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 Wavelets and Applications Semester
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J. Bernoulli
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Semester talks

During the summer semester March-June 2006, research seminars are organized at EPFL.

16 March 2006 at 17:00, room AAC006
Dynamic Imaging of Cyclically Moving Embryonic Structures: Combining Fast Confocal Microscopes and Wavelet-Based Reconstruction Techniques
Michael Liebling, California Institute of Technology (Pasadena (CA), USA)
Assessing the influence on embryonic development of fast biomechanical processes, such as those induced by blood flow in the embryonic heart, requires the ability to acquire dynamic three- dimensional data with high temporal resolution. Despite the availability of confocal laser scanning microscopes that can acquire hundreds of two-dimensional optical sections per second, direct three- dimensional imaging, which is 2-3 orders of magnitude slower, does not yield satisfactory results. However, for objects whose motion is cyclic, we were able to build dynamic three-dimensional reconstructions by sequentially acquiring nongated slice sequences at high speed and at increasing depths then retrospectively synchronizing them. In this talk, I will present the acquisition and synchronization procedures that we developed for that purpose. The reconstruction is based on (non)uniform temporal registration algorithms, which, in turn, rely on the minimization of the intensity difference between adjacent slice-sequence pairs. The challenges are the considerable amount of data and typical fluorescence imaging caveats (e.g. low photon count and photo-bleaching) combined with requirements for a fast and reproducible approach. We have naturally selected wavelets as the tool of choice because of their ability to yield sparse, hierarchical data representations via fast and flexible time- frequency analysis transforms. Through in vivo imaging of zebrafish embryos we were able to extract both qualitative and quantitative information that should contribute to reach a better understanding of the mechanisms that drive heart development. This is joint work with A. S. Forouhar, J. Vermot, M. Gharib, S. E. Fraser, and M. E. Dickinson.

23 March 2006 at 17:00, room AAC006
Signal Reconstruction from Limited Number of Measurements: Theory and Algorithms
Minh N. Do, University of Illinois (Urbana-Champaign, USA)
In a wide range of inverse problems arising from electromagnetic and acoustic wave propagation and scattering, due to physical constraints or timing requirements only a limited number of measurements can be acquired from the unknown object. Recently, there has been a series of remarkable results showing that one can effectively exploit the knowledge that the unknown signal has a sparse representation in a certain basis as a prior information for signal reconstruction from limited number of measurements. We propose a geometric framework to study this problem via a sampling theory for signals from union of subspaces. In this framework, the sampling operator is viewed as projecting the signals into a low dimensional space, while still preserves all the information. We find the necessary and sufficient conditions for such sampling operators to exist, and find the minimum sampling requirement for several cases. Furthermore, in many multiscale bases (e.g. wavelets), signals of interest (e.g. piecewise-smooth signals) not only have few significant coefficients, but also those significant coefficients are well-organized in trees. We present algorithms to exploit the tree-structured sparse representation as additional prior information that lead to more accurate reconstruction and faster computation than existing methods.

30 March 2006 at 17:00, room AAC006
Shift and Rotation Invariant Image Parameters derived from Complex Wavelets
Nick Kingsbury, University of Cambridge (Cambridge, UK)
The Dual-Tree Complex Wavelet Transform (DT CWT) is now a reasonably well-known technique for image analysis. It overcomes the problems of shift dependency and poor directional selectivity of the conventional Discrete Wavelet Transform (DWT), and produces a multiscale directionally selective filterbank that quite closely imitates the V1 cortical filters in the vision systems of many mammals, including humans. In this talk we will discuss our recent work which has derived new sets of parameters from the outputs of the DT CWT. These parameters exhibit much better invariance to shift (particularly in their phases) than the basic DT CWT coefficients and we will discuss several applications for them including object feature detection and denoising. We will also show how similar ideas can lead to a rotation invariant feature detector. Finally we shall consider whether these developments provide any hints as to how the human vision system performs so effectively.

13 April 2006 at 17:00, room AAC006
Some new results about dual-tree wavelet decompositions
Jean-Christophe Pesquet, Université de Marne la Vallée (Marne la Vallée France)
In this talk, an overview of our recent work on dual-tree wavelet decompositions is given. In particular, a generalization to the M-band case of these decompositions is proposed. The construction of the dual basis and the resulting 2D directional analysis is studied. The problems involved in the implementation of the decompositions are also discussed. A new estimator for image denoising using a 2D dual-tree M-band wavelet transform is then presented which extends existing block-based wavelet thresholding methods by exploiting simultaneously the coefficients in the two M-band wavelet trees.

