- Email: [email protected]
Correlation fringe pattern of ESPI generated method based on the orientation partial differential equation
Optics Communications 310 (2014) 85–89
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Correlation fringe pattern of ESPI generated method based on the orientation partial differential equation Fang Zhang a, Zhitao Xiao a,n, Lei Geng a, Jun Wu a, Hongqiang Li a, Jiangtao Xi b, Jinjiang Wang c,d a
School of Electronics and Information Engineering, Tianjin Polytechnic University, Tianjin 300387, China School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, Wollongong, NSW 2522, Australia c College of Precision Instrument and Opto-electronics Engineering of Tianjin University, Tianjin 300072, China d Key laboratory of Opto-electronic Information Technology, Ministry of Education (Tianjin University), Tianjin 300072, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 12 March 2013 Received in revised form 23 July 2013 Accepted 26 July 2013 Available online 7 August 2013
Fringe-generation is one of the most exciting problems in electronic speckle pattern interferometry. We present an efficient correlation fringe pattern generation method based on the orientation partial differential equation, which derives the fringe pattern with smooth fringes without speckle noise from two original speckle patterns. The average calculations in the correlation method are replaced by the anisotropic filter provided by the oriented partial differential equation. Because the filter along fringe orientation for the entire image can be simply realized based on the oriented partial differential equation, our method does not spoil the phase distribution, and needs not calculate each gray average in the corresponding filter window pixel by pixel. The experimental results show that it is an effective fringe pattern generation method for electronic speckle pattern interferometry. & 2013 Elsevier B.V. All rights reserved.
Keywords: Electronic speckle pattern interferometric fringe Correlation fringe pattern Orientation partial differential equation
1. Introduction Electronic speckle pattern interferometry (ESPI) is a wellknown, nondestructive, whole-field technique for applying to vibration measurement, measurement of displacements and their derivatives, and three-dimensional object reconstruction [1,2]. In order to extract the information from speckle patterns, subtraction, addition or multiplication between two original speckle patterns with a phase difference are usually used to generate a fringe pattern, which is full of strong grain-shape random noise and is of low contrast. This leads to heavy restrain to phase value extraction. Schmitt proposed a direct correlation method to obtain fringe patterns from original speckle patterns [3]: the correlation coefficients of each pixel between two original speckle patterns with a phase difference are calculated within rectangular windows and mapped the fringe patterns. The important advantage of the direct correlation is that the illumination over the surface of the object need not be perfectly uniform, because the correlation coefficient calculated is automatically normalized over the entire fringe pattern. Therefore the method can be used under poor illumination conditions. Although the direct correlation method gives the
n
Corresponding author. Tel.: +86 22 83955418. E-mail address: [email protected] (Z. Xiao).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.07.072
high contrast fringe patterns than the classical subtraction method, the resultant fringe patterns still have some large size noises, which spoil the fine phase distribution. The reason of this question is that, to avoid the impact of noise when calculating correlation, gray average needs to be computed in the rectangular window, but the real phase is not a constant inside the calculation window [3,4]. Yu proposed the contoured correlation fringe pattern method [4]. They first establish fringe contour windows coinciding with fringe contours of equal-phase from the corresponding subtraction speckle fringe pattern (seen from Fig. 1) and then the correlation is performed only on these contour windows instead of rectangular windows. With the contoured correlation method, the ESPI fringe patterns have good quality. However, it is noticed that to calculate the correlation coefficient of each pair of pixels between two speckle patterns, a contour window for the given point must be established along the fringe orientation and the correlation is performed in the established contour window. It is needed to move to the next point and repeat the above main steps, pixel by pixel. As noise in speckle patterns affects correlation fringe and gray average of the rectangular window spoils the phase distribution, the filter method based on the phase distribution should be adopted to reduce the noise. In recent years, partial differential equation (PDE) image processing method has been studied as a useful tool for ESPI image denoising [5–8]. In Ref. [5], some classical PDEs denoising models are first introduced for the ESPI fringe patterns. Ref. [6] evaluated the performance of a few
86
F. Zhang et al. / Optics Communications 310 (2014) 85–89
Denote β ¼ ϕr ϕo and
The contour window along the fringe orientation
o cos β 4 mn ¼ o cos ðβ þ φÞ 4 mn ¼ 0
ð5Þ
o I 1 4 mn ¼ 2 o I 4 mn
ð6Þ
o I 2 4 mn ¼ 2 o I 4 mn
ð7Þ
Since I o , I r and β are independent from each other, their product's average-value equals to their average-value's product, i.e. o I 2 4 mn ¼ 2 o I 4 2mn
ð8Þ
Therefore,
pffiffiffiffiffiffiffi o I 1 I 2 4 mn ¼ o ðI r þ I o Þ2 þ 2ðI r þ I o Þ I r I o cos β pffiffiffiffiffiffiffi þ2ðI r þ I o Þ I r I o cos ðβ þ φÞ þ4I r I o cos β cos ðβ þ φÞ 4 mn
fringe
Fig. 1. The contour window along the fringe orientation.
