# Image-Based Crack Detection Using Total Variation Strain DVC Regularization
## 1 Introduction
While some methods can compute sharp displacement fields with discontinuities [15; 16], the sharpness of the resulting strain field may not be preserved. This is due to the spatial mathematical differentiation of the estimated displacement field to compute the strain field, which amplifies small, high-frequency variations, noise, and errors in the displacement field. To mitigate this issue, various solutions have been proposed, including post-processing techniques such as smoothing the displacement field before differentiation. Common approaches, summarized in Table 1, include using average, median, and Gaussian mean filters [17; 18; 19; 20]. Gaussian filters are applied by replacing each pixel's value with the average of the neighboring pixels, weighted using a Gaussian kernel. However, Gaussian filters are not suitable for interface-related issues as they may inadvertently average the pixel intensities from both sides of the interface. Average filters substitute each value in the region of interest with the average (or weighted average) of the values within the neighborhood specified by the filter mask. Average filters substitute each value in the region of interest with the average (or weighted average) of the values within the neighborhood specified by the filter mask. This approach results in information loss at the edges and borders of the filtered data. Therefore, this method is also not suitable for interface-related issues. Median filters are effective in smoothing strain fields while preserving edges. This filtering technique involves replacing each pixel's value with the median value of its neighborhood. The size of the selected window dimension affects the locality of the solution [10]. Another alternative is to compute the strain from the noisy displacement field first and then apply a filter to the strain field [17; 21; 22]. These techniques are applied in many DIC/DVC open-source and commercial software, such as those offered by [18; 19; 22]. In some cases, alternatives to filtering methods have been employed such as planar approximation (2D) [23], point-wise least-squares (PLS) algorithm, both in 2D and 3D [24; 25] and Geers' method in 3D [26]. These methods are implemented in MatchID [23], Ncorr [27], and SPAM [22]. The PLS method involves the definition of a window around the voxels where the strain needs to be computed, if the window is small enough the strain can be approximated as linear. Geers' method computes strains from the displacements in a discrete set of points in the neighborhood. The local differences in the displacements have been expanded in a Taylor series, which was truncated after the second-quadratic term [26]. Both these methods reduce the locality of the solution. A trade-off between noise and spatial resolution exists in the computation of the strain field, which leads to a reduction in the resolution despite having a sharp displacement field. Fractures are usually associated with localized discontinuities in the displacement field and peaks in the strain [10]. Figure 1: Scheme of direct (**a**) and indirect (**b**) crack-detection methods. The proposed TV-reg method falls under the indirect category. To overcome the challenge of obtaining a sharp and low noise strain field, we propose a novel DVC regularization method. Our approach is based on minimizing the spatial total variation of the strain field (TV-reg). By minimizing the total variation, our method promotes a piecewise constant strain profile that allows for discontinuities in the strain field. The constraint of piecewise constant strain field inside the distinct structures that compose the material leads to a strain field with a higher signal-to-noise ratio.