Detection of boundary curves on the piecewise smooth boundary surface of three dimensional solids

Suppose that Ω is a three-dimensional solid with boundary surface S=S1∪⋯∪Sq, where each Sr is a smooth surface with boundary curve Γr. Multiscale directional representation systems (e.g., shearlets) are able to capture the essential geometry of Ω by precisely identifying the boundary setN={(p,nr(p)):p∈Sr,r=1,…,q}, where nr(p) denotes the normal vector to the surface Sr 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 Γr 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.