations,” Opt. Lett. 33,183–185 (2008).
    [Crossref] [PubMed]
  16. C. Tang, F. Zhang, B. Li, and H. Yan, “Performance evaluation of partial differential equation models in electronic speckle pattern interferometry and δ-mollification method of phase map,” Appl. Opt. 457392–7400 (2006).
    [Crossref] [PubMed]
  17. C. Tang, L. Han, H. Ren, D. Zhou, Y. Chang, X. Wang, and X. Cui, “Second-order oriented partial-differential equations for denoising in electronic-speckle-pattern interferometry fringes,” Opt. Lett. 332179–2181 (2008).
    [Crossref] [PubMed]
  18. L. Hong, Y. Wan, and A. Jain, “Fingerprint Image Enhancement: Algorithm and Performance Evaluation,” IEEE Transactions on pattern analysis and machine intelligence, 20, 777–789 (1998).
    [Crossref]
  19. U. Schnars and W. P. O. Jueptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002)
    [Crossref]

2008 (2)

2007 (1)

2006 (2)

2002 (2)

Qifeng Yu, X. Sun, and X. Liu, “Spin filtering with curve windows for interferometric fringe patterns,” Appl. Opt. 412650–2654 (2002).
[Crossref] [PubMed]

U. Schnars and W. P. O. Jueptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002)
[Crossref]

2001 (1)

Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, “Smoothing and edge detection by time-varying coupled nonlinear diffusion equations,” Computer Vision and Image Understanding, 82, 85–100 (2001).
[Crossref]

2000 (1)

Y. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
[Crossref]

1999 (2)

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[Crossref]

M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, and R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
[Crossref]

1998 (1)

L. Hong, Y. Wan, and A. Jain, “Fingerprint Image Enhancement: Algorithm and Performance Evaluation,” IEEE Transactions on pattern analysis and machine intelligence, 20, 777–789 (1998).
[Crossref]

1995 (1)

1992 (1)

L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 845–866 (1992).
[Crossref]

1990 (1)

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE TPAMI. 12, 629–639 (1990).
[Crossref]

1986 (1)

D. W. Robinson and D. C. Williams, “Digital phase stepping speckle interferometry,” Opt. Commun. 57, 26–30 (1986).
[Crossref]

1985 (2)

Aebischer, H. A.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[Crossref]

Alvarez, L.

L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 845–866 (1992).
[Crossref]

Barcelos, C. A. Z.

Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, “Smoothing and edge detection by time-varying coupled nonlinear diffusion equations,” Computer Vision and Image Understanding, 82, 85–100 (2001).
[Crossref]

Cai, Y.

Chang, Y.

Chen, S.

Chen, Y.

Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, “Smoothing and edge detection by time-varying coupled nonlinear diffusion equations,” Computer Vision and Image Understanding, 82, 85–100 (2001).
[Crossref]

Chen, Z.

Creath, K.

Cuevas, F. J.

Cui, X.

Han, L.

Han, Lin

Hong, C. K.

Hong, L.

L. Hong, Y. Wan, and A. Jain, “Fingerprint Image Enhancement: Algorithm and Performance Evaluation,” IEEE Transactions on pattern analysis and machine intelligence, 20, 777–789 (1998).
[Crossref]

Jain, A.

L. Hong, Y. Wan, and A. Jain, “Fingerprint Image Enhancement: Algorithm and Performance Evaluation,” IEEE Transactions on pattern analysis and machine intelligence, 20, 777–789 (1998).
[Crossref]

Jueptner, W. P. O.

U. Schnars and W. P. O. Jueptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002)
[Crossref]

Kaveh, M.

Y. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
[Crossref]

Li, B.

Lim, H. C.

Lions, P.-L.

L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 845–866 (1992).
[Crossref]

Liu, X.

Lu, W.

Mairz, B. A.

Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, “Smoothing and edge detection by time-varying coupled nonlinear diffusion equations,” Computer Vision and Image Understanding, 82, 85–100 (2001).
[Crossref]

Malacara, D.

