1240 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 31, NO. 6, JUNE 2012 About the Geometry of Asymmetric Fiber Orientation Distributions Marco Reisert*, Elias Kellner, and Valerij G. Kiselev Abstract—Fiber orientation distributions (FODs) based on diffusion-sensitized magnetic resonance imaging are usually sym- metric, primarily due to the nature of the diffusion. In contrast, the underlying fiber configurations are not, as bending or fanning con- figurations are inherently asymmetric. We propose to dismiss the symmetry of the FOD to additionally encode the asymmetry of the underlying fiber configuration. This is of particular importance for low resolution images that are common in diffusion weighted imaging. We set up the mathematical foundations and geometric interpretations of asymmetric FODs and show how one can benefit from these considerations. We infer a continuity condition that is used as a prior during FOD estimation by constrained spherical deconvolution. This new prior shows superior performance in comparison to other spatial regularization strategies in reliability and accuracy. Index Terms—Curvature, diffusion tensor imaging (DTI), fiber orientation distribution, high angular resolution diffusion imaging (HARDI), left-invariant diffusion, magnetic resonance imaging (MRI), partial differential equation (PDE), spatial regularization, spherical deconvolution, tractography. I. INTRODUCTION M AGNETIC resonance imaging (MRI) has the potential to visualize noninvasively the fibrous structure of human brain white matter [1]. White matter fibers connect the data-processing parts, and therefore, subject-specific represen- tations of these connections are of great interest in fundamental neuroscience and medicine. Attempts to realize this potential gave rise to a thriving research field termed tractography or fiber tracking. Accurate and reliable processing and estimation of fiber ori- entation distribution (FODs) is very important, particularly for deterministic tractography algorithms [2], [3], where decisions during tracking are typically based on the local maxima of the FOD. Moreover, for voxel-wise statistics [4] and diffusion based integrity measures [5], a reliable preprocessing of the data is also a major requirement. There are numerous methods for estimating symmetric ori- entation distributions: classical Q-ball imaging [6], constrained spherical deconvolution [7], proper probability density estima- tion [8]–[11], and spatially regularized density estimations of Manuscript received November 16, 2011; revised January 24, 2012; accepted February 07, 2012. Date of publication February 15, 2012; date of current ver- sion May 29, 2012. M. Reisert is indebted to the Baden-Württemberg Stiftung for the support of this research project by the Eliteprogramme for Postdocs. The work of E. Kellner was supported by Deutsche Forschungsgemeinschaft (DFG) under Grant KI 1089/3-1. Asterisk indicates corresponding author. *M. Reisert is with the Department of Radiology, University Medical Center Freiburg, 79106 Freiburg, Germany. E. Kellner and V. G. Kiselev are with the Department of Radiology, Univer- sity Medical Center Freiburg, 79106 Freiburg, Germany. Digital Object Identifier 10.1109/TMI.2012.2187916 FODs and tensor-valued images [12]–[17]. However, literature extracting asymmetric fiber orientation distributions (AFOD) is limited. Previous work in [18]–[20] implemented asymmetric smoothing kernels to introduce the asymmetry. The key idea to obtain AFODs is to consider the local surrounding of a voxel and use intervoxel information. Asymmetric FODs are of par- ticular importance when the spatial resolution of the measure- ment is low, that is, when the curvature of the underlying fiber bundles is in the range of the voxel size. In this work, we an- alyze these relations in detail and derive from these considera- tions an asymmetric fiber continuity condition that can be used as a prior term in FOD estimation. This differs from previous work in [18]–[20], where the AFODs are obtained by a post- processing step. The proposed condition will be of differential nature, hence our solution is based on the numerical solution of partial differential equations (PDE). This is similar to the contour enhancement kernel [20] and the fiber continuity as- sumption [13]. Both can be motivated from a Tikhonov-regu- larized optimization problem. Both do not infer any asymmetric information if the underlying data is symmetric. The so-called contour completion kernel [20] is able to produce asymmetric FODs, but is neither hermitian nor positive definite, and hence, cannot be inferred from an optimization problem. In contrast, this work will provide a regularization term that is able to infer AFODs. The regularization term will be deduced from a geo- metric framework that gives a rigorous mathematical interpreta- tion of the corresponding AFODs, defining them as probability measures on the space of fiber configurations. It relates fiber curvature and the resolution of the measurement and gives an interpretation of the angle under which a curved fiber appears in the FOD. From an application oriented point of view, our approach aims for FOD estimation on the basis of clinical diffu- sion-weighted MR-imaging protocols. Usually, clinical pro- tocols only allow for diffusion tensor estimation and do not support the estimation of more complex orientation distri- butions. Our goal is to use the prior knowledge during FOD estimation to resolve even complex fiber configurations on the basis of ‘low-quality’ clinical measurements. Neverthless, one must be cautious, as a small amount of over-regularization may introduce a bias on the estimate ([13], [20]). The prior term in- troduced here is less restrictive than those proposed previously, which manifests in the allowance for an anti-symmetric part. In fact, we will illustrate that the linearization bias found in [13], [20] disappears. II. METHOD We denote FODs by functions , where denotes the spatial coordinate and the orientation 0278-0062/$31.00 © 2012 IEEE