Abstract

This paper presents a method to reconstruct the nonuniform background for camera-based quantitative evaluation of thin-layer chromatography (TLC). After analyzing the concave distribution feature of illumination produced by a linear light source on a plane, the paper then makes use of the feature and convex hull algorithm to find points belonging to the background. After that, B-spline is employed to reconstruct the background. An experiment is also made to test the performance of the method, which shows that the correlation coefficient between the linear samples is 0.9949 after removing the estimated background.

© 2006 Optical Society of America

1. Introduction

Thin-layer chromatography (TLC) or planar chromatography is one of the most popular methods used in chemical analysis. The most common detection method in situ is slit-scanning densitometry which is performed by measurement of the absorbance of fluorescence of the separated zones in the chromatogram tracks using an optical densitometic scanner with a fixed sample light beam in the form of a rectangular slit. Disadvantages of scanning with a moving light spot include problems with data processing, lower spatial resolution, and unfavorable error propagation. It was suggested that the breakthrough leading to greater use of two-dimensional planar separations would come from the use of optical imaging quantification systems [1, 2]. One way is to use a flatbed scanner to obtain a TLC image, but it is hard to use different light sources for different substances. Digital camera is becoming more and more popular, and the most potential one to be used to make the quantitative evaluation of TLC. But the main limitation based on a digital camera is that no solution has been found for illuminating a TLC plate uniformly with monochromatic light [3, 4]. Although we can use a big plane light source, it is too expensive and only useful for transmission cases.

One possible way to solve the problem is to estimate the nonuniform image background caused by the light sources, and then normalize the illumination intensity for every pixel and make the quantitative evaluation. But it is difficult to reconstruct the background, because the developed samples make their edges unclear which is hard to make clear where it should be to separate the samples from the background. An easy way to estimate the background is to take a picture of a blank TLC plate as the background. This is possible provided that the camera and its lens are fixed and the exact position of the TLC plate containing materials has to be known first. This paper will use the fact that the illumination distribution from a linear light source can be concave along one direction parallel to the light source in a certain area. So, when taking pictures of TLC plate in that area, the intensity of the background would be concave too along that direction. After finding some background points line by line under the guide of the concave feature and convex hull algorithm, control points are selected and the background reconstruction is completed by interpolation. The lens-camera is also a factor leading to the nonuniform problem. But compared to the illumination, this factor is quite small, so we will not take it into account in this paper.

2. Illumination distribution of a linear light source

We will analyze the illumination distribution feature first, so that we can find the right method. Light plays an important role in imaging. Linear light source is a kind of popular light model used in our daily life, and it is also the light source used in most camera-based TLC imaging system. So, this paper will only consider linear light sources. Many papers have studied the light source models and their illumination [5, 6]. We will try to find out the convexity of the fluorescence intensity distribution on a plane excited by the linear light source.

Suppose a linear light source is above a plane, as shown in Fig. 1(a). Let the length of the linear light source be L, its luminous intensity I, and the distance to the plane h. Also let r stand for the distance from light source to P, and θ the incident angle. Thus, according to the Square and Cosine Law of light, the illumination at point P is

p(x,y)=L/2L/2Icosθr2dl=L/2L/2Ihr3dl=L/2L/2Ih(x2+(ly)2+h2)3dl.

According to the theorems determining the convexity of a function by the second derivative, we will derive the second partial derivative of p(x, y) with respect to y.

Since

p(x,y)y=Ih((y+L/2y)2+x2+h2)3Ih((yL/2)2+x2+h2)3,

thus, we have

2p(x,y)y2=3Ih(L/2+y)((L/2+y)2+x2+h2)5+3Ih(yL/2)((yL/2)2+x2+h2)5.

Let 2p(x,y)y2=0 then we obtain

y+L/2((L/2+y)2+x2+h2)5=yL/2((L/2+y)2+x2+h2)5.

The solution of Eq. (4) is

x2=(yL/2)2/5((y+L/2)2h2)(y+L/2)2/5((yL/2)2+h2)(y+L/2)2/5(yL/2)2/5.

