Abstract

we propose the oriented spatial filter masks for filtering in electronic speckle pattern interferometry (ESPI) phase fringe patterns. We establish the oriented derivative operator that only highlights noise without edges of an image. The noise in the image can be removed while still preserving the edges simply by subtracting the oriented derivative image from original image, which can be implemented with one pass of the oriented spatial filter mask. Further, we make an improvement on the oriented spatial filter mask for enhancing the smoothness. The performance of the oriented spatial filter masks is demonstrated via application to a simulated speckle phase fringe pattern and an experimentally obtained phase fringe pattern and comparison with other directional filtering methods.

©2010 Optical Society of America

1. Introduction

Electronic speckle pattern interferometry (ESPI) is a well-known, whole-field technique for measurement. Accurate extraction of phase value is of fundamental importance because it is related with a physical quantity. The phase-shifting technique is one of the most commonly used methods for phase extraction because of its high accuracy [1,2]. However, owing to speckle decorrelation effects, the resultant phase fringe patterns are strongly affected by noise. Moreover, the easily blurred high density regions in fringe patterns increase the difficulty of removing the noise while preserving the features [3]. In order to make the phase unwrapping process easier, the noise presented in wrapped phase patterns must somehow be dealt with.

Filtering methods along the fringe orientation have been demonstrated to be a powerful tool for filtering in ESPI fringe patterns, especially for patterns with high fringe density. In the past few years some works have been developed for directional filtering such as the contoured window filter [4] and our tangent fitting filter [5], in which the small filtering window is designed along the fringe orientation, pixel by pixel, and a filtering method is performed in this directional filtering window. Recently, some efforts have been made toward the directional filtering methods without having to laboriously establish the small filtering window along the fringe orientation. In our former works [6,7], we have proposed the second-order oriented partial differential equations (PDEs) filtering models and the oriented-couple PDE models respectively, which make the diffusion only along fringe orientation. In the oriented PDEs-based filtering methods, filtering along fringe orientation for the entire image is simply realized through solving the oriented PDEs numerically. However, when the oriented PDEs-based filtering methods are applied to the phase patterns that are seriously noisy, they usually require many iterations [7]. Furthermore, single-direction diffusion is not sufficient in the regions with low fringe density [3]. For solving this problem, Wang et al. [3] improved the second-order oriented PDEs filtering models, and proposed an application of coherence-enhancing diffusion to fringe-pattern denoising. It smoothed a fringe pattern along directions both parallel and perpendicular to fringe orientation with suitable diffusion speeds, and could give the better results in both high and low fringe density. However, for implementing the diffusion, the distance map is needed, which seem to be inconvenient. J. Villa et al. [8] used a different and powerful mathematical tool for directional filtering. They constructed the oriented, regularized quadratic-cost function. The minimization of this cost function is equivalent to smoothing the image only along the fringe’s tangent direction, which is easily implemented using iterative methods.

In this paper, we derive the new directional filtering method with better performance, based on the spatial masks. It is well-known that the spatial masks have been widely used in smoothing and sharpening of an image [9], and the processing based on spatial masks may be straightforward. Unfortunately, not all previous spatial filter masks take into account fringe orientation, and they cannot carry out filtering along fringe orientation. Inspired by image sharpening using the Laplace operator, we establish the oriented derivative operator that only highlights noise without edges of an image. The noise in the image can be removed while still preserving the edges simply by subtracting the oriented operator image from original image, which can be implemented with one pass of the mask (we name the oriented spatial filter mask). Further, we make an improvement on the oriented spatial filter mask for enhancing the smoothness in the regions with low fringe density. We test the proposed method on the simulated and experimentally obtained ESPI phase fringe pattern, respectively, and comparison with the oriented-couple PDE method and the oriented, regularized quadratic-cost function method. In all cases, our method can give desired results. In the following sections, we will first review image sharpening based on the Laplace operator briefly and then describe in detail the derivation of our oriented spatial filter masks. Section 3 contains experiments and discussion. Section 4 provides the summary for this paper.

