Affine Registration of label maps in Label Space

Two key aspects of coupled multi-object shape analysis and atlas generation are the choice of representation and subsequent registration methods used to align the sample set. For example, a typical brain image can be labeled into three structures: grey matter, white matter and cerebrospinal fluid. Many manipulations such as interpolation, transformation, smoothing, or registration need to be performed on these images before they can be used in further analysis. Current techniques for such analysis tend to trade off performance between the two tasks, performing well for one task but developing problems when used for the other. This article proposes to use a representation that is both flexible and well suited for both tasks. We propose to map object labels to vertices of a regular simplex, . the unit interval for two labels, a triangle for three labels, a tetrahedron for four labels, etc. This representation, which is routinely used in fuzzy classification, is ideally suited for representing and registering multiple shapes. On closer examination, this representation reveals several desirable properties: algebraic operations may be done directly, label uncertainty is expressed as a weighted mixture of labels (probabilistic interpretation), interpolation is unbiased toward any label or the background, and registration may be performed directly. We demonstrate these properties by using label space in a gradient descent based registration scheme to obtain a probabilistic atlas. While straightforward, this iterative method is very slow, could get stuck in local minima, and depends heavily on the initial conditions. To address these issues, two fast methods are proposed which serve as coarse registration schemes following which the iterative descent method can be used to refine the results. Further, we derive an analytical formulation for direct computation of the "group mean" from the parameters of pairwise registration of all the images in the sample set. We show results on richly labeled 2D and 3D data sets.