## 3 Results and Discussion
### Images Alignment and 3D Reconstruction  
All the images of each data set were successfully aligned. The quality of the reconstructed 3D points can be examined and compared for both cameras based on the resulting parameters presented in Table 7. The number of features detected by the SIFT algorithm is not a reliable criterion based on which the data sets of both cameras can be compared. This is due to the difference in the cameras' resolution and their covered area. For the digital camera, its higher resolution results in more detected features than the smartphone. However, the smartphone's images were captured with a relatively shorter focal length, resulting in a larger captured area, which results in a higher number of features, Table 2. Nevertheless, the number of matched points can be used to compare the data sets since this number presents only the key points successfully reconstructed within the reconstruction bounding box. As indicated, the matched points are higher for the digital camera data. This is mainly due to the higher resolution associated with the digital camera. Figure 5: Radial distortion profiles as a function of the distance in pixels from the image center for: (**a**) the digital camera; (**b**) the smartphone camera. The pixel size of the digital camera is 4.04 μm and 1.06 μm for the smartphone. Figure 6: Decentering distortion profiles as a function of the distance in pixels from the image center for: (**a**) the digital camera; (**b**) the smartphone camera. The pixel size of the digital camera is 4.04 μm and 1.06 μm for the smartphone. Furthermore, the quality of the reconstruction process can be evaluated and compared for both cameras based on the following parameters:  
* The number of projections, which represents the total number of projections projected from all overlapping images to compute and construct all the matched points. The number of projections is correlated to the number of points successfully matched and constructed. This correlation is given by the tie multiplicity parameter. * Tie multiplicity (i.e., image redundancy)--that, is the average number of projections or images contribute to calculating a given 3D point. It can be estimated by the following ratio: \[m_{ave}=\frac{1}{S}\ \sum_{i=1}^{S}(P_{i})\ \approx\ \frac{N^{\circ}\ \text{of projections}}{N^{\circ}\ \text{of matched points}}\] (7) where \(P_{i}\) is the number of projections used to reconstruct point (\(i\)), and \(S\) is the total number of reconstructed points (i.e., sparse cloud size). An average tie multiplicity value of 2.396 indicates that an average of 2.396 images were used to compute and reconstruct a given 3D point in the bundle adjustment step by triangulating this point \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Specimen**} & **Capturing** & **N\({}^{\circ}\) of Points** & **N\({}^{\circ}\) of Points** & **N\({}^{\circ}\) of Points** & **Average** & **RMSE (pix)** \\  & **Device** & **Detected** & **Matched** & **Projections** & **Multiplicity** & **RMSE (pix)** \\ \hline \multirow{2}{*}{Square Beam} & DC & 226,189 & 194,535 & 457,680 & 2.396 & 0.607 \\  & SP & 199,752 & 84,572 & 210,912 & 2.472 & 1.2 \\ \multirow{2}{*}{Cylinder} & DC & 107,530 & 91,634 & 229,964 & 2.531 & 0.79 \\  & SP & 129,469 & 82,189 & 232,109 & 2.768 & 1.4 \\ \multirow{2}{*}{Sand Pile} & DC & 132,618 & 115,289 & 273,129 & 2.466 & 0.656 \\  & SP & 143,124 & 78,321 & 211,129 & 2.561 & 1.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The 3D reconstruction parameters for both cameras’ data sets. Figure 7: Image total distortion across the area of: (**a**) the digital camera sensor; (**b**) the smartphone camera sensor. The scale bar represents 50 pixels. The pixel size is 4.04 μm for the digital camera and 1.06 μm for the smartphone camera. The magnification factor is \(\times 11\) in both plots. from those images into the 3D space.