INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS Inverse Problems 19 (2003) 1031–1046 PII: S0266-5611(03)56256-6 Persistent angular structure: new insights from diffusion magnetic resonance imaging data Kalvis M Jansons 1 and Daniel C Alexander 2 1 Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK 2 Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK E-mail: PASMRI@Kalvis.com Received 14 November 2002, in final form 13 June 2003 Published 22 August 2003 Online at stacks.iop.org/IP/19/1031 Abstract We determine a statistic called the (radially) persistent angular structure (PAS) from samples of the Fourier transform of a three-dimensional function. The method has applications in diffusion magnetic resonance imaging (MRI), which samples the Fourier transform of the probability density function of particle displacements. The PAS is then a representation of the relative mobility of particles in each direction. In PAS-MRI, we computethe PAS in each voxel of an image. This technique has biomedical applications, where it reveals the orientations of microstructural fibres, such as white-matter fibres in the brain. Scanner time is a significant factor in determining the amount of data available in clinical brain scans. Here, we use measurements acquired for diffusion-tensor MRI, which is a routine diffusion imaging technique, but extract richer information. In particular, PAS-MRI can resolve the orientations of crossing fibres. We test PAS-MRI on human brain data and on synthetic data. The human brain data set comes from a standard acquisition scheme for diffusion-tensor MRI in which the samples in each voxel lie on a sphere in Fourier space. 1. Introduction Diffusion magnetic resonance imaging (MRI) measures the displacements of particles that are subject to Brownian motion within a sample of material. The microstructure of the material determines the mobility of these particles, which is normally directionally dependent. The anisotropy of particle displacements gives information about the anisotropy, on a microscopic scale, of the material in which the particles move. The probability density function p of particle displacements (in three-dimensional space) reflects the anisotropy of particle displacements. 0266-5611/03/051031+16$30.00 © 2003 IOP Publishing Ltd Printed in the UK 1031