## 3 Results and Discussion
### Images Alignment and 3D Reconstruction
Higher multiplicity values propose greater reliability of the computed 3D points, given that the more images that contribute to constructing a 3D point minimizes its positional error. Nevertheless, if the reprojection error associated with a given image is higher relative to the other contributed images, it will result in a higher positional error. Therefore, the tie multiplicity value by itself is not sufficient to judge the reliability of the computed 3D points. * RMS reprojection error--that is, the root mean square of normalized reprojection error (d), Figure 8. A tie point gets reconstructed as a 3D point by triangulating its corresponding 2D point from all the images sharing that point to compute its relative position. When the 3D point is reconstructed, it is reprojected back on each image that contributed to its reconstruction initially. The reprojected position on the corresponding image does not perfectly match the actual position of the original 2D point on the same image. The Euclidean distance between the two positions (i.e., actual and reprojected) in the image plane represents the reprojection error (d) in that image, Equation (8). The reprojection error varies within the contributed images sharing the same matched point; therefore, the average error is expressed as the root mean square error in all those images. It is calculated as the following:  
\[d_{i}=\sqrt{\ \left(x_{i}^{\prime}-x_{i}\right)^{2}+\left(y_{i}^{\prime}-y_{i} \right)^{2}} \tag{8}\]  
\[\text{RMSE}=\sqrt{\ \frac{1}{N}\ \sum_{i=1}^{N}[\left(x_{i}^{\prime}-x_{i} \right)^{2}+\left(y_{i}^{\prime}-y_{i}\right)^{2}]} \tag{9}\]  
where \(d_{i}\) is the reprojection error on image (_i_) (i.e., Euclidean distance between both positions), (\(x\), \(y\)) is the actual position of the matched point on the corresponding image plane, (\(x^{\prime}\), \(y^{\prime}\)) is the reprojected position of the reconstructed 3D point on the corresponding image plane, RMSE is the root mean square reprojection error, and \(N\) is the number of images sharing the detected point. Based on the values provided in Table 7, the images captured with the digital camera result in almost half the RMSE associated with those of the smartphone camera. This  
Figure 8: Reprojection error (d) associated with a target point, as an illustrating example. can be further demonstrated by plotting the image residuals across the sensor area as valued vectors from each sensor cell for both cameras, Figure 9. The image residuals provided in the figure were generated based on the first data set (i.e., beam images) for both cameras. Both plots are presented with the same magnification factor of \(\times\)398 and have the same scale bar of 1 pixel. The image residuals associated with the smartphone data are significantly higher than those of the digital camera data. A higher RMSE value of the reconstructed point cloud indicates a higher error in the overall geometric estimation. This is not only because of the positional error associated with all the reconstructed 3D points in the cloud. However, it is mainly due to the error associated with reconstructing the target points used to scale the 3D data, Figure 8. The RMSE associated with the reconstructed targets' points contributes significantly to the overall geometrical measurement error. Out of all the reconstructed 3D points, the accuracy of reconstructing the targets' center points is crucial since the RMSE of a target point represents a positional error that might cause inaccuracy in the scaling distance between every two targets. Therefore, they must be examined individually to evaluate their reliability in terms of their image redundancy and RMSE. Figure 10a provides the average projection percentage of the four targets for each specimen. The projection percentage indicates the percentage of images that contribute to computing a given target point out of the total input images.