Performance Criteria

Performance was assessed by comparing the correct translation value, $T_{true}$, used to generate the measured image, to the recovered translation parameter estimate $T_{MAP,j}$ in the $j$th trial.

A measure of the robustness of the estimated translation is:

$$M_R = \frac{1}{N_{trials}} \; \char93 \{ T_{MAP,j} \; \, \vert \; \, ( T_{MAP,j} - T_{true} )^2 > \Delta^2 \}$$

where $\char93 \{ A \}$ represents the number of elements in set $A$, and $\Delta$ is a moderate tolerance, taken to be 2mm in this case. This measure represents the fraction of solutions which were not considered ``close'', where $\Delta$ sets the threshold by which something is considered ``close''.

A measure of the accuracy of the estimated translation is:

$$M_A = \frac{1}{\Delta} \left( \frac{1}{N_{trials}} \sum_{j=1}^{N_... ...\left( (T_{MAP,j} - T_{true})^2, \Delta^2 \right) - M_R \Delta^2 \right)^{1/2}$$

This measure is a modified RMS error measurement, where any errors greater than $\Delta$ are replaced by $\Delta$. In this way it downweights the contribution for very large errors which a non-robust technique can produce and gives a normalised measure between 0 and 1.

For an ideal method, both measures, $M_R$ and $M_A$, would be zero.