The Wavelet IDR Center Research Subteam

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It is often tacitly assumed that redundant representations are "worse" than nonredundant ones; for instance, an orthonormal basis is considered to give a better representation of data than a frame, because it is maximally nonredundant. We believe that a systematic effort to turn this view upside down can have major benefits. We are already familiar with the benefits of redundant representations for the detection of features in data of for the denoising of signals; we intend to explore their enormous potential for other applications, taking advantage of technical advances in the understanding and accessibility of redundant systems. 

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Redundant representations in a time-frequency or multiresolution setting have come up in a variety of settings. They offer many advantages: 

Invariance: Insisting on orthogonality or some other notion of nonredundancy generally leads to a variety of artifacts that pose difficulty for applications. By allowing redundancy, one can enable translation invariance, modulation invariance, direction invariance, and thereby avoid such artifacts. 

Design Freedom: Nonredundant representations are in a sense rigid; given a nonredundant dictionary, there is essentially only one way to obtain the coefficients of a representation. Redundant representations, in contrast, are flexible: for a given dictionary there may exists many different ways to get coefficients. One may exploit this to design coefficient functionals obeying additional side constraints (smoothness, localization, ease of computation). 

Rapid Impact: Redundant representations are far easier to construct that nonredundant ones. This means that in attacking a new problem area, it is a better strategy to build a redundant representation than a nonredundant one. Hence, if one is interested in propagating Computational Harmonic Analisys (CHA) ideas into new scientific/engineering problem domains, it is obvious that one will go farther and faster by using redundant representations than if one insists on nonredundant representations. 

Nonlinearity: Redundant representations allow the selection of an `ideal' or `best' representation for the data at hand. This a form of nonlinearity since the best representation will depend on the data. The success of many specific algoirthms in data processing stem from the flexibility provided by this nonlinearity. Examples include the `best-basis' approach of Coifman and Wickerhauser, Synchosqueezed wavelet transform of Daubechies and Maes, the Wedgelet representations of Donoho, and the nonlinear wavelet transform of Donoho and Yu. 

Impossibility of Nonredundant Representation: There are even settings in which nonredundant representations don't exist. (For example: critical Gabor phenomena.) 

Particular redundant representations that are used in practice and that we want to explore further include: wavelet and Gobor frames, libraries of bases (such as wavelet packets, localized Fourier transforms), oversampling for bandlimited functions.  

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Particularly nice tight wavelet frames: Recent work by A. Ron and Z. Shen has led to a better understanding of the structure of certain types of wavelet frames, and has led to the construction of tight frames of wavelet with many desirable properties [RS97a], [RS97b], [RS98].  

A few examples of such frames are given in the following figure. 

LEFT: Two "Mother wavelets" that generate a dyadic tight wavelet frame. Each of these functions is a piecewise linear polynomial supported on [-1,1]. RIGHT: Four "Mother wavelets" that generate a dyadic tight wavelet frame. Each of these functions is a piecewise cubic polynomial supported on [-2,2].

Three "Mother wavelets" that generate a bivariate tight frame using the quincunx dilation. Each of these functions is a C1, piecewise quadratic polynomial on 4-directional mesh.

 

Quantization for oversampled bandlimited functions:Bandlimited functions can be reconstructed from their Nyquist samples. In analog-to-digital conversion it turns out that rather than digitize finely the Nyquist samples, it is easier, in practice, to oversample significantly and to replace the samples by coarsely quantized values; in one extreme, these quantized values can be 1 or -1 only. Recent work by I. Daubechies and R. DeVore constructs explicit schemes of this nature that allow approximation of the original bandlimited function faster than any inverse polynomial in the oversampling rate, but still far from the exponential rate that one would hope is achievable. Note that this is a special case of a frame expansion, where one trades off the redundancy of the frame for the possibility of using only a very small set of coefficients. 
Reference: I. Daubechies and R. DeVore, Reconstructing a Bandlimited Function from Very Coarsely Quantized Data: I. A Family of Stable Sigma-Delta Modulators of Arbitrary Order. 

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Quantization: We do not know, as yet, how to quantize efficiently coefficients in redundant wavelet or windowed Fourier expansions. Identifying good strategies, and proving bounds for them, is one big open area. 

Best expansions with respect to frames: The "standard dual frame" provides, for a give frame, the expansion that uses coeffiicents with the smallest norm in the sense of square summability. Very often this is not the appropriate quantity to minimize. It is highly likely, moreover, that for some purposes, the "best" frame expansion coefficients are not the result of a linear procedure. This too is a wide open area.