20 April 2006 at 17:00, room AAC006
Multifractal analysis at work: The wavelet leaders contribution
Patrice Abry, Ecole Normale Supérieure de Lyon (Lyon, France)
Multifractal analysis has become a popular tool for empirical data analysis used in numerous and various applications such as turbulence, Internet traffic, Heart beat variability to name but a few. It mainly consists in estimating scaling exponents from the power law behaviors of the time average of its wavelet coefficients with respect to scales. It has recently been shown that multifractal analysis should be based on wavelet leaders instead of on wavelet coefficients. In the present talk, we define wavelet leaders and illustrate and explain how and why they are better suited to multifractal analysis. Mainly, we show that they enable to correctly analysis the entire range of the multifractal spectrum and that they correctly handles the potential existence of oscillating singularities in data. We illustrate the performance of wavelet based leader analysis both on synthetic data and on actual empirical data obtained from various applications.

27 April 2006 at 17:00, room AAC006
Sparse representations and statistical analysis of functional magnetic resonance images of the human brain
Jalal Fadili, ENSI Caen (Caen, France)
Sparse representations, and wavelets in particular, have become increasingly used for analysis of biological signals and images, such as functional magnetic resonance images (fMRI) of the human brain. This kind of data often demonstrate fractal or scale invariant properties. This work is devoted to exploiting key properties of these representations in order to analyse fMRI spatio-temporal data.
We focus on three applications in particular: (i) wavelet coefficient resampling or "wavestrapping" of 1-D time series, 2- to 3-D spatial maps and 4-D spatiotemporal processes; (ii) sparse representation-based estimators for signal and noise parameters of semiparametric time series; and (iii) wavelet shrinkage in frequentist and Bayesian frameworks to support multiresolution hypothesis testing on spatially extended statistic maps. We conclude that the sparse representations, and wavelets in particular, are rich sources of new concepts and techniques to enhance the power of statistical analysis of human fMRI data.

4 May 2006 at 17:00, room AAC006
Approaches to complex splines
Brigitte Forster-Heinlein, TU München (München, Germany)
We give approaches to complexify classical splines, while keeping the nice properties such as smoothness and decay of their real-valued fellows. Extensions to more general corpses are also considered. Our motivation comes from piecewise polynomials as e.g. Schoenberg's cardinal B-Splines or Duchon's polyharmonic splines, which have proved to be adequate tools for many analysis problems. They also showed to fit perfectly into the concept of multiresolution bases and wavelets. However, from a mathematical point of view, it is unnecessary to restrict spline bases to real valued function. Moreover, it is known that phase information is important for may applications in signal and image processing. This gives rise to consider more general forms of splines.

11 May 2006 at 16:00, room AAC006
Multivariate wavelets with vanishing moments
Maria Skopina, Saint Petersburg University (Saint Petersburg, Russia)
We present a polyphase criterion for vanishing moments of wavelets with a matrix dilation. In contrast to the "sum rule" and other known criterions, our one allows to give a general form for all masks providing vanishing moments of dual wavelet bases. We also study construction of tight and dual wavelet frames with vanishing moments, for which situation is essentially different. In contrast to the bases, two pairs of dual wavelet frames may be generated by the same refinable functions and have different number of vanishing moments. We found a necessary condition and a sufficient condition under which a given pair of refinable functions generates dual wavelet systems (potential frames) with a given number of vanishing moments. Explicit methods for construction of such frames are described.

11 May 2006 at 17:00, room AAC006
Sampling Moments and Reconstructing Signals with Finite Rate of Innovation: Shannon meets Strang-Fix
Pier Luigi Dragotti, Imperial College (London, UK)
Consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, piecewise polynomial or piecewise sinusoidal signals, and call the number of degrees of freedom per unit of time the rate of innovation. Classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited. In this talk, we show that many signals with finite rate of innovation can be sampled and perfectly reconstructed using kernels of compact support and a local reconstruction algorithm. The class of kernels that we can use is very rich and includes functions satisfying Strang-Fix conditions, Exponential Splines and functions with rational Fourier transforms. Extension of such results to the 2-dimensional case are also discussed and an application to image superresolution is presented.

