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We will leave aside the question of how the registration process itself is implemented. The ANTs algorithm is mathematically rather sophisticated. Algorithms simpler to understand include affine registration , or , for spline-based nonlinear algorithms such as used in Elastix (see the manual), both of which are based on optimizing some image metric (potentially penalized in some way) with respect to a finite set of parameters using gradient-based optimization methods, or discrete methods such as the ART algorithm described in Ardekani et al. (2011) , or the Elastix algorithm (see the manualwhich uses local grid search. We simply note that one common feature of image registration methods is the use of a spatial pyramid, i.e. repeatedly performing the same registration at increasingly fine spatial scales (using blurring and/or downsampling in the earlier stages). Instead, we will concern ourselves with the question of what is a transform, and how is it applied?
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