¼ oðI r þ I o Þ2 þ 4I r I o cos β cos ðβ þ φÞ 4 mn representative second-order PDEs denoising models quantitatively. Ref. [7] proposed the oriented PDE model which made the diffusion only along the fringe orientation and protected the boundary of the fringe during filtering. In Ref. [8], the coherence-enhancing diffusion is presented for fringe pattern denoising, which smoothed a fringe pattern along directions both parallel and perpendicular to fringe orientation with different diffusion speeds to more effectively reduce noise. Although the PDE processing method has popular application of denoising for the ready fringe, it is never used to generate fringe patterns. In this paper, we propose the correlation fringe pattern of ESPI generated method based on orientation partial differential equation, which derives the fringe pattern with smooth fringes without speckle noise from two original speckle patterns. When calculating the correlation coefficient, the calculation for gray average is substituted by the oriented PDE filter in our implementation. Our method has two main advantages. Firstly, because the filter is permitted only along the fringe orientation, it does not spoil the phase distribution. Secondly, it need not compute the filter window pixel by pixel, and the calculation for curve follow is substituted by the coordinate transformation which can be simply realized based on PDE for the entire image.
2. Direct correlation method to obtain fringe patterns In ESPI, two original speckle patterns with a phase difference φðx; yÞ are first acquired by a digital imaging system. The intensities of the two speckle patterns can be represented as follows: pffiffiffiffiffiffiffi I 1 ¼ I o þ I r þ 2 I o I r cos ðϕr ϕo Þ ð1Þ I2 ¼ Io þ Ir þ 2
pffiffiffiffiffiffiffi I o I r cos ðϕr ϕo þ φÞ
ð2Þ
where I o ðx; yÞ and I r ðx; yÞ are the intensities of the object's scattered light and reference beam, respectively. ϕo ðx; yÞ and ϕr ðx; yÞ are the phases of random speckles, and φðx; yÞ is the phase difference between the two speckle patterns resulting from some displacement. The correlation between the two speckle patterns can be calculated by [4] Cðx; yÞ ¼ h
o ðI 1 o I1 4 mn ÞðI 2 o I 2 4 mn Þ 4 mn i1=2 h i1=2 oðI 2 o I 2 4 mn Þ2 4 mn
o ðI 1 o I 1 4 mn Þ2 4 mn
ð3Þ where o I 1 4 mn , o I 2 4 mn are the average values of I1, I2 in the m n window, respectively. According to the speckle statistic theory, assume oI r 4 mn ¼ oI o 4 mn ¼ o I 4 mn
ð4Þ
¼ 6 o I 2 4 mn þ o 2I r I o ð cos ð2β þ φÞ þ cos φÞ 4 mn 6 o I 2 4 mn þ 2 oI 2 4 mn cos φ
ð9Þ
o ðI 1 o I 1 4 mn ÞðI 2 o I 2 4 mn Þ 4 mn ¼ oI 1 I 2 4 mn o I 1 4 mn o I 2 4 mn 2 o I 2 4 mn þ 2 oI 2 4 mn cos φ
ð10Þ
o ðI 1 o I 1 4 mn Þ2 4 mn ¼ oI 21 2I 1 o I 1 4 mn þ oI 1 4 2mn 4 mn ¼ 4 oI 4 2mn
ð11Þ
o ðI 2 o I 2 4 mn Þ2 4 mn ¼ oI 22 2I 2 o I 2 4 mn þ oI 2 4 2mn 4 mn ¼ 4 oI 4 2mn ð12Þ Under the condition that the phase term of φ is a constant over the whole operation window of m n, the sign ‘ ’ in Eqs. (9) and (10) can be substituted by the sign ‘ ¼’. Substituting Eqs. (10)–(12) into Eq. (3), the correlation coefficients can be rewritten as follows Cðx; yÞ ¼
2 oI 2 4 mn þ 2 o I 2 4 mn cos φ ð cos φ þ 1Þ ¼ 2 4 o I 4 2mn
ð13Þ
This relation indicates that the correlation fringe pattern is a cosine fringe pattern, which is the same as normal interferometric fringe patterns. When ϕðx; yÞ ¼ 2nπ, I1(x,y) ¼ I2(x,y) at the point (x,y), C(x,y) ¼1 and bright fringes are obtained. When ϕðx; yÞ ¼ ð2n þ 1Þπ, that means no correlation, C(x,y)¼0 and it forms dark fringes [4]. Based on the correlation coefficient C, the normalized correlation fringe pattern I cor can be obtained, i.e. 1 cos φ ð14Þ I cor ¼ 255ð1CÞ ¼ 255 2 Eq. (14) maintains the correlation fringe pattern has the same phase field as that of subtraction speckle fringe pattern. It is noticed that during above deduction, the phase term of ϕðx; yÞ is assumed as a constant within the correlation windows. When the correlation is performed within rectangular windows, this assumption is no valid. This is the reason why the direct correlation method brings certain error. The contoured correlation method performs correlation only on the fringe contour windows on which the phase term is a constant, and obtains good fringe patterns [4]. However, it needs to move the contour window pixel by pixel and repeat the tedious calculations. 3. The orientation partial differential equation filter method As noise in speckle patterns affects correlation fringe and gray average of the rectangular window spoils the phase distribution,
F. Zhang et al. / Optics Communications 310 (2014) 85–89
y
conditions. In model (19), the diffusion is made only along the fringe orientation, in which the phase term of ϕðx; yÞ is a constant. Therefore the orientated PDE can restrain noise of the original speckle patterns and bring no error for correlation. In order to solve Eq. (19) numerically, the equation has to be discretized. The image is represented by M N matrices of intensity values. So, for any function (i.e., image) u(x, y), we let ui,j denote u(i, j) for 1 oio M, 1 oj oN. The evolution equations give images at times t n ¼ nΔt. We denote uði; j; t n Þ by uni;j . The time derivative ut at (i, j, tn) is approximated by the forward difference
fringe tangent direction T
fringe normal direction N
θ
x
fringe
ðut Þni;j ¼
Fig. 2. The relationship between the rectangular coordinate system (x,y) and the !! inner coordinate system ( T , N ).