We plot the solution of Eq. (5) in Fig. 1(b). From Fig. 1(b) we observe that the plane is divided into three parts, the central part and two side parts. For a fixed x, the central part is a concave area where 2p(x,y)y2<0 alon y, and the two side parts are convex areas where 2p(x,y)y2>0 along y. It is obvious that the central part is more useful, because the illumination becomes weak in the too side areas. So we will use the concave distribution property of illumination. One of the properties of convexity tells us that the sum of two concave functions is still concave. Therefore, we can figure out that if two linear light sources are placed in parallel, there would be an area between them in which the illumination distribution feature along y is still concave. This property will be used in our later experiment.

In order to make difference between a TLC plate and the samples on it, the TLC plate is usually covered with special substances to absorb ultraviolet and emit fluorescence, but the samples to be tested usually couldn’t. When the camera takes pictures of the TLC plate within the concave area, the intensity of the image background in vertical direction will be concave too, because the fluorescence intensity is proportional to the inducing light intensity [7].

 

Fig. 1. (a). Linear light source. (b). Illumination distribution feature on the plane of (a) along y.

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3. Reconstruction method

In TLC, since the chemical samples don’t emit fluorescence as mentioned above, so they are darker than the background. That means that the samples are below the concave background in density value. Therefore, we can use convex hull algorithm to find the background. After locating points belonging to the background, an interpolation method can be employed to reconstruct or estimate the background.

Finding the convex hull of a set of points is the most elementary interesting problem in computational geometry. Its main purpose is to find the smallest convex polygon containing all the points of a set. For a TLC image, we can process line by line or column by column along the direction with respect to the illumination distribution feature to find the points belonging to the background. The position and the value of pixels of one line or column of an image composite a set of points. When the convex hull of a line is found, the points on the upside part of the convex hull are the background points of that line. Figure 2 illustrates how a line or column of an image is composed and the steps of finding the background points by convex hull algorithm.

Once the points belonging to the background are found, we can choose some of them as the control points for interpolation. First we simply appoint the position for the control points by uniform intervals both in horizontal and vertical directions. However, if the position appointed is not the position where a background point found exists, we should find nearest one around that position to be the control point. Based on the control points chosen, we employ bi-cubic B-spline to reconstruct the background, because B-spline can be thought as a low-pass filter [9], it always lies in the convex hull of the control points but changes smoother than the control points.

 

Fig. 2. (a). Concave background. (b). Background with signal. (c). Convex hull of (b). (d). Background points found.

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4. Experiment

We have made evaluation experiment to test the performance of the method. The experimental instrumentation is based on a light source apparatus (Transilluminator 2020D, Cold Spring) shown in Fig. 3(a). It consists of two parallel linear ultraviolet light sources, UV Light1 and UV Light2. They are all 10cm high above a plane on which a TLC plate would be placed. Their wavelength is 254nm, the geometric length of them is L=14cm. The distance between the two light sources is W=27cm. The light sources are contained within a light-proof cabinet. A digital camera, Canon Powershot G1, is mounted above the cabinet for capturing the image of TLC plate. The camera is set to work at auto-work station, and a filter mounted in front of the lens is used to keep out of the ultraviolet, although the cabinet is designed to be not highly reflective in UV.

Astragaloside, a compound which has been certified by the National Institute for the Control of Pharmaceutical and Biological Products (Beijing, China), is used as the samples to do the experiment. The experiment is performed on the high performance thin layer silica gel G plates (Qingdao Ocean Chemical Plant, China). Different volume of samples, 1μL, 2μL, 3μL, 4μL, 5μL, and 6μL, are sampled respectively on the different places of the plate [4]. After developing, the plate is put into the light source apparatus to be captured by the camera, as shown in Fig. 3(a). Figure 3(b) is an example of images captured by the camera. It is should be pointed out that the camera’s horizontal line is parallel to the direction x of the plane, thus the intensity of every column’s background would be concave in vertical direction.

According to the additional property of convexity discussed in Section 2, there would be an area between the two parallel linear ultraviolet light sources UV Light1 and UV Light2 where the illumination distribution feature along y1 and y2 is concave. Figure 3(c) illustrates that concave area. Note that there are areas in which the convexity is not sure after the simple addition of two light sources without exact calculation, but that will not affect our experiment. Figure 3(d) is one column of Fig. 3(b) from which we see its background is concave. We will take this column for example to show the process of finding background points. But from Fig. 3(d) we notice that noises exist and will affect the precision of reconstruction, so a low-pass filter has to be applied to the original image so as to suppress the noises. Weiner filter is used by this paper.

