Information-Theoretic Approach for Automated White Matter Fiber Tracts Reconstruction

Abstract

Fiber tracking is the most popular technique for creating white matter connectivity maps from diffusion tensor imaging (DTI). This approach requires a seeding process which is challenging because it is not clear how and where the seeds have to be placed. On the other hand, to enhance the interpretation of fiber maps, segmentation and clustering techniques are applied to organize fibers into anatomical structures. In this paper, we propose a new approach to automatically obtain bundles of fibers grouped into anatomical regions. This method applies an information-theoretic split-and-merge algorithm that considers fractional anisotropy and fiber orientation information to automatically segment white matter into volumes of interest (VOIs) of similar FA and eigenvector orientation. For each VOI, a number of planes and seeds is automatically placed in order to create the fiber bundles. The proposed approach avoids the need for the user to define seeding or selection regions. The whole process requires less than a minute and minimal user interaction. The agreement between the automated and manual approaches has been measured for 10 tracts in a DTI brain atlas and found to be almost perfect (kappa > 0.8) and substantial (kappa > 0.6). This method has also been evaluated on real DTI data considering 5 tracts. Agreement was substantial (kappa > 0.6) in most of the cases.

Appendix: 2D Image Partitioning Algorithm

Rigau et al. (2004) proposed an information-theoretic partitioning algorithm. In this algorithm, the partitioning of a 2D image is guided by the maximization of mutual information gain and is constructed from an information channel X → Y between the random variables X (input) and Y (output), which represent, respectively, the set of regions $\mathcal{X}$ of an image and the set of intensity bins $\mathcal{Y}$. The basic notions of information theory can be found in Cover and Thomas’s book (Cover and Thomas 1991).

To describe the method, first, we define the information channel and, then, we review the partitioning algorithm. The channel X → Y is defined by a conditional probability matrix p(Y|X) which expresses how the pixels corresponding to each region of the image are distributed into the histogram bins. Note that the capital letters X and Y as arguments of p() are used to denote probability distributions. For instance, while p(X) represents the input distribution of the regions, p(x) denotes the probability of a single region x.

Given an image with N pixels, the three basic elements of the channel X → Y are:

• The conditional probability matrix p(Y|X), which represents the transition probabilities from each region of the image to the bins of the histogram, is defined by $p(y|x)=\frac{n(x,y)}{n(x)}$, where n(x) is the number of pixels of region x and n(x,y) is the number of pixels of region x corresponding to bin y. Conditional probabilities fulfill $\sum_{y \in \mathcal{Y}} p(y|x)=1$, $\forall x \in \mathcal{X}$.

• The input distribution p(X), which represents the probability of selecting each image region, is defined by $p(x)=\frac{n(x)}{N}$ (i.e., the relative area of region x).

• The output distribution p(Y), which represents the normalized frequency of each bin y, is given by $p(y)=\sum_{x \in \mathcal{X}} p(x)p(y|x)=\frac{n(y)}{N}$, where n(y) is the number of pixels corresponding to bin y.

The mutual information (MI) between Y and X is given by

$$ I(X,Y) =\sum\limits_{x\in\mathcal{X}}p(x) \sum\limits_{y\in\mathcal{Y}}p(y|x) \frac{p(y|x)}{p(y)} $$

(2)

and represents the shared information or correlation between X and Y. It is important to note that the maximum MI that can be achieved in the partitioning process is the entropy H(Y) of the histogram, given by

$$ H(Y) = - \sum\limits_{y\in\mathcal{Y}} p(y) \log p(y). $$

(3)

The entropy H(Y) represents the information content of the image.

The partitioning algorithm is a greedy top-down procedure which partitions the image in quasi-homogeneous regions. The partitioning strategy takes the full image as the unique initial partition and progressively subdivides it with vertical or horizontal lines chosen according to the maximum MI gain for each partitioning step. This algorithm produces a binary space partition (BSP) driven by the maximum information gain and stops when a given mutual information ratio $MIR=\frac{I(X,Y)}{H(Y)}$ or a predefined number of regions is achieved.