2. The derivation of our oriented spatial filter masks

2.1 A brief review of the Laplacian for sharpening

The principal objective of sharpening is to highlight fine details in an image or to enhance details that have been blurred. Sharpening can be accomplished by spatial differentiation [9]. The Laplacian is a simple, widely used isotropic derivative operator, which is implemented using the Laplace masks. Its use highlights gray-level discontinuities in an image and deemphasizes regions with slowly varying gray levels. Background features can be recovered while still preserving the sharpening effect of the Laplacian operation simply by adding the original and Laplacian image. There are several ways to define the Laplace masks. As noted, it is very important to keep in mind which mask is used. If the mask used has a negative center coefficient, then we subtract, rather than add, to the Laplacian image to obtain a sharpened result. Thus, the basic way in which the Laplacian is used for image enhancement is [9]:

g(x,y)=f(x,y)α2f(x,y)
Where α is a constant.

2.2 The derivation of the oriented spatial filter mask

It is well-known that filtering and sharpening are two distinct problems. Therefore, it is logical to conclude that we should subtract Laplacian image from the original for filtering. If the mask used has a negative center coefficient, then we add, rather than subtract for filtering. Contrary to Eq. (1), the basic way in which the Laplacian is used for image filtering is:

g(x,y)=f(x,y)+α2f(x,y)
However, as we can see that the Laplacian image for a noisy image includes both noise and image edges. Thus, the filtering method based on Eq. (2) will blur image edges seriously. For solving this problem, we propose a oriented derivative operator, which, for an image f(x,y), is defined as:
2ρf=2fρ2
Where ρis a length coordinate in the direction of the fringe or edge orientation. 2f/ρ2 is the second order partial derivative of f with respect to ρ. It is obvious that the gray values of an image along the image edge have little change. Unlike the Laplacian, the oriented derivative at image edges still has small values. Therefore, the oriented derivative image for a noisy image mainly contains noise without edges. The noise in the image can be removed while still preserving the edges simply by substituting ρ2f for 2fin Eq. (2). Thus, the basic way in which the oriented derivative is used for image filtering is:
g(x,y)=f(x,y)+αρ2f
It is noted the parameter α in Eqs. (1)-(4) have the opposite sign as center coefficient of Laplacian mask or oriented derivative mask.Since, the first order partial derivative of f(x,y) with respect to ρ is:
fρ=fρ=fxcosθ+fysinθ
Where fxand fyare the first order partial derivatives of f(x,y), and θ(x,y) denotes the angle between the orientation of ρ at pixel (x,y)and the x coordinate axis, which must be evaluated in advance. Here the spatial variables(x,y) are omitted for simplicity.Hence, the oriented derivative can be expressed as:
2ρf=fxxcos2θ+2fxycosθsinθ+fyysin2θ
Where fxxand fyyare the second order partial derivatives of f(x,y), fxyis the mixed partial derivative of f(x,y).

In order to be useful for digital image processing, Eq. (6) needs to be expressed in discrete form. Here we use the following fourth-order finite differences, derived by standard collocated formulas [10],

fx=f(x2,y)8f(x1,y)+8f(x+1,y)f(x+2,y)12 (7-a)
fy=f(x,y2)8f(x,y1)+8f(x,y+1)f(x,y+2)12 (7-b)
fxx=f(x2,y)+16f(x1,y)30f(x,y)+16f(x+1,y)f(x+2,y)12 (7-c)
fyy=f(x,y2)+16f(x,y1)30f(x,y)+16f(x,y+1)f(x,y+2)12 (7-d)
fxy=fx(x,y2)8fx(x,y1)+8fx(x,y+1)fx(x,y+2)12 (7-e)
Inserting Eq. (7) into Eq. (6), the discrete scheme of the oriented derivative ρ2fis obtained. Further, Eq. (4) can be expressed as:
g(x,y)=s=22t=22For(s,t)f(x+s,y+t)
Where Foris the oriented spatial filter mask,

For=α72(cosθsinθ8cosθsinθ6cos2θ8cosθsinθcosθsinθ8cosθsinθ64cosθsinθ96cos2θ64cosθsinθ8cosθsinθ6sin2θ96sin2θ72/α18096sin2θ6sin2θ8cosθsinθ64cosθsinθ96cos2θ64cosθsinθ8cosθsinθcosθsinθ8cosθsinθ6cos2θ8cosθsinθcosθsinθ)

