Current students

MANDELLI SARA | Cycle: XXXII |

Section:

**Telecommunications**

Tutor:

**MONTI-GUARNIERI ANDREA VIRGILIO**

__Major Research topic__:

**Regularization of Inverse Problems: Improvements in Geophysics and Image Processing**

Advisor:

**TUBARO STEFANO**

*Abstract:*

The project goal is to investigate the use of regularization on inverse problems for applications in image processing and geophysics. In particular, we improve over state of the art by focusing on: the effect of different regularizers; the correct choice of their penalty weights; the selection of good iterative solution strategies.

The ability of solving inverse problems is paramount in a wide variety of fields. Indeed, the core of many applications is the estimation of a set of unknown parameters given a set of observations and the theoretical law that links parameters to observations. However, the vast majority of these problems is typically ill-posed or ill-conditioned. This is either due to lack of data (e.g., broad or sparse sampling not providing enough information for inversion), presence of noise (e.g., corrupted pixels in images), the use of approximated models (e.g., linearization of physical laws), or even other factors. It is therefore essential to include additional prior information in the inversion process to get a reliable estimate of the sought parameters. As inverse problems are typically solved by minimizing properly defined cost functions, a common way to exploit additional priors is the use of regularization strategies. Therefore, the final inversion result does not depend only on the observed data but it is the consequence of a trade off between measurements and prior information. However, selecting and correctly tuning a regularization strategy is far from being an easy task. Indeed, different regularization rules lead to different solutions. The choice of the correct weight is fundamental as selecting the wrong penalty term can lead to strongly unstable and completely incorrect results. Moreover, as regularized problems are typically non-linear, iterative minimization strategies must be applied.

The goal of this project is a deep study of regularization effects on inverse problems, aimed at inferring a taxonomy of rules to accurately select and tune regularizers for different applications. In particular, we tailor our investigation to applications in geophysical data processing and image processing. This choice is motivated by the complementarity of these two fields that allows us to cover a broad variety of cases (e.g., from low to high dimensional data, different sets of constraints and priors, etc.). The first task of the project consists in studying the effect of different regularizers on the estimated parameters given a set of priors. First, we study the effect of combining in a synergistic fashion some of the aforementioned regularizers and a set of novel ones within a unique cost function. Second, we investigate the possibility of exploiting these concepts into geophysical scenarios in order to take into account prior information and constraints given by the physical and geological worlds (e.g., well data, migration results, dip information and geologists’ insights). The next step will consist in more detailed investigations about limits and theoretical properties of these weights. Once a regularization rule and its penalty have been chosen for a specific problem, the issue of minimizing the derived cost function still remains. In the light of this, our final task consists in studying the behavior of commonly used minimization techniques. The right choice of a stopping criterion (e.g., number of iterations, threshold on the achieved error, etc.) is essential.

In addition to the theoretical aspects, it is worth noting also the envisioned practical implications of our research. Concerning geophysics, we can count on the collaboration between ISPG and Eni. This allows us to perform a significant amount of tests on different applications in real-world scenarios, thus further validating the proposed techniques.

The ability of solving inverse problems is paramount in a wide variety of fields. Indeed, the core of many applications is the estimation of a set of unknown parameters given a set of observations and the theoretical law that links parameters to observations. However, the vast majority of these problems is typically ill-posed or ill-conditioned. This is either due to lack of data (e.g., broad or sparse sampling not providing enough information for inversion), presence of noise (e.g., corrupted pixels in images), the use of approximated models (e.g., linearization of physical laws), or even other factors. It is therefore essential to include additional prior information in the inversion process to get a reliable estimate of the sought parameters. As inverse problems are typically solved by minimizing properly defined cost functions, a common way to exploit additional priors is the use of regularization strategies. Therefore, the final inversion result does not depend only on the observed data but it is the consequence of a trade off between measurements and prior information. However, selecting and correctly tuning a regularization strategy is far from being an easy task. Indeed, different regularization rules lead to different solutions. The choice of the correct weight is fundamental as selecting the wrong penalty term can lead to strongly unstable and completely incorrect results. Moreover, as regularized problems are typically non-linear, iterative minimization strategies must be applied.

The goal of this project is a deep study of regularization effects on inverse problems, aimed at inferring a taxonomy of rules to accurately select and tune regularizers for different applications. In particular, we tailor our investigation to applications in geophysical data processing and image processing. This choice is motivated by the complementarity of these two fields that allows us to cover a broad variety of cases (e.g., from low to high dimensional data, different sets of constraints and priors, etc.). The first task of the project consists in studying the effect of different regularizers on the estimated parameters given a set of priors. First, we study the effect of combining in a synergistic fashion some of the aforementioned regularizers and a set of novel ones within a unique cost function. Second, we investigate the possibility of exploiting these concepts into geophysical scenarios in order to take into account prior information and constraints given by the physical and geological worlds (e.g., well data, migration results, dip information and geologists’ insights). The next step will consist in more detailed investigations about limits and theoretical properties of these weights. Once a regularization rule and its penalty have been chosen for a specific problem, the issue of minimizing the derived cost function still remains. In the light of this, our final task consists in studying the behavior of commonly used minimization techniques. The right choice of a stopping criterion (e.g., number of iterations, threshold on the achieved error, etc.) is essential.

In addition to the theoretical aspects, it is worth noting also the envisioned practical implications of our research. Concerning geophysics, we can count on the collaboration between ISPG and Eni. This allows us to perform a significant amount of tests on different applications in real-world scenarios, thus further validating the proposed techniques.