## 5 Point Cloud Uncertainty Evaluation: Gaussian Process Error Models
### Gaussian Process Models Incorporating Spatial Correlation  
Gaussian process (GP) models [32] can be used to develop empirical models of behaviour that do not explicitly involve sets of basis functions such as polynomials or splines. Spatial or temporal correlation associated with data points \((\mathbf{x}_{i},e_{i})\) takes the form \(\mathrm{corr}(\mathrm{e},\mathrm{e}^{\prime})=k(\mathbf{x},\mathbf{x}^{\prime}|\mathbf{ \sigma})\) where \(k\) is a correlation kernel depending on statistical parameters \(\mathbf{\sigma}\). Often \(k\) depends on \(\mathbf{x}\) and \(\mathbf{x}^{\prime}\) through \(\|\mathbf{x}-\mathbf{x}^{\prime}\|\), e.g.,  
\[\mathrm{cov}(e,e^{\prime})=k(\mathbf{x},\mathbf{x}^{\prime}|\sigma_{E},\lambda_{E})= \sigma_{E}^{2}\exp\{-\|\mathbf{x}-\mathbf{x}^{\prime}\|^{2}/\lambda_{E}^{2}\}. \tag{19}\]  
The strength of the correlation between \(e\) and \(e^{\prime}\) depends on the distance between \(\mathbf{x}\) and \(\mathbf{x}^{\prime}\): the closer \(\mathbf{x}\) is to \(\mathbf{x}^{\prime}\), relative to \(\lambda\), the stronger the correlation between \(e\) and \(e^{\prime}\). MPE and Spatial Correlation  
The use of spatially correlated error models can be motivated in terms of consequences of the statement of maximum permissible error (MPE) in measuring length. In measuring length along a single axis, say the \(x\)-axis, suppose the basic statistical model for the measurement of two points is  
\[x_{1}=x_{1}^{*}+e_{1}+\epsilon_{1},\quad x_{2}=x_{2}^{*}+e_{2}+\epsilon_{2},\]  
with  
\[e_{1},e_{2}\in\mathcal{N}(0,\sigma_{E}^{2}),\quad\epsilon_{1},e_{2}\in \mathcal{N}(0,\sigma_{R}^{2}).\]The MPE statement implies that the measured length \(\hat{d}_{12}\) is related to the true length \(d_{12}\) according to  
\[|\hat{d}_{12}-d_{12}|\leq A+d_{12}/B.\]  
The MPE implies  
\[|e_{2}-e_{1}+e_{2}-e_{1}|\leq A+d_{12}/B\]  
so that for \(2\sigma_{R}^{2}\leq A^{2}\) and for \(\sigma_{R}\ll\sigma_{E}\), the similarity of \(e_{1}\) and \(e_{2}\) depends on the _spatial separation_\(d_{12}\). We can also use a model of the form  
\[x=x^{*}+e(x^{*})+\epsilon\approx x^{*}+e(x)+\epsilon,\quad\epsilon\in\mathcal{ N}(0,\sigma_{R}^{2})\]  
where \(e(x)\) is an error function with \(|e(x)|\leq 2\sigma_{E}\) that encodes the local scale error. (The same concept can be applied to straightness errors, etc.). The MPE implies  
\[|e(x_{2})-e(x_{1})+e_{2}-e_{1}|\leq A+|x_{2}-x_{1}|/B\]  
so for \(\sigma_{R}\ll\sigma_{E}\) and \(x_{1}\
eq x_{2}\),  
\[\left|\frac{e(x_{2})-e(x_{1})}{x_{2}-x_{1}}\right|\leq\frac{A}{|x_{2}-x_{1}|} +\frac{1}{B}.\]  
In other words, the slope of the error function \(e(x)\) cannot be too large, and so some measure of smoothness on the error function \(e(x)\) is required in order for it to be consistent with the MPE statement. From this point of view, in the MPE statement, \(A\) quantifies effects over short length scales, while \(d/B\) controls the size and smoothness of effects over longer length scales. Figures 1-3 give examples of spatially correlated error functions generated using the correlation kernel in Equation (19) with \(\sigma_{E}=0.005\) mm and \(\lambda_{E}=1000\) mm, \(500\) mm, \(200\) mm and \(100\) mm. These error functions were evaluated at \(200\) equally spaced points \(0<x_{i}<1000\). Given \(\mathbf{x}\), \(\sigma_{E}\) and \(\lambda_{E}\), each error function is a random draw from the distribution \(\mathcal{N}(\mathbf{0},V(\mathbf{x}|\sigma_{E},\lambda_{E})\), where \(V(\mathbf{x}|\sigma_{E},\lambda_{E})\) is the spatially correlated variance matrix. Each error function was centred so that \(e(x=0)=0\), simulating the convention that the error at \(x=0\) is zero. Each error function represents plausible behaviour, given the spatial correlation. The spatial correlation length \(\lambda_{E}\) controls the degree of smoothness of the error functions. Figure 1: Eight examples of spatially correlated error functions generated using the correlation kernel in Equation (19) with \(\sigma_{E}=0.005\) mm and \(\lambda_{E}=1000\) mm.