the filter method based on the phase distribution should be adopted to reduce the noise. Fig. 2 shows the inner coordinate !! system ( T , N ) established along the fringe tangent direction and the fringe normal direction. Let Iðx; yÞ : Ω-R represents the initial noisy image, where Iðx; yÞ is the gray-level value and Ω is an image domain. Introducing an artificial time t. The oriented PDE filter makes the diffusion only along the fringe orientation and protects the boundary of the fringe during filtering, therefore the image deforms according to ∂u ∂2 u ¼ 2; ∂t ∂T
uðx; y; 0Þ ¼ Iðx; yÞ
ð15Þ
where uðx; y; tÞ : Ω ½0; þ1Þ-R is the evolving image, ∂u ∂t is the ∂2 u first-order partial derivatives of u about time coordinate, ∂T 2 is the second-order partial derivatives of u along the fringe tangent direction. In mathematics, the rectangular coordinate (x,y) and the inner !! coordinate ( T , N ) can be connected by the coordinate transformation, i.e. ∂ ∂ ∂ ¼ cos θ þ sin θ ∂T ∂x ∂y ∂2 ∂T
2
¼
∂2 ∂2 ∂2 2 cos θ sin θ cos 2 θ þ 2 sin θ þ 2 ∂x∂y ∂x2 ∂y
ð16Þ ð17Þ
where θ denotes the angle between the fringe orientation and the x coordinate. For fringe patterns, the fringe orientation θ for point (x,y) is estimated within its neighborhood of a small window as follows [9]: θðx; yÞ ¼
∑k;l 2ux ðk; lÞuy ðk; lÞ 1 tan 1 2 ∑k;l ðu2y ðk; lÞu2x ðk; lÞÞ
ð18Þ
where k and l are the subscripts of the pixel point in this window. Ref. [9] presents the error between the true orientation and the extracted orientation with different neighborhood window size in Eq. (18). The conclusion is that when the window size is larger than three times of the local fringe width, the orientation error is less than 0.2. (In Ref. [9], the error is the average value of j sin ½θðx; yÞθT ðx; yÞj of the calculated area, where θðx; yÞ and θT ðx; yÞ are the calculated orientation and the true orientation respectively.) Therefore, the oriented PDE model can be expressed as following [7] ∂u ¼ uxx cos 2 θ þ uyy sin 2 θ þ 2uxy sin θ cos θ ∂t
87
ð19Þ
where uxx, uyy and uxy are the second-order partial derivatives of u about the rectangular coordinate. The numerical solutions of the PDE give the filtered images with the original image I(x,y) as initial
n unþ1 i;j ui;j
ð20Þ
Δt
where Δt is step-size in time. The spatial derivatives are as follows ðuxx Þni;j ¼ uniþ1;j 2uni;j þ uni1;j
ð21aÞ
ðuyy Þni;j ¼ uni;jþ1 2uni;j þ uni;j1
ð21bÞ
ðuxy Þni;j ¼ ðuniþ1;jþ1 uni1;jþ1 uniþ1;j1 þ uni1;j1 Þ=4
ð21cÞ
Then the diffusion term is expressed by the following: 2
¼ uni;j þ Δt½ðuxx Þni;j cos 2 θi;j þ ðuyy Þni;j sin θi;j unþ1 i;j þ2ðuxy Þni;j
sin θi;j cos θi;j
ð22Þ
4. Correlation method based on orientation partial differential equation to obtain fringe patterns In this paper, we propose to calculate the correlation based on the oriented PDE model. The average calculations in the correlation method are replaced by the anisotropic filter provided by the oriented PDE. The correlation can be obtained from Cðx; yÞ ¼
o ðI 1 o I 1 4 f ÞðI 2 o I 2 4 f Þ 4 f ½ oðI 1 oI 1 4 f Þ2 4 f 1=2 ½ oðI 2 oI 2 4 f Þ2 4 f 1=2
ð23Þ
where o U 4 f denotes the filtered result by the oriented PDE. To illuminate the main processing steps and the result of the proposed method, two groups of speckle patterns are given. Let us take Fig. 3 for example. Fig. 3(a) and (b) shows the undeformed speckle pattern and the deformed speckle pattern, respectively. The main processing steps of the proposed method are as follows. (1) The two original speckle patterns (Fig. 3(a) and (b)) with a phase difference are subtracted from each other to form a subtraction speckle fringe pattern (Fig. 3(c)), and then the fringe orientation (Fig. 3(d)) is determined from the subtraction speckle fringe pattern based on Eq. (18). (2) Performing the anisotropic filter for I1, I2, ðI 1 o I 1 4 f Þ2 , ðI 2 o I 2 4 f Þ2 and ðI 1 o I 1 4 f ÞðI 2 o I 2 4 f Þ based on the oriented PDE (i.e. Eq. (19)). (3) Calculating correlation C(x,y) conforming to Eq. (23). (4) Generating the correlation fringe pattern with I cor ¼ 255 ð1CÞ (Fig. 3(e)). Because the filter along fringe orientation for the entire image can be simply realized by the coordinate transformation based on PDE, our method need not calculate each gray average in the corresponding filter window pixel by pixel. We tested the proposed algorithm on two speckle patterns. Fig. 3 shows the comparison results of the subtraction fringe pattern (SFP) and the correlation fringe pattern based on orientation partial differential equation (OPDE-CFP) on the usual speckle pattern. Fig. 4 shows the similar comparison results on the usual speckle pattern. The fringe orientation is calculated within 31 31 pixels window, which is about three times of the bigger local
F. Zhang et al. / Optics Communications 310 (2014) 85–89
y (Grays)
88
200
100
0
0
50
100
150
200
150
200
y (Grays)
x (Pixels) 200
100
0
0
50
100
x (Pixels) Fig. 3. Comparison of two fringe pattern generated method on the usual speckle pattern. (a) The undeformed speckle pattern; (b) The deformed speckle pattern; (c) The subtraction fringe pattern; (d) The orientation map of the fringe pattern; (e) The correlation fringe pattern generated based on orientation partial differential equation; (f) Gray-distribution curve of (c) (upper) and (e) (lower) of the middle intersection (from the left dot to the right dot).
