## 5 Point Cloud Uncertainty Evaluation: Gaussian Process Error Models
### Gaussian Models for Location Errors (ET)  
We can apply a GP model for CMM behaviour, as follows, with  
\[\mathbf{x}_{i}=\mathbf{x}_{i}^{*}+\mathbf{e}_{i}, \tag{20}\]  
where the systematic effects are spatially (and sometimes temporally) correlated with the systematic effects \(\mathbf{e}_{i}\). In general, the covariance applies only to the same coordinates,  
Figure 3: As Figure 1 but with \(\lambda_{E}=100\) mm.  
Figure 2: As Figure 1 but with \(\lambda_{E}=200\) mm.  
with the \(x\)-, \(y\)- and \(z\)-coordinates of \(\mathbf{e}\) mutually independent. The covariance of \(e_{x}\) with \(e_{x}^{\prime}\) could be modelled as  
\[\text{cov}(e_{x},e_{x}^{\prime})=k(\mathbf{x},\mathbf{x}^{\prime}|\sigma_{ET,x},\lambda _{ET,x})=\sigma_{ET,x}^{2}\exp\left\{-\|\mathbf{x}-\mathbf{x}^{\prime}\|^{2}/\lambda_{ ET,x}^{2}\right\}, \tag{21}\]  
for example, where \(\lambda_{ET,x}\) defines the length scale for the correlation in the \(x\)-coordinate. Note that in this model, the strength of the correlation in the effects \(e_{x}\) depends on the distance \(\|\mathbf{x}-\mathbf{x}^{\prime}\|\) in 3D, not the distance along the \(x\)-axis.  
Let \(D\) be the \(m\times m\) matrix of distances with \(D_{ij}=\|\mathbf{x}_{i}-\mathbf{x}_{j}\|\). The variance contribution \(V_{XT}\) from \(\mathbf{e}_{1:m}\) to the \(x\)-coordinates of \(\mathbf{x}_{i:m}\) is given by  
\[V_{XT,x}=\sigma_{ET,x}^{2}\exp\left\{-D^{2}/\lambda_{ET,x}^{2}\right\},\]  
where the calculations associated with \(D\) are made element-wise. The contribution to the \(y\)- and \(z\)-components are of exactly the same form. The matrix \(V_{XT}\) is assembled from \(V_{XT,x}\)\(V_{XT,y}\) and \(V_{XT,z}\), with all other elements zeros since we assume that the systematic effects associated with the \(x\)-coordinates are independent from those associated with the \(y\)- and \(z\)-coordinates. We assume that the GP model relates to uncorrected kinematic errors that are not likely to have a significant correlation between axes, even if the kinematic errors themselves are likely to produce such a correlation. While the GP model assumes there is no inter-axes correlation, the GP model will model successful behaviour that does have such a correlation.  
If it can be assumed that the systematic effects along each axis have the same behaviour, so that \(\sigma_{ET,x}=\sigma_{ET,y}=\sigma_{ET,z}=\sigma_{ET}\), etc., then the model is specified by two statistical hyper-parameters \(\sigma_{ET}\) and \(\lambda_{ET}\).  
Gaussian Process Models for Location Errors Incorporating Multiple Probes  
Suppose that the point cloud \(\mathbf{x}_{1:m}\) is gathered using multiple probes with offsets \(\mathbf{p}_{k}\), \(k=1,\ldots,n_{P}\). The measured coordinates \(\mathbf{x}_{i}\) are related to the true point coordinates \(\mathbf{x}_{i}^{*}\) through a model of the form  
\[\mathbf{x}_{i}=\mathbf{x}_{i}^{*}+\mathbf{e}_{i}+\mathbf{p}_{k(i)}+\mathbf{e}_{i},\]  
where \(\mathbf{p}_{k(i)}\) denotes the probe configuration associated with the \(i\)th measurement, etc. For this case, it is important to note that the spatial correlation is dependent on \(\|\mathbf{x}_{i}^{*}-\mathbf{x}_{j}^{*}\|\), not \(\|\mathbf{x}_{i}-\mathbf{x}_{j}\|\). For different probe configurations, we have  
\[\|\mathbf{x}_{i}^{*}-\mathbf{x}_{j}^{*}\|\doteq\|(\mathbf{x}_{i}-\mathbf{p}_{k(i)})-(\mathbf{x}_ {j}-\mathbf{p}_{k(j)})\|.\]  
Similar considerations apply to the model for scale and squareness errors with multiple probe offsets, Section 4.3, and for GP models for rotation errors, Section 5.4, below.