SIJIMat Seminar

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Seminar
December 01, 2022

Place: Aula petita CRM.

Time: 12h

Algebraic Geometry and line reconstruction

in Computer Vision

ABSTRACT

The 3D image reconstruction problem aims to create a 3D model of a scene or object starting from 2D images. This process is done in four stages: Feature identification in the images, point and line matching, camera estimation and triangulation, and construction of the 3D model. Stages two and three of the process deal mainly with geometric information that can be studied algebraically. This is precisely where Algebraic Geometry comes into play. In this talk I will introduce the 3D image reconstruction problem and the algebraic tools that allow to model the cameras and image features, I will also define the point and line Multiview varieties, and finally mention some algebraic results on the geometry of points and lines that can be reconstructed effectively.

SPEAKER

Angélica Torres

Angélica Torres

Postdoctoral Researcher at CRM

About me:

I am a postdoctoral researcher in the Mathematical Biology group working on Phylogenetics under the supervision of Marta Casanellas.

Previously I was a postdoctoral researcher at the School of Engineering Sciences of KTH Royal Institute of Technology in Stockholm where I worked with Kathlén Kohn. I finished my PhD at the University of Copenhagen in the MBIO group lead by Elisenda Feliu.

My research lies in the field of Applied Algebraic Geometry, and I am interested in the use of algebraic and geometric tools to understand problems in other areas. I have projects in Chemical Reaction Networks, Computer vision, and now Phylogenetics!

past sessions

18/11/22 | Mar Giralt-Miron (UPC) | Chaotic dynamics, exponentially small phenomena and Celestial Mechanics

ABSTRACT

A fundamental problem in dynamical systems is to prove that a given model has chaotic dynamics. One of the methods employed to prove this type of motions is to verify the existence of transversal intersections between the stable and unstable manifolds of certain objects. Then, there exists a theorem (the Smale-Birkhoff homoclinic theorem) which ensures the existence of chaotic motions.

In this talk we present a method to analyze the distance and transversality between certain stable and unstable manifolds when a small perturbation is added to an integrable system. In particular, we consider the case where the distance between manifolds is exponentially small. This implies that this difference cannot be detected by expanding the manifolds into a series with respect to the small perturbation parameter. Therefore, classical perturbation theory cannot be applied.

Finally, we apply these techniques to a celestial mechanics problem. In particular, we study the Lagrange point L 3 in the restricted planar circular 3-body problem.