Abstract

Filtering methods based on oriented partial differential equations (OPDEs) have been demonstrated as a powerful tool for denoising while preserving all fringes. In this paper, we first briefly review the existing OPDEs and then derive numerous possible OPDE filtering models based on the variational methods. These models include a class of new single OPDE models, a class of new selective OPDE models, a class of new coupled OPDEs, and a class of new double OPDEs. We also investigate the performance of main OPDE models, including the choices of parameters and the influences of fringe orientation and diffusion control function on filtering results via our extensive experiments. Finally, we summarize the performance of these OPDEs.

© 2013 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  4. 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]
  5. Y. Wang, X. Ji, and Q. Dai, “Fourth-order oriented partial-differential equations for noise removal of two-photon fluorescence images,” Opt. Lett. 35, 2943–2945 (2010).
    [CrossRef]
  6. C. Tang, L. Han, H. Ren, T. Gao, Z. Wang, and K. Tang, “The oriented-couple partial differential equations for filtering in wrapped phase patterns,” Opt. Express 17, 5606–5617 (2009).
    [CrossRef]
  7. H. Wang, Q. Kemao, W. Gao, F. Lin, and H. S. Seah, “Fringe pattern denoising using coherence enhancing diffusion,” Opt. Lett. 34, 1141–1143 (2009).
    [CrossRef]
  8. C. Tang, N. Yang, H. Yan, and X. Yan, “The new second-order single oriented partial differential equations for optical interferometry fringes with high density,” Opt. Lasers Eng. 51,707–715 (2013).
    [CrossRef]
  9. Y. Chen, B. C. Vemuri, and L. Wang, “Image denoising and segmentation via nonlinear diffusion,” Comput. Math. Appl. 39, 131–149 (2000).
    [CrossRef]
  10. C. Tang, L. Wang, and H. Yan, “Overview of anisotropic filtering methods based on partial differential equations for electronic speckle pattern interferometry,” Appl. Opt. 51, 4916–4926 (2012).
    [CrossRef]
  11. Wikipedia., “Euler–Lagrange equation,” http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation .
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    [CrossRef]
  13. P. Zhou and K. E. Goodson, “Subpixel displacement and deformation gradient measurement using digital image/speckle correlation (DISC),” Opt. Eng. 40, 1613–1620 (2001).
    [CrossRef]

2013

C. Tang, N. Yang, H. Yan, and X. Yan, “The new second-order single oriented partial differential equations for optical interferometry fringes with high density,” Opt. Lasers Eng. 51,707–715 (2013).
[CrossRef]

X. Zhu, Z. Chen, C. Tang, Q. Mi, and X. Yan, “Application of two oriented partial differential equation filtering models on speckle fringes with poor quality and their numerically fast algorithms,” Appl. Opt. 52, 1814–1823 (2013).
[CrossRef]

2012

2011

2010

H. Wang and Q. Kemao, “Coherence-enhancing diffusion and windowed Fourier filtering for fringe pattern denoising (II),” AIP Conf. Proc. 1236, 52–56 (2010).
[CrossRef]

Y. Wang, X. Ji, and Q. Dai, “Fourth-order oriented partial-differential equations for noise removal of two-photon fluorescence images,” Opt. Lett. 35, 2943–2945 (2010).
[CrossRef]

2009

2008

2001

P. Zhou and K. E. Goodson, “Subpixel displacement and deformation gradient measurement using digital image/speckle correlation (DISC),” Opt. Eng. 40, 1613–1620 (2001).
[CrossRef]

2000

Y. Chen, B. C. Vemuri, and L. Wang, “Image denoising and segmentation via nonlinear diffusion,” Comput. Math. Appl. 39, 131–149 (2000).
[CrossRef]

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

Chang, Y.

Chen, Y.

Y. Chen, B. C. Vemuri, and L. Wang, “Image denoising and segmentation via nonlinear diffusion,” Comput. Math. Appl. 39, 131–149 (2000).
[CrossRef]

Chen, Z.

Cui, X.

Dai, Q.

Gao, T.

Gao, W.

Goodson, K. E.

P. Zhou and K. E. Goodson, “Subpixel displacement and deformation gradient measurement using digital image/speckle correlation (DISC),” Opt. Eng. 40, 1613–1620 (2001).
[CrossRef]

Han, L.

Ji, X.

Kaveh, M.

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

Kemao, Q.

H. Wang and Q. Kemao, “Coherence-enhancing diffusion and windowed Fourier filtering for fringe pattern denoising (II),” AIP Conf. Proc. 1236, 52–56 (2010).
[CrossRef]

H. Wang, Q. Kemao, W. Gao, F. Lin, and H. S. Seah, “Fringe pattern denoising using coherence enhancing diffusion,” Opt. Lett. 34, 1141–1143 (2009).
[CrossRef]

Lin, F.

Mi, Q.

Qian, K.

