## Hybrid split

As I mentioned in the previous post a hybrid split between the linear and the logarithmic split can be a good idea because when the linear splitting scheme falls short the logarithmic one can be used and the other way around.

Because when multiple layers contain the same information artifacts may occur, the criteria for choosing the ratio between linear and logarithmic splitting is so that it produces consecutive layers as different from each other as possible. Or put in another way each new layer should bring new information. In terms of computer vision this can be translated to having the mutual information between these images as small as possible.

Two techniques of measuring mutual information were tested: sum of absolute differences and cross-correlation coefficient.

The sum of absolute differences is pretty straight forward to compute and it involves adding the absolute value of the difference between each two corresponding pixels from the two images.

The cross-correlation coefficient represents the ratio between the covariance of two images and the product of their standard deviation, and can be computed using the following formula:

where *Ī*(•) is the mean of image *I. *Another useful property about correlation is that it has values on a scale ranging from [-1, 1] and it gives a linear indication of the similarity between images.

As expected from the findings in the previous post the mutual information is smaller when choosing a more linear split for sparse objects and a more logarithmic one for denser objects (*Figure 1*).

*Figure 1* Plot generated using gnuplot. Linear splitting corresponds to a split ratio of 0 while logarithmic splitting maps to the value of 1.

Because, as we can see from *Figure 1*, the cross-correlation coefficient (shown in green) covers a wider range of values, giving better estimate for each density value, it was chosen as the default method of computing the mutual information. The cross-correlation probably performs better due to the influence of standard deviation, which is completely neglected for the sum of absolute differences (shown in red).