## The oriented-couple partial differential equations for filtering in wrapped phase patterns

Optics Express, Vol. 17, Issue 7, pp. 5606-5617 (2009)

http://dx.doi.org/10.1364/OE.17.005606

Acrobat PDF (1195 KB)

### 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

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

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. **41**2650–2654 (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]

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]

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]

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]

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. **45**7392–7400 (2006). [CrossRef] [PubMed]

*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. **33**2179–2181 (2008). [CrossRef] [PubMed]

*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*cos

_{xx}^{2}

*θ*+

*u*sin

_{yy}^{2}

*θ*+ 2

*u*sin

_{xy}*θ*cos

*θ*) 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

**82**, 85–100 (2001). [CrossRef]

*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:

*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]

*u*–

*I*)

^{2}enforces the fidelity of the smoothed image

*u*to the original image

*I*.

*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.

*v*in Eq. (10). The choice of

*v*plays an important role in the results [12

**82**, 85–100 (2001). [CrossRef]

_{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]

*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*cos

_{xx}^{2}

*θ*+

*v*sin

_{yy}^{2}

*θ*+ 2

*v*sin

_{xy}*θ*cos

*θ*) (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. **33**2179–2181 (2008). [CrossRef] [PubMed]

*v*:

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. 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. **33**2179–2181 (2008). [CrossRef] [PubMed]

*u*is the numerical solution, the subscripts

^{n}_{i,j}*i*,

*j*denote the pixel position in a discrete two-dimensional grid, the superscript

*n*denotes iteration time, then the discrete time

*t*=

_{n}*n*Δ

*t*, Δ

*t*is time step. In all mentioned models,

*g*is calculated by

^{n}_{i,j}*k*is a constant parameter, and the gradient of

*v*is approximated by the upwind scheme [12

**82**, 85–100 (2001). [CrossRef]

^{-}

_{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

*θ*of fringe orientation of each pixel has to be evaluated before the implementation of our models. In Ref. [17

**33**2179–2181 (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]

- Divide fringe pattern
*u*into blocks of size*w*_{1}×*w*_{1}. - Calculate the
*x*and*y*directional gradients*u*and_{x}*u*of each pixel within the block using a gradient operator._{y} - Calculate the orientation angle
*θ*of the (*i*,*j*) centered*w*_{1}×*w*_{1}sized block using the following equationwhere*k*and*l*are the subscripts of the pixel point in this*w*_{1}×*w*_{1}window,*i*–(*w*_{1}−1)/2 ≤*k*≤*i*+ (*w*_{1}−1)/2 and*j*−(*w*_{1}−1)/2 ≤*l*≤*j*+ (*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 Φ_{x}and Φ_{y}respectively:and - Smooth the continuous vector fields Φ
_{x}and Φ_{y}by a low-pass filter,andwhere*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

*u*and

_{x}*u*in Eq. (16) must be chosen based on the width of ESPI fringes. In Ref. [17

_{y}**33**2179–2181 (2008). [CrossRef] [PubMed]

*u*and

_{x}*u*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

_{y}6. Qifeng Yu, X. Sun, and X. Liu, “Spin filtering with curve windows for interferometric fringe patterns,” Appl. Opt. **41**2650–2654 (2002). [CrossRef] [PubMed]

*u*and

_{x}*u*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,

_{y}*x*and

*y*are coordinates and

*a*,

*b*,

*c*are the plain coefficients.

*u*and

_{x}*u*are equal to the coefficients

_{y}*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},

## 3. Experiments and discussion

*i*,

*j*) pixel of the four phase-shifted original speckle patterns are then given by

*I*

_{0,i,j}and

*I*are the intensities of the object and the reference beams, respectively.

_{r,i,j}*φ*is the random interferometric phase of the speckle field, and

_{i,j}*ψ*is the phase change that is due to deformation of the surface of the tested object.

_{i,j}*n*

_{0,i,j}is the random noise. The deformation phase

*ψ*is calculated from

_{i,j}*φ*,

*I*,

_{o}*I*and

_{r}*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*], respectively, where

_{n}*I*,

_{m}*ρ*and

*I*are the constant parameters.

_{n}*π*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]

*t*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, Δ

*t*= 0.8, the chosen parameters in the previous second-order oriented PDE model are

*k*=0.0001, Δ

*t*= 0.5, and the chosen parameters in the new oriented couple PDE models are

*α*= 0.40,

*k*= 0.0001, Δ

*t*= 0.8 .