 

Fig. 3. (a). Experimental instrumentation. (b) An image obtained by the camera in (a). (c) Illumination distribution feature of (a) along;;. (d) One column of (b).

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Figure 4 shows the concrete procedure how to reconstruct the background. Figure 4(a) shows the step to find the convex hull of Fig. 3(d). Figure 4(b) illustrates what points are left to be the background points. All the background points found in this way for the whole image are marked as brightest pixels in Fig. 4(c). Figure 4(d) displays the control points found, namely the cross points. Figure 4(e) is the three-dimensional (3D) plot of the grid built up by the control points. Figure 4(f) is the background reconstructed by B-spline surface. Figure 4(g) is the 3D plot of Fig. 3(a), and Fig. 4(h) is the image in 3D plot of Fig. 3(a) with background removed.

So far we have reconstructed the background. Next, we make a rough quantitative evaluation to test the performance of the method. In TLC, it is said that the intensity integration of the sample is proportional to its concentration [3]. So, if the concentrations f of the samples are arranged in linearity, then their intensity integrations g should be linear too. Therefore, we make image segmentation by subtracting the estimated background from the original image first, then simply calculate the intensity integration for each simple, and finally use the following formula to calculate the correlation coefficient R

R=1n1i=1n(fif̅Sf)(gig̅Sg),

where n is the number of samples, fi and gf are the concentration and intensity integration of sample i respectively, and are the average values of f and g respectively, and Sf and Sg are the standard deviations of f and g respectively. For Fig. 4(h), the correlation coefficient R is 0.9949. Compared with 0.9955, the coefficient given by Chau [8] who used a flatbed scanner to obtain TLC images and made the segmentation manually, the result of the proposed method is almost on the same level. In addition to the removal of the uneven background, if we consider the effect acted on the samples by the nonuniform illumination, the coefficient would be higher. It should be pointed out that the compression of the image output by the camera would slightly affect the evaluation too.

 

Fig. 4. (a). Finding the convex hull of Fig. 3(d). (b) Background points found for Fig. 3(d). (c) All the background points found for Fig. 3(b) (the white pixels). (d) The control points selected (cross points). (e) 3D form of the grid in (d). (f) The background reconstructed by B-spline surface. (g) 3D plot of Fig. 3(b). (h) (f) subtracting (g).

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5. Conclusions

This paper takes advantage of the concave illumination distribution feature of the linear light source and B-spline to reconstruct the background for TLC. The proposed method can solve the nonuniform illumination problem which is thought to be the limitation of camera-based quantitative evaluation in TLC. The evaluation experiment shows that the method is satisfied. Although the idea using the concave feature of illumination comes from TLC, it can also be used to other optical devices, such as protein gel electrophoresis, another popular experimental tool in biomedical and pharmaceutical fields, etc.

Acknowledgments

Mr. Shiwei Zhang prepared the experiment for this work. This work was supported by Natural Science Foundation of China under Grant 60572087. The manuscript has benefited greatly from the constructive comments from the journal editor and the anonymous reviewer, many thanks to them.

References and links

1. C. F. Poole, “Pannar chromatography at the turn of the century” J. Chromatogr. A 856, 399–427 (1999). [CrossRef]   [PubMed]  

2. J. Sherma, “Planar chromatography,” Anal. Chem. 78, 3841–3852 (2006). [CrossRef]   [PubMed]  

3. I. Vovk and M. Prosek, “Reproducibility of densitometric and image analyzing quantitative evaluation of thin-layer chromatograms,” J. Chromatogr. A 768, 329–336 (1997). [CrossRef]  

4. L. Zhang and X. G. Lin, “Quantitative evaluation of thin-layer chromatography with image background estimation based on charge-coupled device imaging,” J. Chromatogr. A 1109, 273–278 (2006). [CrossRef]   [PubMed]  

5. B. K. P Horn and R. W. Sjober, “Calculating reflectance map,” Appl. Opt. 18, 1770–1779 (1979). [CrossRef]   [PubMed]  

6. T. Nishita, I. Okamura, and I. Nakamae, “Shading models for point and linear sources,” ACM T. Graphic. 4, 124–146 (1985). [CrossRef]  

