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Recently, digital image correlation as a tool for surface deformation measurements has found widespread use and acceptance in the field of experimental mechanics. The method is known to reconstruct displacements with subpixel accuracy that depends on various factors such as image quality, noise, and the correlation algorithm chosen. However, the systematic errors of the method have not been studied in detail. We address the systematic errors of the iterative spatial domain crosscorrelation algorithm caused by gray-value interpolation. We investigate the position-dependent bias in a numerical study and show that it can lead to apparent strains of the order of 40% of the actual strain level. Furthermore, we present methods to reduce this bias to acceptable levels.

a b s t r a c tThe effect of out-of-plane motion (including out-of-plane translation and rotation) on two-dimensional (2D) and three-dimensional (3D) digital image correlation measurements is demonstrated using basic theoretical pinhole image equations and experimentally through synchronized, multi-system measurements. Full-field results obtained during rigid body, out-of-plane motion using a singlecamera vision system with (a-1) a standard f55mm Nikon lens and (a-2) a single Schneider-Kreuznach Xenoplan telecentric lens are compared with data obtained using a two-camera stereovision system with standard f55mm Nikon lenses.Results confirm that the theoretical equations are in excellent agreement with experimental measurements. Specifically, results show that (a) a single-camera, 2D imaging system is sensitive to out-of-plane motion, with in-plane strain errors (a-1) due to out-of-plane translation being proportional to DZ/Z, where Z is the distance from the object to the pin hole and DZ the out-of-plane translation displacement, and (a-2) due to out-of-plane rotation are shown to be a function of both rotation angle and the image distance Z; (b) the telecentric lens has an effective object distance, Z eff , that is 50 Â larger than the 55 mm standard lens, with a corresponding reduction in strain errors from 1250 ms/mm of outof-plane motion to 25 ms/mm; and (c) a stereovision system measures all components of displacement without introducing measurable, full-field, strain errors, even though an object may undergo appreciable out-of-plane translation and rotation.

Basic concepts in probability are employed to develop analytic formulae for both the expectation (bias) and variance for image motions obtained during subset-based pattern matching. Specifically, the expectation and variance in image motions in the presence of uncorrelated Gaussian intensity noise for each pixel location are obtained by optimising a least squares intensity matching metric. Results for both 1D and 2D image analyses clearly quantify both the bias and the covariance matrix for image motion estimates as a function of: (a) interpolation method, (b) sub-pixel motion, (c) intensity noise, (d) contrast, (e) level of uniaxial normal strain and (f) subset size. For 1D translations, excellent agreement is demonstrated between simulations, theoretical predictions and experimental measurements. The level of agreement confirms that the analytical formulae can be used to provide a priori estimates for the 'quality' of local, subset-based measurements achievable with a given pattern. For 1D strain with linear interpolation, theoretical predictions are provided for the expectation and co-variance matrix for the local displacement and strain parameters. For 2D translations with bi-linear interpolation, theoretical predictions are provided for both the expectation and the co-variance matrix for both displacement components. Theoretical results in both cases show that the expectations for the local parameters are biased and a function of: (a) the interpolation difference between the translated and reference images, (b) magnitude of white noise, (c) decimal part of the motion and (d) intensity pattern gradients. For 1D strain, the biases and the covariance matrix for both parameters are directly affected by the strain parameter p 1 as the deformed image is stretched by (1 + p 1 ). For 2D rigid body motion case, the covariance matrix for measured motions is shown to have coupling between the motions, demonstrating that the directions of maximum and minimum variability do not generally coincide with the x and y directions.KEY WORDS: digital image correlation/matching, error assessment, expectation and variance for image motion, intensity interpolation, intensity pattern noise, probabilistic formulation 1 The muted effect of quantisation on the measurement bias for an 8-bit signal is highlighted in the Discussion section.

ABSTRACT--Digital image correlation techniques are commonly used to measure specimen displacements by finding correspondences between an image of the specimen in an undeformed or reference configuration and a second image under load. To establish correspondences between the two images, numerical techniques are used to locate an initially square image subset in a reference image within an image taken under load. During this process, shape functions of varying order can be applied to the initially square subset 9 Zero order shape functions permit the subset to translate rigidly, while first-order shape functions represent an affine transform of the subset that permits a combination of translation, rotation, shear and normal strains.In this article, the systematic errors that arise from the use of undermatched shape function, i.e., shape functions of lower order than the actual displacement field, are analyzed 9 It is shown that, under certain conditions, the shape functions used can be approximated by a Savitzky-Golay low-pass filter applied to the displacement functions, permitting a convenient error analysis 9 Furthermore, this analysis is not limited to the displacements, but naturally extends to the higher-order terms included in the shape functions. This permits a direct analysis of the systematic strain errors associated with an undermatched shape function. Detailed numerical studies are presented for the case of a second-order displacement field and first-and second-order shape functions 9 Finally, the relation of this work to previously published studies is discussed 9