2.3 The improvement of the oriented spatial filter mask

Using above oriented spatial filter mask, filtering can be made only along fringe orientation. As above-mentioned, the single-direction smoothing is not sufficient, and the level of smoothness may be low in the regions with low fringe density [3]. For improving the performance of the oriented spatial filter mask, we add a Gaussian filter FG(x,y)to the right side of Eq. (4)

g(x,y)=f(x,y)+α2ρf(x,y)+βFG(x,y)
Where β is a positive constant. FG(x,y) can be expressed as
FG(x,y)=s=22t=22MG(s,t)f(x+s,y+t)
Where MGis the Gaussian mask,
MG=152(1121112421248421242111211).
Inserting Eq. (11) into Eq. (10), we have
g(x,y)=s=22t=22Fim(s,t)f(x+s,y+t)
Where Fimis the improved, oriented spatial filter mask,
Fim=(αcosθsinθ72+β52αcosθsinθ9+β52αcos2θ12+β26αcosθsinθ9+β52αcosθsinθ72+β52αcosθsinθ9+β528αcosθsinθ9+β264αcos2θ3+β138αcosθsinθ9+β26αcosθsinθ9+β52αsin2θ12+β264αsin2θ3+β1315α2+2β134αsin2θ3+β13αsin2θ12+β26αcosθsinθ9+β528αcosθsinθ9+β264αcos2θ3+β138αcosθsinθ9+β26αcosθsinθ9+β52αcosθsinθ72+β52αcosθsinθ9+β52αcos2θ12+β26αcosθsinθ9+β52αcosθsinθ72+β52).
The complete filtered image is obtained with one pass of this mask, or by applying Eq. (13) for x=0,1,,M1and y=0,1,,N1, where M and N are sizes of image.

For implementing our filtering method, the orientation angle θ of each pixel has to be evaluated in advance. Recently, we proposed an approach for estimating orientation field of optical fringes with poor quality, which is based on the Fourier transform [11]. What follows is a brief synopsis of the method used in this study.

A window with its center point(x,y)is considered. Let(k,l)denote pixel position in the window. The orientation angle θ(x,y) of this point can be estimated by

θ(x,y)=12tan1(klE(ωk,ωl)sin(2θk,l)klE(ωk,ωl)cos(2θk,l))
where E(ωk,ωl)is power spectrum in the window. θk,l is predefined as by θk,l=tan1(k,l). Then the estimated orientation is smoothed by a Gaussian lowpass filter [11]. Here the sizes of the window are chosen as 27×27pixels.

3. Experiments and discussion

In this section, we test the improved, oriented spatial filter mask (IOSFM) on a simulated and a real ESPI phase image respectively, and compare with the oriented-couple PDE model (OCPDE) [7] and the oriented, regularized quadratic-cost function (ORQCF) [8]. As noted, applying directly a filter to original phase image often smears out the 2πdiscontinuities. This problem can be solved by the sine/cosine filter method [7]. The sine and cosine of the wrapped phase image are firstly calculated, and then are filtered individually. The arctangent of the ratio of the filtered sine image to the filtered cosine image is the denoised phase image.

Figure 1(a) is a simulated noisy ESPI phase pattern. Figures 1(b)-1(d) show the filtered images by ORQCF, OCPDE and IOSFM, respectively. For comparing, we take the same iterations in the three methods. We use 30 iterations in this case. Further, to quantitatively evaluate the performance, we calculate the image fidelity f, which is defined as f=1(I0I)2/I02, where I0 and I are the noiseless wrapped phase image and the estimated, wrapped phase image, respectively. A high fidelity value will indicate that the processed image is very similar to the noiseless one. The value f in the respective figure (Figs. 1(b)1(d)) is 0.8457, 0.8716, and 0.9048. Figure 2 shows a real ESPI phase pattern and its filtered images obtained by the three methods with 80 iterations.

