There are some cases where you might want to

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There are some cases where you might want to transform an image using onlya subset of the possible linear transformations, which corresponds to an affinetransformation with fewer than 12 parameters. For example, in motion correc-tion we assume that the head is moving over time without changing its size orshape. We can realign these images using an affine transform with only six param-eters (three translations and three rotations), which is also referred to as arigid
192.3 Spatial transformations-2-1-31023-2-1-31023-2-1-31023-2-1-31023-3-2-10123Translation-3-2-10123Rotation-3-2-10123Scaling-3-2-10123ShearingFigure 2.3.Examples of linear transforms. In each figure, the black dots represent the original coordinatelocations, and the blue points represent the new locations after the transformation is applied.body transformationbecause it does not change the size or shape of the objects inthe image.2.3.1.2 Piecewise linear transformationOne extension of affine transformations is to break the entire image into severalsections and allow different linear transforms within each of those sections. This isknown as apiecewise lineartransformation. Piecewise linear transformations wereemployed in one of the early methods for spatial normalization of brain images,developed by Jean Talairach (which will be discussed in more detail in Chapter4).2.3.1.3 Nonlinear transformationsNonlinear transformations offer much greater flexibility in the registration ofimages than affine transformations, such that different images can be matchedmuch more accurately. There is a very wide range of nonlinear transformationtechniques available, and we can only scratch the surface here; for more details, seeAshburner & Friston(2007) andHolden(2008). Whereas affine transformationsare limited to linear operations on the voxel coordinates, nonlinear transfor-mations allow any kind of operation. Nonlinear transforms are often describedin terms ofbasis functions, which are functions that are used to transform the
20Image processing basicsBox 2.3.1Mathematics of affine transformationsAffine transformations involve linear changes to the coordinates of an image,which can be represented as:Ctransformed=TCorigwhereCtransformedare the transformed coordinates,Corigare the original coor-dinates, and T is the transformation matrix. For more convenient applicationof matrix operations, the coordinates are often represented ashomogenouscoordinates, in which theN-dimensional coordinates are embedded in a (N+1)-dimensional vector. This is a mathematical trick that makes it easier to performthe operations (by allowing us to writeCtransformed=TCorigrather thanCtransformed=TCorig+Translation). For simplicity, here we present an exam-ple of the transformation matrices that accomplish these transformations fortwo-dimensional coordinates:C=CXCY1whereCXandCYare the coordinates in theXandYdimensions, respectively.

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