Registration

The registration problem studied here is to find the best geometric alignment of two (volumetric brain) images. Call the two images the reference ($Y$) and floating ($X$) images. More precisely, the registration problem seeks that transformation which, when applied to the floating image, maximises the ``similarity'' between this transformed floating image and the reference image. A standard, and common, way of formulating this as a mathematical problem is to construct a cost function which quantifies the dissimilarity between two images, and then search for the transformation ($T^*$) which gives the minimum cost. In mathematical notation this is:

$$T^* = \arg\min_{T \in S_T} C(Y,T(X))$$ (1)

where $S_T$ is the space of allowable transformations, $C(I_1,I_2)$ is the cost function and $T(X)$ represents the image $X$ after it has been transformed by the transformation $T$. In this paper we shall only consider linear registration so that $S_T$ is either the set of all affine transformations or some subset of this (such as the set of all rigid-body transformations).