Malik, J.

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE TPAMI. 12, 629–639 (1990).
[Crossref]

Marroguin, J. L.

Morel, J.-M.

L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 845–866 (1992).
[Crossref]

Nakadate, S.

Perona, P.

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE TPAMI. 12, 629–639 (1990).
[Crossref]

Ren, H.

Robinson, D. W.

D. W. Robinson and D. C. Williams, “Digital phase stepping speckle interferometry,” Opt. Commun. 57, 26–30 (1986).
[Crossref]

Rodriguez-Vera, R.

Ryu, H. S.

Saito, H.

Schnars, U.

U. Schnars and W. P. O. Jueptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002)
[Crossref]

Servin, M.

Sun, X.

Tang, C.

Waldner, S.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[Crossref]

Wan, Y.

L. Hong, Y. Wan, and A. Jain, “Fingerprint Image Enhancement: Algorithm and Performance Evaluation,” IEEE Transactions on pattern analysis and machine intelligence, 20, 777–789 (1998).
[Crossref]

Wang, G.

Wang, W.

Wang, X.

Williams, D. C.

D. W. Robinson and D. C. Williams, “Digital phase stepping speckle interferometry,” Opt. Commun. 57, 26–30 (1986).
[Crossref]

Witkin, A. P.

A. P. Witkin, “Scale-space filtering,” in Proceedings of IJCAI, (Karlsruhe, 1983), pp. 1019–1021.

Yan, H.

You, Y.

Y. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
[Crossref]

Yu, Qifeng

Zhang, F.

Zhang, Z.

Zhou, D.

Appl. Opt. (7)

Computer Vision and Image Understanding, (1)

Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, “Smoothing and edge detection by time-varying coupled nonlinear diffusion equations,” Computer Vision and Image Understanding, 82, 85–100 (2001).
[Crossref]

IEEE TPAMI. (1)

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE TPAMI. 12, 629–639 (1990).
[Crossref]

IEEE Trans. Image Process. (1)

Y. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. 9, 1723–1730 (2000).
[Crossref]

IEEE Transactions on pattern analysis and machine intelligence, (1)

L. Hong, Y. Wan, and A. Jain, “Fingerprint Image Enhancement: Algorithm and Performance Evaluation,” IEEE Transactions on pattern analysis and machine intelligence, 20, 777–789 (1998).
[Crossref]

Meas. Sci. Technol. (1)

U. Schnars and W. P. O. Jueptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002)
[Crossref]

Opt. Commun. (2)

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[Crossref]

D. W. Robinson and D. C. Williams, “Digital phase stepping speckle interferometry,” Opt. Commun. 57, 26–30 (1986).
[Crossref]

Opt. Lett. (3)

SIAM J. Numer. Anal. (1)

L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. 29, 845–866 (1992).
[Crossref]

Other (1)

A. P. Witkin, “Scale-space filtering,” in Proceedings of IJCAI, (Karlsruhe, 1983), pp. 1019–1021.

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Figures (3)

Fig. 1.
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.

Fig. 2.
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.

Fig. 3.
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.

Tables (2)

