ASSESSMENT OF DIGITAL IMAGE CORRELATION PERFORMANCES

Transcription

1 ASSESSMENT OF DIGITAL IMAGE CORRELATION PERFORMANCES Workgrop Metrology of CNRS research network 2519 a MCIMS, Presented by Larent Robert and Jean-José Orte b a b CROMeP, Ecole des Mines d Albi-Carmax, ALBI Cedex 9, France larent.robert@enstimac.fr ABSTRACT Optical fll-field measrement methods sch as Digital Image Correlation (DIC) are increasingly sed in the field of experimental mechanics, bt they still sffer from a lack of information abot their metrological performances. In order to assess the performance of DIC techniqes, some collaborative work has been carried ot by the Workgrop Metrology of the French CNRS research network 2519 MCIMS 1. Basically, the stdy is based on displacement error assessment from synthetic speckle images. First, some series of synthetic images with random patterns and sbmitted to sinsoidal displacements with varios freqencies and amplitdes have been generated. Then displacements are evalated by several DIC packages that are based on varios formlations. Different correlation window sizes and speckle pattern sizes are tested. Displacements fond are finally compared with the exact imposed vales. Reslts show that the overall RMS error seems to be mainly controlled by the first order difference between the real transformation and the local transformation of the correlation window assmed by the DIC algorithm. Two limiting sitations have been observed: on one hand, displacements varying at a low spatial freqencies for which the classical error observed for rigid motions is recovered. On the other hand, displacement fields with high spatial freqencies for which the error is eqal to the RMS of the displacement itself (this sitation does not allow DIC evalation). Introdction Optical fll-field measrement techniqes are very promising tools for the experimental analysis of the mechanical properties of materials and strctres. While they are more and more widely sed, they still sffer from the lack of a complete metrological characterisation. Sch techniqes rely on complex measrement chains, and the error sorces of each of its elements reqire proper evalations before a global assessment of the measrement. The collaborative work carried ot by the members of the workgrop Metrology of the French CNRS research network 2519, coordinated by Y. Srrel (Visol Technologies) and F. Brémand (Univ. Poitiers), aims at contribting to a systematic and qantitative approach to these qestions [1]. Digital Image Correlation (DIC) techniqe is among the most poplar optical methods, becase of the availability of commercial packages, the constantly shrinking cost of digital cameras and compters, and the general apparent simplicity of sample preparation and optical setp. An important, bt not sole, element of the measrement procedre is the image analysis software package spposed to provide an apparent 2-D displacement field that maps a so-called reference image to a deformed image, provided by an appropriate optical system, at a discrete set of positions, according to some principle of optical flow conservation. The analysis of rigid body motions (both on real or simlated images) shows that the ltimate resoltion of this mapping, in pixels, is bonded by the actal dynamic range of the digital images (noise divided by the fll grey levels range). This can be jstified by the fact that a motion smaller than this limit does not generate any significant modification in the images. In practice, resoltions close to 1 over 100 pixels can be achieved with noiseless 8 bits images. Note however that sch vales can only be reached when the speckle pattern sed by the matching algorithm exhibits appropriate characteristics in terms of 1 MCIMS: Mesres de Champs et Indentification en Mécaniqe des Solides / Fll-Field measrements and Identification in Solid Mechanics. People directly involved in this work are M. Grédiac (Univ. Clermont-Ferrand), Y. Srrel (Visol Technologies), P. Vacher (Univ. of Savoie), S. Misto (ENIT), J.C. Dpré and F. Brémand (Univ. of Poitiers), F. Hild (ENS Cachan), M. Bornert, (EP), B. Wattrisse (Univ. of Montpellier), and L. Robert and J.J. Orte (EMAC).

