# Panoramic Digital Image Correlation for 360-Deg Full-Field Displacement Measurement
## 2 Materials and Methods
### Epipolar Geometry-Based Extrinsic Parameters Determination
#### 2.3.5. Point Registration Based Scale Information Determination  
Scale information is a key factor in determining the actual shape and actual displacement values and can, in principle, be determined in three ways: (1) the physical distance between two known points on the specimen, (2) the known shape of the specimen, and (3) the known displacement of the specimen. In order to calibrate the scale information, the following method is proposed based on laser scanning point clouds and ICP algorithms [45]. The three-dimensional point cloud contains rich geometric information in the spatial domain. According to the invariability of the relative position of the center of gravity and the centroid of the point cloud in the rigid body transformation, the distance ratio between the center of gravity and the centroid of the point cloud at different scales is defined as the scale factor, and the scale information model is established. Combined with the ICP algorithm, the minimum registration error function is defined to refine the scale information. Details are given below. Firstly, filtering the point cloud data. The original-scale three-dimensional point cloud data of the test specimen and the normalized-scale three-dimensional reconstructed point cloud data were obtained by laser scanning equipment and the aforementioned method, respectively. The original-scale point cloud data obtained by laser scanning is dense, thus, it needs to be filtered to reduce the amount of point cloud data and improve the efficiency of alignment. The normalized-scale point cloud data obtained by using above three-dimensional reconstruction will contain noise and outliers, which will affect the accuracy of the calculation of center of centroid and center of gravity as well as the final alignment results, thus, the normalized-scale point cloud data are filtered to reduce or even eliminate the influence of noise and outliers. Next, establishing a scale factor calculation model. If the center of gravity and the centroid of the original-scale point cloud data \(\mathbf{P}\) and normalized-scale point cloud data \(\mathbf{Q}\) are denoted as \(p_{g}\), \(p_{c}\) and \(q_{g}\) and \(q_{c}\), respectively, and point \(p_{g}\), \(p_{c}\), \(q_{g}\) and \(q_{c}\) can be expressed as \(p_{g}=\frac{1}{M}\sum_{i=1}^{M}p_{i}\), \(p_{c}=\sum_{i=1}^{M}m_{i}p_{i}\), \(q_{g}=\frac{1}{N}\sum_{i=1}^{N}q_{i}\) and \(q_{c}=\sum_{i=1}^{N}m_{i}q_{i}\). Where \(p_{i}\) and \(q_{i}\) are the points on point cloud data \(\mathbf{P}\) and \(\mathbf{Q}\), respectively. \(M\) and \(N\) are the number of points in the point cloud data, and \(m_{i}=r_{i}^{-2}/\sum r_{i}^{-2}\) is the centroid factor, satisfying \(\sum m_{i}=1\). The scale factor \(s\) has form of  
\[s=\frac{d\big{(}p_{g},p_{c}\big{)}}{d\big{(}q_{g},q_{c}\big{)}}=\frac{\big{|}p _{g}-p_{c}\big{|}}{\big{|}q_{g}-q_{c}\big{|}} \tag{13}\]  
where function \(d\) defines the Euclidean distance between two points. The scale factor calculated by Equation (13) is taken as the initial value, the normalized-scale point cloud is scaled up equally to obtained the point cloud \(s\mathbf{Q}\), which is used for the subsequent iterative optimization of scale factor. Thirdly, refining the scale factor. The point cloud \(s\mathbf{Q}\) is very close to the original-scale point cloud. In this step, the accuracy of the scale factor is improved further through the ICP iterative calculating method. For each point \(q_{i}\) in the normalized-scale point cloud data \(\mathbf{Q}\), the pairwise point \(p_{i}\) is searched by taking the closest point of Euclidean distance on the original-scale point cloud data \(\mathbf{P}\) as the corresponding relationship.