Applicable Harmonic Analysis

In spite of great success of applicable harmonic analysis in the last decade, there are still many important problems to be investigated. It seems appropriate to organize a workshop on applicable harmonic analysis with emphasis on wavelet analysis. The purpose of the workshop is to evaluate the current state of the area, to find interesting new directions for further research, and to disseminate useful research results for wide applications. The workshop will bring together first-rate senior experts in the theory and application of wavelet analysis along with promising young researchers. The workshop will also emphasize combination of theoretical development with practical applications.

The mode of the workshop will be plenary lectures by established researchers, seminar talks by younger investigators, and panel discussions involving all participants. The workshop will emphasize open problems from theory, case studies in applications needing further analysis, and promising new algorithms.

The workshop will generally cover the following topics:

-- nonlinear approximation associated with wavelet expansion and its application to signal compression and signal denoising -- optimal wavelet bases, wavelet frames, and wavelet packets, and their application to a variety of problems including signal and image processing -- unconditional bases for function spaces and their applications to numerical solutions of partial differential equations -- multivariate refinable functions and their applications to fractal geometry -- cascade algorithms and their applications to computer aided geometric design.

However, to focus some of discussion, we plan to have up to four related subgroup focus areas; two of a more theoretical nature, and two which are more applied in scope. We have asked appropriate experts to lead each of the four subgroups. The four focus areas are: Nonlinear approximation, redundant systems and frames, biosignal analysis, and fluid flows and turbulence. These subgroups complement each other Bnd represent areas where future development will have great impact. Below we give a brief commentary on these four areas.

Non-linear Approximation: Non-linear approximation techniques have played a major role in the applications that involve wavelet methods. The idea is that the elements used in approximation do not come from a fixed linear space but are allowed to depend on the function in a non-linear manner. This has proved extremely useful in both theory and practice for adaptive PDE solvers, compression of images and signals, statistical classification, and in development of efficient encoders . The classical method is the so-called $n$-term approximation: approximate a given function by a linear combination of $n$-terms from a selected basis; for example, choosing the appropriate $n$-terms from a wavelet basis to obtain good compression of data. Much of the theoretical work was to relate rates of approximation to the smoothness of the approximated function, the smoothness space providing the norms or metric in which to do the approximation. For example, when the smoothness space is a Besov space, and one is approximating by $n$-terms from an orthogonal system with a metric determined by an $L^p$ space, the theory developed uses deep results from finite dimensional geometry, and provides an example of the interaction between $n$-term nonlinear approximation and modern functional analysis. Perhaps surprisingly, researchers have found a relationship between Besov spaces and rate distortion performance in compression techniques that helps to establish rate-distortion estimates and to identify performance limits in compression algorithms.

On the practical side, the goal is to develop algorithms which will achieve best, or near best approximation using $n$-term approximation. Work has been carried out in a surprisingly wide variety of contexts important for the development of applications using computational harmonic analysis methods. Recent research into the general development of such algorithms has been targeted to include redundant systems (frames, dictionaries of functions etc). There has emerged a necessity to look at some very highly nonlinear systems. Approximation of the type of interest here relates to adaptive basis selection and adaptive pursuit or `greedy' algorithms, The flexibility of allowing redundancy offers great promise for improvements in approximation and applications (see some of the remarks below), but it presents very different, highly non-trivial theoretical and practical problems. This should be a promising and fruitful area of research for both theory and practice over the next few years and a discourse between the theoreticians and practitioners will be extremely useful. Professors Ron DeVore and Vladimir Temlyakov of the University of South Carolina, two leading researchers in the nonlinear approximation, have agreed to organize this subgroup.

