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Since there is no single subject brain scan capable of representing a whole population and its large anatomical inter-subject variability, probabilistic atlases, constructed from a set of images, have emerged as the tool of choice in the representation, analysis and interpretation of population-based imaging studies. The optimal unbiased probabilistic atlas most representative of the dataset is the one that requires the minimum amount of deformation to all individual atlas images of the dataset. In the simplest case, a probabilistic atlas can be computed as the mean image of a collection of images, thereby representing the average anatomy of the population. Within the context of this definition, a population atlas refers to a 3D average of cross-sectional data without any notion of time or age of the subjects.

In early methods, the anatomy of a single subject was used as a template for normalisation. All other population images were then brought into this same space. A limitation of this approach is that the choice of template introduces a bias towards the shape of its anatomy. One solution towards the use of unbiased brain atlases for determining differences in anatomical patterns was proposed by Thompson and Toga [209]. A target image is reg- istered to all existing atlases and the resulting deformation fields are used to

determine the probability distribution of corresponding points. This allows the calculation of the likelihood of the targets region to be in a similar spa- tial location to the corresponding atlas brains. Due to the warping of the target to all atlas images, all of them contribute equally. In order to avoid the registrations between each atlas and a new target, an unbiased atlas can be calculated by simultaneously registering all of the subjects to their aver- age coordinate system [33]. This makes the selection of a reference subject unnecessary. In this approach the computation of the arithmetic average of small displacement fields (vector fields) is well defined. This is not the case in the large deformation setting where we operate in the high dimensional group of diffeomorphisms. Based on the theory of large deformation diffeo- morphisms, which allows the estimation of the displacement by integrating a velocity field, a distance or similarity measure is defined and the problem can be stated as finding the image and associated coordinate system that requires the least amount of deformation to align with every subject of the population [113]. The solution for this minimisation problem can be estimated with a greedy fluid algorithm by iteratively updating the transformation for each image and updating the template as the voxel-wise arithmetic mean. The fi- nal deformation can then be composed of the transformations gained at each iteration. One drawback of this method is that the average arithmetic mean of different tissues that are not in correspondence might lead to biologically wrong structures.

In order to solve this problem, SPM [14, 16] first requires approximate alignment of the images to pre-defined tissue probability maps. Once in the same space, tissue probability maps for GM, WM and CSF can be con- structed for each of the individual images. These individual subject maps are iteratively registered to their average followed by the construction of a new average. The initial template is calculated as the intensity average of the GM and WM maps. The similarity measure for the registration is based on the SSD, which considers the difference between the likelihood of the in- dividual GM probability maps and the GM mean, between the likelihood of

the individual WM maps and the WM mean and the likelihood of the rest (1-prob(WM)-prob(GM)). Due to this procedure, the template construction and registration is not directly based on the image intensity values but rather on the tissue probability maps. Compared to the first mean, which is very smooth and blurry, the template becomes crisper with every iteration. The registration of each image provides a diffeomorphic deformation field to the average. On the one hand, the initial alignment to the pre-defined tissue probability maps and the concurrent use of the maps does not introduce false structures when averaging them. On the other hand, this initial align- ment restricts the anatomical space, i.e. the space used for normalisation is not the average of the individual images.

In the advanced normalisation tools (ANTS) package, a method for the symmetric-group-wise normalisation (SyGN) independent from an individu- als space and unbiased by an individuals shape was proposed [23]. The tem- plate creation facilitates the use of the previously described SyN algorithm for registration. Diffeomorphisms with symmetric properties are computed for the alignment of the individual images and the algorithm does not re- quire an initial template estimate. Thereby, both shape and appearance of the constructed template are unbiased by the individual images. The goal is stated as finding the template and transformations that increase image similarity and reduce the path length of the diffeomorphisms as stated by an energy function. The diffeomorphisms that map a template to the individual images and the template shape as given by another diffeomorphism are ini- tialised as identity. Then each part of the energy function is optimised while the rest are kept constant. First the diffeomorphisms between each image and the fixed template are re-estimated. Then the template appearance is updated with fixed shape and diffeomorphisms. And finally the template shape is updated. The template used in the first iteration is the affinely registered average appearance image [194]. In more detail in one iteration the set of transformations between each image and template is computed. Then the template appearance is iteratively calculated by deforming each of

the images with their corresponding inverse transformation. The gradients of the similarity term are calculated for each deformed image and averaged, which is followed by an update of the template. The shape is updated by averaging the diffeomorphisms between the template and each image. The new diffeomorphism can be applied to the template deforming its shape more towards the new average.