1 June 2006 at 17:00, room AAC006
Constrained Total Variation Minimization for Solving Ill-posed Inverse Problems in Fourier or Wavelet Domains.
Jacques Froment, Université de Bretagne Sud (Vannes, France)
Many ill-posed inverse problems in signal and image processing may be solved by minimizing the total variation of the data, subject to a constraint. By preserving sharp transitions, the total variation functional provides a probable solution while the constraint ensures that this solution satisfies all the conditions stated by the problem. When the constraint can be formulated in a transform domain, the transformation being linear and orthogonal, the approach may be implemented by a simple and fast algorithm. Examples involving Fourier and Wavelet transforms are presented in the context of JPEG restoration, super-resolution, computerized tomography and denoising via wavelet shrinkage.

8 June 2006 at 17:00, room AAC006
Multiscale Geometry for Images and Textures
Gabriel Peyré, Ecole Polytechnique (Palaiseau, France)
The geometry of images is multiscale, because edges of natural images are often blurry and textures contain a broad range of geometric structures. This geometry can be constructed directly over a multiscale domain and corresponds to a grouping process of wavelet coefficients. The resulting adaptive representations are discrete, orthogonal and allow a multiscale description of the geometric content of an image. This leads to the construction of orthogonal bandelet bases, for which the grouping process is locally defined using a best orientation. These orthogonal bases improve over state of the art schemes for images and surfaces compression and for the inversion of the tomography operator. In order to understand and model the complex geometry of turbulent textures, we design an association field that is able to capture long range interactions. This allows a statistical modelling of the geometry of natural textures. We apply this construction to geometric texture synthesis.

15 June 2006 at 17:00, room AAC006
Using Wavelet-Based Priors for Image Segmentation
Mario Figueiredo, Instituto Superior Tecnico (Lisboa, Portugal)
Wavelets have been widely used for several tasks in image processing/analysis and computer vision, but not as priors for image segmentation. The main obstacle to using wavelet-based priors for segmentation is that they're aimed at representing real (or vector) valued signals/images, rather than the discrete labels needed for segmentation. In this talk, I will describe a formulation which allows using wavelet-based priors for image segmentation. The main motivation is to exploit the well-known ability of wavelet-based priors to encourage piece-wise smoothness, which underlies state-of-the-art methods for denoising, coding, and restoration, as well as the availability of fast algorithms for wavelet-based processing. This new formulation completely avoids the combinatorial nature of standard Bayesian approaches to segmentation and can be used for supervised, unsupervised, or semi-supervised segmentation.

22 June 2006 at 17:00, room AAC006
Polyharmonic (pre-)wavelets
Christophe Rabut, INSA (Toulouse, France)
Polyharmonic splines present many advantages as approximating functions, most of them being a consequence of their minimizing property: if σ is a m-harmonic spline, and if |.|m is the semi norm defined for any f in D-m(Rd) by |f|2m=∫Rd |Dm f(x)|2 dx (where Dmf is the vector of all mth-order derivatives of f), then for all f∈ D-m(Rd) meeting f(xi)=σ(xi) for all xi "centres" of the spline, we have |σ|m≤ |f|m; appropriate extension of this is now classical for non-integer order polyharmonic splines.
After a short presentation of some of these properties and of their importance for multivariate approximation, we present two different ways for building polyharmonic wavelets (on regular grids). The first one is by the simple formula : ψ=Δm L2m where L2m is the Lagrangean m-harmonic spline, ie the 2m-harmonic spline meeting L2m(0)=1, and for all j∈ Zd-{0}, L2m(j)=0. The second one is by using classical way to build wavelets starting from a scaling function, and by using the Lagrangean function Lm as the scaling function. We so obtain a pre-wavelet which is exponentially decaying and has a numerical support smaller than the one obtained by the first method. Appropriate extension is shown for non-integer order polyharmonic pre-wavelet.
We show some examples of using these pre-wavelets and the associated multiresolution analysis.

 
 NEWS
July 13, 2006
The pictures are on line.
June 20, 2006
The conference program is online now! More than eighty poster presentations and series of talks by leading scientists.
June 1, 2006
The registration deadline (June 15) is rapidly approaching! Don't miss the opportunity to participate to this event. Take a look at the preliminary program.
March 23, 2006
The registration pages for WavE are open now!
February 27, 2006
The semester talks are scheduled now.
December 19, 2005
Due to popular demand, the deadline for submissions with application for fellowship travel grants is extended to January 22, 2006.
December 8, 2005
Online submissions for Abstracts and Fellow Travel Grants are welcome now!
October, 2005
First Call for Abstracts
 LINKS
epfl.ch
EPFL, Lausanne, Switzerland
bigwww.epfl.ch
Biomedical Imaging Group
lcavwww.epfl.ch
Laboratoire de Communications AudioVisuelles
bernoulli.epfl.ch
Bernoulli Center
wavelet.org
The Wavelet Digest