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Data Preprocessing

This chapter describes how the data is processed from output of the scanner to the actual diffusion tensor representation.

Solving the Basic Equations

LSDI_recon is a program that solves the Stejskal Tanner Equation 3.18 presented in Section 3.3. As input the program requires the images produced by the scanner as well as a parameter file. The eight images provide eight equations for $M$ in each voxel which are solved in a least-squares sense for the 6+1 unknowns: the six independent components of the symmetric diffusion tensor $D$ and $M_0$. Besides solving the equations, LSDI_recon reorients and interpolates the diffusion tensor data so that the resulting output has a standardized orientation and shape. The output is saved as *.pic files of size $256\times256$ with short precision and represents the eigenvalue and eigenvector decomposition of the tensor. There is no header and therefore no additional information of the data is stored. Table 5.1 describes the content of the single images that are produced by LSDI_recon.

Table 5.1: File extensions and content of the files generated by LSDI_recon. xx represents the slice number.

File extension	Content
file-SxxL1e.pic	first eigenvalue
file-SxxL1x.pic	first eigenvector
file-SxxL1y.pic
file-SxxL1z.pic
file-SxxL2e.pic	second eigenvalue
file-SxxL2x.pic	second eigenvector
file-SxxL2y.pic
file-SxxL2z.pic
file-SxxL3e.pic	third eigenvalue
file-SxxL3x.pic	third eigenvector
file-SxxL3y.pic
file-SxxL3z.pic
file-SxxAAI.pic	Absolute anisotropy index
file-SxxADC.pic	Apparent diffusion coefficient
file-SxxDWI.pic	Diffusion weighted image
file-SxxRAI.pic	Relative anisotropy $=\frac{\mathrm{AAI}}{\mathrm{ADC}}$
file-SxxT2W.pic	T2 weighted image

Conversion to Tensor Fields

In this step the files generated by LSDI_recon are converted into floating point precision data structures representing tensor- and eigenfields. Additional images produced by LSDI_recon are considered as background images and are also collected into a single data structure. An additional xml-file containing information about the data is generated. The xml-file and its Document Type Definition (DTD) are described in Section 9.3. Most important it stores the field dimension, voxel dimensions and the location of the data without any file extension. Table 5.2 describes the file extensions used and their meaning. Some of the files are generated in later steps and described when they first appear.

Table 5.2: File extensions and content of the files generated and used in this application

File extension	Content
.xml	the xml-header with information of the data
.ten	the tensorfield data
.eig	the eigenfield data
.bg	the background images
.mask.0xx	the mask for the data (MRI file format)
.masked.xml, .ten, .eig, .bg	masked data
.final-param-file	text file with parameters for the rigid transformation
.aligned.xml, .ten, .eig, .bg	the aligned data
.matched.xml, .ten, .eig, .bg	the matched data
.matched.dx	displacement field in x-direction
.matched.dy	displacement field in y-direction
.matched.dz	displacement field in z-direction

Orthogonalization

Despite the fact that LSDI_recon does a lot of enhancement of the data the output cannot directly be used as tensor data. Most importantly the simpler Equation 4.8 for conversion from an eigensystem decomposition to a matrix cannot be applied since it was found that the eigenvectors produced by LSDI_recon do not build an orthonormal basis. As the tensor has to be symmetric it would be incorrect to apply the general Equation 4.7 to build a matrix from the eigensystem. Making the matrix orthogonal after transformation by computing $D' = 0.5(D+ D^T)$, which is the best approximation without any prior knowledge when using the norm $\Vert D\Vert^2 = \sum_{i,j} (D_{ij})^2$, deforms the tensor in an unwanted manner as the off-diagonal elements often have opposite signs.

To built an orthonormal basis with the eigenvectors given, it is assumed that the signal-to-noise ratio is best for the largest eigenvalue and therefore also for the corresponding eigenvector. It then makes sense to keep the eigenvector corresponding to the largest eigenvalue and correct the remaining two eigenvectors. The second eigenvector is projected into the plane normal to the first eigenvector. This plane is described by

$$\hat{\mathbf{e}}_{1x}x + \hat{\mathbf{e}}_{1y}y + \hat{\mathbf{e}}_{1z}z = 0$$ (5.1)

where $\hat{\mathbf{e}}_1$ is the first eigenvector. The projection into the plane is then given by

$$\hat{\mathbf{e}}_2' = s(\alpha\hat{\mathbf{e}}_1 + \hat{\mathbf{e}}_2)$$ (5.2)

where

$$\alpha = -\frac{\hat{\mathbf{e}}_{1x}\hat{\mathbf{e}}_{2x} +... ...f{e}}_{1z}\hat{\mathbf{e}}_{2z}}{\Vert\hat{\mathbf{e}}_1\Vert}$$ (5.3)

and $s$ is a scaling parameter to normalize the result to length 1.

The third eigenvector is computed as $\hat{\mathbf{e}}_3' = \hat{\mathbf{e}}_1\times \hat{\mathbf{e}}_2'$. In case the distance $\Vert\hat{\mathbf{e}}_3' - \hat{\mathbf{e}}_3\Vert$ is smaller when taking the negative value $-\hat{\mathbf{e}}_3'$, then the sign of the third eigenvalue is changed. Figure 5.1 visualizes this process. The mean distance between $\Vert\hat{\mathbf{e}}_2' - \hat{\mathbf{e}}_2\Vert$ for older data sets was found to be about 0.1 while $\Vert\hat{\mathbf{e}}_3' - \hat{\mathbf{e}}_3\Vert$ usually was 0.2 For newer data sets these values were around 0.05 respectively 0.1. These errors are output when orthogonalizing the eigenfield.

Figure 5.1: Orthogonalization of the eigenvectors

Filtering

Older data sets show some one-pixel wide strips in the border to the background respectively inside the brain in the border to the cerebrospinal fluid (CSF). The tensors in these strips have huge eigenvalues compared to their neighborhood. They certainly do not provide any meaningful information and should be removed completely. Therefore a simple 3 by 1 filter is incorporated in this processing step which acts like a Median filter only on locations where the left and right voxel belong to the background. No other smoothing or filtering of the data is performed at this step.