 

Fig. 1 A simulated phase pattern and its filtered images. (a) Initial image. (b) ORQCF with λ=0.01,μ=12. (c) OCPDE with time step Δt=0.8. (d) IOSFM withα=0.35,β=1/33.

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Fig. 2 A real phase pattern and its filtered images. (a) Initial image. (b) ORQCFλ=0.01,μ=10. (c) OCPDE with time step Δt=0.8.(d) IOSFM withα=0.3,β=1/10.

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As we can see, the two original phase patterns are seriously noisy; especially for the real ESPI phase pattern, and the fringe density of the test images is variable. Obviously, our results are even better than the ones obtained with ORQCF and OCPDE. The consequence is accordant with the quantitative evaluation result. Moreover, our directional filter is easily implemented, in which the fringe orientation angle of each pixel is first estimated, and then the filtered image is obtained by applying Eq. (13).

Like OCPDE and ORQCF, there isn’t the explicit formula or a method to determine the parameters α and β in Eq. (10). They should be set according to the fringe density. Generally, the higher the fringe density is, the larger value α is and smaller value β is. According to our own experience in this method, the appropriate absolute value of αis somewhere between 0.25 and 0.36, and β is between 1/50 and 1/10.

4. Conclusion

We propose the oriented spatial filter masks for directional filtering of ESPI phase patterns. We test our IOSFM on a simulated and a real ESPI phase fringe pattern that contain variable fringe density, and heavy noise, and compare with the OCPDE and the ORQCF. Simulation and experimental results show that our IOSFM outperforms ORQCF and OCPDE.

This work is supported the by National Natural Science Foundation of China (NNSFC) (grant 60877001).We thank the anonymous reviewers for their constructive and helpful comments.

References and links

1. S. Nakadate and H. Saito, “Fringe scanning speckle-pattern interferometry,” Appl. Opt. 24(14), 2172–2180 (1985). [CrossRef]   [PubMed]  

2. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24(18), 3053–3058 (1985). [CrossRef]   [PubMed]  

3. H. Wang, Q. Kemao, W. Gao, F. Lin, and H. S. Seah, “Fringe pattern denoising using coherence-enhancing diffusion,” Opt. Lett. 34(8), 1141–1143 (2009). [CrossRef]   [PubMed]  

4. Q. Yu, X. Sun, X. Liu, and Z. Qiu, “Spin filtering with curve windows for interferometric fringe patterns,” Appl. Opt. 41(14), 2650–2654 (2002). [CrossRef]   [PubMed]  

5. C. Tang, W. Wang, H. Yan, and X. Gu, “Tangent least-squares fitting filtering method for electrical speckle pattern interferometry phase fringe patterns,” Appl. Opt. 46(15), 2907–2913 (2007). [CrossRef]   [PubMed]  

6. 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(19), 2179–2181 (2008). [CrossRef]   [PubMed]  

7. 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(7), 5606–5617 (2009). [CrossRef]   [PubMed]  

8. J. Villa, J. A. Quiroga, and I. De la Rosa, “Regularized quadratic cost function for oriented fringe-pattern filtering,” Opt. Lett. 34(11), 1741–1743 (2009). [CrossRef]   [PubMed]  

9. R. C. Gonzalez, and R. E. Woods, Digital image processing.2nd ed. (Publishing House of Electronics Industry, Beijing, 2002).

10. N. A. Kampanis and J. A. Ekaterinaris, “A staggered grid, high-order accurate method for the incompressible Navier–Stokes equations,” J. Comput. Phys. 215(2), 589–613 (2006). [CrossRef]  

11. C. Tang, Z. Wang, L. Wang, J. Wu, T. Gao, and S. Yan, “Estimation of fringe orientation for optical fringe patterns with poor quality based on Fourier transform,” Appl. Opt. 49(4), 554–561 (2010). [CrossRef]   [PubMed]  