Tables Icon

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

Tables Icon

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

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

{ u t = α g ( v ) u div ( u u ) + α ( g ( v ) ) u β ( u I ) u v t = a ( t ) div ( v v ) b ( v u )
u t = g ( u ) ( u xx cos 2 θ + u yy sin 2 θ + 2 u xy sin θ cos θ )
E ( u ) = Ω { 1 2 g ( v ) u ρ 2 + 1 2 β ( u I ) 2 } dxdy
= Ω { 1 2 g ( v ) ( u x cos θ + u y sin θ ) 2 + 1 2 β ( u I ) 2 } dxdy
f u x ( f u x ) y ( f u y ) = 0
f = 1 2 g ( v ) ( u x cos θ + u y sin θ ) 2 + 1 2 β ( u I ) 2
f u = β ( u I )
x ( f u x ) = g cos θ ( u xx cos θ + u xy sin θ )
y ( f u y ) = g sin θ ( u yy sin θ + u xy cos θ )
αg ( v ) ( u xx cos 2 θ + u yy sin 2 θ + 2 u xy sin θ cos θ ) + β ( u I ) u = 0
u t = αg ( v ) ( u xx cos 2 θ + u yy sin 2 θ + 2 u xy sin θ cos θ ) β ( u I ) u
v t = a ( t ) v v div ( v v ) b ( v u )
v t = a ( t ) v ( v xx cos 2 θ + v yy sin 2 θ + 2 v xy sin θ cos θ ) b ( v u )
u x y 0 = I x y , v x y 0 = I x y
{ u i , j n + 1 = u i , j n + Δ t g i , j n [ ( u xx ) i , j n cos 2 ( θ i , j ) + ( u yy ) i , j n sin 2 ( θ i , j ) + 2 ( u xy ) i , j n cos ( θ i , j ) sin ( θ i , j ) ] β ( u i , j n I ) ( u ) i , j n v i , j n + 1 = v i , j n + Δ t a ( t ) ( v ) i , j n [ ( u xx ) i , j n cos 2 ( θ i , j ) + ( u yy ) i , j n sin 2 ( θ i , j ) + 2 ( u xy ) i , j n cos ( θ i , j ) sin ( θ i , j ) ] b ( v i , j n u i , j n )
g i , j n = 1 ( 1 + k ( v i , j n ) 2 )
v i , j n = ( ( max ( Δ i v i , j n , 0 ) ) 2 + ( min ( Δ i + v i , j n , 0 ) ) 2 + ( max ( Δ j + v i , j n , 0 ) ) 2 + ( min ( Δ j v i , j n , 0 ) ) 2 ) 1 2
θ i j = 1 2 tan 1 k , l 2 u x k l u y k l k , l ( u x 2 k l u y 2 k l )
Φ x i j = cos ( 2 θ i j )
Φ y i j = sin ( 2 θ i j )
Φ x i j = k = w 2 1 2 w 2 1 2 l = w 2 1 2 w 2 1 2 F k l Φ x i + k j + l
Φ y i j = k = w 2 1 2 w 2 1 2 l = w 2 1 2 w 2 1 2 F k l Φ y i + k j + l
θ i j = 1 2 tan 1 Φ y i j Φ x i j
u x y = a + bx + cy
b = u x = x , y u x y x x , y x 2
c = u y = x , y u x y y x , y y 2
I 1 , i , j = I o , i , j + I r , i , j + 2 I o , i , j I r , i , j cos φ i , j + n 0 , i , j
I 2 , i , j = I o , i , j + I r , i , j + 2 I o , i , j I r , i , j cos ( φ i , j + π 2 ) + n 0 , i , j
I 3 , i , j = I o , i , j + I r , i , j + 2 I o , i , j I r , i , j cos ( φ i , j + ψ i , j ) + n 0 , i , j
I 4 , i , j = I o , i , j + I r , i , j + 2 I o , i , j I r , i , j cos ( φ i , j + ψ i , j + π 2 ) + n 0 , i , j
ψ i , j = 2 tan 1 ( I 4 , i , j I 2 , i , j + I 3 , i , j I 1 , i , j I 4 , i , j + I 2 , i , j I 3 , i , j I 1 , i , j )
ψ i , j = 3 π [ ( 6 × i 150 300 ) 2 + ( 6 × j 150 300 ) 2 ]
ψ i , j = ψ i , j 1 + ψ i , j 2
ψ i , j 1 = 80 [ exp ( ( 1 85 ) 2 + ( j 85 ) 2 3500 ) + exp ( ( i 295 ) 2 + ( j 295 ) 2 3500 ) ]
ψ i , j 2 = 100 ( i 190 222 ) 2 10 ( i 190 222 ) ( j 190 222 ) + 40 ( j 190 222 ) 2
f = 1 ( I 0 I ) 2 I 0 2

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