Characterization and Analysis of Edges Using the Continuous Shearlet Transform

Abstract

This paper shows that the continuous shearlet transform, a novel directional multiscale transform recently introduced by the authors and their collaborators, provides a precise geometrical characterization for the boundary curves of very general planar regions. This study is motivated by imaging applications, where such boundary curves represent edges of images. The shearlet approach is able to characterize both locations and orientations of the edge points, including corner points and junctions, where the edge curves exhibit abrupt changes in tangent or curvature. Our results encompass and greatly extend previous results based on the shearlet and curvelet transforms which were limited to very special cases such as polygons and smooth boundary curves with nonvanishing curvature.

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

“…The proof is completed using the fact thatψ 1 is decreasing and applying the following Lemma, which is also found in [7].…”

“…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.…”