Contact: Adrian Jarret (LCAV)
Synopsis: Solving Inverse problem for Sparse-plus-Smooth composite signals
Levels: Bachelor, Master
Sections: IC
Description: When performing experiences, researchers are rarely able to directly measure the data of interest, but rather side effects of the phenomenon, referred to as indirect measurements. The stereotypical examples are tomography (MRI imaging) that uses magnetic fields to detect organs in a body, or radar speed guns that detect reflection of a given radio wave to estimate the velocity of an object. Solving an inverse problem amounts to estimate the source behind such measurements. This may be modelled as a linear measurement equation Ax = y.
A common strategy to do so consists in formulating an optimization problem, the solution of which is an estimate of the looked-after source. Among other advantages, introducing a penalty term in the objective function of the problem allows to introduce some prior on the shape of the solution. A square norm penalty tends to favour smooth solutions with weight relatively spared over the search space, whereas absolute value penalty gives sparse solution with localized information.
In this project, we want to study the case when the solutions lie at the the boundary between these two scenarios, being composed of a sparse component as well as a smooth one. This framework has already been formalized using a composite penalty term and a representer theorem exists to characterize the shape of the solutions [1]. We want to use this theorem as a leverage to implement a numerical solver and study the performances of the model on simulated data in a periodic context.
A potential application of this framework would be celestial objects detection in RA. Indeed, some sources can be approximated as sparse point sources, whereas others are smooth extended sources. The indirect measurements are obtained with radio interferometry, which is a linear process and thus fits within the framework. Solving such a problem would then allow to both reconstruct the map of celestial sources and to distinguish between localized and extended sources.
Deliverables:
- Python code that implements simulation and solver for the composite problem
- Project report that presents the theoretical context and summarizes experience and findings
Prerequisites:
- Basic knowledge in linear algebra (and functional analysis)
- Basic experience in numerical computing with Python (Numpy, Matplotlib)
- Any experience in (convex) optimization is a plus
Type of Work :
- ~60% implementation: Python coding
- ~40% theory: representer theorem, B-splines
(This repartition might be adapted according to the interest of the student.)
Reference:
[1] Debarre, T., Aziznejad, S., and Unser, M., “Continuous-Domain Formulation of Inverse Problems for Composite Sparse-Plus-Smooth Signals”, <i>arXiv e-prints</i>, 2021.