Ren, H.

Seah, H. S.

Tang, C.

Tang, K.

Vemuri, B. C.

Y. Chen, B. C. Vemuri, and L. Wang, “Image denoising and segmentation via nonlinear diffusion,” Comput. Math. Appl. 39, 131–149 (2000).
[CrossRef]

Wang, H.

Wang, L.

Wang, X.

Wang, Y.

Wang, Z.

Yan, H.

C. Tang, N. Yang, H. Yan, and X. Yan, “The new second-order single oriented partial differential equations for optical interferometry fringes with high density,” Opt. Lasers Eng. 51,707–715 (2013).
[CrossRef]

C. Tang, L. Wang, and H. Yan, “Overview of anisotropic filtering methods based on partial differential equations for electronic speckle pattern interferometry,” Appl. Opt. 51, 4916–4926 (2012).
[CrossRef]

Yan, X.

X. Zhu, Z. Chen, C. Tang, Q. Mi, and X. Yan, “Application of two oriented partial differential equation filtering models on speckle fringes with poor quality and their numerically fast algorithms,” Appl. Opt. 52, 1814–1823 (2013).
[CrossRef]

C. Tang, N. Yang, H. Yan, and X. Yan, “The new second-order single oriented partial differential equations for optical interferometry fringes with high density,” Opt. Lasers Eng. 51,707–715 (2013).
[CrossRef]

Yang, N.

C. Tang, N. Yang, H. Yan, and X. Yan, “The new second-order single oriented partial differential equations for optical interferometry fringes with high density,” Opt. Lasers Eng. 51,707–715 (2013).
[CrossRef]

You, Y.

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

Zhou, D.

Zhou, P.

P. Zhou and K. E. Goodson, “Subpixel displacement and deformation gradient measurement using digital image/speckle correlation (DISC),” Opt. Eng. 40, 1613–1620 (2001).
[CrossRef]

Zhu, X.

AIP Conf. Proc.

H. Wang and Q. Kemao, “Coherence-enhancing diffusion and windowed Fourier filtering for fringe pattern denoising (II),” AIP Conf. Proc. 1236, 52–56 (2010).
[CrossRef]

Appl. Opt.

Comput. Math. Appl.

Y. Chen, B. C. Vemuri, and L. Wang, “Image denoising and segmentation via nonlinear diffusion,” Comput. Math. Appl. 39, 131–149 (2000).
[CrossRef]

IEEE Trans. Image Process.

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

Opt. Eng.

P. Zhou and K. E. Goodson, “Subpixel displacement and deformation gradient measurement using digital image/speckle correlation (DISC),” Opt. Eng. 40, 1613–1620 (2001).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

C. Tang, N. Yang, H. Yan, and X. Yan, “The new second-order single oriented partial differential equations for optical interferometry fringes with high density,” Opt. Lasers Eng. 51,707–715 (2013).
[CrossRef]

Opt. Lett.

Other

Wikipedia., “Euler–Lagrange equation,” http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation .

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Figures (7)

Fig. 1.
Fig. 1.

Computer-simulated noisy fringe pattern.

Fig. 2.
Fig. 2.

Iterations dependence of PSNR. (a) For F1; (b) for F2; (c) for F3; and (d) for F4.

Fig. 3.
Fig. 3.

Filtering results of Fig. 1. (a) By F1; (b) by F2; (c) by F3; and (d) by F4.

Fig. 4.
Fig. 4.

(a) Iterations dependence of PSNR for F1θ and (b) filtered image by F1θ.

Fig. 5.
Fig. 5.

Filtered images of Fig. 1. (a) By (F2+F3) and (b) by (F3+F4).

Fig. 6.
Fig. 6.

Computer-simulated fringe pattern, its filtered images and skeletons. (a) Initial image; (b) the ideal skeletons of the light fringes; (c), (e), (g), and (i) filtering results of (a) by DOPDEs (13), (41)–(43); (d), (f), (h), and (j) corresponding skeletons of the light fringes.

Fig. 7.
Fig. 7.

Real fringe pattern and its skeletons. (a) Initial image; (b), (d), (f), and (h) filtering results of (a) by DOPDEs (13), (41)–(43); (c), (e), (g), and (i) Corresponding skeletons of the light fringes.