7. Y. Inoue, H. Hori, T. Sakurai, Y. Tokitomo, J. Saito, and T. Misonou, “Measurement of fluorescence quantum yield of ultraviolet absorbing substance extracted from red alga: porphyra yezoensis and its photothermal spectroscopy,” Opt. Rev. 9, 75–87 (2002). [CrossRef]  

8. F. T. Chau, T. P. Chan, and J. Wang. “TLCQA: quantitative study of thin-layer chromatography,” Bioinformatics 14, 540–541 (1998). [CrossRef]   [PubMed]  

9. M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Proc. Mag 16, 22–38 (1999). [CrossRef]  

References

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  1. C. F. Poole, "Pannar chromatography at the turn of the century" J. Chromatogr. A 856, 399-427 (1999).
    [CrossRef] [PubMed]
  2. J. Sherma, "Planar chromatography," Anal. Chem. 78, 3841-3852 (2006).
    [CrossRef] [PubMed]
  3. I. Vovk and M. Prosek, "Reproducibility of densitometric and image analyzing quantitative evaluation of thin-layer chromatograms," J. Chromatogr. A 768, 329-336 (1997).
    [CrossRef]
  4. L. Zhang and X. G. Lin, "Quantitative evaluation of thin-layer chromatography with image background estimation based on charge-coupled device imaging," J. Chromatogr. A 1109, 273-278 (2006).
    [CrossRef] [PubMed]
  5. B. K. P Horn and R. W. Sjober, "Calculating reflectance map," Appl. Opt. 18, 1770-1779 (1979).
    [CrossRef] [PubMed]
  6. T. Nishita, I. Okamura, and I. Nakamae, "Shading models for point and linear sources," ACM T.Graphic. 4, 124-146 (1985).
    [CrossRef]
  7. Y. Inoue, H. Hori, T. Sakurai, Y. Tokitomo, J. Saito, and T. Misonou, "Measurement of fluorescence quantum yield of ultraviolet absorbing substance extracted from red alga: porphyra yezoensis and its photothermal spectroscopy," Opt. Rev. 9, 75-87 (2002).
    [CrossRef]
  8. F. T. Chau, T. P. Chan, and J. Wang. "TLCQA: quantitative study of thin-layer chromatography," Bioinformatics 14, 540-541 (1998).
    [CrossRef] [PubMed]
  9. M. Unser, "Splines: a perfect fit for signal and image processing," IEEE Signal Proc. Mag 16, 22-38 (1999).
    [CrossRef]

2006 (2)

J. Sherma, "Planar chromatography," Anal. Chem. 78, 3841-3852 (2006).
[CrossRef] [PubMed]

L. Zhang and X. G. Lin, "Quantitative evaluation of thin-layer chromatography with image background estimation based on charge-coupled device imaging," J. Chromatogr. A 1109, 273-278 (2006).
[CrossRef] [PubMed]

2002 (1)

Y. Inoue, H. Hori, T. Sakurai, Y. Tokitomo, J. Saito, and T. Misonou, "Measurement of fluorescence quantum yield of ultraviolet absorbing substance extracted from red alga: porphyra yezoensis and its photothermal spectroscopy," Opt. Rev. 9, 75-87 (2002).
[CrossRef]

1999 (2)

C. F. Poole, "Pannar chromatography at the turn of the century" J. Chromatogr. A 856, 399-427 (1999).
[CrossRef] [PubMed]

M. Unser, "Splines: a perfect fit for signal and image processing," IEEE Signal Proc. Mag 16, 22-38 (1999).
[CrossRef]

1998 (1)

F. T. Chau, T. P. Chan, and J. Wang. "TLCQA: quantitative study of thin-layer chromatography," Bioinformatics 14, 540-541 (1998).
[CrossRef] [PubMed]

1997 (1)

I. Vovk and M. Prosek, "Reproducibility of densitometric and image analyzing quantitative evaluation of thin-layer chromatograms," J. Chromatogr. A 768, 329-336 (1997).
[CrossRef]

1985 (1)

T. Nishita, I. Okamura, and I. Nakamae, "Shading models for point and linear sources," ACM T.Graphic. 4, 124-146 (1985).
[CrossRef]

1979 (1)

Chan, T. P.

F. T. Chau, T. P. Chan, and J. Wang. "TLCQA: quantitative study of thin-layer chromatography," Bioinformatics 14, 540-541 (1998).
[CrossRef] [PubMed]

Chau, F. T.

