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Rigid Registration

Before any local comparison of data values between subjects can be done, the data sets should be rigidly aligned so that the shapes of the objects are as close to each other as possible while keeping the single shapes unchanged. This involves three basic transformations:

1. Translation
2. Rotation
3. Scaling

P. Woods [29] gives a very detailed explanation on how to apply rigid transformations. Here existing programs are used to find the transformations and apply them to the data. They are then extended to process tensor data.

The basic characteristics of rigid transformations are linearity and preservation of internal distances and angles.

Masking

First, the shape of the object of interest has to be extracted as binary data where the background is assigned the value $o$ and the object the value $\bar{o}$. In the case of baby brains this could easily be done by thresholding the T2 weighted background image. After the threshold has been set to best extract the brain this mask can be applied to all the remaining data, i.e. the tensorfield, the eigenfield, and the other background images. The mask is stored in the standard MRI-file format for further processing (see Table 5.2 page ).

Alignment of Grayscale Data

The task is now to find the nine transformation parameters (3 rotation angles, 3 translation values and 3 scale factors) that best align one mask with the other. Two functions are utilized. First a rough estimation is computed. This is then improved by small variations of the first approach. optimizing the result. The resulting transformation can be applied directly to all background images and the single components of the tensor field.

By applying a transformation to scalar data, the value at a voxel position $(i,j,k)$ is displaced by a vector $U = (u,v,w)^T = (u(i,j,k),v(i,j,k),w(i,j,k))^T$. This vector will usually not be a multiple of the respective voxel size so that the original position is displaced into a position between raster points. As the resulting image has a discrete raster like the original one, the value at the destination raster position has to be estimated. Two possible estimations are implemented: Nearest neighborhood or linear interpolation. The difference of the methods can be seen in Figures 6.2 and 6.3, where the synthetic square of Figure 6.1 was rotated by 45$^\circ$ degrees.

Figure 6.1: Original synthetic square

Figure 6.2: Rigid rotation of synthetic example with nearest neighbor interpolation

Figure 6.3: Rigid rotation of synthetic example with linear interpolation

Alignment of Tensor Data

In the first step the tensors are decomposed. For each component, a scalar field is built. These scalar fields are then transformed according to the previous section and the resulting scalar fields are merged into a new, rigidly transformed tensorfield.

As tensors have an internal structure that is related to the body they belong to, any transformation has to consider local changes of this structure. That is, after merging the scalar fields, the resulting tensors have to be further processed.

For the case of a translation, this local change is trivial since the neighborhood of the tensor under inspection is displaced by the same amount. Any rotation of the body leads to a local rotation of the tensor by the same amount as the body was rotated. In the tensor-domain this leads to

$$D' = R^T D R$$ (6.1)

where $R$ is the rotation matrix applied to the body and $det(R) = 1$. In the eigen-domain this is equivalent to rotating the eigenvectors by the rotation matrix:

$$\hat{\mathbf{e}}_i' = R\hat{\mathbf{e}}_i$$ (6.2)

It is not evident how to apply the scaling of the body locally on the tensor. Mathematically this would be

$$D' = S^T D S$$ (6.3)

in the tensor-domain, where $S$ is a diagonal matrix. In the eigen-domain this would lead to a scaling of the eigenvalues

$$\lambda_i' = s_i\lambda_i$$ (6.4)

where $s_i$ is a linear combination of the global scale parameters $a_i$ according to the direction of the corresponding eigenvector $\hat{\mathbf{e}}_i$.

When considering an application like the alignment of brains, it is no longer obvious if the scaling of the tensor as a meaningful operation. Consider a small subject that is aligned to a larger one, say twice as large. Enlarging the tensors by a factor of two would mean that the diffusion in the corresponding tissue is now twice as large as before the transformation. Fibers probably do not change their diffusion properties in such a manner when the subject is growing or in different subjects. Missing values are interpolated component-wise when the rigid transformation is applied on each component, as described above, so that the consistency of the diffusion data is guaranteed otherwise. Therefore it is suggested here, that local scaling of the tensors not be applied in rigid registration. Figures 6.4 and 6.5 show the same example as in the previous section, but now with local transformation of the tensors.

Figure 6.4: Rigid rotation of synthetic example with nearest neighbor interpolation and local transformation of tensors

Figure 6.5: Rigid rotation of synthetic example with linear interpolation and local transformation of tensors