2 grey level histograms, typical size with respect to pixel and correlation window size, spectral content, etc. Isses relative to the definition of an optimal pattern are not addressed in this presentation [2], even if they are part of the research grop discssions. Sch qantitative evalations of the errors of DIC measrements are sally limited to sitations with homogeneos mechanical transformations, namely, simple niform translation, in plane rotations, or ot of plane rigid body motions, which reslt in apparent (almost) affine transformations of the 2-D image [3,4]. Very few stdies [2,4,5] address sitations with spatially flctating displacement fields, which need to be investigated for a qantitative assessment of the spatial resoltion of sch techniqes. Since it is very difficlt, and in practice impossible, to experimentally generate non niform deformation fields, with precisely prescribed strains (some athors have investigated this way, e.g. [6]), it is necessary to perform the analysis on simlated images, obtained with algorithms that mimic as closely as possible the generation of images in a real camera. In Ref. [5] qadratic displacement fields have been considered; the present approach extends the analysis to in-plane sinsoidal displacements exhibiting varying spatial freqencies. The proposed methodology is similar to that leading to the Modlation Transfer Fnction classically sed to characterize optical devices. The RMS errors of the displacements obtained with varios DIC softwares are evalated as fnctions of the spatial freqency and the amplitde of the displacement field, for varios correlation window sizes and DIC formlations. Methodology Two sets of synthetic speckle painting images have been generated. In the first set, Wima (provided by S. Codert, ENSAM Paris and S. Misto, ENIT) the images are defined as random distribtions of circlar spots with a Gassian local variation of the grey levels. The second set of synthetic images (provided by L. Robert and J.J. Orte, EMAC) has been obtained sing the TexGen software [7]. The method is based on sccessive transformations of Perlin's coherent noise fnction to prodce the desired pattern aspect. It is worth noting that no bias or interpolation errors are indced by the image generation becase the speckle is deformed in the real space before being mapped in the object space. In both sets, parameters were selected sch that the radis r at half height of the atocorrelation fnction of the reference image is 3 pixels. Deformed images are obtained assming an in-plane sinsoidal displacement. In a first approach, only tensile/compressive displacements along the x direction with a zero y displacement are considered. The displacement along x is given by: 2 X x ( X, Y ) psin (1) p where p is the period in pixels and is the amplitde of the flctation of the xx component of the displacement gradient. The xx component of the first and second displacement gradients are given by Eq. (2) and (3) respectively: x 2 x, ( X, Y ) 2 cos X p x, ( X, Y ) sin p xx X p (2) (3) Grey levels of the pixels in the deformed images are obtained according to the optical flow conservation rle: g( x) f ( 1 ( x)) (4) where (X) = X+(X) is the transformation map and g (resp. f) is the grey level in the deformed (resp. reference) image. Note that appropriate procedres have been sed in the TexGen software in order to mimic the (perfect or imperfect) spatial integration of light performed by a real image sensor. All generated sets contain 1024 x 1024 or 512 x 512 pixel image and varios deformed configrations ( {0.1, 0.05, 0.02, 0.01, 0.005, 0.001} and p {10, 20, 30, 60, 130, 260, 510} pixels). The set considered in the present work have been generated sing the TexGen software. It contains one reference 512 x 512 pixel image and 28 deformed configrations obtained with {0.02, 0.01, 0.005, 0.001} and 10, 20, 30, 60, 130, 260, 510} pixels. Note that vales of the maximm of the displacement gradient are max X,X =2 12%, 6.3%, 3.1%, 0.63% respectively. At this stage only the 8 bit (256 grey levels) images have been processed so far. Figre 1 presents, for the considered set of images, a sb-image (512 x 100 pixel image) of the reference image and deformed images obtained for p = 130 pixels (for periods) and = 0.02, 0.05 and 0.1 respectively.

3 (a) (b) (c) Figre 1. Illstration of simlated synthetic images: reference (a) and deformed (b - d) images (second set, p = 130, (b) = 0.02, (c) = 0.05 and (d) = 0.1). In order to stdy the inflence of the speckle pattern size on the displacement ltimate resoltion (resoltion for low freqency deformation field), the speckle pattern mean size (radis r at half height of the atocorrelation fnction of the ref. image) can be adjsted in order to prodce fine (r/2), medim (standard, r = 2.2 pixels) or coarse (2r) patterns, as presented in Figre 2. Images have been processed with seven DIC packages, namely 7D (P. Vacher, Univ. Savoie) [8], Aramis 2D (S. Misto, ENIT) [9], Correla (J.C. Dpré/F. Brémand, Univ. Poitiers) [10], Correli (F. Hild, ENS Cachan) [11], CorrelManV (M. Bornert, EP) [12], KelKins (B. Wattrisse, Univ. Montpellier) [4] and Vic-2D (L. Robert/J.J. Orte, EMAC) [13]. (d) Intensity 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0-0,1 Atocorrelation fnction TexGen-fine TexGen-medim TexGen-coarse WIMA Pixels Figre 2. Illstration of synthetic speckle patterns (fine, medim and coarse) (left) and the associated atocorrelation fnction radis at half height (right). Packages are based on the minimisation of a correlation fnction (e.g. cross-correlation coefficient) C( ) [14] in the real or Forier space. Several definitions of C( exist, depending on the software, as well as descriptions of the local transformation of the sbset (the shape fnctions that relate image coordinates in the reference image to coordinate in the deformed image), mainly translation (two coefficients), rigid transformation (3 coefficients), affine (6 coefficients), bi-linear (8 coefficients), qadratic (12 coefficients) and bi-qadratic (18 coefficients) transformations. Fll local optimisation is carried ot directly

