Seminar Details
2018-11-27 (14:00) : An introduction to sub-Riemannian geometry in SE(2) for vessel analysis and roto-translation equivariant deep learning
At Euler building (room A.002)
Organized by Mathematical Engineering
Speaker :
B. Erik Ydstie (Carnegie Mellon University
)
Abstract :
This presentation focusses on the application of sub-Riemannian geometry and group theory in 2D medical image analysis. In the discussed image analysis applications, 2D images are lifted to 3D functions on the coupled space of positions and orientations. Lifting is done via a (learned) wavelet-type transformation, leading to a neat organization of image data based on position and orientation. In the extended domain, which we identify with the Lie group SE(2), a sub-Riemannian (SR) geometry is recognized. Here, "sub" refers to the restriction that tangent vectors of naturally lifted curves are contained in a sub-space of the full tangent space. A parallel can be drawn between these curves and the paths of moving cars: a car can only move forward and change direction (2 controls) in a 3D state-space (2D position and orientation), but is not able to move sideways.
In this presentation the following items are discussed:
- A introduction to the Lie group SE(2) (group product, algebra, exponential/logarithmic map and representations)
- An explanation of the sub-Riemannian geometry on SE(2)
- The benefit of using sub-Riemannian geometry for vessel tracking
- The benefit of using the Lie group structure of SE(2) for equivariant or invariant deep learning