## Nonlinear Spatial Transformation Models

Once linear constraints are abandoned, many different spatial transformation models are possible. In certain instances, nonlinear models are used to correct for nonlinear distortions imparted by the equipment used for image acquisition (see the chapters entitled "Physical Basis of Spatial Distortions in Magnetic Resonance Images" and "Physical and Biological Bases of Spatial Distortions in Positron Emission Tomography Images"). In these instances, the mathematical form of the nonlinear spatial transformation model may be simple and dictated by the physical processes that underlie the distortions. To the extent possible, these kinds of corrections should be made at the time of image reconstruction, but in some cases, parameters necessary to describe the distortions can only be estimated when at least two misregistered images are available.

More often, nonlinear spatial transformation models are used for intersubject registration. As discussed in the chapter "Biological Underpinnings of Anatomic Consistency and Variability in the Human Brain," biological intersubject variability is highly unconstrained and varied. Nonlinear models are used to improve upon the results traditionally achieved with linear approaches such as the Talairach transformation discussed in the chapter "Talairach Space as a Tool for Intersubject Standardization in the Brain," but it is implicitly understood that the spatial transformation model that is being used is only an approximation that may simultaneously overly constrain certain types of distortions while inadequately constraining others. In most cases, the choice of nonlinear spatial transformation model turns out to be dictated by the availability of a cost function and minimization strategy that can estimate the model's parameters in a rapid and reproducible manner. These models can be loosely classified by the number of degrees of freedom that they require. At the lower end of this spectrum, low-order polynomial warps provide a straightforward extension of the linear models discussed already in this chapter. A second-order polynomial warp is characterized by the equations x' = £00 + %x + w + £03 z + e04x2 + ^xy + £)6*z + W2 + Wz +

y' = e10 + e11x + e12y + e13z + e14x2 + e15xy + e16xz + e17y2 + e18yz + e19z2

z' = £20 + e21x + e^y + ez3z + e24x2 + e^xy + e^xz + e27y2

and extension to higher order models is straightforward. One nice feature of the polynomial warps is that they constitute a closed set under affine transformation. Consequently, changing the affine shape or orientation of either of the two images being registered will not alter the types of distortions available using the polynomial spatial transformation model. Another advantage is that the polynomials can be analytically differentiated locally to give a local affine warp. This can be useful for shape analysis or for efficiently inverting the transformation numerically. A theoretical disadvantage of polynomial warps is that they can give rise to local spatial inversions since no formal construct is included for preventing the local affine warp from having a negative determinant.

At the opposite end of the spectrum from polynomial warps are methods that provide one or more parameters for every voxel in the images. Although often described as a vector field mapping each point to its presumed homologue, these models usually have other constraints. A common constraint is that local spatial inversions are explicitly forbidden. The spatial transformation models used for intersubject warping are discussed further in the chapter "Warping Strategies for Intersubject Registration.''