SCIENTIFIC PROGRAMS AND ACTIVITIES

Upcoming Seminars

June 4-6
10:00 a.m.-12:00 p.m. and
2:00 p.m.-3:00 p.m.
Fields Institute, Room 230

Ugo Boscain, CNRS and Ecole Polytechnique
Minicourse on Sub-Riemannian Geometry and Reconstruction of Images

Plan of the Lectures:

Part 1
- Definition of Sub-Riemannian manifold.
- Minimizers, Pontryagin Maximum Principle.
- The intrinsic volume and the intrinsic Laplacian.
- Small time heat kernel asymptotics.

Part 2
- A model for the visual cortex due to Petitot-Citti and Sarti.
- Geodesics on the bundle of directions of the plane.
- Inpaiting via the sub-elliptic diffusion equation.

March 25
1:10 p.m.
Fields Institute, Room 210

Gunay Dogan, National Institute of Standards and Technology (NIST)
Variational Algorithms for Fast and Robust Shape Reconstruction in Imaging

In many image processing problems, the end goal or an essential ingredient is the detection and reconstruction of a shape or a set of shapes (curves in 2d or surfaces in 3d) from the given data. Examples of such problems are image segmentation, surface regularization, geometric interpolation of data point clouds and multiview stereo reconstruction. Often these tasks can be expressed as energy minimization problems in which the free variable is the sought shape. In this work, we adopt a continuous shape optimization approach to such problems and develop a Lagrangian minimization framework that can handle a diverse collection of such shape energies. We discretize the continuous formulation with the finite element method. By formulating well-behaved gradient descent velocities for shape evolutions and by employing spatial adaptivity, we obtain an efficient and robust shape reconstruction algorithm. We demonstrate the effectiveness of our method with several examples from image processing.

March 18
1:10 p.m.
Fields Institute, Stewart Library

Peter Gibson, York University
Stable Polynomials and the Identification of Minimum-phase Preserving Operators

It is claimed by seismologists that causal signals in the form of elastic waves transmitted through layers of rock end up being minimum phase, meaning that their energy is maximally concentrated at the front. We analyze the mathematical implications of this claim. It turns out that the underlying operators are limited to a precisely defined class which can be identified using specific test signals. The key insight comes from the theory of stable polynomials and what are known as Polya-Schur problems

Monday, Mar. 11
1:10 p.m.
Fields Institute, Room 210

Boris Khesin, University of Toronto
A Nonholonomic Moser Theorem and Diffeomorphism Groups

We discuss the following nonholonomic version of the classical Moser theorem: given a bracket-generating distribution on a connected compact manifold (possibly with boundary), two volume forms of equal total volume can be isotoped by the flow of a vector field tangent to this distribution. The subriemannian heat equation turns out to be a gradient flow on the "nonholonomic" Wasserstein space with the potential given by the Boltzmann relative entropy functional. This is joint work with Paul Lee.

Jeff Calder, University of Michigan
The circular area signature for graphs of periodic functions

The representation of curves by integral invariants is an important step in shape recognition and classification in the computer vision community. Integral invariants are preferred over their differential counterparts due to their robustness with respect to noise. However, in contrast to differential invariants, it is currently unknown whether integral invariants offer unique representations of curves. An important example is the circular area signature, which is of particular interest due to its relation to curvature. The circular area signature is computed by placing a ball of fixed radius at each point on the curve and measuring the area of the portion of the ball that lies inside the region bounded by the curve. In this talk, I will sketch the proof of a new uniqueness result for the circular area signature for graphs of periodic functions, and discuss the difficulties with obtaining a similar result for simple closed curves.

Allen Tannenbaum, Comprehensive Cancer Center/ECE, UAB
Monge-Kantorovich for Problems in Signal Processing and Control

Monge-Kantorovich (also known as Optimal Mass Transport) is an important problem with applications for numerous disciplines including econometrics, fluid dynamics, automatic control, transportation, statistical physics, shape optimization, expert systems, and meteorology. In this talk, we will discuss certain computational approaches to this problem and their applications in signal and image processing. Further via gradient flows defined by the Wasserstein 2-metric from optimal transport theory, we will relate some key concepts in information theory with mass transport that impacts problems in control, statistical estimation, and image analysis. A goal of this program is to understand the apparent relationship between mass transport, conservation laws, entropy functionals, on one hand and probability and power distributions and related metrics on the other.

Abdol-Reza Mansouri, Queen's University
Short-time behaviour of hypoelliptic heat kernels on Lie groups

In recent work, Boscain et al. have shown how sub-Riemannian ideas can be used in imaging applications. In particular, a hypoelliptic heat equation on SE(2) and its corresponding heat kernel lead to an image diffusion algorithm which implements image inpainting. Heat kernels are of independent interest as well, since they encode geometric information. In particular, the sub-Riemannian distance can be recovered from the short-time behaviour of the heat kernel, just as in Riemannian geometry. In this talk, we will describe recent work on computing the short-time behaviour of hypoelliptic heat kernels on unimodular Lie groups, and we will illustrate it for the case of the Cartan, and Engel sub-riemannian geometries.

Xiteng Liu, Toronto
System Compression: A New Computational Phenomenon
(Abstract)

Stephen Marsland, Massey University, New Zealand
Shape Metrics, Correspondence, and the Role of the Connection

Metrics on shape space are used to describe deformations that take one shape to another, and to determine a distance between them. I will present a family of elastic metrics and demonstrate its use on shape space. Following this, I will argue that such metrics are not always useful, and that the data should inform the choice of connection, and hence metric, and I will present one possible way to do this.

Thursday, Oct. 4
2:10
Math Dept,. Bahen Centre BA 6290

Klas Modin, University of Toronto
On the geometric approach to image diffusion

I will discuss two geometric approaches to image diffusion, and their relation to evolution equations on spaces of diffeomorphisms. The first approach is "Beltrami diffusion", where images are represented as 2-dimensional submanifolds, called image manifolds, embedded in a higher dimensional Euclidean space. The Laplace-Beltrami operator, obtained by pulling back the Euclidean metric, is then used to smoothen (or sharpen) the image manifold. The second approach is "hypoelliptic diffusion", in which images are transformed by subelliptic diffusion equations on the projective tangent bundle of the plane.