Optical measurement of a 3D displacement field using one ... · Optical measurement of a 3D displacement field using one camera D. Vavrik1,2, J. Bryscejn1, J. Jakubek2 & J. Valach1

Optical measurement of a 3D displacement field using one camera

D. Vavrik1,2, J. Bryscejn1, J. Jakubek2 & J. Valach1 1Institute of Theoretical and Applied Mechanics, Prague, Czech Republic 2Institute of Experimental and Applied Physics, Prague, Czech Republic

Abstract

The optical measurement of flat specimen deformation is a frequent task. In some applications it is necessary to measure both the in-plane and out-of-plane displacement or strain fields respectively. To achieve this goal a variety of methods have been developed in the past. This paper describes a novel technique based on the use of only one camera which employs a combination of two different optical methods to achieve 3D displacement measurements. The optical grid method for the measurement of the in-plane strain field and the proposed Coded Photometric Stereo method for the measurement of the out-of-plane displacement can be applied simultaneously. This approach reduces hardware part costs. The combined method is full field (all data acquired at once), non-contact and non-destructive. Keywords: full field strain measurement, optical method.

1 Introduction

The finite element method is widely used for numerical simulations of solid object behaviour in many applications questioning in some cases the necessity of physical experiments. Available databases of material behaviour are quite sufficient for simulations of linear systems. However for the simulations of nonlinear systems, it is still necessary to have experimental information. Unfortunately, the feedback based on strain measurements in several points only can lead to accurate but an imaginary numerical solution. Full field strain measurement is routinely performed today. Let’s reduce the class of the analyzed object onto flat specimens only. Dependency on loading type and studied feature, either the in-plane or out-of-plane experimental method is usually chosen. One

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Computational Methods and Experiments in Material Characterisation II 229

of the highly non-linear problems is the failure of the ductile specimen with stress concentrator especially when failure is accompanied by the development of a damage zone in the vicinity of the stress concentrator. Corresponding damage models have a number of parameters which it is necessary to determine. The feedback based on in-plane strain measurement only can lead to a non-unique solution as well as without prior knowledge of material damage type. For this reason both in-plane and out-of-plane displacement measurement is required. On the other hand monitoring the out-of-plane displacement can detect the onset of intensive damage zone development [1]. It is possible to employ the combination of two different optical methods using only one camera to achieve 3D displacement measurement. The proposed Coded Photometric Stereo method for the measurement of the out-of-plane displacement and the optical grid method for the measurement of the in-plane strain field can be applied simultaneously. The shape from shading (SFS) method [2, 3] originally developed for astronomy (moon crater relief reconstruction for instance) allows topographic reconstruction of the object surface using its optical image. The successive Photometric stereo (PS) method [4, 5] based on similar principles overcomes some restrictions of the SFS method. Both methods utilize the close relation between the relative lightness of a uniformly illuminated surface and the angle to the direction of light. It means that knowing the illumination geometry we can determine the slope (normal) of the surface at every point. Surface topography is obtained as an integration of this normal field. This fact can be used for the purpose of out-of-plane measurements of loaded object deformations. SFS and PS differ in the number of images used for topography reconstruction. Only one image of the studied surface is necessary for SFS. The ambiguity of the SFS method is avoided by using rather strong assumptions that the whole surface is of the same colour and that the surface has to be cylindrical. Three images required by PS using three lights illuminating surface from different directions are sufficient for determination of both x and y slopes at any studied point of the surface regardless of its colour variations. Knowledge of both slopes is advantageous for the reconstruction integration especially in the case of locally irregular surfaces. For this reason three lights have been used in our experiment. Coded Photometric Stereo (CPS) is the enhanced PS method utilizing Red-Green-Blue (RGB) lights coding. Monochromatic RGB lights positioned around the observed area produce directional illumination. Three different scenes coded in one composed image can be separated using standard RGB colour channels of the digital camera we use. Out-of-plane displacement is measured by CPS using these scenes. The composed image serves for the measurement of the in-plane displacement field by the grid method. The grid method analyzes the deformation of the surface measuring grid. The method of Interpolated Ellipses (MIE) has been selected for this purpose [6]. The MIE is a technique based on the optical monitoring of deformations during loading processes of hexagonal grids of dots deposited on the surface of the monitored specimen. Loading the


230 Computational Methods and Experiments in Material Characterisation II

specimen deforms a circle on the surface into an ellipse. Each ellipse is interpolated by six neighbouring dots of the hexagonal grid. Knowledge of the ellipse parameters directly yields the magnitude and the direction of principal strains on the specimen surface. The out-of-plane displacement is measured by CPS using three scenes coded in one RGB image. The composed RGB image serves for the measurement of the in-plane displacement field by the MIE. As the two displacement fields can be determined from one image the method is also suitable for dynamical experiments.

