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 mappingf , 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 n=1 f n χ Ωn 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.

Introduction to Shearlets

“…In particular, consider a function f = χ B ⊂ L 2 (R 2 ), where B ⊂ R 2 is a planar region with piecewise smooth boundary. Then SH ϕ ,ψ,ψ f characterizes both the location and orientation of the boundary edge ∂ B by its decay at fine scales [32,38]. This property is very useful in applications which require the analysis or detection of edge discontinuities.…”

Image Processing Using Shearlets

“…In fact, the discrete shearlet transform which was presented above for image denoising, produces large sidelobes around prominent edges 3 which interfere with the detection of the edge location. By contrast, the special discrete shearlet transform introduced in [90,91] is not affected by this issue since the analysis filters are chosen to be consistent with the theoretical results in [44,45], which require that the shearlet generating function ψ satisfies certain specific symmetry properties in the Fourier domain (this is also discussed in Chapter 3 of this volume).…”

Analysis and Identification of Multidimensional Singularities Using the Continuous Shearlet Transform

“…It was shown in [6] (by refining an incomplete proof in [17]) that the continuous shearlet transform of B R exactly characterizes the curve ∂ D R .…”

“…On the other hand, the ability of the continuous shearlet transform to detect the geometry of the singularity set goes far beyond the continuous wavelet transform and is its most distinctive feature. As a particular manifestation of this ability, we will shows that the continuous shearlet transform provides a very general and elegant characterization of step discontinuities along 2D piecewise smooth curves, which can summarized as follows (see [7,6]). Let B = χ S , where S ⊂ R 2 and its boundary ∂ S is a piecewise smooth curve.…”