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The analysis and detection of edges is a central problem in applied mathematics and image processing. A number of results in recent years have shown that directional multiscale methods such as continuous curvelet and shearlet transforms offer a powerful theoretical framework to capture the geometry of edge singularities, going far beyond the capabilities of the conventional wavelet transform. The continuous shearlet transform, in particular, provides a precise geometric characterization of edges in piecewise constant functions in R 2 and R 3 , including corner points. However, a question has been raised frequently: What happens if the function is piecewise smooth and not just piecewise constant? Clearly, a piecewise smooth function is a much more realistic model of images with edges. In this paper, we extend the characterization results previously known and show that, also in the case of piecewise smooth functions, the continuous shearlet transform can detect the location and orientation of edge points, including corner points, through its asymptotic decay at fine scales. The new proof introduces innovative technical constructions to deal with the more challenging problem. The new results set the theoretical groundwork for the application of the shearlet framework to a wider class of problems from image processing. [ABSTRACT FROM AUTHOR]

Suppose that Ω is a three-dimensional solid with boundary surface S = S 1 ∪ ⋯ ∪ S q , where each S r is a smooth surface with boundary curve Γ r . Multiscale directional representation systems (e.g., shearlets) are able to capture the essential geometry of Ω by precisely identifying the boundary set N = { ( p , n r ( p ) ) : p ∈ S r , r = 1 , … , q } , where n r ( p ) denotes the normal vector to the surface S r at p . This property has resulted in the successful application of multiscale directional methods in a variety of image processing problems, since edges and boundary sets are usually the most informative features in many types of multidimensional data. However, existing methods are ill-suited to capture those edge-type singularities in the three-dimensional setting resulting from the intersection of piecewise smooth boundary surfaces. In this paper, we introduce a new multiscale directional system based on a modification of the shearlet framework and prove that the associated continuous transform has the ability to precisely identify both the location and orientation of the boundary curves Γ r from the solid Ω . This paper extends a number of results appeared in the literature in recent years to the challenging problem of extracting curvilinear singularities in three-dimensional objects and is motivated by image analysis problems arising from areas including biomedical and seismic imaging and astronomy. [ABSTRACT FROM AUTHOR]

Abstract: Directional multiscale transforms such as the shearlet transform have emerged in recent years for their ability to capture the geometrical information associated with the singularity sets of bivariate functions and distributions. One of the most striking features of the continuous shearlet transform is that it provides a very simple and precise geometrical characterization for the boundary curves of general planar regions. However, no specific results were known so far in higher dimensions, since the arguments used in dimension do not directly carry over to the higher dimensional setting. In this paper, we extend this framework for the analysis of singularities to the 3-dimensional setting, and show that the 3-dimensional continuous shearlet transform precisely characterizes the boundary set of solid regions in by identifying both its location and local orientation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 , where the analyzing elements form an affine system of well localized functions at continuous scales , locations , and oriented along lines of slope 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 where the functions are smooth and the sets have smooth boundaries, then one can use the asymptotic decay of , as (fine scales), to exactly characterize the location and orientation of the boundaries . 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. [Copyright &y& Elsevier]

It is well known that the wavelet transform provides a very effective framework for analysis of multiscale edges. In this paper, we propose a novel approach based on the shearlet transform: a multiscale directional transform with a greater ability to localize distributed discontinuities such as edges. Indeed, unlike traditional wavelets, shearlets are theoretically optimal in representing images with edges and, in particular, have the ability to fully capture directional and other geometrical features. Numerical examples demonstrate that the shearlet approach is highly effective at detecting both the location and orientation of edges, and outperforms methods based on wavelets as well as other standard methods. Furthermore, the shearlet approach is useful to design simple and effective algorithms for the detection of corners and junctions. [ABSTRACT FROM AUTHOR]