Gaussian pre-filtering for uncertainty minimization in digital image correlation using numerically-designed speckle patterns
Introduction
The design and implementation of effective speckle patterns on two-dimensional measurement surfaces are key to enhance the accuracy of digital image correlation (DIC), along with suitable displacement and strain field estimation algorithms [1], [2]. The accuracy of DIC measurements was studied as a function of mean speckle size and subset size, for which desirable ranges were reported [3], [4], [5]. Several techniques have been used to create speckle patterns, depending on the specimen dimensions and materials. Spray paint or toner powders are typically used for larger specimens, whereas lithography is preferred for smaller patterns [6]. The resulting speckle patterns are characterized by non-repetitiveness and high contrast between light and dark areas. As shown by Wang et al. [7] for translation in two planar directions, x and y, the form of the covariance matrix for the displacement vector, d, is written in Eq. (1):where: d is the displacement vector, (u,v), in the x and y direction, respectively; σI is the standard deviation in the intensity pattern noise (gray levels); and I=I(x,y) is the reconstructed deformed intensity pattern (gray levels). If the gradients in both directions are independent, then the off-diagonal term tends to zero, and the matrix is approximately diagonal. In this case, the standard deviation in each displacement component, σu and σv, can be written per Eq. (2):where “high contrast” corresponds to the summation of high gradients in intensity within a subset, increasing the denominator and reducing variability in the measured displacement. With maximum range between brightest and darkest regions, smooth transitions in intensity across the camera’s dynamic range can be accurately reconstructed by interpolation algorithms, offering the potential for high accuracy when performing subset matching with DIC algorithms. Thus, the gray level distribution within the speckle pattern can be used as a measure of the effectiveness of a speckle pattern [8]. Schreier et al. [9] proposed the implementation of low-pass image filters in the pre-processing stage to produce blurring, either by defocusing the camera’s optics prior to image acquisition or by applying digital filters on the acquired image data. The latter option is more attractive as it allows for better control of the parameters selected to produce blurring. In fact, digital filters are commonly used in image processing. For example, Berg et al. [10] and Cantatore et al. [11] implemented digital filters to produce image blurring, thereby improving the accuracy of algorithms for edge detection.
The effect of digital image pre-filtering on the uncertainty in two-dimensional DIC measurements is discussed in this paper for the specific case of high-contrast speckle patterns whose particle shape, mean size and on-center spacing are designed for use in efficient patterning of large areas (“designer patterns”). This case represents instances where numerically-designed speckle patterns are applied to the measurement surface through different techniques, such as using laser-printed adhesive coatings on fiber-reinforced polymer composite coupons [12], or spray-painting through a flexible polymer stencil placed against the surface of concrete and masonry specimens [13] as demonstrated in Fig. 1. These solutions are especially appealing and practical for large regions of interest (i.e., having sides of the order of meters) on full-scale specimens such as structural concrete and masonry walls (Fig. 12) or portions of bridge girders, when spray-painting or using toner powders is less practical and may pose aliasing problems whereas using relatively large speckles (e.g., through time-consuming manual painting) may result in an insufficient spatial resolution [13]. The resulting “designer patterns” are characterized by speckles with well-defined edges and consistent shape and spacing, making their frequency content fundamentally different from that of typical spray-painted patterns. In the latter case, pre-processing image blurring can be effective in reducing the bias error [9], [14]. However, the concurrent reduction in noise level and intensity pattern gradients (i.e., numerator and denominator for σu and σv in Eq. (2), respectively) may result in a negligible change or even an increase in measurement uncertainty.
The methodology followed in this study employs numerical simulations where images are pre-processed using Gaussian low-pass (blurring) filters [15]. First, the effect of blurring is examined on a numerically built speckle pattern as a function of the standard deviation of the Gaussian kernel (i.e., filter cut-off frequency). The resulting patterns are used to quantify the DIC measurement uncertainty for the case of constant, linear, quadratic and cubic displacement fields and the associated strain fields. The robustness of the simulation procedure is verified through experiments where a planar specimen with a “designer pattern” is subjected to a constant displacement. For comparison purposes, the effect of image blurring is also assessed on a speckle pattern that is representative of typical spray-painted ones [16], [17] subjected to constant displacements. Finally, the stability of the relation between Gaussian standard deviation and measurement uncertainty is tested via numerical simulations using different levels of image noise representative of real-case scenarios, subset sizes, and frequency contents in the “designer pattern”.
Section snippets
Methodology
The effect of pre-processing image blurring on DIC measurement uncertainty is investigated via numerical simulations on a predefined speckle pattern, as recently demonstrated by Zappa et al. [18] for the case of dynamic applications. The methodology is summarized in the flow chart in Fig. 2. The simulations are implemented using Matlab Image Processing Toolbox (The MathWorks, Inc., Natick, MA). A 4000×4000 pixel array with eight-bit quantization is numerically constructed and an ordinate grid of
Effect of image filter pre-processing
The results for the following cases are presented and discussed separately: (a) constant horizontal (along x) displacement and zero strain; and (b) higher-order (linear, quadratic and cubic) displacement functions with non-zero horizontal strain fields (zero vertical strain is imposed). In particular, the simulation of cubic displacement fields aims at testing the filters when the subset matching cannot be exact, since the DIC software used implements a second-order matching shape function [6].
Interpretation of results
The parametric study presented in Section 3 shows that pre-processing image blurring by means of Gaussian filters with a well-defined range of standard deviations (approximately 0.75–1.25 pixel) results in the overall reduction in DIC measurement errors, irrespective of the degree of the polynomial displacement and strain functions, when using “designer patterns” having well-defined speckles similar to those in Fig. 1(b) and Fig. 3(b) and (c). For strain fields with εmax ~103 µε, a significant
Stability of effect of image filter pre-processing
Following the methodology in Fig. 2(a) used for the numerical study presented, numerical simulations are performed for the representative case of linear displacement and constant strain (1000, 4000 and 20,000 µε) fields. Here, the effects of image intensity pattern noise (which is always present in actual measurements), subset size, and frequency content are evaluated. The strain measurement uncertainty is quantified by means of RMSEε in Eq. (7) accounting for all 15×15 pixel subsets for each
Conclusions
The first part of this paper presents a numerical and experimental study of the effect of Gaussian pre-filtering on the uncertainty of DIC measurements for the specific case of high-contrast “designer patterns” having speckles with well-defined and consistent shape, size and spacing. These patterns can be numerically designed and applied on the measurement surface using printed coatings or pre-cut stencils, and are ideal when using standard-resolution cameras to acquire images of large regions
Acknowledgements
The collaborative research presented herein was made possible through the financial and logistic support of the University of South Carolina and the Politecnico di Milano, and partial funding from the US National Science Foundation under grant no. CMMI-1049483. These sources of support are gratefully acknowledged. Special thanks are extended to Correlated Solutions, Inc. (Columbia, SC) for providing the software Vic-2D free of charge.
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