Ten lectures on wavelets
Interpolatory subdivision schemes and wavelets
Journal of Approximation Theory
Stationary Subdivision
Journal of Computational and Applied Mathematics - Special issue: Approximation theory, wavelets, and numerical analysis
The contourlet transform: an efficient directional multiresolution image representation
IEEE Transactions on Image Processing
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We consider a family of d 脳 d matrices W e indexed by e驴驴驴E where (E, μ) is a probability space and some natural conditions for the family (W e ) e驴驴驴E are satisfied. The aim of this paper is to develop a theory of continuous, compactly supported functions $\varphi: {{\mathbb R}}^d \to {\mathbb{C}}$ which satisfy a refinement equation of the form $$ \varphi (x) = \int_E \sum\limits_{\alpha \in {{\mathbb Z}}^d} a_e(\alpha)\varphi\left(W_e x - \alpha\right) d\mu(e) $$ for a family of filters $a_e : {{\mathbb Z}}^d \to {\mathbb{C}}$ also indexed by e驴驴驴E. One of the main results is an explicit construction of such functions for any reasonable family (W e ) e驴驴驴E . We apply these facts to construct scaling functions for a number of affine systems with composite dilation, most notably for shearlet systems.