Joint Histogram Apodization

A more general weighting scheme is required for apodizing the joint histogram required for the entropy-based cost functions. This is because the number of entries in each histogram bin becomes discontinuous as the intensity at a floating image location (calculated using interpolation) passes through the threshold value between intensity bins. As it is the intensity passing through a threshold value that causes the discontinuities for the joint histogram, we propose a weighting function that is determined by the intensities and applied to every location. We choose, once again, a linear weighting function (as shown in Figure 4) where $w_k$ is the weight for bin $k$, $I$ is the intensity at the location under consideration, $T_k$ is the intensity threshold between bins $k-1$ and $k$, and $\Delta$ is the smoothing threshold -- the equivalent of $D$ in the preceding section. This weight is then applied to the accumulation of intensity within the joint histogram bin, as well as the number of entries in the bin, which is no longer an integer. That is for a point of intensity $I$, the updating equations for bin $k$ are:

$$N_k \rightarrow N_k + w_k(I)$$

$$S_k \rightarrow S_k + I . w_k(I)$$

where $w_k(\cdot)$ is the weighting function for the $k$th bin (see figure 4), $N_k$ is the occupancy of bin $k$ (a non-integer version of the number of elements), and $S_k$ is the sum of intensities in the bin.

Figure 4: Weighting function used for apodization of the entropy-based cost functions. This weighting function is equivalent to a fuzzy-bin membership function based on the intensity $I$.

This approach is effectively fuzzy-binning, where each intensity bin no longer has sharp thresholds, but fuzzy membership functions. It also means that a given location can influence more than one bin entry. Because of the way the weighting function is calculated though, each location contributes equally, since the sum of weights for all bin entries is equal to one. As changing overlap will still create discontinuities, both the geometrical weighting and the fuzzy-binning need to be applied to have a continuous joint histogram. Moreover, the parameter $\Delta$ will give a continuous joint histogram for any value greater than zero, although the value should not exceed the intensity bin width. The smoothing capacity of $\Delta$ is shown in Figure 5 for the Mutual Information cost function. Results for the Normalised Mutual Information cost function are very similar. Note that, in general, a value of $\Delta = 0.5$ together with $D$ equal to the resolution scale (e.g. 8mm), gives a desirably smooth cost function. Note that the Partial Volume Interpolation introduced by Maes [13] also creates continuous joint histogram estimates if used in conjunction with the geometrical weighting (applied to the reference locations instead). However, as the name suggests, PVI is more than just a apodization scheme -- in fact it functions as an interpolation method too. Therefore, different interpolation methods cannot be used in conjunction with PVI, whereas for fuzzy-binning the interpolation method used can be freely chosen. Furthermore, the fuzzy-binning scheme provides an adjustable parameter, $\Delta$, which controls the amount of smoothing of the cost function, allowing for different degrees of smoothing, as desired. Finally, it can be seen that fuzzy-binning can be made fully symmetric with respect to the two images (see [2] for a discussion of symmetry in general registration cost functions). That is, both floating and reference intensities can have fuzzy-binning applied to them. However, there is an inherent asymmetry in the way that interpolation is applied only to the floating image. Therefore, although such a symmetric approach appears initially attractive, the simpler and faster approach of only using fuzzy-bins for the floating image was adopted in practice.

Figure 5: Plots of the Mutual Information cost functions versus rotation at different values for the fuzzy-binning apodization threshold, $\Delta$. The values of $\Delta$ shown are 0.0 for the top row and 0.1, 0.3 and 0.5 for the three columns (left to right respectively) in the bottom row. In (a) no geometric apodization is applied, whilst in (b) to (e), the geometric apodization with $D=8$mm is applied together with fuzzy-binning for (c) to (e). The resolution of the image in each case is 8mm. Each plot was generated by measuring the cost functions between two real images, about the estimate of the global minimum, which was a non-identity transformation and so did not correspond to maximum overlap of the FOVs. This clearly shows that both apodization methods are required to have the desired smoothing effect on the cost function, eliminating large discontinuities. In general, a value of $\Delta = 0.5$ together with $D$ equal to the resolution scale (e.g. 8mm), gives a desirably smooth cost function.