Redundant systems: Frames, Dictionaries, etc. The notions that have led to the recent work on redundant systems go back more than fifty years to Gabor and Duffin and Schaffer, but it was the context of multiresolution analysis and the wavelet revolution in applied harmonic analysis that has thrust frames and other means of generating redundant systems into the forefront. In recognition that different bases have different properties, and thus in turn extract different information from signals and images, researchers turned to using ``dictionaries'' of bases, and to the development of algorithms that would adaptively select the proper basis from the dictionary to perform the task at hand. Many of the bases were found by adapting classical transforms, e.g. the windowed Fourier transform, mixed with wavelet methods, or extending the wavelet transform to wavelet packets from which an adaptive selection would give the decomposition for the function (signal). By using several bases, redundancy in representations are introduced that should allow greater flexibility to achieve better approximation or isolation of desired features. Frames are a system of functions that mimic many of the properties of orthogonal systems in the sense that the sum of the squares of the coefficients in a frame expansion of a function is equivalent to the Hilbert norm of the function. However, frames may not be a basis, even when the frame bounds are tight and the norm is given exactly by the sum of the squares of the coefficients. The added redundancy allows for the possibility that in discarding some coefficients, one may not entirely lose all information associated with the coefficients. Over the last ten years, the theory of frames has matured and now theoretical and practical frameworks for systematic constructions now exist. Over the same time, many researchers have developed ad hoc constructions of frames and redundant systems to suit particular applications with concomitant algorithms for their implementation. It would be an appropriate time to bring theorists and the practitioners together to assess the directions for future theory, and the implications for applications. Amos Ron (Wisconsin) and Zuowei Shen (Singapore) have agreed to lead the subgroup on redundant systems.

Two of the most important contemporary applications of harmonic analysis are the processing of complicated and noisy biosignals, and the extraction and computation of features in fluid flows. They are mathematically related: both applications treat high-dimensional sampled data that is well modeled by smooth though nonstationary stochastic processes. Both applications also test the limits of existing computers. Successful denoising, feature extraction, accelerated computation, modeling and visualization of biosignals and flows, using the wavelet and multiscale methods developed in the past few years, suggests that bringing together investigators specializing in the two areas would benefit both.

Biosignal analysis: Biosignal analysis is a burgeoning area of biomedical engineering. Signals from biological systems are typically weak, noisy, and transient. Examples include: radiographs and tomographs, whose poor signal to noise ratios cannot be improved due to dose and flux limits; MRIs, contrast-limited by magnetic field strengths and spin relaxation times; EEGs, vulnerable to myelitic noise; and evoked otoacoustic emissions which at best are 60 dB weaker than the stimulus with which they are mixed. A vast array of new ideas and methods have been published to improve the diagnostic value of such sigals. These depart from traditional engineering for noise reduction and statistical signal estimation by exploring novel decompositions. Judging their reliability requires good rate of approximation estimates for compact representations, including many taken from computational harmonic analysis, such as: matching pursuit with time-frequency dictionaries; various thresholded wavelet estimators; local nonlinear discriminant analysis; and independent components analysis. Some of these ideas and methods are poorly understood, from a mathematical perspective, and their usefulness is not demonstrated outside of case studies. Our goal is to expose some energetic young harmonic analysts to the algorithm developers and users, in the presence of experienced senior investigators, to address the gaps and advance the usefulness. Metin Akay has agreed to lead this focus area.

Fluid Flows and Turbulence: Wavelet transform methods began to be used in numerical simulations of fluid flows and turbulence in the 1990s, to overcome poor scaling of computation costs at large Reynolds numbers. Interest in the global existence and uniqueness problem for the Navier-Stokes evolution received a big boost when this was announced as one of the seven Millennium Challenge Prize problems. There have been some surprising advances in both the computational and theoretical arena arising from better representations of the solution spaces. For example, an approximate diagonalization of the Navier-Stokes evolution in a wavelet basis, which seems to separate a few large driving components from many small noise components, led to M. Cannone's local existence and uniqueness result from unexpectedly rough initial conditions. Global existence and uniqueness would follow from an appropriate rate of approximation inequality in the same basis. Simulations by M. Farge and collaborators indicate that the driving components have other important properties, and since the extraction of a few large wavelet component ``features'' can be done automatically from simulated or measured flows, there is a possibility of flow prediction with lower computational cost and equivalent accuracy. Our goal is to mix together theoretical and computational fluid dynamicists, users of similar decomposition techniques, to advance the state of each others' arts. Marie Farge has agreed to lead this subgroup.

Confirmed Participants

Workshop Schedule

Videos

Final Report (pdf file 72kb)