Innovative Three Dimensional Digital Image Correlation Technology for Dynamical Measurement

Abstract

The three-dimensional measurements of objects are very important for the manufacturing processes in mechanical engineering as well as detecting and the monitoring of civil engineering projects. Particularly, the application of laser scanning technology can achieve a comprehensive measurement results for monitoring of large civil engineering projects. But, the laser scanning technology can not be applied to dynamic measurements. The digital image correlation method (DIC) is a non-contact measurement technology which can provide the displacement field as well as strain field of the object to be inspected. Presently, the three-dimensional digital image correlation method using more than two video capture devices is also capable of the high accuracy of the DIC method and a three-dimensional measurement capability. However, the two cameras can not ensure the synchronization of the circumstances and the error in a dynamic environment is serious. This paper introduces an innovative three-dimensional DIC developed from the DIC research team, Taiwan, which uses a digital camera and a specially shaped prism. As a result of the use of a single image capturing device, there is no drive time asynchronous problem. The principles of this proposed technology and its verification results are presented in this study.

Keywords: digital image correlation, dynamic three-dimensional digital image

correla, prism.

1 Introduction

The digital camera has been widely used all over the world. Traditionally, more than one camera is widely applied for conducting three-dimensional image process. In this study, the practicability of the technology developed in this research using a single camera to detect 3-D images for analyzing the surface geometry or 3-D motion of an object was verified. The digital image correlation (DIC) method is an

Paper 114

Innovative Three Dimensional Digital Image Correlation

Technology for Dynamical Measurement

M.H. Shih1_{, S.H. Tung}2 _{and W.P. Sung}3 1 _{Department of Civil Engineering}

National Chi Nan University, Nantou, Taiwan

2 _{Department of Civil and Environmental Engineering}

National University of Kaohsiung, Taiwan

3 _{Department of Landscape Architecture}

National Chin-Yi University of Technology, Taichung, Taiwan

©Civil-Comp Press, 2011

Proceedings of the Thirteenth International Conference on Civil, Structural and Environmental Engineering Computing, B.H.V. Topping and Y. Tsompanakis, (Editors),

emerging image-based technology that can provide a low-cost digital image correlation coefficient method based on advanced digital cameras and high-performance computers. It is also a non-contact optical measurement method to conduct the movement measurement and calculate displacement filed and strain filed without disturbing the test samples. The Digital Image Correlation is a method proposed Peter et al [1] in 1982 for obtaining surface movement and strain using computers to analyze the digital images of an object subject to external force. The problem of noise that interferes with the results obtained using the DIC method can be reduced to 0.01 pixel with the method proposed by Chu et al. [2] in 1989. In 1989, Bruck et al. [3] suggested the use of Newton-Raphson to replace the coarse-fine method for obtaining the optimal deformation parameters. Sutton, Turner, Bruck and Chae[4] suggested a method for diminishing the analysis noise. Lu et al [5] proposed a high-order interpolation function for simulating the movement field to study the influence of the slope of high-order movement terms on the application of DIC for strain analyses. Vellinga et al. [6] in 2000 proposed a method to combine scanning electronic microscope (SEM) and DIC to detect small fractures. Dost et al. [7,8] took SEM images in nanometer scale, and combined it with the DIC technology for observing nano-scale cracks.

The research team of Digital Image Correlation Method, Taiwan has developed and applied this DIC method for observing cracks developed in brick walls, microscopically observing metal anisotropic behavior, observing warp cracks developed in reinforced concrete, testing steel plate damages mechanically, studying cracks developed in brittle material, observing warp cracks developed in light aggregate concrete, monitoring bridge deformation under traffic loads and monitoring the structural dynamic response of structure under excitation of earthquake forces[9-11]. Our research team acquires very good experimental results of 2-D test. Also a static 3-D DIC method using only one camera was developed in 2010[12].

Three dimensional DIC is nowadays widely promoted technology in fields of mechanical engineering and material science. The common 3-D DIC uses simultaneously two or more cameras for capturing parallax images. Synchronization of the shutter is usually not guaranteed. Thus the application of such technology in a dynamic three-dimensional measurements, often lead to serious errors. The purpose of this study proposes a 3-D DIC method using a single camera to simultaneously capture images with parallax. The technology makes the accurate dynamic three-dimensional measurement possible.