fringe width. The PDE filter parameters include the time step Δt ¼ 0:2, the iteration time n ¼50 for I1, I2 and n ¼250 for ðI 1 oI 1 4 f Þ2 , ðI 2 o I 2 4 f Þ2 , ðI 1 o I 1 4 f ÞðI 2 oI 2 4 f Þ. From Figs. 3(c) and 4(c), one can see that the noise in SFP is high, while OPDE-CFP is smooth and has no speckle noise at all, neither the high-density fringe nor the low-density fringe. Figs. 3 (f) and 4(f) show the intensity distribution of the middle intersection (from the left dot to the right dot) of Fig. 3(c) and (e) and Fig. 4(c) and (e), respectively. It is noticed that the SFP is very noisy, which is shown on the upper part of Figs. 3(f) and 4(f), while the OPDE-CFP is smooth and its intensities and amplitudes are general uniform, which is shown on the lower part of Figs. 3(f) and 4(f). This result indicates that the proposed method is suitable for the complicated applications of the fringe patterns with varying fringe density and different fringe orientation.
5. Conclusion We present an efficient correlation fringe pattern generation method based on the orientation partial differential equation, which derives the fringe pattern with smooth fringes without speckle noise from the two original speckle patterns. First of all, two original speckle patterns I1, I2 with a phase difference are subtracted from each other to form a subtraction speckle fringe pattern and then the fringe orientation is determined from the subtraction speckle fringe pattern. Then, performing the anisotropic filter for I1, I2 based on the oriented partial differential equation. Finally, calculating correlation and generating correlation fringe pattern. The proposed method has two main advantages. Firstly, it does not spoil the phase distribution. Secondly, it need not calculate
y (Grays)
F. Zhang et al. / Optics Communications 310 (2014) 85–89
89
200
100
0
0
50
100
150
200
150
200
y (Grays)
x (Pixels) 200
100
0
0
50
100
x (Pixels) Fig. 4. Comparison of two fringe pattern generated method on the shear speckle pattern. (a) The undeformed speckle pattern; (b) The deformed speckle pattern; (c) The subtraction fringe pattern; (d) The orientation map of the fringe pattern; (e) The correlation fringe pattern generated based on orientation partial differential equation; (f) Gray-distribution curve of (c) (upper) and (e) (lower) of the middle intersection (from the left dot to the right dot).
each gray average in the corresponding filter window pixel by pixel, and the calculation for curve follow is substituted by the coordinate transformation which can be simply realized based on PDE for the entire image. The experimental results show that our method is capable of significantly improving the quality of the fringe patterns and helpful for subsequent phase extraction.
