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Optics Express

  • Vol. 17, Iss. 7 — Mar. 30, 2009
  • pp: 5606–5617
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The oriented-couple partial differential equations for filtering in wrapped phase patterns

Chen Tang, Lin Han, Hongwei Ren, Tao Gao, Zhifang Wang, and Ke Tang  »View Author Affiliations


Optics Express, Vol. 17, Issue 7, pp. 5606-5617 (2009)
http://dx.doi.org/10.1364/OE.17.005606


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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

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

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

] 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

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

]. 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

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]

]. However, the second-order PDE models tend to cause the processed image to look “blocky” [11

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

] and can’t remove impulse noise [12

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]

]. Thus some more complex models, such as the fourth-order PDE model [11

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

] and the coupled nonlinear PDEs filtering model [12

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]

] have been proposed. In Ref. [12

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]

], 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

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]

], and also evaluated the performance of a few representative second-order PDE models for filtering in ESPI quantitatively [16

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]

]. 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)
(1)

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

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]

],

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

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.

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

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]

]. 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
(3)

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

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

]. 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
(4)

where

f=12g(v)(uxcosθ+uysinθ)2+12β(uI)2
(5)

We can easily derive that

fu=β(uI)
(6)
x(fux)=gcosθ(uxxcosθ+uxysinθ)
(7)
y(fuy)=gsinθ(uyysinθ+uxycosθ)
(8)

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
(9)

Consequently, the corresponding evolution equation is

ut=αg(v)(uxxcos2θ+uyysin2θ+2uxysinθcosθ)β(uI)u
(10)

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

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]

].

We rearrange Eq. (1-b) as

vt=a(t)vvdiv(vv)b(vu)
(11)

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

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]

]). 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

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]

]).

We take the following form for v:

vt=a(t)v(vxxcos2θ+vyysin2θ+2vxysinθcosθ)b(vu)
(12)

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

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]

, 17

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]

]).

{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)
(13)

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)
(14)

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

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]

],

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
(15)

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

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]

], we described a method to obtain the smoothed orientation field of ESPI fringes based on the least mean square algorithm [18

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]

]. 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)
    (16)

    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)
    (17)

    and

    Φyij=sin(2θij)
    (18)

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

    Φxij=k=w212w212l=w212w212FklΦxi+kj+l
    (19)

    and

    Φyij=k=w212w212l=w212w212FklΦyi+kj+l
    (20)

    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
    (21)

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

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]

], 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

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

] 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
(22)

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
(23)

and

c=uy=x,yuxyyx,yy2
(24)

3. Experiments and discussion

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
(25-a)
I2,i,j=Io,i,j+Ir,i,j+2Io,i,jIr,i,jcos(φi,j+π2)+n0,i,j
(25-b)
I3,i,j=Io,i,j+Ir,i,j+2Io,i,jIr,i,jcos(φi,j+ψi,j)+n0,i,j
(25-c)
I4,i,j=Io,i,j+Ir,i,j+2Io,i,jIr,i,jcos(φi,j+ψi,j+π2)+n0,i,j
(25-d)

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)
(26)

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

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]

]. 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.

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]
(27)

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
(28)

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)]
(29)
ψi,j2=100(i190222)210(i190222)(j190222)+40(j190222)2
(30)

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.

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

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]

]. 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
(31)

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.

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. 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.

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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4. Conclusion

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]

OCIS Codes
(110.6150) Imaging systems : Speckle imaging
(120.6160) Instrumentation, measurement, and metrology : Speckle interferometry

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: December 11, 2008
Revised Manuscript: January 21, 2009
Manuscript Accepted: January 27, 2009
Published: March 25, 2009

Citation
Chen Tang, Lin Han, Hongwei Ren, Tao Gao, Zhifang Wang, and Ke Tang, "The oriented-couple partial differential equations for filtering in wrapped phase patterns," Opt. Express 17, 5606-5617 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-5606


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References

  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]
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