(October 13, 2000) Manos Papadakis: Generalized frame multiresolution analysis of abstract Hilbert spaces and the construction of univariate wavelets of L^(R) - The Generalized frame multiresolution analysis of an abstract Hilbert space (GFMRA) is an increasing chain of closed susbpaces of a Hilbert space generated by the action of a cyclic unitary group refered to as the Dilation group, on a particular element of the GFMRA, called Core subspace. The core subspace has a frame generated by the action of another abelian unitary group, refered to as Translation group, on a countable set of vectors in the core subspace. This particular set of vectors is called set of frame multiscaling vectors. We prove that for every GFMRA a set of frame multiwavelet vectors associated with this GFMRA can be constructed. We will present the main ideas of two different constructions giving perhaps different sets of frame multiwavelet vectors. Also, we develop the generalization of the concept of Quadratic Mirror filters and we characterize all the sets of frame multiwavelet vectors associated with given GFMRA. Finaly, we show how ALL orthonormal wavelets of L^2(R) are associated with GFMRAs. Our techniques are based on characterizations of commutants of certain Von Neumann algebras. We correlate these characterizations with the so-called fiberization technique of Ron and Shen to derive our results, which generalize the first foundamental classical results of multiresolution theory developed by Mallat and Meyer.
(October 23) Chris Heil: Gabor expansions in L^p - Gabor expansions in L^p
(October 30) Yura Lyubarskii: "Interpolation in spaces of analytic functions and singular operators" - The purpose of the talk is to survey some recent results on interpolation in spaces of entire function and to show their relation to problems of boundedness of the weighted Hilbert transform as well as to application in signal analysis.
(November 9) George Kostakis: Universal Taylor series - Universal Taylor series (U.T.S) are Taylor series, so that subsequences of their partial sums have nice approximation properties outside the circle of convergence.For universal Taylor series defined on the open unit disk we prove a great type Picard theorem.Universal Taylor series may also be defined on some doubly connected sets, although this is not always the case.We would also like to discuss some methods that have been developed in order to prove the existence or not of U.T.S in some doubly connected sets.Our basic tools in our approach (related to the existence questions) are Baire and Mergelyan's theorems.To prove a great type Picard theorem we use essentially a result about the growth of coefficients of U.T.S
(November 30) Lizhong Peng: WAVELETS ON THE HEISENBERG GROUP - In the theory of wavelets, if one goes beyond the Euclidean space $\bold R^n$, the first, or perhaps the most important case is the Heisenberg group $\bold H^n$. In this talk we will discuss: (1) The admissible (continuous) wavelets on the Heisenberg group. The irreducible decompositions of $L^2$ function spaces and the reproducing kernels (H. Liu and L. Peng). (2) The orthogonal wavelets on the Heisenberg group. In fact we discover a method to construct the discrete wavelets on Heisenberg group from one dimensional wavelets which is also valid for biorthogonal wavelets and multiwavelets (preprint by B. Jawerth and L. Peng).
(December 7) Jean-Pierre Leduc: A group-theoretic approach to analyze spatio-temporal transformations - A theoretical framework will be developed to analyze, estimate and track motion transformations in signals. The approach relies on Lie algebras that describe geometry and motion. The construction proceeds to Lie groups and their representations in the function spaces of signals.This method further derives continuous wavelets and all the related tools necessary for motion analysis. This theoretical framework will be presented with examples taken in the Galilei group of velocity, and in the group of rotational motion. Illustrations will demonstrate the efficiency of this approach for signal processing.
(April 12) Vasily Strela:
Denoising Using a Gaussian Scale Mixture Model in the Wavelet Domain -
We describe a statistical model for images decomposed in an
overcomplete wavelet pyramid. Each subband of
the pyramid is modeled as the pointwise product of two independent
random fields: a Gaussian Markov random field, and a hidden multiplier
with a marginal log-normal prior. The latter thus modulates the local
variance of the coefficients. We assume additive Gaussian noise of
known covariance, and compute a MAP estimate of each multiplier
variable based on observation of a local neighborhood of coefficients.
Then, conditioned on this multiplier, we estimate the subband
coefficients with a local Wiener estimator.
Unlike previous models, we 1) motivate empirically
our choice for the prior on the multiplier; 2) use the full covariance
of signal and noise in the estimation; 3) include adjacent scales in
the conditioning neighborhood.
To our knowledge, the results are the best in the
literature, both visually and in terms of MSE.
We extend our ideas to the case of semiregular meshes
and present the results of our surface denoising algorithm.