Contemporary Mathematics

Volume 247, 1999

Reconstruction of vector and tensor fields from sampled

discrete data

Akram Aldroubi and Peter Basser

ABSTRACT. We construct atomic spaces S C L2(JRn, Tl) that are appropriate

for the representation and processing of discrete tensor field data. We give

conditions for these spaces to be well defined, atomic subspaces of the Wiener

amalgam space W(C, L2(JRn, Tf )) which is locally continuous and globally £2.

We show that the sampling or discretization operator R from S to

b(Zn,

Tf)

is a bounded linear operator. We introduce the dilated spaces

S~:;.

=

D~:;.

S

parametrized by the coarseness

~.

and show that the discretization operator

is also bounded with a bounded inverse for any

~

E

zn.

This allows us to

represent discrete tensor field data in terms of continuous tensor fields in S

~:;.,

and to obtain continous representations with fast filtering algorithms.

1.

Introduction

Modern imaging systems, (e.g., Magnetic resonance image scanner) acquire

discrete sets of data and store them as arrays of numbers. In many new imaging

modalities, the acquired images are no longer a set of scalar values representing the

gray levels voxels (spatial positions on some three dimensional lattice). Instead,

the images are vector or tensor-valued functions. Prototypical example (and the

motivation behind this mathematical development) is Diffusion Tensor Magnetic

Resonance Imaging (DT-MRI) which provides a measurement ofthe effective diffu-

sion tensor of water in each voxel of an image volume (see Figure 1). These tensor

images can be used to elucidate the three dimensional fiber architectural features

of anisotropic fibrous tissues and organs in vivo, and provide microstructural infor-

mation noninvasively and nondestructively in materials sciences applications [4, 3].

However, the measured tensor in each voxel is inherently a noisy, discrete, and

volume-averaged quantity. Thus, one goal of this work is to develop mathematical

methods to ameliorate these problems. More generally, we are interested in devel-

oping a general mathematical framework that enables us to analyze, process and

compress these data sets. We show we can do this by constructing a smooth, con-

tinuous representation of the diffusion tensor field. Moreover, the algorithms that

1991 Mathematics Subject Classification. Primary 42A38, 42C15, 47837.

The first author was supported in part by NSF Grant# DMS-9805483.

©

1999 American Mathematical Society

1

http://dx.doi.org/10.1090/conm/247/03794