Table 3.

Measurement functions and associated uncertainties.

	Pose($X$, $Y$, $\Theta$)		Motion($\Delta X$,$\Delta Y$,$\Delta \Theta$)
Measurand	Measurement	Uncertainty	Measurement	Uncertainty
$X_u$, $\Delta X_u$	$\frac{1}{\eta }\sum _{i=1}^{\eta }{x}_i$	$\frac{1}{\eta }\sqrt{\sum _{i=1}^{\eta }{\sigma }_i^2}$	$\frac{1}{\eta }\sum _{i=1}^{\eta }(x_{k+1,i}-x_{k,i})$	$\sqrt{\frac{1}{{\eta }^2}\sum _{i=1}^{\eta }({\sigma }_{k,i}^2+{\sigma }_{k+1,i}^2)}$
$X_w$, $\Delta X_w$	$\frac{\sum _{i=1}^{\eta }\frac{x_i}{{\sigma }_i^2}}{\sum _{\mathrm{i}=1}^{\eta }\frac{1}{{\sigma }_i^2}}$	$\sqrt{\frac{1}{\sum _{i=1}^{\eta }\frac{1}{{\sigma }_i^2}}}$	$\frac{\sum _{i=1}^{\eta }\frac{x_{k+1,i}-x_{k,i}}{{\sigma }_{k,i}^2+{\sigma }_{k+1,i}^2}}{\sum _{\mathrm{i}=1}^{\eta }\frac{1}{{\sigma }_{k,i}^2+{\sigma }_{k+1,i}^2}}$	$\sqrt{\frac{1}{\sum _{i=1}^{\eta }\frac{1}{{\sigma }_{k,i}^2+{\sigma }_{k+1,i}^2}}}$
${\Theta }_u$, $\Delta {\Theta }_u$	$\frac{\sum _{i=1}^{\eta }{r}_i{\widehat{s}}_i({\widehat{\gamma }}_i-{\theta }_i)}{\sum _{i=1}^{\eta }{r}_i{\widehat{s}}_i}$	$\frac{\sqrt{\sum _{i=1}^{\eta }{\sigma }_i^2{r}_i^2}}{\sum _{i=1}^{\eta }{r}_i^2}$	$\frac{\sum _{i=1}^{\eta }{r}_{k,i}{r}_{k+1,i}({\theta }_{k+1,i}-{\theta }_{k,i})}{\sum _{i=1}^{\eta }{r}_{k,i}{r}_{k+1,i}}$	$\frac{\sqrt{\sum _{i=1}^{\eta }(r_{k+1,i}^2{\sigma }_{k,i}^2+r_{k,i}^2{\sigma }_{k+1,i}^2)}}{\sum _{i=1}^{\eta }{r}_{k,i}{r}_{k+1,i}}$
${\Theta }_w$, $\Delta {\Theta }_w$	$\frac{\sum _{i=1}^{\eta }\frac{r_i{\widehat{s}}_i({\widehat{\gamma }}_i-{\theta }_i)}{{\sigma }_i^2}}{\sum _{i=1}^{\eta }\frac{r_i{\widehat{s}}_i}{{\sigma }_i^2}}$	$\sqrt{\frac{1}{\sum _{i=1}^{\eta }{(\frac{r_i}{{\sigma }_i})}^2}}$	$\frac{\sum _{i=1}^{\eta }\frac{r_{k,i}^2{r}_{k+1,i}^2({\theta }_{k+1,i}-{\theta }_{k,i})}{r_{k+1,i}^2{\sigma }_{k,i}^2+r_{k,i}^2{\sigma }_{k+1,i}^2}}{\sum _{i=1}^{\eta }\frac{r_{k,i}^2{r}_{k+1,i}^2}{r_{k+1,i}^2{\sigma }_{k,i}^2+r_{k,i}^2{\sigma }_{k+1,i}^2}}$	$\sqrt{\frac{1}{\sum _{i=1}^{\eta }\frac{r_{k,i}^2{r}_{k+1,i}^2}{r_{k+1,i}^2{\sigma }_{k,i}^2+r_{k,i}^2{\sigma }_{k+1,i}^2}}}$
Expressions for $x$-axis pose and motion are isomorphic to those for $y$-axis pose and motion.

$\eta$ is the number of point sources used in the measurement; $x_i$, $x_{k,i}$, $x_{k+1,i}$ the estimated $x$ coordinate of the ith point in the only image, image $k$, and image $k+1$; ${\sigma }_i$, ${\sigma }_{k,i}$, ${\sigma }_{k+1,i}$ the single-axis uncertainties of the $i^{\text{th}}$ point in the only image, image $k$, and image $k+1$; $r_i$, $r_{k,i}$, $r_{k+1,i}$ the estimated $r$ coordinate of the $i^{\text{th}}$ point in the only image, image $k$, and image $k+1$; ${\theta }_i$, ${\theta }_{k,i}$, ${\theta }_{k+1,i}$ the estimated $\theta$ coordinate of the $i^{\text{th}}$ point in the only image, image $k$, and image $k+1$; ${\widehat{s}}_i$, ${\widehat{\gamma }}_i$ the true centroid-offset position of the $i^{\text{th}}$ point in polar coordinates; $X_u$, $X_w$ the uniformly weighted and optimally weighted estimates of $x$-axis centroid position; ${\Theta }_u$, ${\Theta }_w$ the uniformly weighted and optimally weighted estimates of orientationl; ${\Delta X}_u$, ${\Delta X}_w$ the uniformly and optimally weighted estimates of $x$-axis centroid displacement; ${\Delta \Theta }_u$, ${\Delta \Theta }_w$ the uniformly weighted and optimally weighted estimates of rotation.