Large Translation

In this case the intersection of the volume domains becomes small and generally any tissue in one volume corresponds only to background in the other volume. More specifically, consider the case where there is a single, small amount of tissue contained in either volume, with the proportion of voxels containing tissue to total number of voxels in the intersection volume being $\chi_1$ for X and $\chi_2$ for Y(see also figure 1). Furthermore, without loss of generality, take the background intensity as zero and the tissue intensity as B. The cost functions are then given by:

$$\approx (1 - \chi_1) \frac{\sqrt{\frac{\chi_2}{1-\chi_1} B^2 - (\frac{\chi_2}{1-\chi_1} B)^2}} {\frac{\chi_2}{1-\chi_1} B}$$ C^W

$$\quad \rightarrow 0 \quad \mathrm{as} \quad \chi_1 \rightarrow 0$$ (17)

$$\quad \rightarrow \infty \quad \mathrm{as} \quad \chi_2 \rightarrow 0$$ (18)

$$\approx (1-\chi_1) \left( \frac{\frac{\chi_2}{1-\chi_1} B^2 - (\frac{\chi_2}{1-\chi_1} B)^2} {\chi_2 B^2 - \chi_2^2 B^2} \right) \rightarrow 1$$ C^CR

$$\quad \quad \quad \mathrm{as} \quad \chi_1, \chi_2 \rightarrow 0$$ (19)

$$H(X,Y) \approx - \chi_1 \log (\chi_1) - \chi_2 \log (\chi_2)$$ C^JE =

$$\quad - (1 - \chi_1 - \chi_2) \log (1 - \chi_1 -\chi_2) \rightarrow 0$$

$$\quad \quad \quad \mathrm{as} \quad \chi_1, \chi_2 \rightarrow 0$$ (20)

C^MI = H(X,Y) - H(X) - H(Y)

$$\approx - \chi_1 \log (\chi_1) - \chi_2 \log (\chi_2)$$

$$\quad - (1 - \chi_1 - \chi_2) \log (1 - \chi_1 -\chi_2)$$

$$\quad + \chi_1 \log (\chi_1) + (1-\chi_1) \log (1-\chi_1)$$

$$\quad + \chi_2 \log (\chi_2) + (1-\chi_2) \log (1-\chi_2) \rightarrow 0$$

$$\quad \quad \quad \mathrm{as} \quad \chi_1, \chi_2 \rightarrow 0$$ (21)

$$\frac{H(X,Y)}{H(X) + H(Y)}$$ C^NMI =

$$\quad \rightarrow \frac{ -\chi_1 \log (\chi_1) - \chi_2 \log(\chi_2)} { - \chi_1 \log(\chi_1) - \chi_2 \log(\chi_2)} = 1$$

$$\quad \quad \quad \mathrm{as} \quad \chi_1, \chi_2 \rightarrow 0.$$ (22)

  

Figure 1: Example of an asymptotically large transformation between images X and Y where the overlap volume becomes small. The proportion of non-background voxels in the overlap volume (shaded) is denoted by $\chi_1$ for Xand $\chi_2$ for Y.

Thus the Woods function (as $\chi_1 \rightarrow 0$) and Joint Entropy actually approach the minimum values which indicates, erroneously, that the registration is good. The other three cost functions approach their maximum values, indicating poor registrations.