Tables (7)

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Table 1. Best Possible Restoration Results of F1, F2, F3, and F4 in PSNR with the Chosen Δt and n

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Table 2. Best Possible Restoration Results of F1θ, F2θ, F3θ, and F4θ in PSNR with the Chosen Δt and n

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Table 3. Best Possible Restoration Results of F1g, F2g, F3g, and F4g in PSNR with the Chosen Δt and n

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Table 4. Best Possible Restoration Results of the Combined Single OPDEs in PSNR with the Chosen Δt and n

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Table 5. Best Possible Restoration Results of Combined Single OPDEs in PSNR with the Chosen Δt and n

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Table 6. Computational Time of Various Models for Single Iteration

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Table 7. Parameters Used in DOPDEs (13), (41)–(43)

Equations (48)

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E(u)=Ω12|uρ|2dxdy,
ut=2uρ2=uxxcos2θ+uyysin2θ+2uxysinθcosθ=F1,
ut=g(·)(uxxcos2θ+uyysin2θ+2uxysinθcosθ)=F1g,
g(|u|)=11+k|u|2,
E=Ω12|2uρ2|2dxdy.
ut=(uxxxxcos4θ+uyyxxsin2θcos2θ+2uxyxxsinθcos3θ+uyyyysin4θ+uxxyysin2θcos2θ+2uxyyysin3θcosθ),
ut=(uxxxxcos4θ+uyyyysin4θ+uyyxxcos2θsin2θ+uxxyycos2θsin2θ+2uxyxxcos3θsinθ+2uxyyycosθsin3θ+2uxxxycos3θsinθ+2uyyxycosθsin3θ+4uxyxysin2θcos2θ)=F2.
E(u)=Ω(|uρ|+|2uρ2|)dxdy.
ut=(s1cosθ)x+(s1sinθ)y2(s2cos2θ)x22(2s2sinθcosθ)xy2(s2sin2θ)y2,
s1=sign(uρ)=sign(uxcosθ+uysinθ)
s2=sign(2uρ2)=sign(uxxcos2θ+uyysin2θ+2uxysinθcosθ).
{ut=αg(|v|)2uρ2β(uI)|u|vt=a(t)|v|2vρ2b(vu),
ut=λ12uρ2+2uρ2=λ1(uxxsin2θ+uyycos2θ2uxysinθcosθ)+(uxxcos2θ+uyysin2θ+2uxysinθcosθ),
λ1={αnN0max(α,((dthr)/max(d)))N0<nN,
ut=λ12uρ2+λ22uρ2+r(ux,uy,θ,θx,θy)=λ12uρ2+2uρ2+θx(λ2λ1)(uysin2θuxcos2θ)+θy(1λ1)(uxcos2θ+uysin2θ),
E(u)=Ω|uρ|dxdy.
E(u)=Ω|2uρ2|dxdy.
Fux(Fux)y(Fuy)=0,
F=|uρ|=s1uρ=s1(uxcosθ+uysinθ).
Fu=0,Fux=s1cosθ,Fuy=s1sinθ.
(s1cosθ)x(s1sinθ)y=0.
ut=s1xcosθ+s1ysinθ=F3.
ut=2s2x2cos2θ2s2y2sin2θ22s2xysinθcosθ=F4,
F=12(uxcosθ+uysinθ)2.
x(Fux)=x(uxcos2θ+uysinθcosθ)y(Fuy)=y(uxsinθcosθ+uysin2θ).
ut=(uxcos2θ)x+(uysinθcosθ)x+(uxsinθcosθ)y+(uysin2θ)y=F1θ.
ut=(uxxcos4θ+uyysin2θcos2θ+2uxysinθcos3θ)x2(uxxcos2θsin2θ+uyysin4θ+2uxysin3θcosθ)y2(2uxxcos3θsinθ+2uyysin3θcosθ+4uxysin2θcos2θ)xy=F2θ,
ut=(s1cosθ)x+(s1sinθ)y=F3θ,
ut=2(s2cos2θ)x22(s2sin2θ)y22(2s2sinθcosθ)xy=F4θ.
E(u)=Ω(|uρ|+|2uρ2|)dxdy,
E(u)=Ω(|2uρ2|+12|uρ|2)dxdy,
E(u)=Ω(|uρ|+|2uρ2|+12|uρ|2)dxdy.
ut=F3+F4,
ut=F1+F4,
ut=F1+F3+F4.
ut=F1θ+F3θ
ut=F1θ+F3.
ut=F3θ+F4θ.
ut=g(·)(2s2x2cos2θ2s2y2sin2θ22s2xysinθcosθ)=F4g.
ut=g(·)((s1cosθ)x+(s1sinθ)y)=F3g,θ.
{ut=αg(|v|)(s1(u)xcosθ+s1(u)ysinθ)β(uI)|u|vt=a(t)|v|(s1(v)xcosθ+s1(v)ysinθ)b(vu).
ut=λ12uρ2+F1θ.
ut=λ12uρ2+(F3+F4).
ut=λ12uρ2+(F1θ+F4).
I(x,y)=A(x,y)+B(x,y)cosφ(x,y)+N(x,y),
φ(x,y)=2π{(9(xm/2)m)2+(9(yn/2)n)2}.
N(x,y)=14πa2I0k=1s[erf(xxka)erf(x+1xka)]×[erf(yyka)erf(y+1yka)],
PSNR=20log10(2551M×Ni=1Mj=1N(ui,ju0i,j)2),

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