References

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  1. S. Nakadate and H. Saito, “Fringe scanning speckle-pattern interferometry,” Appl. Opt. 24(14), 2172–2180 (1985).
    [Crossref] [PubMed]
  2. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24(18), 3053–3058 (1985).
    [Crossref] [PubMed]
  3. H. Wang, Q. Kemao, W. Gao, F. Lin, and H. S. Seah, “Fringe pattern denoising using coherence-enhancing diffusion,” Opt. Lett. 34(8), 1141–1143 (2009).
    [Crossref] [PubMed]
  4. Q. Yu, X. Sun, X. Liu, and Z. Qiu, “Spin filtering with curve windows for interferometric fringe patterns,” Appl. Opt. 41(14), 2650–2654 (2002).
    [Crossref] [PubMed]
  5. C. Tang, W. Wang, H. Yan, and X. Gu, “Tangent least-squares fitting filtering method for electrical speckle pattern interferometry phase fringe patterns,” Appl. Opt. 46(15), 2907–2913 (2007).
    [Crossref] [PubMed]
  6. 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(19), 2179–2181 (2008).
    [Crossref] [PubMed]
  7. 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(7), 5606–5617 (2009).
    [Crossref] [PubMed]
  8. J. Villa, J. A. Quiroga, and I. De la Rosa, “Regularized quadratic cost function for oriented fringe-pattern filtering,” Opt. Lett. 34(11), 1741–1743 (2009).
    [Crossref] [PubMed]
  9. R. C. Gonzalez, and R. E. Woods, Digital image processing.2nd ed. (Publishing House of Electronics Industry, Beijing, 2002).
  10. N. A. Kampanis and J. A. Ekaterinaris, “A staggered grid, high-order accurate method for the incompressible Navier–Stokes equations,” J. Comput. Phys. 215(2), 589–613 (2006).
    [Crossref]
  11. C. Tang, Z. Wang, L. Wang, J. Wu, T. Gao, and S. Yan, “Estimation of fringe orientation for optical fringe patterns with poor quality based on Fourier transform,” Appl. Opt. 49(4), 554–561 (2010).
    [Crossref] [PubMed]

2010 (1)

2009 (3)

2008 (1)

2007 (1)

2006 (1)

N. A. Kampanis and J. A. Ekaterinaris, “A staggered grid, high-order accurate method for the incompressible Navier–Stokes equations,” J. Comput. Phys. 215(2), 589–613 (2006).
[Crossref]

2002 (1)

1985 (2)

Chang, Y.

Creath, K.

Cui, X.

De la Rosa, I.

Ekaterinaris, J. A.

N. A. Kampanis and J. A. Ekaterinaris, “A staggered grid, high-order accurate method for the incompressible Navier–Stokes equations,” J. Comput. Phys. 215(2), 589–613 (2006).
[Crossref]

Gao, T.

Gao, W.

Gu, X.

Han, L.

Kampanis, N. A.

N. A. Kampanis and J. A. Ekaterinaris, “A staggered grid, high-order accurate method for the incompressible Navier–Stokes equations,” J. Comput. Phys. 215(2), 589–613 (2006).
[Crossref]

Kemao, Q.

Lin, F.

Liu, X.

Nakadate, S.

Qiu, Z.

Quiroga, J. A.

Ren, H.

Saito, H.

Seah, H. S.

Sun, X.

Tang, C.

Tang, K.

Villa, J.

Wang, H.

Wang, L.

Wang, W.

Wang, X.

Wang, Z.

Wu, J.

Yan, H.

Yan, S.

Yu, Q.

Zhou, D.

Appl. Opt. (5)

J. Comput. Phys. (1)

N. A. Kampanis and J. A. Ekaterinaris, “A staggered grid, high-order accurate method for the incompressible Navier–Stokes equations,” J. Comput. Phys. 215(2), 589–613 (2006).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Other (1)

R. C. Gonzalez, and R. E. Woods, Digital image processing.2nd ed. (Publishing House of Electronics Industry, Beijing, 2002).

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Figures (2)

Fig. 1
Fig. 1 A simulated phase pattern and its filtered images. (a) Initial image. (b) ORQCF with λ = 0.01 , μ = 12 . (c) OCPDE with time step Δ t = 0.8 . (d) IOSFM with α = 0.35 , β = 1 / 33 .
Fig. 2
Fig. 2 A real phase pattern and its filtered images. (a) Initial image. (b) ORQCF λ = 0.01 , μ = 10 . (c) OCPDE with time step Δ t = 0.8 .(d) IOSFM with α = 0.3 , β = 1 / 10 .

