Shearlets and Microlocal Analysis
Philipp Grohs · Applied and numerical harmonic analysis · 2012
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.