Analysis and Applied Analysis Seminar
Department of Mathematics, Yale University
442 Dunham Lab., 10 Hillhouse Ave., New Haven Ct 06511
Organizers: Matthew Feiszli
(matthew.feiszli@yale.edu), Triet Le (triet.le@yale.edu) and
Peter Jones (jones@math.yale.edu).
Usual Time: Wednesdays 3-4pm
If you are interested in giving a talk in the seminar, please contact
one of the organizers to arrange the date.
Fall 2009:
- Week 1:
- Week 2:
- Week 3:
- Week 4: September 28 - October 2. There will be two talks
this week.
- Matthew Feiszli: Department of Mathematics, Yale university.
- Time: Wednesday 3-4pm in LOM 215
- Title: Multi-scale Metrics on Plane Curves
- Abstract: We will present several families of metrics on plane curves, each
of which
is based on some multi-scale representation and is equivalent to a Sobolev-type norm.
These metrics arise when trying to characterize local regularity of functions and curves.
The underlying techniques are borrowed from harmonic and complex analysis. We will
present theoretical and experimental results.
- Triet Le: Department of
Mathematics, Yale university.
- Time: Thursday 4-5pm in LOM 215.
- Title: Local Scales in
oscillatory patterns and on plane curves.
- Abstract:
In this talk, we study the problem of extracting local
scales of oscillatory patterns in images and on plane curves. In the
first case, Given a multi-scale representation {u(t)} of an image f,
we are
interested in automatically picking out a few choices of t_i(x), which
we call local scales, that better represent the multi-scale structure
of f at x. We will characterize local scales coming from the Gaussian
kernel. In the second case, we propose an approach to extracting
local scales on curves to segment objects with irregular boundaries.
Theory and experimental results will be presented with applications to
image decomposition/denoising.
- Week 5: October 5-9.
- Week 6: October 12-26.
- Speaker: Facundo Memoli (memoli@math.stanford.edu), Department of Mathematics, Stanford University.
- Time: Wednesday 3-4pm in LOM 215.
- Title: A Spectral notion of Gromov-Wasserstein
distances
- Absrtact: We introduce a spectral notion of distance between shapes
(closed
Riemannian manifolds) and study
its theoretical properties. We show that our distance satisfies the
properties of a metric on the class of isometric shapes, which means,
in particular, that two shapes are at 0 distance if and only if they
are isometric. Our construction is similar to the recently proposed
Gromov-Wasserstein distance, but rather than viewing shapes merely as
metric spaces, we define our distance via the comparison of
heat kernels.
This allows us to relate our distance to previously proposed spectral
invariants used for shape comparison, such as the spectrum of the
Laplace-Beltrami operator and statistics of diffusion distances. In
addition, the heat kernel provides a
natural notion of scale, which is useful for multi-scale shape
comparison. We also prove a hierarchy of lower bounds for our
distance, which provide increasing discriminative power at the cost of
increase in computational complexity.
- Week 7: October 19-23
- Week 8: October 26-30
- Week 9: November 2-6.
- Speaker: William Allard (wka@math.duke.edu), Department of Mathematics, Duke University.
- Time: Thursday 3-4pm in LOM 215.
- Title:
Some new results on total variation regularization for image processing
- Absrtact:
Total variation regularization has been used for image denoising for about twenty years now.
These regularizations have other uses as well; in particular, they can be used to detect
differences in scales in data. I have been studying the geometric and regularity properties of
minimizers for the associated variational problems with the goal of better understanding them.
In this talk I will describe some new results in this area. Some of these results describe
minimizers where total variation is defined using an anisotropic norm for the gradient; the
motivation for this work, which is joint with Kevin Vixie, is that some computational schemes
for computing minimizers naturally use a polygonal approximation to the standard Euclidean
metric to define total variation.
- Week 10: November 9-13.
- Speaker: Arthur Szlam (aszlam@courant.nyu.edu), Department of
Computer Science, NYU.
- Time: Wednesday 4:15-5:00pm in AKW 500.
- Title:
Total variation, Cheeger ratio cuts, and graph clustering
- Absrtact:
I will discuss a continuous relaxation of the Cheeger cut problem on a
weighted graph, and show how the relaxation is actually equivalent to the
original problem. Then I will introduce an algorithm which experimentally
is very efficient at approximating the solution to this problem on some
clustering benchmarks. I will also give a heuristic variant of the
algorithm which is faster but often gives just as accurate clustering
results. This is joint work with Xavier Bresson, inspired by recent
papers of Buhler and Hein, and Goldstein and Osher, and by an older
paper of Strang.
- Week 11: November 16-20
- Week 12: November 23-27
- Week 13: November 30 - December 4.
Spring 2009: