Continuous shearlet tight frames (Q719691)

The author constructs a system of bivariate functions with the following desirable properties: {\parindent=6,5mm \begin{itemize}\item[(a)] Directionality. The geometry of the set of singularities of a tempered distribution \(f\) can be accurately described in terms of the interaction between \(f\) and the elements of the system. \item[(b)] Tightness. The system forms a tight frame of \(L^2(\mathbb R^2)\). \item[(c)] Locality. The representation is local, that is, the representation can also be interpreted as a representation with respect to a non-tight frame and its dual frame such that both of these frames only consist of compactly supported functions. \end{itemize}} The importance of these criteria is obvious: First, it is widely agreed that a large part of the information that a function carries lies in its singularities (for instance, the edges in an image). Secondly, any transform should possess a stable analysis and reconstruction operation. This is encoded in the tight frame property. And finally, in many cases it is easier to work with a local transform than with a non-local one. This is especially true when working with functions over bounded domains like in image processing or numerical PDE theory.