*I*= 180,

_{m}*ρ*= 0.2,

*I*= 20 .The phase

_{n}*ψ*is calculated from

_{i,j}*I*=150,

_{m}*ρ*= 0.4,

*I*= 20. The phase

_{n}*ψ*is calculated from

_{i,j}*ψ*

_{i,j1}and

*ψ*

_{i,j2}are the exponential phase and the polynomial phase, respectively,

*u*and

_{x}*u*in Eq. (16), whereas the fringes shown in Fig. 3(a) are wide, we use the plane-fit method to calculate the gradients

_{y}*u*and

_{x}*u*in Eq. (16) based on Eqs. (23) and (24).

_{y}*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. **45**7392–7400 (2006). [CrossRef] [PubMed]

*f*, which is a parameter that quantifies how good image details are preserved after noise removal, is defined as

*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.

*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

## Acknowledgments

## References and links

1. | S. Nakadate and H. Saito, “Fringe scanning speckle-pattern interferometry,” Appl. Opt. |

2. | K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. |

3. | D. W. Robinson and D. C. Williams, “Digital phase stepping speckle interferometry,” Opt. Commun. |

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. |

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. |

6. | Qifeng Yu, X. Sun, and X. Liu, “Spin filtering with curve windows for interferometric fringe patterns,” Appl. Opt. |

7. | H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. |

8. | A. P. Witkin, “Scale-space filtering,” in |

9. | P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE TPAMI. |

10. | L. Alvarez, P.-L. Lions, and J.-M. Morel, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal. |

11. | Y. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEE Trans. Image Process. |

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, |

13. | C. Tang, F. Zhang, and Z. Chen, “Contrast enhancement for electronic speckle pattern interferometry fringes by the differential equation enhancement method,” Appl. Opt. |

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. |

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. |

16. | C. Tang, F. Zhang, B. Li, and H. Yan, “Performance evaluation of partial differential equation models in electronic speckle pattern interferometry and |

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. |

18. | L. Hong, Y. Wan, and A. Jain, “Fingerprint Image Enhancement: Algorithm and Performance Evaluation,” IEEE Transactions on pattern analysis and machine intelligence, |

19. | U. Schnars and W. P. O. Jueptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. |

**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

- S. Nakadate and H. Saito,"Fringe scanning speckle-pattern interferometry," Appl. Opt. 24,2172-2180 (1985). [CrossRef] [PubMed]
- K. Creath, "Phase-shifting speckle interferometry," Appl. Opt. 24,3053-3058 (1985). [CrossRef] [PubMed]
- D. W. Robinson and D. C. Williams, "Digital phase stepping speckle interferometry," Opt. Commun. 57,26-30 (1986). [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]
- 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]
- QifengYu , X. Sun, and X. Liu, "Spin filtering with curve windows for interferometric fringe patterns," Appl. Opt. 412650-2654 (2002). [CrossRef] [PubMed]
- 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]
- A. P. Witkin, "Scale-space filtering," in Proceedings of IJCAI, (Karlsruhe, 1983), pp. 1019-1021.
- P. Perona and J. Malik, "Scale-space and edge detection using anisotropic diffusion," IEEE TPAMI. 12, 629-639 (1990). [CrossRef]
- 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]
- Y. You and M. Kaveh, "Fourth-order partial differential equations for noise removal," IEEE Trans. Image Process. 9, 1723-1730 (2000). [CrossRef]
- Y. Chen, C. A. Z. Barcelos, and B. A. Mairz, "Smoothing and edge detection by time-varying coupled nonlinear diffusion equations," Computer Vision Image Understand. 82,85-100 (2001). [CrossRef]
- 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]
- C. Tang, W. Lu, S. Chen, Z. Zhang, B. Li, W. Wang, and L. 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]
- 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]
- C. Tang, F. Zhang, B. Li, and H. Yan, "Performance evaluation of partial differential equation models in electronic speckle pattern interferometry and -mollification method of phase map," Appl. Opt. 45,7392-7400 (2006). [CrossRef] [PubMed]
- 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. 33, 2179- 2181 (2008). [CrossRef] [PubMed]
- 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]
- U. Schnars and W. P. O. Jueptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002) [CrossRef]

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