Since VBM compares the whole brain volumes between two groups of subjects, achieving an accurate inter-subject registration seems to be a necessary request. For this reason the Dieomorphic Anatomical Registration using Exponentiated Lie algebra (DARTEL) algo- rithm represents a fundamental step to process the segmentation output. This procedure implements a dieomorphic warping of each GM segmented image to a study-speci c tem- plate generated within the same algorithm. The SPM8 package oers a speci c tool to implement at the same time the study-speci c template generation and the dieomorphic image registration. This procedure is extensively illustrated in the paper of Ashburner [26], of which we will remark the main elements.
Image registration is based on the estimation of a smooth continuous mapping between the points in one image and those in another. Then, the parameters characterizing this mapping can be used to determine the relative shapes of the images. The aim is to obtain the single best set of values for these parameters. We can distinguish between two ways of modelling such mappings:
• The small-deformation model does not necessarily preserve topology and parameter- izes a displacement eld (u), which is simply added to an identity transform (x):
Φ(x) = x + u(x) (3.15) In this case, the inverse transformation is sometimes approximated by subtracting the displacement, which fails for larger deformations.
• The large-deformation model generates deformations (dieomorphisms) that possess the important property of enforcing the preservation of topology. A dieromorphism represents a one-to-one smooth and continuous mapping with derivatives that are invertible (i.e. nonzero Jacobian determinant). If the mapping is not dieomorphic, then topology is not necessarily preserved.
The DARTEL algorithm regards the latter framework, which shows the positive character- istic of consistency under composition of deformations. In fact, if these are dieomorphic, the result of the composition operation will also be dieomorphic. However, as deformations are represented discretely by a nite number of parameters, there may be some small viola- tions. Instead, perfect (i.e. in nitely dimensional) dieomorphisms form a Lie group under the composition operation, because they satisfy the requirements of closure, associativity, inverse and identity.
If u(t) is a velocity eld at time t, then the dieomorphism Φ evolves as follows:
dΦ dt = u
(t) Φ(t). (3.16)
According to the paper we are referring to [26], the DARTEL procedure considers a single ( ow) velocity eld, which remains constant over unit time. Registration involves simul- taneously minimising a measure between the image and the warped template, while also minimising an energy measure of the deformations used to warp the template. This energy can be computed by integrating the energy of the velocity elds over unit time. The xed velocity eld employed has to describe the whole trajectory of an evolving dieomorphism, therefore it may be forced to assume complicated and high energy trajectories in order to achieve good correspondence between images. In Group theory, the ow eld may be con- sidered as a member of the Lie algebra, which is exponentiated to produce a doformation, which is a member of a Lie group. Because the Jacobian of a deformation that adapts to an
exponentiated ow eld is always positive, we can be sure that the mapping is dieomor- phic and, implicitly, that the forward and inverse transformations can be generated from the same ow eld:
Φ = Exp(u). (3.17)
3.4.1 The DARTEL optimization procedure
This dieomorphic image registration procedure uses a model which contains a number of unknown parameters that describe how the image is deformed to match with another. The optimization procedure recurs to a discrete parametrization of the velocity eld, u(x), that can be written as a combination of basis functions:
u(x) =X
i
νiρi(x), (3.18)
where ν is a vector of coe cients and ρi(x) is the i-th basis function at position x. The
purpose of the optimization procedure is to individuate the single best set of values that parameterize the velocity eld. We can write the posterior probability of the parameters given the image data (D):
p(ν|D) = p(D|ν)p(ν)
p(D) . (3.19)
that is proportional to the likelihood, i.e. the probability of the image data given the parameters (p(D|ν)) and to the priors of the parameters (p(ν)). The probability of the data is a constant (p(D)). The eq. (3.19) is the objective function has to be maximized to obtain a maximum a posteriori estimate for the parameters. Normally, the objective function is represented by the logarithm of the a posteriori probability (in which case it is maximized) or the negative logarithm (in which case it is minimized). It may be seen as the sum of two terms: a prior term and a likelihood term.
− log p(ν|D) = log p(ν) − log p(D|ν), (3.20) or
(ν) = 1(ν) + 2(ν). (3.21)
Nevertheless, in practice there are many technical di culties that can impede a simple Bayesian interpretation of the problem. The procedure adopted to solve this problem con- sists in a local optimization, that is practically actuated through the Lavenberg-Marquardt (LM) algorithm [27]. The necessary matrix solutions are obtained in reasonable time using a multigrid method.
3.4.2 Group-wise registration
Until now, we have discussed about how to match a pair of images through dieomorphic image registration. Nevertheless, our initial purpose was to warp together multiple subjects' images. For this reason, a study-speci c template is usually generated and then the defor- mation elds that warp each image to this average template are estimated. The scheme that the DARTEL algorithm follows to construct the study-speci c template involves an iterative procedure. Therefore, the DARTEL is used to map the scans to their average, and then the warped images are used to compute a new average template. This cycle is re- peated until the spatial precision required has been achieved. An example of output images implementing DARTEL algorithm in SPM8 is shown in gure3.7.
This procedure has been applied to intersubject registration of 471 whole brain im- ages, and the resulting deformations were evaluated in terms of how well they encode the
(a) GM segmented and DARTEL-imported image. (b) Study-speci c template.
(c) Flow eld that warps the3.7ainto the3.7b.
Figure 3.7: An example of ow eld estimation through the DARTEL algorithm. The T1- weighted image of one subject has been GM segmented and then spatially nor- malized to a lower resolution space (i.e. the segmented image has been DARTEL- imported (3.7a)). The ow eld that warps the latter to the study-speci c template (3.7b), both computed through the DARTEL algorithm, is shown in3.7c.
shape information necessary to separate male and female subjects and to predict the age of the subjects [26]. As explained in the latter paper, cross-validation was used to assess classi cation accuracy, showing a performance better than a small-deformation approach.