2 Analysis methods

The DIC method compares two images of an object before and after deformation or movement. The comparison results can be used to analyze the relative displacement and deformation of the object between the two moments. The DIC method can be divided into two categories, namely, two- and three-dimensional. There are several main hypotheses for the two-dimensional DIC method. The distance between the image acquisition instrument and the specimen is constant during the test; the specimen is planar; the planar specimen is parallel to the image sensor plane; and the

specimen undergoes in-plane displacement/strain. Therefore, it is suitable for the plane strain test or the test without obvious out-of-plane displacement. If there exists a large out-of-plane displacement or the specimen surface is not a plane, then the three-dimensional DIC method is necessary for the measurement. The experimental set-up of the two-dimensional DIC is shown as figure 1. In fact, the three-dimensional DIC utilizes the same image identification algorithm as the two-dimensional DIC method. The space coordinates of the surface of the object are calculated by utilizing proper transformation. The principle of the two-dimensional DIC method and the method to determine the three-dimensional coordinates of the specimen surface are introduced below.

Figure 1: The experimental set-up for 2-D Digital Image Correlation method

2.1 Principle of Two-Dimensional DIC Method

The main spirit of this method is to find the best mapping relationship of two images. In case the real mapping relation is too complicate, sub-images which contain only a small rectangular region of the image will be taken. So the lower ordered mapping function can be used to produce precise measurement result. For most cases the mapping functions are chosen as below:

* * ( , ) ( , ) x x u x y y y v x y = + = + (1)

Where (_x*,_y*)_{denote the coordinates in the image after deformation of the} point ( yx, ) in the image before deformation. u( yx, ) and v( yx, ) are the displacement function in x- and y-direction respectively. For example, the bi-linear displacement field can be:

xy b y b x b b y x v xy a y a x a a y x u 4 3 2 1 4 3 2 1 + + + = + + + = ) , ( ) , ( (2) The parameters of the displacement field (a₁" ,a₄ b₁"b₄) are to be optimized. The correlation coefficient of the reference image and deformed image is used to quantify the degree of agreement

Computer Camera/CCD

Figure 2: Relative location of sub-images of deformed and un-deformed images on surface

As shown in Figure 2, the central point of sub-image A prior to deformation is point P, and then moved to point P* after deformation while the corresponding sub-image is B. The correlation between the grayscale-pattern of sub-sub-image A and B can be defined as below: 2 2 ij ij ij ij g g COF g g Σ = Σ ⋅ Σ   (3)

where g and _ij g is grayscale of sub-image A on coordinate _ij

( )

( )

( )

( )

2.2 Three-Dimensional Digital Image Correlation Method

The measurement principle of conventional three-dimensional DIC is as follows: (1) Using two cameras to capture two images of an object from two different

locations.

(2) These two parallax images are regarded as the images before and after deformation, the relative displacement field between these two images can thus be determined using two-dimensional DIC.

(3) According to the spatial parameters between these two cameras and the camera parameters, the spatial coordinate transformation relationship and the obtained relative displacement field can be applied to calculate the 3D coordinates of the object surface.

The above described measurement principle can be effectively applied in static situations. However, this principle is not suitable for dynamic measurements. The key point to obtain accurate measurement results is that these two cameras have to capture images of object in the same situation. If the object under test is in motion, then these two images might not be captured in the same situation. Thus, applying above described principle to dynamic measurements could result in significant errors.

This research proposes a new technology to conduct three-dimensional measurement. A spine-type prism is used to separate the view axis of the digital camera into two axes which are inclined to each other. Therefore, digital cameras can receive two images with parallax in a single capture action. These two images with parallax are stored in the right and left part of the sensor of the camera. The working principle is shown in Figure 3. The major advantage of technology is the mobility. As long as the prism is installed on the camera, the camera can be used normally to capture images with parallax.

Assuming the prism is a thin lens, the mirror phase shift of the light can be ignored. This phase shift will cause a slight nonlinear error, and the error can be eliminated by distortion calibration.

For illustration, we first define the measurement coordinate systems. Reference to Figure 4, there are physical coordinate, right virtual coordinate and left virtual coordinate systems. Object Virtual image L Image Prism Camera Virtual camera R Virtual camera L Virtual image R

(1) Physical coordinate system: The physical coordinate system is a Cartesian coordinate system with the origin at the center of the lens of the camera. The y-axis denote the Object Distance.

(2) Right virtual coordinate system: The left part of the sensor receives the light comes from the hand side of the lens. In theory only the points at the right-hand side of the longitudinal axis (ξ_R) will format image on the left part of the sensor. The origin is located at the center of the lens of the right virtual camera. There is an inclination angle (θ ) between the longitudinal axes of the physical and right virtual coordinate systems.