Acknowledgment Sponsored by the National Nature Science Foundation of China under grant no.61102150, and Open Foundation of Key laboratory of Opto-electronic Information Technology of Ministry of
Education (Tianjin University) under grant no.2012KFKT003. We thank the referees for their valuable suggestions. References [1] V. Bavigadda, R. Jallapuram, E. Mihaylova, V. Toal, Optics Letters 35 (2010) 3273. [2] K. Genovese, L. Lamberti, C. Pappalettere, Optics and Lasers in Engineering 42 (2004) 543. [3] D.R. Schmitt, R.W. Hunt, Applied Optics 36 (1997) 8848. [4] Q. Yu, S. Fu, X. Yang, X. Sun, X. Liu, Optics Express 12 (2004) 75. [5] C. Tang, F. Zhang, H. Yan, Z. Chen, Optics Communications 260 (2006) 91. [6] C. Tang, F. Zhang, B. Li, H. Yan, Applied Optics 45 (2006) 7392. [7] C. Tang, L. Han, H. Ren, Z. Tao Gao, Wang, K. Tang, Optics Express 17 (2009) 5606. [8] H. Wang, Q. Kernao, W. Gao, F. Lin, H.S. Seah, Optics Letters 35 (2009) 1141. [9] X. Yang, Q. Yu, S. Fu, Optics Communications 273 (2007) 60.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Correlation fringe pattern of ESPI generated method based on the orientation partial differential equation Fang Zhang a, Zhitao Xiao a,n, Lei Geng a, Jun Wu a, Hongqiang Li a, Jiangtao Xi b, Jinjiang Wang c,d a
School of Electronics and Information Engineering, Tianjin Polytechnic University, Tianjin 300387, China School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, Wollongong, NSW 2522, Australia c College of Precision Instrument and Opto-electronics Engineering of Tianjin University, Tianjin 300072, China d Key laboratory of Opto-electronic Information Technology, Ministry of Education (Tianjin University), Tianjin 300072, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 12 March 2013 Received in revised form 23 July 2013 Accepted 26 July 2013 Available online 7 August 2013
Fringe-generation is one of the most exciting problems in electronic speckle pattern interferometry. We present an efficient correlation fringe pattern generation method based on the orientation partial differential equation, which derives the fringe pattern with smooth fringes without speckle noise from two original speckle patterns. The average calculations in the correlation method are replaced by the anisotropic filter provided by the oriented partial differential equation. Because the filter along fringe orientation for the entire image can be simply realized based on the oriented partial differential equation, our method does not spoil the phase distribution, and needs not calculate each gray average in the corresponding filter window pixel by pixel. The experimental results show that it is an effective fringe pattern generation method for electronic speckle pattern interferometry. & 2013 Elsevier B.V. All rights reserved.
Keywords: Electronic speckle pattern interferometric fringe Correlation fringe pattern Orientation partial differential equation
1. Introduction Electronic speckle pattern interferometry (ESPI) is a wellknown, nondestructive, whole-field technique for applying to vibration measurement, measurement of displacements and their derivatives, and three-dimensional object reconstruction [1,2]. In order to extract the information from speckle patterns, subtraction, addition or multiplication between two original speckle patterns with a phase difference are usually used to generate a fringe pattern, which is full of strong grain-shape random noise and is of low contrast. This leads to heavy restrain to phase value extraction. Schmitt proposed a direct correlation method to obtain fringe patterns from original speckle patterns [3]: the correlation coefficients of each pixel between two original speckle patterns with a phase difference are calculated within rectangular windows and mapped the fringe patterns. The important advantage of the direct correlation is that the illumination over the surface of the object need not be perfectly uniform, because the correlation coefficient calculated is automatically normalized over the entire fringe pattern. Therefore the method can be used under poor illumination conditions. Although the direct correlation method gives the
n
Corresponding author. Tel.: +86 22 83955418. E-mail address: [email protected] (Z. Xiao).
0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.07.072
high contrast fringe patterns than the classical subtraction method, the resultant fringe patterns still have some large size noises, which spoil the fine phase distribution. The reason of this question is that, to avoid the impact of noise when calculating correlation, gray average needs to be computed in the rectangular window, but the real phase is not a constant inside the calculation window [3,4]. Yu proposed the contoured correlation fringe pattern method [4]. They first establish fringe contour windows coinciding with fringe contours of equal-phase from the corresponding subtraction speckle fringe pattern (seen from Fig. 1) and then the correlation is performed only on these contour windows instead of rectangular windows. With the contoured correlation method, the ESPI fringe patterns have good quality. However, it is noticed that to calculate the correlation coefficient of each pair of pixels between two speckle patterns, a contour window for the given point must be established along the fringe orientation and the correlation is performed in the established contour window. It is needed to move to the next point and repeat the above main steps, pixel by pixel. As noise in speckle patterns affects correlation fringe and gray average of the rectangular window spoils the phase distribution, the filter method based on the phase distribution should be adopted to reduce the noise. In recent years, partial differential equation (PDE) image processing method has been studied as a useful tool for ESPI image denoising [5–8]. In Ref. [5], some classical PDEs denoising models are first introduced for the ESPI fringe patterns. Ref. [6] evaluated the performance of a few
86
F. Zhang et al. / Optics Communications 310 (2014) 85–89
Denote β ¼ ϕr ϕo and
The contour window along the fringe orientation
o cos β 4 mn ¼ o cos ðβ þ φÞ 4 mn ¼ 0
ð5Þ
o I 1 4 mn ¼ 2 o I 4 mn
ð6Þ
o I 2 4 mn ¼ 2 o I 4 mn
ð7Þ
Since I o , I r and β are independent from each other, their product's average-value equals to their average-value's product, i.e. o I 2 4 mn ¼ 2 o I 4 2mn
ð8Þ
Therefore,
pffiffiffiffiffiffiffi o I 1 I 2 4 mn ¼ o ðI r þ I o Þ2 þ 2ðI r þ I o Þ I r I o cos β pffiffiffiffiffiffiffi þ2ðI r þ I o Þ I r I o cos ðβ þ φÞ þ4I r I o cos β cos ðβ þ φÞ 4 mn
fringe
Fig. 1. The contour window along the fringe orientation.
