Michael Unser is Professor and Director of EPFL's Biomedical Imaging Group.
His
main research area is biomedical image processing. He has a strong interest in
sampling theories, multiresolution algorithms, wavelets, and the use of splines
for image processing. He is the author of over 150 published journal papers in
these areas, and is one of ISI’s Highly Cited authors in Engineering.

The purpose of this talk is to present a modern, unifying perspective of
sampling, while demonstrating that the research in this area is still alive and
well. We concentrate on the traditional setup where the samples are taken on a
uniform grid, but we explicitly take into account the non-ideal nature of the
acquisition device and the fact that the measurements may be corrupted by
noise.
We argue in favor of a variational formulation where the optimal signal
reconstruction is specified via a functional optimization problem. The cost to
minimize is the sum of a discrete data term and a regularization functional
that
penalizes non-desirable solutions. We show that, when the regularization is
quadratic, the optimal signal reconstruction (among all possible functions) is
a
generalized spline whose type is tied to the regularization operator. This
leads
to an optimal discretization and an efficient signal reconstruction in terms of
generalized B-spline basis functions. A possible variation is to penalize the
L1-norm of the derivative of the function (total variation), which can also be
achieved within the spline framework via a suitable knot deletion process.
The theory of compressed sensing provides an alternative approach to sampling
that is qualitatively similar to total-variation regularization. Here the idea
to favor solutions that have a sparse representation in a wavelet basis.
Practically, this is achieved by imposing a regularization constraint on the
l1-norm of the wavelet coefficients. We show that the corresponding inverse
problem can be solved efficiently via a multi-scale variant of the ISTA
algorithm (iterative skrinkage-thresholding). We illustrate the method with two
concrete imaging examples: the deconvolution of 3-D fluorescence micrographs,
and the reconstruction of magnetic resonance images from arbitrary
(non-uniform)
k-space trajectories.