| 11-11-2022 |
Deep learning para reducción de artefactos en tomografía dental
En la tomografía dental los implantes metálicos, como tapaduras, producen artefactos visuales que limitan la utilidad de estas imágenes en la práctica. La teoría matemática de la tomografía permite proponer un método para reducir estos artefactos. Sin embargo, la implementación de este método en la práctica es complejo. Este proyecto tiene por objetivo evaluar el uso de herramientas de aprendizaje profundo para implementar este método, o bien para desarrollar uno con un mejor desempeño a partir de datos sintéticos. Trabajo conjunto entre Prof. Carlos Sing Long (IMC) y Prof. Benjamin Palacios (MAT).
Prerequisitos:
no tiene.
Tiene un método de evaluación Nota 1-7, con 10 créditos y tiene 1/2 vacantes disponibles |
Mentor(es): Ver en la plataforma |
| 26-12-2022 |
Prerequisites:
IMT2113
Evaluation method: Nota 1-7, with 0/1 available vacants |
Mentor(s): Open in the plataform |
| 11-11-2022 |
Prerequisites:
None.
Evaluation method: Nota 1-7, with 1/2 available vacants |
Mentor(s): Open in the plataform |
| 17-06-2022 |
Inverse problems and RKHS
There is a class of inverse problems with the property of being separable, that is, they are linear in some variables and non-linear in others. A problem in this class can be reformulated as a linear problem in a space of measures on a suitable set. Recently, Bernstein et al analyzed the solvability of this equivalent formulation in terms of a kernel defined on the same set. The objective of this project is to study how the Reproducing Kernel Hilbert Space (RKHS) associated to this kernel yields insight into the properties of the inverse problem. In particular, whether there exist closed subspaces on which the inverse problem is well-posed.
Keywords:
análisis matemático
Problemas Inversos
Prerequisites:
None.
Evaluation method: Nota 1-7, with 0/1 available vacants |
Mentor(s): Open in the plataform |
| 24-01-2022 |
Atomic norm regularization with generic atoms
Atomic norm regularization consists in considering an atomic set that forms the building blocks for a class of objects of interest, to then use the Minkowski functional associated to its convex hull as a regularizer. A problem of the convex hull is masking: any atom in the interior of the hull will never be selected when reconstructing an object. In practice, this is avoided by normalizing the atoms. However, this may destroy the structure of the atomic set. In this iPre, we will study strategies to avoid masking, and we will propose a reconstruction method where each atom has a chance of being selected.
Prerequisites:
IMT2113
Evaluation method: Nota 1-7, with 0/1 available vacants |
Mentor(s): Open in the plataform |
| 29-10-2021 |
Multiresolution analysis and superresolution
Multiresolution analysis consists in constructing a filtration of L2 of closed subspaces Vj such that each one represents functions at a scale 2j. The ortogonal projection onto Vj represents the approximation at scale 2j whereas the difference between the projections onto Vj and Vj+1 represents the details at scale 2j. A typical signal distortion process consists in removing structure at small scales. This is modeled through convolutions and resampling. Is it possible to leverage multiresolution analysis to recover the missing details? In this case we do not want to solve the problem for any function, thus constraining the worst-case, but only for those that are of interest and have been distorted by the process under study. The goal of this iPre is to review the existing literature connecting multiresolution analysis with this problem, and to propose a mathematical model that would allow us to answer this question.
Keywords:
análisis de fourier
superresolución
Prerequisites:
IMT2113
Evaluation method: Nota 1-7, with 0/1 available vacants |
Mentor(s): Open in the plataform |
| 08-04-2021 |
Mathematical methods for the deconvolution problem
One of the main properties of an optical system is its resolution. This is defined as the minimum separation between two ideal point sources so that they can be distinguished from one another when observed through the system. In practice, the diffraction of light imposes a physical limit to the resolution of the system. For a linear system, this process is typically modeled by a convolution by the Point Spread Function (PSF). For this reason, a technique that improves the resolution of the system can be interpreted as a deconvolution method. The objective of this project is to study mathematical methods proposed in the literature in the past decade, which combine applied Fourier analysis, convex optimization, and probability, for which there exists conditions that ensure they solve the superresolution problem in a computationally efficient manner.
Prerequisites:
IMT2113
Evaluation method: Nota 1-7, with 0/1 available vacants |
Mentor(s): Open in the plataform |
| 11-07-2019 |
Evolution of NAFLD from chromatography and MR spectroscopy data
One of the most common liver diseases is Non-Alcoholic Fatty Liver Disease (NAFLD). It is believed to be a manifestation of metabolic syndrome, and that the progression of the disease towards steatohepatitis (NASH) and cirrhosis manifests itself in changes in concentrations of fatty acids in the liver. This project has two objectives. First, to determine the empirical relation between chromatography and Magnetic Resonance (MR) spectroscopy data acquired from fatty acids. Second, to determine the geometric structure of the data by estimating the manifold along which they lie.
Prerequisites:
None.
Evaluation method: Nota 1-7, with 0/1 available vacants |
Mentor(s): Open in the plataform |
| 07-12-2018 |
Deconvolution and optimal transport
One of the main properties of an optical system is its resolution. This is defined as the minimum separation between two ideal point sources so that they can be distinguished from one another when observed through the system. In practice, the diffraction of light imposes a physical limit to the resolution of the system. For a linear system, this process is typically modeled by a convolution by the Point Spread Function (PSF). For this reason, a technique that improves the resolution of the system can be interpreted as a deconvolution method. The objective of this project is to study the connection between deconvolution methods and optimal transport, and how the performance of deconvolution methods based on optimal transport compare to the state of the art. Evaluation method: Nota 1-7, with 0/1 available vacants |
Mentor(s): Open in the plataform |