4 (Aramis 2D, Correla, Vic-2D), otherwise the higher order terms are compted from the neighborhood (CorrelManV, KelKins, 7D). The sb-pixel evalation is classically done by direct interpolation of the deformed image. Image interpolation incldes bilinear (Correla, CMV, KelKins), bi-cbic (Correla, 7D) and higher order interpolations (e.g. bi-qintic splines, Vic-2D) as well as interpolation in the Forier space (Correli). Optimization of the correlation coefficient C( is performed by minimization of gradient (7D, Aramis, Correla, CorrelManV, Vic-2D) or by bi-parabolic interpolation (Correli Q1, KelKins). Sqare correlation windows (or sbsets) of different sizes have been sed: 9 or 10, 15 or 16, 21 or 22, 31 or 32 and 63 or 64 pixels. They are denoted CWS in the following. Displacements are evalated at all intersection points of a reglar sqare grid in the initial image, with a pitch sch that correlation windows at adjacent positions do not overlap (i.e., statistical independence of the corresponding errors). Reslts Differences between the evalated and prescribed displacements along x are analysed statistically in terms of RMS errors: global 1 measred imposed 2 (5) = nmber x All positions x This analysis can be performed globally for all points in the image bt also more locally, with error averages compted over points ndergoing the same displacement and the same deformation gradient, i.e. points with same x coordinate. As a general point of view, it is shown that the RMS error is described as = fnction( p,, r,, CWS, interp.,...) where p and describe the imposed transformation, r characterises the speckle size, and, CWS and interp. are relative to the main DIC packages options (shape fnction, sbset size and grey level interpolation). In the following, for a better clarity, the reslts are presented in three sbsections corresponding to the three types of local transformation of the DIC packages sed for the correlation. Rigid transformation (translation): Figre 3 presents the RMS error normalised by as a fnction of the period p. Reslts are given for varios strain amplitdes and local rigid transformations. The whole set of seven DIC packages described above was sed. Figre 3. RMS error normalised by as a fnction of the period p for varios strain amplitdes for a local rigid transformation and for 3 DIC packages. Sbset window size CWS is respectively eqal to 10, 16, 22 and 32 pixels.

