# Image-Based Crack Detection Using Total Variation Strain DVC Regularization
## 4 Results
### Aggregate Grain Compaction Experiment Results  
TV-regularization has been used to detect cracks in \(\mu\)-CT images of an aggregate grain sample, formed as result of a compaction experiment. Moreover, in this case, the results have been compared to the performances of DVC without regularization and filtering operation and with smooth filtering. In order to detect cracks from the strain field a threshold value has been set (\(T_{e}=2\%\)): the peaks of the strain field above \(T_{e}\) are considered to be caused by a crack in the grain. In Figure 6, one slice of the strain field computed without regularization (a), one with smooth regularization (b) and one with TV regularization (c) are shown. Figure 5: (**a**) Simulated strain field. (**b**) Noisy strain field in the proximity of the crack, computed without any filtering steps or regularization. The strain field is shown at three different levels of threshold (\(T1_{e}=10^{-3},T2_{e}=2.2\times 10^{-3},T3_{e}=4\times 10^{-3}\)). (**c**) Strain field obtained by applying a median filter with different windows sizes applying \(T2_{e}\). (**d**) Strain field in the proximity of the crack, computed using TV-reg (\(\lambda_{TV}=0.1\)). The strain field is shown at three different levels of threshold. (**a–d**) the bottom row contains the strain field overlaid with the correspondent slice of the cracked volume. (**e**) Crack profile and noisy strain field profile, the three threshold levels are reported. (**f**) Effect of the variation of the window filter size on the strain profile. (**g**) Crack profile and TV-reg strain profile, the three threshold levels are reported. (**h**) Mean absolute strain error computed between the simulated strain field and the computed strain field by using TV-reg, smooth-reg and no regularization (no-reg. )Figure 6: The figure shows one slice of the volume with cracks (**d**) and the same slice of the strain field (**a–c**) computed using three different methods: DVC without TV-reg regularization (**a**), with smooth-reg (**b**), with TV- regularization (**c**). (**e**) shows the strain profile in the proximity of the crack number 1. (**f**) Details of the cracks. (**g**) Receiver operating characteristic (ROC) to show the FPR and the TPR at different values of thresholds for the three methods: DVC with TV regularization; DVC with smooth filtering; DVC without regularization. In this application, the presence of noise in the strain field leads to numerous false positive detections. The noise peaks not only have similar amplitude to the actual strain, but they also exhibit similar crack openings (5-12 voxels). For instance, the crack labeled as number 1 has an opening of nine voxels, as indicated by the dashed rectangle in Figure 6e. That makes it impossible to distinguish a real crack from the noise. To enhance the sensitivity of the detection, we also evaluated a commonly employed median filtering technique [48; 49] called smooth-reg. For this specific application, a window size of five voxels was chosen for the filter. However, it is important to note that this window size is still insufficient to completely eliminate all noise-induced peaks, and simultaneously compromises the spatial resolution, as it is visible from the strain profile (Figure 6e). The strain field computed using TV regularization demonstrates reduced variations around zero in the crack-free area compared to the strain field computed using the median filtering technique. In Figure 6c, four distinct peaks of high strain are visible, corresponding to the presence of four cracks. Table 2 displays a comparative analysis of the number of voxels representing cracks segmented from the images and the corresponding number of voxels representing high strain areas in the strain field, as computed using the two methods. The impact of the threshold on the sensitivity of TV-reg for crack detection has been evaluated, Figure 6g. Such an analysis has been carried out based on a limited amount of annotated data.