Search

Edges and surface boundaries are often the most relevant features in images and multidimensional data. It is well known that multiscale methods including wavelets and their more sophisticated multidimensional siblings offer a powerful tool for the analysis and detection of such sets. Among such methods, the continuous shearlet transform has been especially successful. This method combines anisotropic scaling and directional sensitivity controlled by shear transformations in order to precisely identify not only the location of edges and boundary points but also edge orientation and corner points. In this paper, we show that this framework can be made even more flexible by controlling the scaling parameter of the anisotropic dilation matrix and considering non-parabolic scaling. We prove that, using 'higher-than-parabolic' scaling, the modified shearlet transform is also able to estimate the degree of local flatness of an edge or surface boundary, providing more detailed information about the geometry of edge and boundary points. [ABSTRACT FROM AUTHOR]

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]

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]

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]