Title:  Options for Shape and Multi-parameter Inverse Problems

Speaker:  Eric Miller

Affiliation:  ECE Dept., Tufts University

Abstract:

The desire to extract information concerning the internal structure of
a medium based upon data collected only at the periphery is a problem
encountered across a broad range of technical fields and application
areas ranging from medical imaging and nondestructive evaluation to
geophysical exploration and environmental remediation. The goal of
this talk is to discuss a number of modeling and estimation options
associated with two subsets of such inverse problems that have been
receiving increased attention in recent years.  

First, we present ideas concerning the recovery of shape information
in an inverse problems context.  Rather than seeking to use limited
and noisy data to recover the values of a large number of pixels in a
region of interest, the goal of a shape-based approach to imaging is
the estimation of the geometric structure of anomalous objects
(tumors, cracks, regions of functional brain activity, ...)  located
in a perhaps uncertain background.  While level set methods have
garnered significant attention in recent years for their ability to
identify even an unknown number of objects, these techniques can be
slow to converge and are far from trivial to implement.  In the first
portion of this talk, we introduce a new approach to modeling shape
that possesses much of the topological flexibility of level sets but
with fewer of the drawbacks.

In the second portion of the talk, we focus on the issues of
recovering multiple parameters of the medium from perhaps multiple
sets of data.  In the geophysical context, for example, one may wish
to characterize the structure of reservoir from electromagnetic as
well as seismic data in which case one seeks information concerning
the electromagnetic and acoustic properties of the medium.  In the
case of breast imaging for cancer screening, there is wide interest in
fusing information from multiple sensors (x-ray, electrical impedance,
diffusion optical, etc) to develop a coherent picture of the state of
the breast.  In these and other cases, the goal of reconstructing
multiple parameters tends to exacerbate the ill-posedness of the
associated inverse problem.  Here then we discuss a couple of options
for regularizing such problems.  The first building on the parametric
level set ideas to achieve regularization via geometry.  The second
approach is appropriate for pixel-based inversions and is based on the
development of regularizing functionals that in some sense enforce
"similarity" across the multiple parameters.