IV.d AAG: Applied algebra and geometry

Head of the group
Name: Tomas Pajdla (Curriculum Vitae)
Phone: +(420) 604 236 022
Email: pajdla(at)cvut.cz

AAG – Applied Algebra and Geometry group works on finding and understanding theoretical elements that make direct impact in solving practical tasks in computer vision, robotics and machine learning.

We study elements of

  1. Geometry of Euclidean and projective spaces
  2. Computational algebraic geometry
  3. Polynomial optimization methods

to apply them in

  1. 3D Reconstruction from photographs and photogrammetry
  2. Camera & robot calibration
  3. 3D object description and recognition
  4. Visual scene understanding and search

We are offering consultations and applied research and development to industrial partners who are looking for new innovative solutions in 2D and 3D measurement, image data analysis, visual inspection, quality control, robotics, photogrammetry, computer vision and machine learning. Please contact Tomas Pajdla (pajdla@ciirc.cvut.cz) to find more about our expertise and results.

We are looking for postdocs and collaborators. Please contact Tomas Pajdla (pajdla@ciirc.cvut.cz) to find more about topics below as well as about other opportunities.

3D Reconstruction from photographs in computer vision, graphics and robotics
3D Reconstruction from photographs is an important problem in computer vision. It has a broad range of applications in computer vision, graphics and robotics, including 3D terrain modelling from aerial photographs, object modelling for cultural heritage preservation, autonomous robot localization and navigation as virtualized reality modelling in movie industry. We investigate geometry and algebra of 3D reconstruction with the emphasis on building robust algorithms based on specialized algebraic computation and polynomial optimization as well as implementing practical technology for carrying out large scale experiments and demonstrations.

Image Based Localization in Large Environments
Image based localization is an important tool for visual navigation using large image collections. It has applications in security, forensics, mapping and technology for mobile devices. It is based on elements of computer vision, robust estimation and machine learning. We will investigate principal elements of visual localization such as feature extraction from images, robust image matching using graph models combined with loose geometrical constraints. Or goal is to formulate rigorous optimization and computation problems and solve them with the current tools of optimization and computational algebraic geometry.

Algebraic Geometry for Bioinformatics
Recent developments in algorithmic algebraic geometry, and in particular in solving systems of polynomial equations, provide new tools for life sciences and bioinformatics. Computational algebraic geometry has been used to determine structures of molecules in biological chemistry or to model gene regulatory networks. We are primarily interested in exploiting algebraic geometry within algebraic statistics application to computational biology. Classical approaches to model estimation in algebraic statistics rely on probabilistic and statistical models, e.g. Bayesian networks, Hidden Markov Models, and tree models. The common problem with these models is that their estimation often fails if data is contaminated by large number of gross errors that can’t be modelled. We propose to borrow the inspiration from recent success of model estimation in computer vision and robotics that builds on randomised data-driven sampling of algebraic candidate models. A large number of candidate models can be generated by drawing minimal (small) number of data points, fast generation of full model descriptions, and efficient calculation of support for the models descriptions by other data.  We plan to investigate this approach in model fitting problems in computational bioinformatics that have to deal with data contaminated by gross errors.

Responsible: Tomas Pajdla