Resolution of the wavefront set using continuous shearlets

By Gitta Kutyniok, Demetrio Labate. 2006

285 citations recorded by the source corpus.

It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unable to describe the geometry of the set of singularities of f and, in particular, identify the wavefront set of a distribution. In this paper, we employ the same framework of affine systems which is at the core of the construction of the wavelet transform to introduce the Continuous Shearlet Transform. This is defined by SH ψ f(a,s,t) = (fψ ast ), where the analyzing elements ψ ast are dilated and translated copies of a single generating function ψ. The dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements {ψ ast } form a system of smooth functions at continuous scales a > 0, locations t ∈ R 2 , and oriented along lines of slope s ∈ R in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution f.

Subfields: Applied Mathematics, Computational Mechanics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/269ed3b8ac9298819eb9e15fed9332e226af6231

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In Applied Mathematics

In Computational Mechanics

In Computer Vision and Pattern Recognition

Citations in the corpus

Reproducing formulas for generalized translation invariant systems on locally compact abelian groups (2014, result): “The above characterization result generalizes results from [24,25, 37].”
Full length article: Compactly supported shearlets are optimally sparse (2011, unknown): “[15] G.”
Transforms and Operators for Directional Bioimage Analysis: A Survey. (2016, background): “Furthermore, they are closely related to a mathematically rigorous notion of the orientation of image features, the so-called wavefront set (Candès and Donoho 2005; Kutyniok and Labate 2009).”
Coorbit spaces and wavelet coefficient decay over general dilation groups (2012, background): “The associated transform has been studied in various papers, e.g. [7,8,26].”
Shearlet Coorbit Spaces: Compactly Supported Analyzing Shearlets, Traces and Embeddings (2011, background): “We concentrate on the two-dimensional case, and we show that • For large classes of weights, variants of Sobolev embeddings exist; • for natural subclasses which in a certain sense correspond to the ‘shearlets on the cone’ [17], there exist embeddings into (homogeneous) Besov spaces: • for the same ”
Shearlet coorbit spaces and associated Banach frames (2009, background): “The corresponding family of continuous shearlets was exploited in [20] to show that the Continuous Shearlet Transform precisely resolves the wavefront set of a distribution.”