¼ oðI r þ I o Þ2 þ 4I r I o cos β cos ðβ þ φÞ 4 mn representative second-order PDEs denoising models quantitatively. Ref. [7] proposed the oriented PDE model which made the diffusion only along the fringe orientation and protected the boundary of the fringe during filtering. In Ref. [8], the coherence-enhancing diffusion is presented for fringe pattern denoising, which smoothed a fringe pattern along directions both parallel and perpendicular to fringe orientation with different diffusion speeds to more effectively reduce noise. Although the PDE processing method has popular application of denoising for the ready fringe, it is never used to generate fringe patterns. In this paper, we propose the correlation fringe pattern of ESPI generated method based on orientation partial differential equation, which derives the fringe pattern with smooth fringes without speckle noise from two original speckle patterns. When calculating the correlation coefficient, the calculation for gray average is substituted by the oriented PDE filter in our implementation. Our method has two main advantages. Firstly, because the filter is permitted only along the fringe orientation, it does not spoil the phase distribution. Secondly, it need not compute the filter window pixel by pixel, and the calculation for curve follow is substituted by the coordinate transformation which can be simply realized based on PDE for the entire image.
2. Direct correlation method to obtain fringe patterns In ESPI, two original speckle patterns with a phase difference φðx; yÞ are first acquired by a digital imaging system. The intensities of the two speckle patterns can be represented as follows: pffiffiffiffiffiffiffi I 1 ¼ I o þ I r þ 2 I o I r cos ðϕr ϕo Þ ð1Þ I2 ¼ Io þ Ir þ 2
pffiffiffiffiffiffiffi I o I r cos ðϕr ϕo þ φÞ
ð2Þ
where I o ðx; yÞ and I r ðx; yÞ are the intensities of the object's scattered light and reference beam, respectively. ϕo ðx; yÞ and ϕr ðx; yÞ are the phases of random speckles, and φðx; yÞ is the phase difference between the two speckle patterns resulting from some displacement. The correlation between the two speckle patterns can be calculated by [4] Cðx; yÞ ¼ h
o ðI 1 o I1 4 mn ÞðI 2 o I 2 4 mn Þ 4 mn i1=2 h i1=2 oðI 2 o I 2 4 mn Þ2 4 mn
o ðI 1 o I 1 4 mn Þ2 4 mn
ð3Þ where o I 1 4 mn , o I 2 4 mn are the average values of I1, I2 in the m n window, respectively. According to the speckle statistic theory, assume oI r 4 mn ¼ oI o 4 mn ¼ o I 4 mn
ð4Þ
¼ 6 o I 2 4 mn þ o 2I r I o ð cos ð2β þ φÞ þ cos φÞ 4 mn 6 o I 2 4 mn þ 2 oI 2 4 mn cos φ
ð9Þ
o ðI 1 o I 1 4 mn ÞðI 2 o I 2 4 mn Þ 4 mn ¼ oI 1 I 2 4 mn o I 1 4 mn o I 2 4 mn 2 o I 2 4 mn þ 2 oI 2 4 mn cos φ
ð10Þ
o ðI 1 o I 1 4 mn Þ2 4 mn ¼ oI 21 2I 1 o I 1 4 mn þ oI 1 4 2mn 4 mn ¼ 4 oI 4 2mn
ð11Þ
o ðI 2 o I 2 4 mn Þ2 4 mn ¼ oI 22 2I 2 o I 2 4 mn þ oI 2 4 2mn 4 mn ¼ 4 oI 4 2mn ð12Þ Under the condition that the phase term of φ is a constant over the whole operation window of m n, the sign ‘ ’ in Eqs. (9) and (10) can be substituted by the sign ‘ ¼’. Substituting Eqs. (10)–(12) into Eq. (3), the correlation coefficients can be rewritten as follows Cðx; yÞ ¼
2 oI 2 4 mn þ 2 o I 2 4 mn cos φ ð cos φ þ 1Þ ¼ 2 4 o I 4 2mn
ð13Þ
This relation indicates that the correlation fringe pattern is a cosine fringe pattern, which is the same as normal interferometric fringe patterns. When ϕðx; yÞ ¼ 2nπ, I1(x,y) ¼ I2(x,y) at the point (x,y), C(x,y) ¼1 and bright fringes are obtained. When ϕðx; yÞ ¼ ð2n þ 1Þπ, that means no correlation, C(x,y)¼0 and it forms dark fringes [4]. Based on the correlation coefficient C, the normalized correlation fringe pattern I cor can be obtained, i.e. 1 cos φ ð14Þ I cor ¼ 255ð1CÞ ¼ 255 2 Eq. (14) maintains the correlation fringe pattern has the same phase field as that of subtraction speckle fringe pattern. It is noticed that during above deduction, the phase term of ϕðx; yÞ is assumed as a constant within the correlation windows. When the correlation is performed within rectangular windows, this assumption is no valid. This is the reason why the direct correlation method brings certain error. The contoured correlation method performs correlation only on the fringe contour windows on which the phase term is a constant, and obtains good fringe patterns [4]. However, it needs to move the contour window pixel by pixel and repeat the tedious calculations. 3. The orientation partial differential equation filter method As noise in speckle patterns affects correlation fringe and gray average of the rectangular window spoils the phase distribution,
F. Zhang et al. / Optics Communications 310 (2014) 85–89
y
conditions. In model (19), the diffusion is made only along the fringe orientation, in which the phase term of ϕðx; yÞ is a constant. Therefore the orientated PDE can restrain noise of the original speckle patterns and bring no error for correlation. In order to solve Eq. (19) numerically, the equation has to be discretized. The image is represented by M N matrices of intensity values. So, for any function (i.e., image) u(x, y), we let ui,j denote u(i, j) for 1 oio M, 1 oj oN. The evolution equations give images at times t n ¼ nΔt. We denote uði; j; t n Þ by uni;j . The time derivative ut at (i, j, tn) is approximated by the forward difference
fringe tangent direction T
fringe normal direction N
θ
x
fringe
ðut Þni;j ¼
Fig. 2. The relationship between the rectangular coordinate system (x,y) and the !! inner coordinate system ( T , N ).
