## Joint Entropy

By considering the two images being registered as the random variables A and B, the joint histogram just described can be normalized by dividing by the total number of voxels, and regarded as a joint probability distribution function or PDF pAB of images A and B. Because of the quantization of image intensity values, the PDF is discrete, and the values in each element represents the probability of pairs of image values occurring together. The joint entropy H(A, B) is therefore given by

The number of elements in the PDF can be determined either by the range of intensity values in the two images, or from a reduced number of intensity "bins". For example, MR and CT images being registered could have up to 4096 (12 bits) intensity values, leading to a very sparse PDF with 4096 by 4096 elements. The use of between 64 and 256 bins is more common. In the foregoing equation, a and b represent either the original image intensities or the selected intensity bins. The joint entropy can be seen to have some intuitive basis as a registration similarity measure if we look again at the example joint histograms shown earlier. With increasing misregistration, the brightest regions of the histograms get less bright, and the number of dark regions in the histogram is reduced. If we interpret the joint histogram as a joint probability distribution, then misregistration involves reducing the highest values in the PDF and reducing the number of zeroes in the PDF, and this will increase the entropy. Conversely, when registering images we want to find a transformation that will produce a small number of PDF elements with very high probabilities and give us as many zero-probability elements in the PDF as possible, which will minimize the joint entropy.

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