F. T. Chau, T. P. Chan, and J. Wang. "TLCQA: quantitative study of thin-layer chromatography," Bioinformatics 14, 540-541 (1998).
[CrossRef] [PubMed]

Hori, H.

Y. Inoue, H. Hori, T. Sakurai, Y. Tokitomo, J. Saito, and T. Misonou, "Measurement of fluorescence quantum yield of ultraviolet absorbing substance extracted from red alga: porphyra yezoensis and its photothermal spectroscopy," Opt. Rev. 9, 75-87 (2002).
[CrossRef]

Horn, B. K. P

Inoue, Y.

Y. Inoue, H. Hori, T. Sakurai, Y. Tokitomo, J. Saito, and T. Misonou, "Measurement of fluorescence quantum yield of ultraviolet absorbing substance extracted from red alga: porphyra yezoensis and its photothermal spectroscopy," Opt. Rev. 9, 75-87 (2002).
[CrossRef]

Lin, X. G.

L. Zhang and X. G. Lin, "Quantitative evaluation of thin-layer chromatography with image background estimation based on charge-coupled device imaging," J. Chromatogr. A 1109, 273-278 (2006).
[CrossRef] [PubMed]

Misonou, T.

Y. Inoue, H. Hori, T. Sakurai, Y. Tokitomo, J. Saito, and T. Misonou, "Measurement of fluorescence quantum yield of ultraviolet absorbing substance extracted from red alga: porphyra yezoensis and its photothermal spectroscopy," Opt. Rev. 9, 75-87 (2002).
[CrossRef]

Nakamae, I.

T. Nishita, I. Okamura, and I. Nakamae, "Shading models for point and linear sources," ACM T.Graphic. 4, 124-146 (1985).
[CrossRef]

Nishita, T.

T. Nishita, I. Okamura, and I. Nakamae, "Shading models for point and linear sources," ACM T.Graphic. 4, 124-146 (1985).
[CrossRef]

Okamura, I.

T. Nishita, I. Okamura, and I. Nakamae, "Shading models for point and linear sources," ACM T.Graphic. 4, 124-146 (1985).
[CrossRef]

Poole, C. F.

C. F. Poole, "Pannar chromatography at the turn of the century" J. Chromatogr. A 856, 399-427 (1999).
[CrossRef] [PubMed]

Prosek, M.

I. Vovk and M. Prosek, "Reproducibility of densitometric and image analyzing quantitative evaluation of thin-layer chromatograms," J. Chromatogr. A 768, 329-336 (1997).
[CrossRef]

Saito, J.

Y. Inoue, H. Hori, T. Sakurai, Y. Tokitomo, J. Saito, and T. Misonou, "Measurement of fluorescence quantum yield of ultraviolet absorbing substance extracted from red alga: porphyra yezoensis and its photothermal spectroscopy," Opt. Rev. 9, 75-87 (2002).
[CrossRef]

Sakurai, T.

Y. Inoue, H. Hori, T. Sakurai, Y. Tokitomo, J. Saito, and T. Misonou, "Measurement of fluorescence quantum yield of ultraviolet absorbing substance extracted from red alga: porphyra yezoensis and its photothermal spectroscopy," Opt. Rev. 9, 75-87 (2002).
[CrossRef]

Sherma, J.

J. Sherma, "Planar chromatography," Anal. Chem. 78, 3841-3852 (2006).
[CrossRef] [PubMed]

Sjober, R. W.

Tokitomo, Y.

Y. Inoue, H. Hori, T. Sakurai, Y. Tokitomo, J. Saito, and T. Misonou, "Measurement of fluorescence quantum yield of ultraviolet absorbing substance extracted from red alga: porphyra yezoensis and its photothermal spectroscopy," Opt. Rev. 9, 75-87 (2002).
[CrossRef]

Unser, M.

M. Unser, "Splines: a perfect fit for signal and image processing," IEEE Signal Proc. Mag 16, 22-38 (1999).
[CrossRef]

Vovk, I.

I. Vovk and M. Prosek, "Reproducibility of densitometric and image analyzing quantitative evaluation of thin-layer chromatograms," J. Chromatogr. A 768, 329-336 (1997).
[CrossRef]

Wang, J.