the filter method based on the phase distribution should be adopted to reduce the noise. Fig. 2 shows the inner coordinate !! system ( T , N ) established along the fringe tangent direction and the fringe normal direction. Let Iðx; yÞ : Ω-R represents the initial noisy image, where Iðx; yÞ is the gray-level value and Ω is an image domain. Introducing an artificial time t. The oriented PDE filter makes the diffusion only along the fringe orientation and protects the boundary of the fringe during filtering, therefore the image deforms according to ∂u ∂2 u ¼ 2; ∂t ∂T
uðx; y; 0Þ ¼ Iðx; yÞ
ð15Þ
where uðx; y; tÞ : Ω ½0; þ1Þ-R is the evolving image, ∂u ∂t is the ∂2 u first-order partial derivatives of u about time coordinate, ∂T 2 is the second-order partial derivatives of u along the fringe tangent direction. In mathematics, the rectangular coordinate (x,y) and the inner !! coordinate ( T , N ) can be connected by the coordinate transformation, i.e. ∂ ∂ ∂ ¼ cos θ þ sin θ ∂T ∂x ∂y ∂2 ∂T
2
¼
∂2 ∂2 ∂2 2 cos θ sin θ cos 2 θ þ 2 sin θ þ 2 ∂x∂y ∂x2 ∂y
ð16Þ ð17Þ
where θ denotes the angle between the fringe orientation and the x coordinate. For fringe patterns, the fringe orientation θ for point (x,y) is estimated within its neighborhood of a small window as follows [9]: θðx; yÞ ¼
∑k;l 2ux ðk; lÞuy ðk; lÞ 1 tan 1 2 ∑k;l ðu2y ðk; lÞu2x ðk; lÞÞ
ð18Þ
where k and l are the subscripts of the pixel point in this window. Ref. [9] presents the error between the true orientation and the extracted orientation with different neighborhood window size in Eq. (18). The conclusion is that when the window size is larger than three times of the local fringe width, the orientation error is less than 0.2. (In Ref. [9], the error is the average value of j sin ½θðx; yÞθT ðx; yÞj of the calculated area, where θðx; yÞ and θT ðx; yÞ are the calculated orientation and the true orientation respectively.) Therefore, the oriented PDE model can be expressed as following [7] ∂u ¼ uxx cos 2 θ þ uyy sin 2 θ þ 2uxy sin θ cos θ ∂t
87
ð19Þ
where uxx, uyy and uxy are the second-order partial derivatives of u about the rectangular coordinate. The numerical solutions of the PDE give the filtered images with the original image I(x,y) as initial
n unþ1 i;j ui;j
ð20Þ
Δt
where Δt is step-size in time. The spatial derivatives are as follows ðuxx Þni;j ¼ uniþ1;j 2uni;j þ uni1;j
ð21aÞ
ðuyy Þni;j ¼ uni;jþ1 2uni;j þ uni;j1
ð21bÞ
ðuxy Þni;j ¼ ðuniþ1;jþ1 uni1;jþ1 uniþ1;j1 þ uni1;j1 Þ=4
ð21cÞ
Then the diffusion term is expressed by the following: 2
¼ uni;j þ Δt½ðuxx Þni;j cos 2 θi;j þ ðuyy Þni;j sin θi;j unþ1 i;j þ2ðuxy Þni;j
sin θi;j cos θi;j
ð22Þ
4. Correlation method based on orientation partial differential equation to obtain fringe patterns In this paper, we propose to calculate the correlation based on the oriented PDE model. The average calculations in the correlation method are replaced by the anisotropic filter provided by the oriented PDE. The correlation can be obtained from Cðx; yÞ ¼
o ðI 1 o I 1 4 f ÞðI 2 o I 2 4 f Þ 4 f ½ oðI 1 oI 1 4 f Þ2 4 f 1=2 ½ oðI 2 oI 2 4 f Þ2 4 f 1=2
ð23Þ
where o U 4 f denotes the filtered result by the oriented PDE. To illuminate the main processing steps and the result of the proposed method, two groups of speckle patterns are given. Let us take Fig. 3 for example. Fig. 3(a) and (b) shows the undeformed speckle pattern and the deformed speckle pattern, respectively. The main processing steps of the proposed method are as follows. (1) The two original speckle patterns (Fig. 3(a) and (b)) with a phase difference are subtracted from each other to form a subtraction speckle fringe pattern (Fig. 3(c)), and then the fringe orientation (Fig. 3(d)) is determined from the subtraction speckle fringe pattern based on Eq. (18). (2) Performing the anisotropic filter for I1, I2, ðI 1 o I 1 4 f Þ2 , ðI 2 o I 2 4 f Þ2 and ðI 1 o I 1 4 f ÞðI 2 o I 2 4 f Þ based on the oriented PDE (i.e. Eq. (19)). (3) Calculating correlation C(x,y) conforming to Eq. (23). (4) Generating the correlation fringe pattern with I cor ¼ 255 ð1CÞ (Fig. 3(e)). Because the filter along fringe orientation for the entire image can be simply realized by the coordinate transformation based on PDE, our method need not calculate each gray average in the corresponding filter window pixel by pixel. We tested the proposed algorithm on two speckle patterns. Fig. 3 shows the comparison results of the subtraction fringe pattern (SFP) and the correlation fringe pattern based on orientation partial differential equation (OPDE-CFP) on the usual speckle pattern. Fig. 4 shows the similar comparison results on the usual speckle pattern. The fringe orientation is calculated within 31 31 pixels window, which is about three times of the bigger local
F. Zhang et al. / Optics Communications 310 (2014) 85–89
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x (Pixels) Fig. 3. Comparison of two fringe pattern generated method on the usual speckle pattern. (a) The undeformed speckle pattern; (b) The deformed speckle pattern; (c) The subtraction fringe pattern; (d) The orientation map of the fringe pattern; (e) The correlation fringe pattern generated based on orientation partial differential equation; (f) Gray-distribution curve of (c) (upper) and (e) (lower) of the middle intersection (from the left dot to the right dot).