(3) Left virtual coordinate system: The right part of the sensor receives the light comes from the hand side of the lens. In theory only the points at the left-hand side of the longitudinal axis (ξ ) will format image on the right part of the _R sensor. The origin is located at the center of the lens of the left virtual camera. There is also an inclination angle (θ ) between the longitudinal axes of the physical and left virtual coordinate systems.

Notations: R η R ξ ξ _L L η ) , ( yx ) , (η_R ξ_R ) , (η_L ξ_L θ θ ) , (η_R q ) , (η_L q q q y x e w P IP

(x,y) physical coordinates, unit=mm

) ,

(η_R ξ_R right virtual coordinates, unit=mm

) ,

(η_L ξ_L lift virtual coordinates, unit=mm

q image distance, unit=mm

e distance between the right and left virtual cameras, unit=mm θ inclination angle of the virtual view axis, unit-radian

λ density of the sensor, unit=pixel/mm

R

η horizontal coordinate of the right virtual image, unit=mm

L

η horizontal coordinate of the left virtual image, unit=mm

w intercept of the line connecting the origins of the virtual coordinate systems, unit=mm

R

X image coordinate of the right image, unit= pixel

L

X image coordinate of the right image, unit= pixel

Based on translational and rotational transformation theory, the physical coordinates of point P can be transformed into the two virtual coordinates, as shown below: ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ − − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ w y e x R R 2 cos sin sin cos θ θ θ θ ξ η (4) ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ w y e x L L 2 cos sin sin cos θ θ θ θ ξ η (5) And by similar triangles theorem, the projected positions on the sensor are:

R R R q ξ η η = (6) L L L q ξ η η = (7) Substitute equation (4) and (5) into equation (6) and (7) respectively, we have the simultaneous equation of the coordinate x and y.

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ + − + − − + + = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − − + ) cos sin ( ) sin cos ( 2 ) cos sin ( ) sin cos ( 2 cos sin sin cos cos sin sin cos θ η θ θ η θ θ η θ θ η θ θ η θ θ η θ θ η θ θ η θ L L R R L L R R q w q e q w q e y x q q q q (8) Because λ η R R X = (9) λ η L L X = (10) Substitute equation (9) and (10) into (8), we have

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ + − + − − + + = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − + ) cos sin ( ) sin cos ( 2 ) cos sin ( ) sin cos ( 2 cos sin sin cos cos sin sin cos θ λ θ θ λ θ θ λ θ θ λ θ θ λ θ θ λ θ θ λ θ θ λ θ L L R R L L R R X q w X q e X q w X q e y x X q X q X q X q (11) Solve the simultaneous equation (11), the physical coordinates( yx, ) are found.

3 Experiment

This article aims to explore the application of a spine-type prism to make a camera or CCD can capture images with a parallax effect characteristics. Therefore, in this order to avoid the problems caused by the dynamic photography (such as dynamic blur and time-difference of scanning and so on) and out of discussion focus, we conduct static experiments to investigate the effects mentioned above. The issues discussed in this include: characteristics of random measurement error of a flat planar specimen and the measurement accuracy of three-dimensional model of shape.

3.1 Characteristics of Random Measurement Error

The images captured by cameras with a spine-type prism cause more serious distortion effect than those without prism. The reason for it is that the object distances of the both virtual cameras vary symmetrically to the central axis, and then result in image distortion. The distortion will affect the accuracy of DIC analysis, and the formation of 3-dimensional measurement error. This impact can be discussed by the quality of 2-dimensional displacement field calculated by 2-D DIC method. The difference between the displacement field and the regressive surface of the image distortion is the random error of the measurement. A good method should cause very small random error, such as less than 0.01Pixel.

Photo 1: experimental setup

Photo 1 shows the experimental setup. A digital camera (Canon EOS 550D) was used in the experiment.

The line of sight will be deflected by the refraction effect of the prism. Because the light waves of different frequency have different relative refractive index, so that different frequencies of light have different degrees of deflection. The three primary colors of sunlight: red, green and blue have wavelengths of 620-740nm, 500-575nm and 445-500nm respectively, as shown in Figure 5. The relative refractive indexes of glass for the light of different colors are also different. The red light has the lowest relative refractive index, the blue light has the highest relative refractive index, and the green is in between. Therefore, the light reflected by the surface of a white board with a black spot will form an image of overlapped spots of yellow, red, black, blue and cyan colors, as shown in Figure 6. This phenomenon will cause a serious error of DIC analysis, and even makes the DIC analysis divergent.