F. T. Chau, T. P. Chan, and J. Wang. "TLCQA: quantitative study of thin-layer chromatography," Bioinformatics 14, 540-541 (1998).
[CrossRef] [PubMed]

Zhang, L.

L. Zhang and X. G. Lin, "Quantitative evaluation of thin-layer chromatography with image background estimation based on charge-coupled device imaging," J. Chromatogr. A 1109, 273-278 (2006).
[CrossRef] [PubMed]

Anal. Chem. (1)

J. Sherma, "Planar chromatography," Anal. Chem. 78, 3841-3852 (2006).
[CrossRef] [PubMed]

Appl. Opt. (1)

Bioinformatics (1)

F. T. Chau, T. P. Chan, and J. Wang. "TLCQA: quantitative study of thin-layer chromatography," Bioinformatics 14, 540-541 (1998).
[CrossRef] [PubMed]

Graphic. (1)

T. Nishita, I. Okamura, and I. Nakamae, "Shading models for point and linear sources," ACM T.Graphic. 4, 124-146 (1985).
[CrossRef]

IEEE Signal Proc. Mag (1)

M. Unser, "Splines: a perfect fit for signal and image processing," IEEE Signal Proc. Mag 16, 22-38 (1999).
[CrossRef]

J. Chromatogr. A (3)

C. F. Poole, "Pannar chromatography at the turn of the century" J. Chromatogr. A 856, 399-427 (1999).
[CrossRef] [PubMed]

I. Vovk and M. Prosek, "Reproducibility of densitometric and image analyzing quantitative evaluation of thin-layer chromatograms," J. Chromatogr. A 768, 329-336 (1997).
[CrossRef]

L. Zhang and X. G. Lin, "Quantitative evaluation of thin-layer chromatography with image background estimation based on charge-coupled device imaging," J. Chromatogr. A 1109, 273-278 (2006).
[CrossRef] [PubMed]

Opt. Rev. (1)

Y. Inoue, H. Hori, T. Sakurai, Y. Tokitomo, J. Saito, and T. Misonou, "Measurement of fluorescence quantum yield of ultraviolet absorbing substance extracted from red alga: porphyra yezoensis and its photothermal spectroscopy," Opt. Rev. 9, 75-87 (2002).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

(a). Linear light source. (b). Illumination distribution feature on the plane of (a) along y.

Fig. 2.
Fig. 2.

(a). Concave background. (b). Background with signal. (c). Convex hull of (b). (d). Background points found.

Fig. 3.
Fig. 3.

(a). Experimental instrumentation. (b) An image obtained by the camera in (a). (c) Illumination distribution feature of (a) along;;. (d) One column of (b).

Fig. 4.
Fig. 4.

(a). Finding the convex hull of Fig. 3(d). (b) Background points found for Fig. 3(d). (c) All the background points found for Fig. 3(b) (the white pixels). (d) The control points selected (cross points). (e) 3D form of the grid in (d). (f) The background reconstructed by B-spline surface. (g) 3D plot of Fig. 3(b). (h) (f) subtracting (g).

Equations (6)

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p ( x , y ) = L / 2 L / 2 I cos θ r 2 dl = L / 2 L / 2 Ih r 3 dl = L / 2 L / 2 Ih ( x 2 + ( l y ) 2 + h 2 ) 3 dl .
p ( x , y ) y = Ih ( ( y + L / 2 y ) 2 + x 2 + h 2 ) 3 Ih ( ( y L / 2 ) 2 + x 2 + h 2 ) 3 ,
2 p ( x , y ) y 2 = 3 Ih ( L / 2 + y ) ( ( L / 2 + y ) 2 + x 2 + h 2 ) 5 + 3 Ih ( y L / 2 ) ( ( y L / 2 ) 2 + x 2 + h 2 ) 5 .
y + L / 2 ( ( L / 2 + y ) 2 + x 2 + h 2 ) 5 = y L / 2 ( ( L / 2 + y ) 2 + x 2 + h 2 ) 5 .
x 2 = ( y L / 2 ) 2 / 5 ( ( y + L / 2 ) 2 h 2 ) ( y + L / 2 ) 2 / 5 ( ( y L / 2 ) 2 + h 2 ) ( y + L / 2 ) 2 / 5 ( y L / 2 ) 2 / 5 .
R = 1 n 1 i = 1 n ( f i f ̅ Sf ) ( g i g ̅ Sg ) ,

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