Analysis with Wavelets, Signals and Geometry

April 4-6, 2001
DIMACS Center, Rutgers University, Piscataway, NJ

Organizers:

Richard Gundy, Rutgers University, gundy@rci.rutgers.edu

Ingrid Daubechies, Princeton University, ingrid@math.princeton.edu

Wil Sweldens, Bell Labs, wim@lucent.com

Guido Weiss, Washington University, guido@math.wustl.edu

Abstracts:

1.

Besov, Bayes and Plato in Multiscale Image Modeling

Richard G. Baraniuk, Rice University

There currently exist two distinct paradigms for modeling
images.  In the first, images are regarded as functions
from a deterministric function space, such as a Besov
smoothness class.  In the second, images are treated
statistically as realizations of a random process.
This talk with review and indicate the links between
the leading deterministic and statistical image models,
with an emphasis on multiresolution techniques and
wavelets.  To overcome the major shortcomings of the
Besov smoothness characterization, we will develop
new statistical models based on mixtures on graphs.
To illustrate, we will discuss applications in image
estimation and segmentation.



2.

Toward Sparse Multiscale Geometric Representation of Images

Albert Cohen, Universite Pierre et Marie Curie

In the first part of this talk, I shall review several results
concerning the sparsity of image representations by wavelet basis,
and their impact on practical applications such as denoising
and encoding.  After pointing out the inherent limitations of 
these representations in terms of coping with the geometry of
edges, I shall present an approach - based on numerical techniques
introduced in the context of shock computations - which aim to 
combine the simple multiscale structure of wavelet algorithms
with a proper treatment of geometry.



3.

Handling Irregular 3D Data in Graphics

Mathieu Desbrun, University of Southern California

When fast, yet robust techniques in modeling or animation
are needed, adaptive sampling is highly desired to reduce
the number of data to proceed while bounding the numerical
errors introduced.  In the case of regularly sampled data,
a lot of tools (from FFT to finite differences) exists,
enabling us to edit or animate virtual objects.  But when
those objects have mesh with arbitrary connectivity, there
is a lack of mathematical foundations to deal with them.  
A long term goal in both animation and modeling is thus to 
be able to manipulate any irregular data with predictive
power.  

In this talk, I will present some recent results
on smoothing of arbitrary data  (3D geometry or tensor
fields), as well as on lossless compression of arbitrary
meshes using single-rate or progressive encoders.



4.

Vladimir Dobric, Lehigh University

An Application of Wavelet Series to Brownian Motion



5.

Plancherel Theory and Continuous Wavelet Transforms

Hartmut Fuehr, Germany

We give an overview of the connection between generalized
continuous wavelet transforms -- or "coherent state transforms"
defined by use of a unitary representation of a locally
compact group, and the Plancherel transform of the underlying
group.  The approach via Plancherel measure allows to decide
for an arbitrary representation whether there exists a 
reconstruction formula involving the Haar measure of the group.
This considerable widens the scope of such transforms; in
particular it allows to study non-irreducible representations.
As an illystration we consider semidirect products and their
quasiregular representations.



6.

Wavelets on the Integers

Philip Gressman, Washington University 

We will introduce the theory of wavelets on the integers.
Our results allow us to identify a larger class of such
wavelets than previously known.  We will also show that
natural MRA structures arise on the integers and these
MRAs correspond in many cases to MRAs on the real line.



7.

Hybrid Meshes

Igor Guskov, Caltech

Hybrid mesh is a novel multiresolution mesh structure which
combines the flexibility of irregular and the simplicity of
(semi)-regular meshes.  I will show how to contruct hybrid
meshes using both regular refinement to build smooth patches
and irregular operations to allow topology changes.  A user
driven procedure for remeshing scanned geometry with hybrid
meshes will be demonstrated.

This is joint work with Andrei Khodakovsky, Peter Schroeder
and Wim Sweldens.



8.

Complete Interpolating Sequences for Classes of
Frequency Band Limited Functions.

W.R. Madych, University of Connecticut

Let PWp be the classical Paley-Wiener space of 
entire functions of exponential type no greater 
than p which are square integrable on the real
line IR. A real complete interpolating sequence
for PWp is a bi-infinite increasing sequence of
real numbers C={xn} which has the property that 
the interpolation problem F|x=y has a unique 
solution F in PWp for every square integrable 
sequence y = {yn}. As a sequence of sampling 
nodes, each such sequence X plays a role 
similar to that played by the integer lattice ZZ; 
for instance, every F in PWp can be recovered from 
samples of its values on C via a Lagrange type 
interpolation series which in the case X = ZZ 
reduces to the classical Whittaker cardinal series.


Every complete interpolating sequence X = {xn} 
for PWp must necessarily satisfy a separateness 
condition: there is a positive constant d, whose 
value may vary from sequence to sequence, such that 
xn+1-xn ³d for all n. On the other hand it is well 
known that functions F in PWp can be unambiguously 
recovered from their values on sampling sequences 
which do not enjoy this condition; the nature of the 
data F|x however will be different. Thus the notion 
of complete interpolating sequence for PWp, as 
defined above, may be too restrictive. In this talk, 
in addition to supplying necessary background, we 
introduce an  extension of this notion which is not 
subject to the aforementioned separateness restriction.


Our method involves extending the notion of complete
interpolating sequence to the class PWp¥ which consists 
of those functions which have a derivative of some order, 
not necessarily the first, in PWp Note that PWp¥ contains 
PWp and, in fact, is significantly larger than PWp.
The corresponding class of suitable sampling sequences
is also significantly larger, indeed its members need 
not satisfy the separateness condition mentioned above.


