Shearlets and microlocal analysis

Abstract

Although wavelets are optimal for describing pointwise smoothness properties of univariate functions, they fail to efficiently characterize the subtle geometric phenomena of multidimensional singularities in high-dimensional functions. Mathematically these phenomena can be captured by the notion of the wavefront set which describes point- and direction-wise smoothness properties of tempered distributions. After familiarizing ourselves with the definition and basic properties of the wavefront set, we show that the shearlet transform offers a simple and convenient way to characterize the wavefront set in terms of the decay properties of the shearlet coefficients.