# Deep Learning-Based Crack Detection on Cultural Heritage Surfaces
## 3 Results
### Crack Length Estimation Analysis  
The GoogleNet model was employed for crack detection, followed by SegNet for segmentation. Crack length estimation was performed using two parallel laser points spaced 15 cm apart as reference markers. Two case studies were conducted to validate this methodology (Figure 6), comparing SegNet-based estimations with actual measurements. Figure 5: Semantic segmentation results of cracks using the SegNet model in (**a**) image A, (**b**) image B, (**c**) image C, and (**d**) image D.  
In Figure 6b, the endpoints of Crack A (black dots) exhibited a pixel difference of 169, while the laser points (yellow dots) showed a difference of 66 pixels, corresponding to a real-world distance of 15 cm. This resulted in a pixel-to-distance ratio of 1 pixel = 0.22 cm, yielding an estimated crack length of 37.18 cm. The actual measured length was 38.4 cm, with a discrepancy of 1.22 cm and an accuracy of 97%. Similarly, in Figure 6c,d, the endpoints of Crack B displayed a pixel difference of 223, while the laser points exhibited a 134-pixel difference, equating to 1 pixel = 0.11 cm. The estimated crack length was 24.53 cm, whereas the actual measured length was 25.3 cm, with a discrepancy of 0.77 cm and an accuracy of 97%. Four primary sources of error contribute to the results presented above: (1) segmentation error from the deep learning-based SegNet model, (2) pixel discretization error arising from manual selection of laser points in the image, (3) laser misalignment error due to imperfect parallelism between the dual-laser beams, and (4) image capture error resulting from non-perpendicular viewpoints during image acquisition. Regarding image capture error, when a horizontal crack is imaged using a forward-tilted smartphone, no distortion occurs if the crack lies at the vertical center of the image. However, if the crack appears in the upper half, it undergoes compression due to perspective distortion. For example, when a crack is located at v = 1 and the camera is tilted by 5\({}^{\circ}\), the measured crack length is underestimated by approximately 0.38%. Conversely, cracks  
Figure 6: Crack length analysis (**a**) original image for Crack A, (**b**) analyzed image for Crack A, (**c**) original image for Crack B, and (**d**) analyzed image for Crack B.  
in the lower half of the image (e.g., v = 256) are stretched, leading to an overestimation of similar magnitude under the same tilt. In the case of vertical cracks distributed symmetrically in the upper and lower halves, the distortion effects tend to cancel out, resulting in negligible overall error. Laser misalignment error occurs when the two laser beams are not perfectly parallel or not aligned on the same plane as the surface of the crack. Such misalignment affects the apparent spacing between laser dots in the image, thereby distorting the reference scale. For instance, a \(\pm 5^{\circ}\) inward tilt of the lasers results in a projected spacing ranging from approximately 14.94 cm to 15.06 cm (a \(\pm 0.4\%\) deviation), leading to a corresponding underestimation of crack length. A pixel discretization error arises from uncertainty in localizing the laser points, particularly when the dots appear slightly blurred or diffused. In the conducted experiments, the pixel distances between laser dots were 66 pixels for Crack A and 134 pixels for Crack B. Assuming an uncertainty of \(\pm 1\) pixel, the resulting relative error is approximately \(\pm 1.52\%\) for Crack A and \(\pm 0.75\%\) for Crack B. These discretization and misalignment errors may either accumulate or partially offset one another. Considering that the overall measurement accuracy in Examples A and B reached approximately 97%, the segmentation error introduced by the SegNet model is estimated to contribute between 0.7% and 3.0% of the total error. These findings support the reliability and applicability of using SegNet in conjunction with laser markers for crack detection and length estimation.