Since functions F in PWp¥ can have polynomial growth on 
the real line they cannot, in general, be recovered from 
their values on sampling sequences X via classical 
Lagrange type interpolation series directly. However 
various summability methods can be used. In particular, if 
sk is the classical piecewise polynomial spline of order 
2k with knots on the sampling nodes X which interpolates
F on X then lim k®¥sk(x)=F(x) uniformly for all x in IR.



9.

Multiplicative Network Modeling and Multifractal
Estimators

Rolf Riedi, Rice University

Recent studines have made it clear that network traffic is
multifractal in nature.  Using moments of (possibly) all
orders multifractal analysis traces and goes beyond the
second order approximations of linear processes.

Multifractal structures are naturally tied to multiplicative
frameworks.  Indeed, a simple stochastic traffice model based
on the Haar wavelet demonstrates the effectiveness of a 
multiplicative approach and its superiority over the typical
additive schemes, both in multifractal statistics capturing
burstiness as well as in network relevant metrics.  After 
making these points this talk explores novel multiscale
multiplicative models and elaborates on wavelet based
estimators of multifractal parameters.



10.

Multi-Resolution Storage of Optimally Encoded,
Adaptively Accessed Data

Robert Sharpley, University of South Carolina

Many commercial and military applications have the goal of
generating fused databases of images (photographic, infrared,
SAR, stereo) and higher dimensional data which may be used
for autonomous navigation, data registration, and the fast 
query and retrieval of information.  For example in the area
of remote sensing, databases of images, digital elevation models
and associated cultural data for specified regions have been
developed and archived, but are not currently amenable to
efficient data navigation or registration.

We discuss how the recently developed theory of Cohen,
Dahmen, DeVore and Daubechies for optimal entropy tree
encoders may be used to address such problems.  The class
of scalable algorithms resulting from this theory yield
optimal (in an information-theoretic sense), progressive,
universal, locally refinable encoders for purposes of
compression, storage and transmission of multiply resolved
data.



11.

Digital Geometry Compression

Wim Sweldens, Bell Labs

Due to rapid progress in scanning technology, digital
geometry is becoming the fourth wave of digital multimedia
after audio, images, and video.  Last year, a two billion
triangle model of Michelangelo's David with sub millimeter
accuracy was built.  As a result, digital geometry compression
has established itself as a new branch of data compression.
Traditionally one observes that meshes consist both of
geometric information (vertex positions) and connectivity 
or graph information.  Compression methods for both geometry
and connectivity have been proposed.  In this work, we observe
that meshes actually consist of three distrinct components;
geometry, parameter, and connectivity information.  More 
importantly, the latter two do not contribute to the reduction
of error in a rate-distortion setting.  We show that using
so-called semi-regular meshes, parameter and connectivity
information can be eliminated.  Combined with wavelet transforms,
zerotree coding, and subdivision, we build a progressive
geometry compression algorithm which improves the standard
methods by 12dB.



12.

Regular Symmetric Wavelets in Higher Dimensions

Yang Wang, Georgia Tech

In this talk, I will discuss the design of orthogonal
filters in higher dimensions from filters in the one
dimension.  The technique is powerful, easy to implement
and can be easily modified for designing other filters
such as biorthogonal filters or tight frame filters.
We use the technique to construct smooth symmetric
wavelets in the higher dimensions.



13.

The Mathematical Theory of Wavelets

Guido Weiss, Washington University

The definitions and properties of "wavelets" and related
notions that are used by researchers in this area are
varied.  There are continuous, discrete, singly generated,
multi and several other types of wavelets.  They are
generated by dyadic and more general matricial dilations
and by several different types of lattices.  There are
:analyzing" and "reconstructing" wavelets, as in the
phi and psi-transform theory and in the case of bi-orthogonal
wavelets.  Other operations, such as modulations, can replace
the role of dilations (as in the case of Gabor systems).
There are group-theoretic approaches for obtaining "wavelet-
like" reproducing forumulae.  There are several ways of 
unifying these different ways of developing this theory. 
This unification not only leads to a better understanding
but it also provides many new ways for the construction 
af wavelet (and related) systems.



14.

Affine Groups Admitting Continuous Wavelets and
their Characterization

Edward Wilson, Washington University

We consider affine groups on R^n.  The most general such 
group G is the semi-direct product of a topological subgroup 
D of GL(n,R) and the translation group R^n.  D is said to 
be admissible if there exists a function on R^n satisfying 
a generalization of the Calderon reproducing formula relative 
to G with any such function said to be a continuous
wavelet relative to G.  We are able to give a nearly complete
characterization of the admissible groups.  If D is admissible, 
then G cannot be unimodular and the stability subgroup for 
the transpose action of D must be compact at almost every 
point of R^n.  Conversely, if G is non-unimodular and at 
almost every point, there is a compact epsilson stabilizer 
for the action of D, then D is admissible.  These conditions
are easily checked in practice and give rise to numerous 
examples of both admissible and non-admissible groups.  
The examples suggest that the compact epsilon stabilizer 
property may be necessary as well as sufficient for admissibility.




15.

Subdivision on Irregular Meshes

Denis Zorin, NYU

The theory of stationary subdivision, closely related 
to wavelet theory, is well developed and numerous results 
are estabilshed on smoothness and approximation properties 
of subdivision schemes. However, most of these results were
obtained for subdivision of regular grids and in functional 
setting.

However, the main practical application of subdivision 
is construction of smooth surfaces on irregular meshes. 
This makes analysis of geometric properties in irregular 
setting particularly important.

In this talk I review the known facts subdivision surfaces: 
necessary and sufficient conditions for tangent plane 
continuity and CÙk continuity, regularity of surfaces 
and approximation properties of subdivision bases. 
Finally, I will discuss open problems and directions 
for future research.

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