2 Coded Photometric Stereo method

The following set of steps describes the method of tested specimen topography with the information extraction procedure. The intensity variations on a homogeneously illuminated surface by collimated light encode the waviness and texture of the surface. This procedure is well known and is central to the shape-from-shading (SFS) method [4, 5] for surface topography reconstruction based on the known distribution of shadows – intensity changes caused by the variation of surface normals direction (also known as ‘needle map’). The assumption of the known relation between illumination, surface normal and viewing directions and the measured light intensity allows deducing field of surface normals or surface gradients defining local surface orientation. The relation between surface normal and observed reflected light intensity is predominantly described in the literature by ‘reflectance maps’ supposed to comply with a Lambertian model of surface. The Lambertian model yields to the image irradiance equation: I(x,y)=R(p(x,y), q(x,y)). It follows that brightness I at pixel (x,y) depends on surface gradients (p,q) at the studied point (x,y) via a so called reflectance map in the following way:

,1

sinsinsincoscos),(22 qp

qpqpR++

−−⋅=

γτγτγρ (1)

where ρ is an albedo (the ratio of reflected to incident light) and τ, γ are azimuth angle (in the surface plane) and elevation angle (in the plane perpendicular to the surface plane) defining the direction of illumination [4]. The drawback of the SFS method is that it represents a mathematically ill-possessed problem which can be solved applying some constraints on the surface properties – it can only cope with cylindrical surfaces of the uniform albedo. In order to deal with a significantly wider class of shapes of varying surface properties, a modification of the SFS method has to be adopted, which is called photometry stereo method [5]. This method uses several images of the investigated surface, each illuminated by a different light source of known direction and position. The set of at least three images acquired under different illumination directions yields enough data to calculate the two perpendicular surface tangents and albedo at every image pixel.



The natural limits of the method are the cases where reflections, self-shadows, translucency of the studied body and secondary reflections occur. Results can be non-unique in these cases. However, as a local problem it may be possible to solve on the other hand. Crack will produce a self-shadow effect for instance: we will not be able to measure the dimension perpendicular to the surface. However the crack length and shape of the surrounding surface will be reconstructed correctly. Reconstructed topography is dimensionless, so dimension calibration must be done. It can be done by one point mechanical measurement of the elevation differences between two points of the studied surface or it is necessary to have some calibration object of known height/depth connected with the surface. As stated above, at least three images are necessary to get sufficient information for surface reconstruction. Switching between the light sources and taking shots by camera would render such a method virtually useless for dynamical testing unless some modification is carried out. The modification allows acquisition of the three images at the same time. Acquiring and storing three independent images in one is made possible by taking advantage of the wavelength selectivity of a red-green-blue (RGB) Bayer filter in front of the camera chip. The idea is that by using three different colours of illumination light sources corresponding to the filter colours, three independent images are coded in one (Coded Photometrical Stereo). The spectra of the illumination light sources do not correspond exactly to the RGB decomposition of the camera Bayer filter in the current experimental setup (e.g. the illumination blue light is partially detected by green filter etc. – this effect is known as ‘colour channel cross-talk’). That is why in order to correctly apply the method without errors, a light separation step has to be performed with the help of calibration images. The light separation relies on a priori knowledge of light source – camera RGB colour components sensitivity values known from a set of calibration images of flat, homogenously white surface properties best suited for the SFS method acquired for each light source separately. The steps involving simultaneous acquisition of the three images and their separation represent an improved technological implementation of the photometrical stereo method in dynamical optical surface topography reconstruction described in the second step. Once the field of normals is known, it is straightforward to reconstruct the surface. The method [7] utilizing wavelet transformation was used, because of its simplicity and availability of source code on the Internet. The reconstruction gives results up to multiplicative constant adjusting heights in the surface topography, which has to be deduced from the known image feature or experimental setting.

3 Method of Interpolated Ellipses

The MIE is a technique based on the optical monitoring of deformations during the loading processes of hexagonal grids of dots deposited on the surface of the monitored specimen. We assume that the surface deformation of a continuous



material is approximately homogeneous inside a circle with initial radius r. Loading the specimen will deform a circle on the surface into an ellipse. Each ellipse is interpolated by six neighbouring dots of a hexagonal grid as shown in Fig. 1. Knowledge of the ellipse parameters yields the magnitude and direction of the principal in-plane strains 21 , εε on the specimen surface as:

rrb

rra −

=−

= 21 , εε (2)

The angle ϕ in Fig. 1 represents the angular rotation of the local coordinate system of the principal strains in the examined element (cell) relative to the global coordinate system XY. The Y axis is oriented in the direction of the loading force. The local coordinate system xy is defined so that the y axis is oriented in the direction of the principal strain 1ε while the x axis follows the

direction of the principal strain 2ε .

Figure 1: Illustration of an ellipse interpolated by a hexagonal grid.

