Frequently Asked Questions - Coregistration
1. What is coregistration?
2. What are the different ways to coregister images?
These days, there are a few, but one de facto standard, which is coregistration by mutual information (MI). In some ways, coregistration is always the same problem - it's just like realignment, but can be between different modalities (PET, MRI, CAT, etc.), and usually you can be slower at it. The problem boils down to finding some function that measures the difference between your two images and then minimizing (or maximizing) it. Minimization/maximization is pretty standard these days; the question is what sort of cost function you use.
In realignment, we just used the sum of the squared difference in intensity between the images, measured voxel-by-voxel. The trouble with these scheme in coregistration is that your images might be different modalities, and hence a tissue type that's very dark in one (say, ventricle in PET) might be very bright in another (say, proton-density-weight MRI). In that case, trying to minimize the intensity differences between the images will give you a horrible registration.
3. What is mutual information, exactly?
Maes et. al, below, give a pretty good summary of it, but here's the nutshell version. If two variables A and B are completely independent, their joint probability Pab (the probability that A comes up a at the same time that B comes up b) is just the product of their respective separate probabilities Pa and Pb; so Pab=Pa*Pb, if A and B are independent. On the other hand, if A and B are completely dependent - that is, knowing what value A takes tells you exactly what value B will have - then their joint probability is exactly the same as their respective separate probabilities; so Pab = Pa = Pb, if A and B are completely dependent. If A and B are dependent a little bit, but not entirely, their joint probability will be somewhere in between there - knowing what value A has tells you a little bit about B, so you can make a good informed guess at what B will be, but not know it exactly.
Mutual information is a way of measuring to what extent A and B are dependent on each other. Essentially, if you can estimate the true joint probability Pab for all a and b, and you know the individual probability distributions Pa and Pb, you can measure how far away the probability distribution Pab is from Pa*Pb, with a Kullback-Leibler statistic that measures the distance between curves. If Pab is much different from Pa*Pb, then you know A and B are dependent to some degree.
4. So how does mutual information help coregistration?
MI coregistration methods work by considering the intensity in one image to be A and the intensity in the other image to be B. The algorithm computes the MI between those two variables - finds the mutual information between the intensity in one image and the intensity in the other - and then attempts to maximize it.
5. How are coregistration and segmentation related?
Fischl et. al (SegmentationPapers) make the point that the two processes operate on different sides of the same coin - each one can solve the other. With a perfect coregistration algorithm, you could be maximally confident that you could line up a huge number of brains and create a perfect probability atlas - allowing you the best possible prior probabilities with which to do your segmentation. In order to do a good segmentation, then, you need a good coregistration. But if you had a perfect segmentation, you could vastly improve your coregistration algorithm, because you could coregister each tissue type separately and greatly improve the sharpness of the edges of your image, which increases mutual information.
Fortunately, MI thus far appears to do a pretty good job with coregistration even in unsegmented images, breaking us out of a chicken-and-egg loop. But future research on each of these processes will probably include, to a greater and greater extent, the other process as well.