# Panoramic Digital Image Correlation for 360-Deg Full-Field Displacement Measurement
## 2 Materials and Methods
### Planar Target-Based IntrinsicParameters Determination  
For a three-dimensional point **P** in the world coordinate system, the coordinates of the corresponding pixel point **u** in the image coordinate system are given by \((u,v)\). Assuming the camera is zero-skew, by mapping the three-dimensional point **P** corresponding to **u** into the homogeneous coordinate system, the undistorted coordinates **x** of **u** with its the components \(x\) and \(y\) are determined. \[x=\left\langle\textbf{RP}+t\right\rangle \tag{1}\]  
where **x** can be expressed in terms of the focal length \(\textbf{f}=\text{diag}\big{(}f_{x},f_{y}\big{)}\) and the principle point \(\textbf{c}=\big{(}c_{x},c_{y}\big{)}\) as \(\textbf{x}=\textbf{f}^{-1}\big{(}\textbf{u}-\textbf{c}\big{)}\). The matrix \(\textbf{R}\in\mathbb{R}^{3\times 3}\) and the vector \(\textbf{t}\in\mathbb{R}^{3\times 1}\) are the rigid-body rotation matrix and translation vector from the world coordinate system of **P** to the camera coordinate system, respectively, and the operator \(\left\langle\cdot\right\rangle\) maps a 3-vector \((x,\ y,\ z)\) to a 2-vector \((x/z,\ y/z)\). The lens distortions, both radial and tangential, typically deteriorate imaging. Since tangential distortion, which can lead to unstable optimization, is negligible for modern camera lenses [31], we only consider the parametric radial distortion model, which can be written as follows:  
\[\mathcal{D}(\textbf{x})=1+k_{1}\left\|\textbf{x}\right\|^{2}+k_{2}\left\| \textbf{x}\right\|^{4} \tag{2}\]  
where \(k_{1}\), \(k_{2}\) are the coefficients of radial distortion. With the model \(\mathcal{D}(\cdot)\), the distorted image coordinates \(\hat{\textbf{x}}\) with its components, \(x\) and \(y\) in the normalized image coordinate frame can be given by:  
\[\hat{\textbf{x}}=\mathcal{D}(\textbf{x})\textbf{x}. \tag{3}\]The basic linear equations for the banded block matrix used to determine the radial distortion model \(\mathcal{D}(\cdot)\) can be reformulated in a component-wise manner by Equation (3) as follows:  
\[ \begin{bmatrix}\ddot{x}-x\\ \ddot{y}-y\end{bmatrix} = \begin{bmatrix}\left\|\mathbf{x}\right\|^{2}x&\left\| \mathbf{x}\right\|^{4}x\\ \left\|\mathbf{x}\right\|^{2}y&\left\|\mathbf{x}\right\|^{4}y\end{bmatrix} \begin{bmatrix}k_{1}\\ k_{2}\end{bmatrix} . \tag{4}\]  
For calibration, we use a checkerboard planar calibration target with \(M\) control points in \(N\) different poses. Based on Equation (4), a system of \(M\times N\) linear equations are established. The closed-form solutions for the radial distortion coefficients \(k_{1}\) and \(k_{2}\) can be solved by superimposing them with the linear least square method. The \(i\)th control point on the calibration target is denoted as \(\mathbf{P}_{i}\) (\(1\leq i\leq M\)), and \(\mathbf{R}_{j}\) and \(\mathbf{t}_{j}\) (\(1\leq j\leq N\)) are denoted as the rotation matrix and translation vector corresponding to the \(j\)th pose of the target with respect to the camera, respectively. Then, the reprojection errors of the control point \(\mathbf{P}_{i}\) in the normalized image domain can be expressed as:  
\[\epsilon_{ij}(\mathbf{f},\mathbf{c},k_{1},k_{2},\mathbf{R}_{j},\ \mathbf{t}_{j}, \mathbf{P}_{i})=\mathcal{D}\big{(}\mathbf{x}_{ij}\big{)}\mathbf{x}_{ij}- \big{\langle}\mathbf{R}_{j}\mathbf{P}_{i}+\mathbf{t}_{j}\big{\rangle} \tag{5}\]  
where \(\mathbf{x}_{ij}\) denotes the normalized undistorted coordinates of \(\mathbf{P}_{i}\) from the \(j\)th calibration pose, which can be expressed as \(\mathbf{x}_{ij}=\mathbf{f}^{-1}\big{(}\mathbf{u}_{ij}-\mathbf{c}\big{)}\).