Quantitative Error Assessment in Pattern Matching: Effects of Intensity Pattern Noise, Interpolation, Strain and Image Contrast on Motion Measurements

Abstract

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.

High Performance Digital Volume Correlation

“…The standard deviation curves in Figure 7.2 also shows a dependence on the translation, with a larger standard deviation near u = 0 and u = 1, and a smaller standard deviation near u = 0.5. This dependence, which we observed experimentally for several cases, was not predicted by the previous standard deviation estimates [71,75] For the PDMS/Skyscan sample, the bias error has a maximum magnitude of 0.05 voxels, which is smaller than the standard deviation of 0.15 voxels, so its effect is difficult to see in our experiments. From these translation tests, we conclude that DVC with this combination of sample, scanner, and parameters is accurate to within 0.2 voxels, accounting for the maximum bias error plus the maximum standard deviation.…”

“…where S is the subset, s is the subset size, N(x) are the eight pixels neighboring Subsequent studies derived specific estimates of σ u based on the image noise σ and gradients of the image gray intensity level [59,75,76]. In particular, Wang et al [75] derive the estimate…”

“…Wang et al [75] also derive expressions for the expected value of the displacement u, which show a bias using bilinear and C 1 bicubic image interpolation. This bias is addressed by Schreier et al [71], who suggest using cubic or quintic splines to reduce the bias.…”

Optimized Patterns for Digital Image Correlation

“…In particular, constraining the patterns to be band-limited prevents the patterns from having high-contrast edges, even though increasing contrast is the most direct way of increasing the intensity gradients in the image, which in turn reduces many of the most important error terms in DIC measurements [2]. It is straightforward to increase the contrast of grayscale patterns, by choosing the new gray scale values as a function of the old gray scale values: where the slope of this function is greater than 1, contrast is increased.…”

“…Quantitative error analysis [2,3,4,5] of these techniques shows that the measurement error depends critically on the presence of large intensity gradients in the image. Therefore, obtaining satisfactory results with the technique often requires a sample preparation procedure to enhance the image texture of the sample.…”