5 The following conclsions can be drawn. First, there are very small differences between DIC packages and one master crve can be obtained, as clearly seen in Figre 3 where each crve is obtained with one of the DIC packages described above. All these reslts obtained with the DIC packages can be gathered in a schematic view that presents the different regimes for the RMS error normalised by the maximm first derivative of the displacement, as a fnction of the period p (see Figre 4). For periods smaller than the sbset size, it is fond that the RMS error is eqal to the RMS of the displacement itself, that is p / 2, and the DIC is not able to evalate any displacement (several periods in a sbset). For periods between CWS and abot 15 CWS, there is a transition regime mainly described by the relationship between the sbset CWS and the speckle size r (the asymptotic regime is obtained faster for small CWS and r). The displacement resoltion can be defined somewhere in this area, and the ltimate spatial resoltion cold be CWS itself. Work is going on for a more precise description of the transition regime. For periods greater than 15 CWS (low freqency strain field), an asymptote is reached and the error is mainly independent of the period and of the sbset size. The behavior is close to those observed for rigid body motions becase the strain is qasi niform inside the sbset: the RMS error is proportional to the strain amplitde ths the error is essentially controlled by the first displacement gradient. Figre 4. Schematic observation of the different regimes for the RMS error normalised by the maximm first derivative of the displacement, as a fnction of period p for a local rigid transformation. It shold be noted, firstly, that for large sbsets (CWS = 32 or 64), the convergence is not completely reached even for a global Max period of 510 pixels. Secondly, the asymptotic error depends highly on the speckle size: the ratio / was fond to X, X be abot 0.35, 0.6 and lies between 0.8 and 1 for respectively the fine (r/2), medim (r) and coarse (2r) speckle pattern presented in Figre 2. Two limitations to the general observations reported in Figre 4 are observed: (i) for small strains and small sbsets, the resoltion is close to the resoltion for a pre translation. The error is transl ( N, CWS, interp.,...) fnct(cws, interp.,...) N where N is the image bit depth, typically less than pixel; (ii) for large / strains and large sbsets, it is observed that the asymptotic vale increases slightly with the sbset. Affine transformation: Figre 5 presents the RMS error normalised by the maximm second derivative of the displacement 4 2 p as a fnction of the period p, for varios strain amplitdes and for varios DIC packages in the case of local affine transformations. In this case, varios crves are observed for the sbset of 10 pixels, and for small strain level ( =0.001) except for large sbsets. Under that consideration, a master crve cold be also considered for almost all DIC packages only for sfficiently large strain and sbset. All these reslts can be gathered in a schematic view that presents the different regimes for the RMS error normalised by the maximm second derivative of the displacement, as a fnction of the period p (see Figre 6). For periods smaller that the sbset size, it is always fond that the RMS error is eqal to the RMS of the displacement itself and DIC is not able to evalate any displacement. For periods between CWS and abot 5 CWS, there is a transition regime shorter than for a rigid transformation. The displacement resoltion can be defined somewhere in this area, and the ltimate spatial resoltion cold be also CWS. Work is going on for a more precise description of the transition regime. For periods larger than 5 CWS (low freqency strain field), an asymptote is reached which increases with the sbset size, as illstrated in Figre 6 and Figre

6 7 on the left, and not with the speckle size r. The error is ths essentially controlled by the second displacement gradient and CWS. For small sbsets and small strains, as it was previosly seen for rigid shape fnction, the asymptotic regime differs and the resoltion is close to the resoltion for a pre translation. In case of affine transformation however, it is observed that the asymptotic vale increases as the sbset size decreases (see Figre 7, right). The asymptotic behavior can be expressed as follows: transl. Sp K(CWS)., (,CWS,interp.,...). The first term is highest for larger strains. It increases with CWS. X, XX N The second term is the highest for smaller strains. It deceases with CWS, showing that an optimal CWS exists. Figre 5. RMS error normalised by 4 p as a fnction of period p for varios strain amplitdes for a local affine transformation and for 6 DIC packages. Sbset window size CWS is eqal to 10, 16, 22 and 32 pixels respectively. Figre 6. Schematic observation of the different regimes for the RMS error normalised by the maximm second derivative of the displacement as a fnction of period p. Case of a local affine transformation.

