arXiv subjects
Publications and source records attributed to Philipp Petersen.
We demonstrate that shearlet systems yield superior $N$-term approximation rates compared with wavelet systems of functions whose first or higher order derivatives are smooth away from smooth discontinuity curves. We will also provide an improved estimate for the decay of shearlet coefficients that intersect a discontinuity curve non-tangentially.
We analyze the detection and classification of singularities of functions $f = χ_B$, where $B \subset \mathbb{R}^d$ and $d = 2,3$. It will be shown how the set $\partial B$ can be extracted by a continuous shearlet transform associated with compactly supported shearlets. Furthermore, if $\partial S$ is a $d-1$ dimensional piecewise smooth manifold with $d=2$ or $3$, we will classify smooth and non-smooth components of $\partial S$. This improves previous results given for shearlet systems with a certain band-limited generator, since the estimates we derive are uniform. Moreover, we will show that our bounds are optimal. Along the way, we also obtain novel results on the characterization of wavefront sets in $3$ dimensions by compactly supported shearlets. Finally, geometric properties of $\partial S$ such as curvature are described in terms of the continuous shearlet transform of $f$.
We provide a statistical analysis of the ability of digitized continuous shearlet systems to detect objects embedded in white noise. We analyze the possibility to subsample the shearlet transform and obtain a subset of significantly reduced cardinality that can still yield statistically optimal detection results.
This paper examines linear independence of shearlet systems. This property has already been studied for wavelets and other systems such as, for instance, for Gabor systems. In fact, for Gabor systems this problem is commonly known as the HRT conjecture. In this paper we present a proof of linear independence of compactly supported separable shearlet systems. For this, we employ a sampling strategy to utilize the structure of an implicitly given underlying oversampled wavelet system as well as the shape of the supports of the shearlet elements.