Diffusion imaging is a variant of magnetic resonance imaging (MRI) that can noninvasively map the nerve fiber tracts of the human brain.^{1} Because fibrous tissue restricts the constant heat motion (i.e., diffusion) of water molecules in a characteristic way, diffusion measurements provide data about the position and orientation of major nerve fiber bundles. Nerve fiber paths reconstructed from diffusion images^{2} are depicted in Figure 1. Such information about the connectivity within the brain is of great interest for medical treatments as well as for understanding both normal brain function and neuronal diseases such as multiple sclerosis and schizophrenia.

The interpretation of diffusion imaging data is relatively well understood for volume elements (voxels) in which a single fiber direction prevails. Unfortunately, voxels are much larger than the individual nerve cells that make up the fiber bundles. Thus, in areas where fibers touch or cross, many voxels can contain two or more distinct fiber populations. Our work proposes a novel method to deal with these difficult cases.^{3}

**Figure 1.** Colored streamlines represent likely paths of nerve fiber bundles. This data was extracted from a diffusion imaging data set.

**Figure 2.** For (a) a given fiber configuration, (b) the orientation distribution function (ODF) indicates the approximate crossing geometry. Our goal is to reconstruct (c) discrete bundle directions.

**Figure 3.** When adding peaks of finite width, as is done with linear deconvolution models, maxima may be (a) shifted or (b) masked. Therefore a linear model is not sufficient for deconvoluting the ODF.

Reconstructing directionally-dependent diffusion behavior requires multiple measurements in different directions. A common way to analyze the resulting data set is called ‘spherical deconvolution’.^{4} This approach produces an orientation distribution function (ODF) that is defined on the unit sphere and takes on high values in the assumed fiber directions. Figure 2(a) illustrates a 60^{°} crossing, whose ODF is shown Figure 2(b). Reconstructing fiber pathways involves extracting discrete fiber orientations from such continuous ODFs as shown in Figure 2(c).

Prior work simply assumed that fiber orientations coincide with the maxima of the ODF. However, spherical deconvolution is a linear model, and in a linear superposition of peaks that have a finite width, the original maxima interfere (see Figure 3). Interpreting the data amounts to finding a set of peaks that total the given ODF. Even though this problem does not have a unique solution, we propose that a plausible resolution can be obtained by making reasonable assumptions. In particular, individual fiber peaks should be narrow, nonnegative, and nonoscillating.

To address this inverse problem, we exploit the fact that higher-order tensors can represent spherical functions via their homogeneous forms. When generalizing the notion of matrix rank to tensors, rank-one tensors correspond to spherical functions that exactly fulfill our requirements: they are narrow, nonnegative, and nonoscillating. Thus, we propose to find a set of *k* fiber contributions that add up to a given fiber ODF by approximating its higher-order tensor representation with a rank-*k* tensor. Even though an optimal rank-*k* approximation generally does not exist,^{5} we present an algorithm that iteratively improves an initial estimate until it is usable in practice.

To validate this method, we conducted experiments on noise-polluted synthetic data and quantitatively compared our results to the ground-truth data. The plots in Figure 4 show that compared to simple extraction of ODF maxima (red), our approach (black) analyzed two- and three-fiber crossings in a wider angular range and at a much reduced bias. We also applied our method to a real diffusion MRI data set. In this case, ground-truth data was not available, but we were able to reconstruct fiber pathways that are known from anatomy textbooks (see Figure 5).

Reconstructing nerve fiber pathways from diffusion images requires the inference of discrete directions from orientation distribution functions. In our work, we have clarified that this step needs explicit treatment as an inverse problem. We suggest a mathematical framework and a practical algorithm to address the problem and demonstrate, both on synthetic and on real data, that this provides a clear improvement over the previous state of the art. In the future, we plan to model the uncertainty in the inferred directions and to address the effect of noise on our tensor approximation.

**Figure 4.** Tensor decomposition (black) reconstructs fiber orientations over a wider angular range and with a smaller bias than traditional maximum extraction (red).

**Figure 5.** Unlike (a) maximum extraction, (b) tensor decomposition successfully reconstructs the path of transcallosal fibers (red).

*The described work was a joint project performed with Hans-Peter Seidel. We would like to thank Alfred Anwander for providing the data set and gratefully acknowledge partial funding provided by the Max Planck Center for Visual Computing and Communication.*

Thomas Schultz

Max-Planck-Institut für Informatik

Saarbrücken, Germany

Thomas Schultz studied medical informatics at the University of Heidelberg and Computer Science at Saarland University, Germany, where he received his Diploma (MSc) degree in 2006. Since then, he has been working as a PhD student at the Max-Planck-Institut für Informatik, concentrating on processing and visualization of diffusion imaging data.

References:

1.

D. Le Bihan, E. Breton, D. Lallemand, P. Grenier, E. Cabanis, M. Laval-Jeantet, MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders, *Radiology* 161, no. 2, pp. 401-407, 1986.