fringe width. The PDE filter parameters include the time step Δt ¼ 0:2, the iteration time n ¼50 for I1, I2 and n ¼250 for ðI 1 oI 1 4 f Þ2 , ðI 2 o I 2 4 f Þ2 , ðI 1 o I 1 4 f ÞðI 2 oI 2 4 f Þ. From Figs. 3(c) and 4(c), one can see that the noise in SFP is high, while OPDE-CFP is smooth and has no speckle noise at all, neither the high-density fringe nor the low-density fringe. Figs. 3 (f) and 4(f) show the intensity distribution of the middle intersection (from the left dot to the right dot) of Fig. 3(c) and (e) and Fig. 4(c) and (e), respectively. It is noticed that the SFP is very noisy, which is shown on the upper part of Figs. 3(f) and 4(f), while the OPDE-CFP is smooth and its intensities and amplitudes are general uniform, which is shown on the lower part of Figs. 3(f) and 4(f). This result indicates that the proposed method is suitable for the complicated applications of the fringe patterns with varying fringe density and different fringe orientation.
5. Conclusion We present an efficient correlation fringe pattern generation method based on the orientation partial differential equation, which derives the fringe pattern with smooth fringes without speckle noise from the two original speckle patterns. First of all, two original speckle patterns I1, I2 with a phase difference are subtracted from each other to form a subtraction speckle fringe pattern and then the fringe orientation is determined from the subtraction speckle fringe pattern. Then, performing the anisotropic filter for I1, I2 based on the oriented partial differential equation. Finally, calculating correlation and generating correlation fringe pattern. The proposed method has two main advantages. Firstly, it does not spoil the phase distribution. Secondly, it need not calculate
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F. Zhang et al. / Optics Communications 310 (2014) 85–89
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x (Pixels) Fig. 4. Comparison of two fringe pattern generated method on the shear speckle pattern. (a) The undeformed speckle pattern; (b) The deformed speckle pattern; (c) The subtraction fringe pattern; (d) The orientation map of the fringe pattern; (e) The correlation fringe pattern generated based on orientation partial differential equation; (f) Gray-distribution curve of (c) (upper) and (e) (lower) of the middle intersection (from the left dot to the right dot).
each gray average in the corresponding filter window pixel by pixel, and the calculation for curve follow is substituted by the coordinate transformation which can be simply realized based on PDE for the entire image. The experimental results show that our method is capable of significantly improving the quality of the fringe patterns and helpful for subsequent phase extraction.
Acknowledgment Sponsored by the National Nature Science Foundation of China under grant no.61102150, and Open Foundation of Key laboratory of Opto-electronic Information Technology of Ministry of
Education (Tianjin University) under grant no.2012KFKT003. We thank the referees for their valuable suggestions. References [1] V. Bavigadda, R. Jallapuram, E. Mihaylova, V. Toal, Optics Letters 35 (2010) 3273. [2] K. Genovese, L. Lamberti, C. Pappalettere, Optics and Lasers in Engineering 42 (2004) 543. [3] D.R. Schmitt, R.W. Hunt, Applied Optics 36 (1997) 8848. [4] Q. Yu, S. Fu, X. Yang, X. Sun, X. Liu, Optics Express 12 (2004) 75. [5] C. Tang, F. Zhang, H. Yan, Z. Chen, Optics Communications 260 (2006) 91. [6] C. Tang, F. Zhang, B. Li, H. Yan, Applied Optics 45 (2006) 7392. [7] C. Tang, L. Han, H. Ren, Z. Tao Gao, Wang, K. Tang, Optics Express 17 (2009) 5606. [8] H. Wang, Q. Kernao, W. Gao, F. Lin, H.S. Seah, Optics Letters 35 (2009) 1141. [9] X. Yang, Q. Yu, S. Fu, Optics Communications 273 (2007) 60.