Edge analysis and identification using the continuous shearlet transform

Abstract

It is well known that the continuous wavelet transform has the ability to identify the set of singularities of a function or distribution f. It was recently shown that certain multidimensional generalizations of the wavelet transform are useful to capture additional information about the geometry of the singularities of f. In this paper, we consider the continuous shearlet transform, which is the mapping f ∈ L 2 ( R 2 ) → SH ψ f ( a , s , t ) = 〈 f , ψ ast 〉 , where the analyzing elements ψ ast form an affine system of well localized functions at continuous scales a > 0 , locations t ∈ R 2 , and oriented along lines of slope s ∈ R in the frequency domain. We show that the continuous shearlet transform allows one to exactly identify the location and orientation of the edges of planar objects. In particular, if f = ∑ n = 1 N f n χ Ω n where the functions f n are smooth and the sets Ω n have smooth boundaries, then one can use the asymptotic decay of SH ψ f ( a , s , t ) , as a → 0 (fine scales), to exactly characterize the location and orientation of the boundaries ∂ Ω n . This improves similar results recently obtained in the literature and provides the theoretical background for the development of improved algorithms for edge detection and analysis.