Principal strains are evaluated independently from the orientation of the global coordinate system without having to determine the shear strain. This is the main advantage of this approach which becomes important especially in the situation when non-uniform strain fields surround the crack tip. The fundamental benefits of the hexagonal grid manifest themselves wherever we are working in the limiting resolution regime of the recording equipment.

4 Experimental

A specimen for experiment was prepared from high-ductile aluminium alloy. This material is well known from other experiments performed in the past, see e.g. [6]. Its elastic modulus E is 70 GPa, Yield stress σys is 296 MPa and Poisson’s ratio µ is 0.315. The experiment was carried out in flat specimens with a pre-machined central “crack" (by spark-out technology. “Crack” width is 0.3 mm.) labelled “CC”; see Fig. 2. The photoresist etching technique was used to apply a hexagonal grid of dots with 0.2 mm pitch and 0.1 mm diameter symmetrically to the crack plane on the



ligament surface (between the crack tip and the specimen edge). The surface of the specimen was illuminated by the three Red, Green and Blue lights. All of them had an elevation angle γ = 45˚ and azimuth angle τ = 0˚ (blue light), 120˚ (green) and 240˚ (red). See Fig. 3 for the composed RGB image of the deformed specimen.

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21

5169

250

50

10

central crack

Figure 2: Geometry of the specimens.

Figure 3: Image of the deformed specimen surface.

The specimen was loaded in uni-axial tension by grips displacement until the first short new 200 µm long crack appeared on the specimen surface. Then the 3D strain field was analyzed. The experimental setup for the testing of the proposed combined method is shown in Fig. 4. The composed image was separated into three R, G, B channels and colour channel cross-talk was corrected. Consequentially three independent scenes were obtained. These scenes served for topographic reconstruction; See Fig. 5 for contour imaging of the out-of-plane displacement uz field of the top specimen face. See Fig. 6 for 3D surface representing the out-of-plane strain field ε3. Note that the extreme value is approximately 0.5 mm away from the crack tip. It



indicates the existence of the intensive damage zone as was confirmed by the X-ray transmission radiography [1]. The half specimen thickness tz = 2.5 mm served as a measuring distance for the principal strain ε3 calculating e.g. ε3 = (uz-tz)/tz. It is necessary to say that for numerical simulation it would be more appropriate to use displacement field uz because real strain value is actually higher because a 5 mm thick specimen does not exhibit pure plane stress behaviour especially in the vicinity of the crack tip.

Figure 4: Experimental setup for a testing of proposed combined method.

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Figure 6: 3D surface, representing principal strain ε3 field.

The composed RGB image served for the calculation of the in-plane principal strains ε1, ε2 fields by the MIE; see Fig. 7.



5 Conclusions

The combined method of the coded photometric stereo and the method of interpolated ellipses was successfully implemented. It was confirmed that it is possible to measure 3D displacement/strain field using one camera only and a set of three R, G, B lights. All necessary information are coded in one image.

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Acknowledgements

This work has been supported in part by Grant Nos. 106/04/0567 and 101/03/0731 of the Grant Agency of the Czech Republic and by the Ministry of Education, Youth and Sports of the Czech Republic under the research project AV0Z20710524. This work has also been supported in part by EU project Histoclean EVK4-CT-2002-30011.

References

[1] Vavrik, D.; Jakubek, J.; Pospisil, S.; Visschers, J.: “Non-destructive Observation of Damage Processes by X-Ray Dynamic Defectoscopy”, Proceedings of the 9th International Conference on the Mechanical Behaviour of Materials, Switzerland, Geneva, May 25-29, 2003.

[2] Horn, B. K. P.: Obtaining shape from shading information, in: The Psychology of Computer Vision (P. H. Winston – Ed.), McGraw Hill, NY, 115-155, (1975).

[3] Horn, B. K. P.; Brooks, M. J.: The variational approach to shape from shading, Computer Vision Graphics Image Processing, 33, 174-208, (1986).

[4] Kim, B.-H.; Park, R.-H.: Multi-image photometric stereo using surface approximation by Legendre polynomials, Patter Recognition, 31, 1033-1047, (1998).



[5] Woodham, R. J.: Photometric method for determining surface orientation from multiple images, Optical Engineering, 19, 139-144, (1980).

[6] Vavrik, D.; Zemankova, J.: “Crack Instability in Ductile Materials Analyzed by the Method of Interpolated Ellipses.” Experimental Mechanics, Vol. 44, August 2004, pp. 327-335, ISSN: 0014-4851.

[7] Kovesi, P. D.: MATLAB functions for computer vision and image analysis, 1996-2003. http://www.csse.uwa.edu.au/~pk/Research /MatlabFns/.



Documents

Optical measurement of a 3D displacement field using one ... · Optical measurement of a 3D displacement field using one camera D. Vavrik1,2, J. Bryscejn1, J. Jakubek2 & J. Valach1