7 Asym ptotic RMS / Ux,xx (a) y = 0,0313x 2 + 0,0159x R 2 = 0, Sbset size CWS (pixel) Asymptotic RMS U ( pixel) 0,018 0,016 0,014 0,012 0,01 0,008 0,006 0,004 0,002 0 (b) Sbset size CWS (pixel) Figre 7. (a) Asymptotic RMS error normalised by the maximm second derivative of the displacement for large strain Asymptotic RMS error for small strain. Case of a local affine transformation, presented for 5 DIC packages. (b) Transformation of higher order: In the case of qadratic or bi-qadratic shape fnctions, the trends are as follows: for p < CWS, no measrement is possible. Then the transition is shorter and the asymptotic regime is qickly obtained (error similar to those of pre translation), bt in this case the RMS error is independent of p and. It is fond that the errors are slightly dependent on the window size CWS, on the degree of grey level interpolation. Conclsion RMS errors of the displacements obtained with 7 DIC softwares on synthetic speckle pattern images have been evalated as fnctions of the spatial freqency and the amplitde of the displacement field, for varios correlation window sizes, speckle size and DIC formlations. Preliminary reslts show that the overall RMS error is mainly controlled by the first order difference between the real transformation and the local transformation of the sbset. Limiting sitations are, on the one hand, displacements varying at low spatial freqencies for which the classical error observed for rigid motions is recovered, and on the other hand, displacement fields with small periods (lower than the sbset size f), for which the RMS error is eqal to the RMS of the displacement itself (this sitation does not allow DIC evalation). The transition between these two regimes depends on CWS and : an asymptotic regime is obtained faster if CWS is smaller and is of higher degree. When is assmed to be a rigid (resp. affine) transformation, the asymptotic error is proportional to the first (resp. second) derivative of the displacement. Moreover, this asymptotic error is independent of CWS bt increases with the speckle size r in the case of a rigid transformation, or increases with CWS in the case of an affine transformation for sfficiently large CWS and strains. For small sbsets and small strains, the asymptotic error decreases with CWS as it is observed for pre translation. For of high order (e.g., qadratic), it is shown that the asymptotic error is independent of p and and slightly dependent on CWS. It is not governed by the third derivative of the displacement, which can be a second order phenomenon. Additional investigations, inclding the analysis of the transition and the inflence of the image noise, are the sbject of ongoing collaborative work and will be addressed dring the presentation. Works focsing on assessment of DIC packages on images sbmitted to shear strain fields, and assessment of strain measrements by DIC, are also in prospect. Acknowledgments The athor and all the participants of this benchmark are gratefl to the French CNRS for spporting this research. References 1. Bornert, M., Resoltion and Spatial Resoltion of Digital Image Correlation Techniqes, in Photomechanics 2006, Clermont-Ferrand, France (2006). 2. Lecompte, D., Smits, A., Bossyt, S., Sol, H., Vantomme, J., Van Hemelrijck, D. and Habraken, A.M., Qality Assessment of Speckle Patterns for Digital Image Correlation, Opt. Lasers Eng., 44, (2006).

8 3. Bergonnier, S., Hild, F. and Rox, S., Digital Image Correlation sed for Mechanical Tests on Crimped Glass Wool Samples, J. Strain Analysis, 40(2), (2005). 4. Wattrisse, B., Chrysochoos, A.,. Mracciole, J.M and Némoz-Gaillard, M., Analysis of Strain Localization dring Tensile Tests by Digital Image Correlation, Exp. Mech., 41(1), (2000). 5. Schreier, H. W. and Stton, M. A., Systematic Errors in Digital Image Correlation de to Undermatched Sbset Shape Fnctions, Exp. Mech., 42(3), (2002). 6. Patterson, E., Brgete, R., Hack, E., Siebert, T. and Whelan, M., Development of an International Standard for Optical Strain Measrement, in Photomechanics 2006, Clermont-Ferrand, France (2006). 7. Orte, J.J., Garcia, D., Robert, L., and Bgarin, F., A Speckle-Textre Image Generator, in Proceedings of Speckle06, edited by P. Slangen and C. Cerrti, SPIE vol (2006). 8. Vacher, P., Dmolin, S., Morestin, F. and Mgil-Tochal, S., Bidimensional strain measrement sing digital images, J. Mech. Eng. Sci. C, 213, (1999). 9. Aramis 2D software, GOM Optical Measring Techniqes, (2006). 10. Germanea, A., Domalin, P. and Dpré, J.C., Fll 3D Measrement of Strain Field by Scattered Light for Analysis of Strctres, to appear in Exp. Mech. (2007). 11. Chevalier, L., Calloch, S., Hild, F. and Marco, Y., Digital Image Correlation sed to Analyze the Mltiaxial Behavior of Rbber-like Materials, Er. J. Mech. A/Solids, 20, (2001). 12. Domalin, P., Bornert, M. and Caldemaison, D., Microextensometry by image correlation applied to micromechanical stdies sing the scanning electron microscopy, in Proceedings of the International Conference on Advanced Technology in Experimental Mechanics, Japan Soc. of Exp. Eng., (1999). 13. Vic-2D software, Correlated Soltions Inc., (2006). 14. Stton, M. A., McNeill, S. R., Helem, J. D. and Chao, Y.J., Advances in Two-Dimensional and Three-Dimensional Compter Vision, in Photomechanics (P. K. Rastogi ed.), Springer, Berlin, (2000).