# Sensitivity Analysis for Gaussian-Associated Features
## 6 Sensitivity Matrix Associated with a Least-Squares Plane Fit to Data  
The calculations associated with an axis given above in Section 2 allow us to evaluate the sensitivity matrix associated with a least-squares plane fit to data. Given a location point \(\mathbf{x}_{A}\) and unit direction vector \(\mathbf{v}_{A}\), the equation of the associated plane can be written as  
\[(\mathbf{x}-\mathbf{x}_{A})^{\top}\mathbf{v}_{A}=0.\]  
The calculations involve a parametrization of the plane in terms of three parameters \(\mathbf{a}\) in which the \(k\)th coordinate of \(\mathbf{v}_{A}\) is held fixed and only the \(k\)th coordinate of \(\mathbf{x}_{A}\) is free.  
Analytical Approximation for Measuring a Rectangular Area  
Suppose data points \(\mathbf{x}_{1:m}\) are distributed approximately uniformly on the plane \(z=0\) with \(-a\leq x_{i}\leq a\) and \(-b\leq y_{i}\leq b\) and \(J\) is the \(m\times 3\) Jacobian matrix associated with fitting a plane to the data with \(J(i,1:3)=(-1,x_{i},y_{i})\). This Jacobian matrix corresponds to parametrization in terms of the \(z\)-coordinate of \(\mathbf{x}_{A}\) and the \(x\)- and \(y\)-coordinates of \(\mathbf{v}_{A}\), \(\mathbf{a}=(z_{A},u_{A},v_{A})^{\top}\), holding \(x_{A}\), \(y_{A}\) and \(w_{A}\) constant. Then, using the principle of Monte Carlo integration (15),  
\[\frac{1}{m}J^{\top}J\approx H_{ab},\]  
where  
\[H_{ab}=\left[\begin{array}{ccc}1&0&0\\ 0&a^{2}/3&0\\ 0&0&b^{2}/3\end{array}\right]. \tag{22}\]  
Set  
\[V_{ab}=H_{ab}^{-1}=\left[\begin{array}{ccc}1&0&0\\ 0&3/a^{2}&0\\ 0&0&3/b^{2}\end{array}\right]. \tag{23}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline
**2\(\mathbf{a}\)/deg** & \(\mathbf{s(x_{0})}\) & \(\mathbf{s(y_{0})}\) & \(\mathbf{s(z_{0})}\) & \(\mathbf{s(r_{0})}\) \\ \hline
360 & 1.73 & 1.73 & 1.73 & 1.00 \\ \hline
270 & 2.20 & 1.57 & 1.73 & 1.13 \\ \hline
180 & 3.46 & 1.73 & 1.73 & 2.00 \\ \hline
160 & 3.85 & 1.85 & 1.73 & 2.36 \\ \hline
140 & 4.22 & 2.02 & 1.73 & 2.74 \\ \hline
120 & 4.50 & 2.26 & 1.73 & 3.09 \\ \hline
100 & 4.66 & 2.62 & 1.73 & 3.36 \\ \hline
80 & 4.69 & 3.19 & 1.73 & 3.53 \\ \hline
60 & 4.64 & 4.16 & 1.73 & 3.62 \\ \hline
40 & 4.56 & 6.15 & 1.73 & 3.65 \\ \hline
20 & 4.50 & 12.19 & 1.73 & 3.66 \\ \hline \(\alpha\leq 5\deg\) & & & & \\ \hline \(\alpha/\text{rad}\) & \(\approx\sqrt{20}\) & \(\approx\sqrt{4.5}/\alpha\) & \(\approx\sqrt{3}\) & \(\approx\sqrt{40/3}\) \\ \hline \end{tabular}
\end{table}
Table 4: Longitudinal segment of a sphere. Square roots \(s(\mathbf{a})\) of the diagonal elements of \(V_{\mathbf{a}}\) in (20). For points \(\mathbf{x}_{1:m}\) approximately uniformly distributed on a segment of the sphere with \(-\alpha\leq\theta_{i}\leq\alpha\leq\pi\) and for point-cloud variance matrix \(\sigma_{R}^{2}I\), the uncertainties \(u(a)=\sigma_{R}s(a)/\sqrt{m}\).  
If the variance matrix associated with \(\mathbf{x}_{1:m}\) is approximated by \(\sigma_{R}^{2}I\), then the standard uncertainties \(u(\mathbf{a})\) associated with the parameters is given by  
\[u(\mathbf{a})=\frac{\sigma_{R}}{\sqrt{m}}(1,\sqrt{3}/a,\sqrt{3}/b)^{\top}. \tag{24}\]  
Thus, the uncertainty in the angle of rotation about the \(x\)-axis scales with \(1/b\) and that associated with rotation about the \(y\)-axis scales with \(1/a\).