# Image-Based Crack Detection Using Total Variation Strain DVC Regularization
## 5 Discussion  
In the previous section, a comparison was made between the results obtained using the proposed method and those obtained using the median filtering technique to reduce noise in the displacement field and, consequently, in the computed strain field. For the simulated dataset, which provided a strain field ground truth, a more comprehensive analysis was conducted. This analysis focused on a typical scenario where the requirements of high strain field resolution and low noise are crucial, particularly in the detection of cracks with multiple branches [10]. Specifically, the investigation encompassed various aspects, including the accuracy of the obtained strain field with and without additional noise in the input images, the data-fit accuracy at different values of lambda (\(\lambda\)), and the accuracy of crack detection. \begin{table}
\begin{tabular}{c|c|c} \hline
**Segmented** & **TV-Reg** & **Smooth-Reg** \\ \hline
336–11 550 & 420–15 129 & 660–23 250 \\ \hline \end{tabular}
\end{table} 
Table 2: The range of the number of voxels attributed to the 28 cracks within the aggregate quartz grain sample. * strain field and crack detection accuracy: The accuracy of the strain field and the effectiveness of the algorithm as a crack detector are interconnected. As anticipated, the strain field computed without regularization and without filtering exhibits high levels of noise. In this case, the noise amplitude is comparable to the strain associated with actual cracks, posing a challenge in determining an appropriate threshold level. The objective is to retain the high strain signal resulting from cracks while excluding high strain values caused by noise. It is noteworthy from the results obtained using the TV-reg method that the impact of noise on crack detection is limited, irrespective of the threshold level employed, Figure 5d,g. Using the median filter, accurate crack detection was achieved only with \(T2\). A similar crack configuration was analyzed by [10], where the authors emphasized the impact of median filter window size on the detection of small crack spacings. Smaller filter sizes lead to a greater ability to detect smaller gaps between cracks. In our experiment, the reduction in detectable crack spacing becomes evident when using a small filter size (w = 7), particularly in the upper section of the image where the distance between the two branches is approximately six voxels, as shown in Figures 3 and 5c. The filtered output with w = 7 produces unrealistic results, including the appearance of additional false branches caused by noise. Decreasing w from 7 to 5, which is smaller than the distance between the two branches, would prevent the two branches from connecting at the top. However, the increased noise would result in even less accuracy compared to w = 7. Additionally, the application of filtering operations alters the magnitude of the strain field, leading to a loss of valuable information regarding the real strain magnitude. Two factors contribute to the degradation of the MAE (Table 3) when median filtering is employed instead of the TV-reg method. Firstly, the presence of noise in the background. Secondly, the actual cracks area appears blurred. This implies that the median filter tends to include some voxels from the background in the crack area. The influence of these two factors varies based on the chosen window size. When the window size is smaller, the noise has a greater impact on the degradation of the MAE. Conversely, with a larger window size, the influence of the noise diminishes, while the impact of blurring becomes more significant (Figure 5, as shown in Table 3). * data-fit: The optimization of the cost function, which consists of a similarity metric and the TV regularization term, involves finding a balance between data fit (measured by the similarity metric) and strain discontinuity. This balance is controlled by the parameter \(\lambda\).