Parabolic Molecules: Curvelets, Shearlets, and Beyond

Parabolic Molecules: Curvelets, Shearlets, and Beyond

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Gitta Kutyniok, Technische Universität Berlin
Fine Hall 214

Anisotropic representation systems such as shearlets and curvelets have had a significant impact on applied mathematics in the last decade. The main reason for their success is their superior ability to optimally resolve anisotropic structures such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shock fronts in solutions of transport dominated equations. By now, a large variety of such anisotropic systems has been introduced, among which we mention second generation curvelets, bandlimited shearlets and compactly supported shearlets, all based on a parabolic dilation operation. These systems share similar sparsity properties, which is usually proven on a case-by-case basis for each different construction. In this talk we will introduce the concept of parabolic molecules which allows for a unified framework encompassing all known anisotropic frame constructions based on parabolic scaling. The main result essentially states that all such systems share similar approximation properties. One consequence we will discuss is that at once all the desirable approximation properties of curvelets can be deduced for virtually any other system based on parabolic scaling. This is joint work with Philipp Grohs (ETH Zurich). Gitta Kutyniok, Technische Universität Berlin