Equations (19)

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g ( x , y ) = f ( x , y ) α 2 f ( x , y )
g ( x , y ) = f ( x , y ) + α 2 f ( x , y )
2 ρ f = 2 f ρ 2
g ( x , y ) = f ( x , y ) + α ρ 2 f
f ρ = f ρ = f x cos θ + f y sin θ
2 ρ f = f x x cos 2 θ + 2 f x y cos θ sin θ + f y y sin 2 θ
f x = f ( x 2 , y ) 8 f ( x 1 , y ) + 8 f ( x + 1 , y ) f ( x + 2 , y ) 12
f y = f ( x , y 2 ) 8 f ( x , y 1 ) + 8 f ( x , y + 1 ) f ( x , y + 2 ) 12
f x x = f ( x 2 , y ) + 16 f ( x 1 , y ) 30 f ( x , y ) + 16 f ( x + 1 , y ) f ( x + 2 , y ) 12
f y y = f ( x , y 2 ) + 16 f ( x , y 1 ) 30 f ( x , y ) + 16 f ( x , y + 1 ) f ( x , y + 2 ) 12
f x y = f x ( x , y 2 ) 8 f x ( x , y 1 ) + 8 f x ( x , y + 1 ) f x ( x , y + 2 ) 12
g ( x , y ) = s = 2 2 t = 2 2 F o r ( s , t ) f ( x + s , y + t )
F o r = α 72 ( cos θ sin θ 8 cos θ sin θ 6 cos 2 θ 8 cos θ sin θ cos θ sin θ 8 cos θ sin θ 64 cos θ sin θ 96 cos 2 θ 64 cos θ sin θ 8 cos θ sin θ 6 sin 2 θ 96 sin 2 θ 72 / α 180 96 sin 2 θ 6 sin 2 θ 8 cos θ sin θ 64 cos θ sin θ 96 cos 2 θ 64 cos θ sin θ 8 cos θ sin θ cos θ sin θ 8 cos θ sin θ 6 cos 2 θ 8 cos θ sin θ cos θ sin θ )
g ( x , y ) = f ( x , y ) + α 2 ρ f ( x , y ) + β F G ( x , y )
F G ( x , y ) = s = 2 2 t = 2 2 M G ( s , t ) f ( x + s , y + t )
M G = 1 52 ( 1 1 2 1 1 1 2 4 2 1 2 4 8 4 2 1 2 4 2 1 1 1 2 1 1 ) .
g ( x , y ) = s = 2 2 t = 2 2 F i m ( s , t ) f ( x + s , y + t )
F i m = ( α cos θ sin θ 72 + β 52 α cos θ sin θ 9 + β 52 α cos 2 θ 12 + β 26 α cos θ sin θ 9 + β 52 α cos θ sin θ 72 + β 52 α cos θ sin θ 9 + β 52 8 α cos θ sin θ 9 + β 26 4 α cos 2 θ 3 + β 13 8 α cos θ sin θ 9 + β 26 α cos θ sin θ 9 + β 52 α sin 2 θ 12 + β 26 4 α sin 2 θ 3 + β 13 1 5 α 2 + 2 β 13 4 α sin 2 θ 3 + β 13 α sin 2 θ 12 + β 26 α cos θ sin θ 9 + β 52 8 α cos θ sin θ 9 + β 26 4 α cos 2 θ 3 + β 13 8 α cos θ sin θ 9 + β 26 α cos θ sin θ 9 + β 52 α cos θ sin θ 72 + β 52 α cos θ sin θ 9 + β 52 α cos 2 θ 12 + β 26 α cos θ sin θ 9 + β 52 α cos θ sin θ 72 + β 52 )
θ ( x , y ) = 1 2 tan 1 ( k l E ( ω k , ω l ) sin ( 2 θ k , l ) k l E ( ω k , ω l ) cos ( 2 θ k , l ) )

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