Abstract :
Inverse problems related to physical processes are of great importance
in practically every field related to signal processing, such as
tomography, acoustics, wireless communications, medical and radar
imaging, to name only a few. At the same time, many of these problems
are quite challenging due to their ill-posed nature.
On the other hand, signals originating from physical phenomena are often
governed by laws expressible through linear Partial Differential
Equations (PDE), or equivalently, integral equations and the associated
Green’s functions. In addition, these phenomena are usually induced by
sparse singularities, appearing as sources or sinks of a vector field.
In this thesis we primarily investigate the coupling of such physical
laws with a prior assumption on the sparse origin of a physical process.
This gives rise to a “dual” regularization concept, formulated either
as sparse analysis (cosparse), yielded by a PDE representation, or
equivalent sparse synthesis regularization, if the Green’s functions are
used instead. We devote a significant part of the thesis to the
comparison of these two approaches. We argue that, despite nominal
equivalence, their computational properties are very different. Indeed,
due to the inherited sparsity of the discretized PDE (embodied in the
analysis operator), the analysis approach scales much more favorably
than the equivalent problem regularized by the synthesis approach.
Our findings are demonstrated on two applications: acoustic source
localization and epileptic source localization in
electroencephalography. In both cases, we verify that cosparse approach
exhibits superior scalability, even allowing for full (time domain)
wavefield extrapolation in three spatial dimensions. Moreover, in the
acoustic setting, the analysis-based optimization benefits from the
increased amount of observation data, resulting in a speedup in
processing time that is orders of magnitude faster than the synthesis
approach. Numerical simulations show that the developed methods in both
applications are competitive to state-of-the-art localization algorithms
in their corresponding areas. Finally, we present two sparse analysis
methods for blind estimation of the speed of sound and acoustic
impedance, simultaneously with wavefield extrapolation. This is an
important step toward practical implementation, where most physical
parameters are unknown beforehand. The versatility of the approach is
demonstrated on the “hearing behind walls” scenario, in which the
traditional localization methods necessarily fail.
Additionally, by means of a novel algorithmic framework, we challenge
the audio declipping problem regularized by sparsity or cosparsity. Our
method is highly competitive against state-of-the-art, and, in the
cosparse setting, allows for an efficient (even real-time)
implementation.
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