Results

Some sample plots of the similarity functions $F_0$, $F_1$ and $F_2$ are shown in figure 4. These show the functions over a relatively large range of translations, and demonstrate two important features: that the secondary (incorrect) peak is downweighted in $F_0$, reducing the chances large mis-matches; and in the close-up results, the improved continuity of the similarity functions (which is smoother and has less discontinuities for $F_0$).

Figure 4: Example similarity functions for $F_0$ (left), $F_1$ (middle) and $F_2$ (right). The top row shows the functions over a large range of translations, the second and third rows show a close-up of the functions, with the actual calculated values represented by circles. The true translation value is -10.2mm for all of these plots.

Table 1 shows the values of $M_R$ and $M_A$ for 3 different SNR values and 2000 trials in each case. The low values of $M_R$ for $F_0$ are a result of the downweighting of the secondary maxima, significantly reducing bad mismatches. Also, $F_0$ was the most accurate (having the smallest value of $M_A$) which is likely to be due to the decrease in discontinuities in $F_0$.

Table 1: Results of robustness and accuracy measures for a simple object (see figure 3b) and a range of SNR values, for each of the three cost functions. $\Delta = 2$mm.

	SNR = 5		SNR = 15		SNR = 50
$F_0$	MR = 0.194	MA = 0.71	MR = 0.065	MA = 0.26	MR = 0.000	MA = 0.20
$F_1$	MR = 1.000	-	MR = 0.909	MA = 0.35	MR = 0.807	MA = 0.97
$F_2$	MR = 0.516	MA = 0.86	MR = 0.494	MA = 0.48	MR = 0.129	MA = 0.54