Figure5: Wavelength of the visible light

This study proposes a solution to overcome the above mentioned problem. A green light filter with a wavelength of 540nm can be attached to the lens of the camera, and allows only the green light to come into the camera. To study the effect of the filter, two tests were conducted in this research. The first test (numbered as PL-PC) uses a spine-type prism with a slope of 0.125 without green-light filter, and the second one (number as PL-PGF) uses the same prism with a green-light filter.

Test results:

The photo 2 shows the images captured by the digital camera. The distance between the front of the lens and the planar specimen is 1389mm. The photo 2(a) and 2(b) are captured by the camera without and with the green-light filter, and the photo 2(c) is the local enlargement of the range indicated by the red rectangle in photo 2(a). The dispersion phenomenon can be observed in photo 2(c).

8 10− ₁₀−6 ₁₀−4 ₁₀−2 Wavelength in meters ₁₀ 10− Frequency 18 10 ₁₀14 ₁₀12 ₁₀10 Visible 50 0nm 0nm60 0nm70 40 0nm

X rays Ultraviolet infrared Microwave s

16

10

Dispersion

There is a slurred belt-shaped region in the center of the Photo 2(a) and 2(b) to mention. This is caused by the overlapping of the images of the two virtual cameras. Figure 7 shows the phenomenon.

Analysis results:

Figure 8 shows the absolute displacement field of the images without and with a green light filter. The smoothness of the displacement field can reflect the quality of the analysis. The smooth displacement field indicates a good analysis quality, while the rough displacement field indicates a significant random error.

Figure 7: Image overlapping at the central Overla d Left part Right

part

(a) PL-PC without filter (b) PL-PGF with filter

(c) detail of PL-PC

(a) Absolute displacement, PL-PC (b) Absolute displacement, PL-PGF Figure 8: Absolute displacement field of 3-D analysis

Observe the surface graph of the displacement fields in the Figure 8, the trend of the displacement field were both like a bowl. The displacement is the smallest at the center and the largest at the corners. However the surface graph of the test PL-PC (Figure (a)) is covered with small irregular fluctuations , while the surface graph of the test PL-PGF (Figure (b)) is smooth. The results indicate that DIC analysis has large random error, and the error can be avoided by simply using a green-light filter.

The standard deviation of error of the test without and with the green-light filter is 0.21 pixels and 0.0076 pixels respectively. It is to say that the adverse consequences of the dispersion are almost perfectly overcome by the use of the green-light filter. Therefore, the present study demonstrated that the use of the spine-type prism will cause stronger distortion than the normal lens, but not result in the random measurement error. The distortion error is the systematic error, and can be eliminated by means of nonlinear calibration.

3.2 Three Dimensional Measurement of a Mask

To verify the feasibility of the proposed method for three-dimensional measurement, a test on a mask was performed. The photo captured by the camera with the spine-type prism and a green light filter is shown in Photo 3.

Figure 9 Shows the displacement field and strain field of the images of the mask. The Figure 9(a) reveals that the proposed method can reflect the very mirror variance of the distance between the mask and the camera. The detail of the mask such as eyebrows, nose, and mouth can be identified by the displacement filed. The strain filed is shown in Figure 9(b). The strain field can more distinct reflect the detail of the mask. These results confirmed that the proposed method is effective for the three dimensional measurement.

Photo 3: Image for three dimensional measurement

(a) Horizontal displacement filed (b) Strain filed ε_x Figure 9: DIC analysis result: displacement field and strain field

4 Conclusions

The research proposed a three-dimensional DIC with very good mobility. The only one hardware requirement is a spine-type prism and eventually a green light filter. In many cases, the green-light filter can be left out, because the digital camera can filter out the red and blue light itself.

The following conclusions are to be drawn:

1. The spine-type prism can make a single camera possible to capture images with parallax.

2. The proposed method perfectly overcomes the synchronization problem of the three-dimensional DIC with two cameras.

3. The displacement and strain fields reveal the detail of the mask. This indicates the feasibility of three-dimensional measurement.

4. Due to the dispersion-effect a digital or hardware filter is necessary. In case a digital filter is used, the image should be stored in the RAW format.

Acknowledgement

The authors would like to acknowledge the support of Taiwan National Science Council through grant No. NSC 99-2625-M-390-001 and NSC 99-2625-M-260-003.

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