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
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