Maximum Likelihood Registration

Given a novel pair of unregistered images of the same modalities as our training images, we assume that when registered, the joint intensity distribution of the novel images should be similar to that of the training data. When mis-registered, one structure in the first image will overlap a different structure in the second image, and the joint intensity distribution will most likely look quite different from the learned model. Given a hypothesis of registration transformation, T, and the Gaussian mixture model, M, we can compute the likelihood of the two images using Equation 2:

$$P(I_1, I_2 \, \vert \, T, M) = \prod_{x} P(I_1(x), I_2(T(x)) \, \vert \, T, M).$$ (7)

We register the images by maximizing the log likelihood of the images, given the transformation and the model, and define the maximum likelihood transformation, T_ML, as follows:

$$\hat{T}_{ML} = \mathop{\rm argmax}_{T} \, \sum_{x} \, \log(P(I_1(x), I_2(T(x)) \, \vert \, T, M))$$ (8)

The likelihood term in this equation can be substituted with either Equation 5 or 6, depending on which joint intensity model is chosen. For the results presented here, the Parzen model is used, as it better explains the intensity relationship between the two modalities. However, the mixture of Gaussians model encodes coarse tissue type classes and thus provides a framework for later incorporating into the registration process prior knowledge of the relative positions and shapes of the various internal structures.

  

Figure: Samples from the negative log likelihood function over various angles and x-shifts. Note that over this range, the function is very smooth and has one distinct minimum, which in this case occurs 0.86 mm away from the correct alignment.

To find the maximum likelihood transformation, T_ML, we use Powell maximization [9] to ascend the log likelihood function defined in Equation 8, finding the best rigid transformation. In both the mixture of Gaussian and Parzen window models of the distribution, the log likelihood objective function is quite smooth. Figure 6 illustrates samples from the negated objective function for various rotation angles (along one dimension) and x position shifts of the transformation. Over this sampled range of $\pm 60$ degrees and $\pm 20$ mm, the function is always concave and has one minimum which occurs within a millimeter of the correct transformation. Computing the registration by maximizing the likelihood of the image pair given the transformation and the model seems to be an efficient, accurate method of registration.