We describe a method for adapting local shift-invariant bases to non-uniform grids via what we call a squeeze map. When the shift-invariant basis is orthogonal there is a squeeze map such that the nonuniform basis is orthogonal and has the same smoothness and same approximation order as the shift-invariant basis. When the smoothness or approximation order is large enough the squeeze map is uniquely determined and may be calculated locally in terms of the ratios of adjacent intervals. Therefore a basis may be rapidly generated for a given grid. Furthermore local changes in a grid (for example knot insertion or deletion) only affect a few of the basis functions. When starting with a refinable scaling vector the squeeze map machinery gives a procedure for generating orthogonal wavelets on semi-regular grids (that is, an arbitrary non-uniform coarse space with uniform refinements) with the same polynomial reproduction and smoothness as the shift-invariant space.
When performing registrations, it is often crucial to maintain certain structure of the template data T - the data being deformed into the subject data S - as well as to keep the deformation field smooth. Current approaches to registration often impose smoothness through heuristic means, but building it into the model has proven to be more difficult due mainly to computational constraints.
Orthogonal bases of piecewise polynomial smooth functions on arbitrary partitions are constructed using techniques developed by the authors for constructing orthogonal multiwavelets. These bases are generated by a small number of functions that are translated and scaled to each of the intervals in the partition.
A family of continuous, compactly supported, bivariate multi-scaling functions have recently been constructed by Donovan, Geronimo, and Hardin using self-affine fractal surfaces.In this paper we describe a construction of associated multiwavelets that uses the symmetry properties of the multi-scaling functions. Illustrations of a particular set of scaling functions and wavelets are provided.
The theory of orthogonal polynomials is used to construct a family of orthogonal wavelet bases of L2(R) which are compactly supported, continuous, and piecewise polynomial and have arbitrary approximation order.
The pyramid algorithm for computing single wavelet transform coefficients is well-known. The pyramid algorithm can be implemented by using tree-structured multirate filter banks. In this paper, we propose a general algorithm to compute multiwavelet transform coefficients, by adding proper pre multirate filter banks before the vector filter banks that generate multiwavelets. The proposed algorithm can be though of as a discrete vector-valued wavelet transform for certain discrete-time vector-valued signals. The proposed algorithm can be also though of as a discrete multiwavelet transform for discrete-time signals. We then present some numerical experiments to illustrate the performance of the algorithm, which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms.
The construction of smooth, orthogonal compactly supported wavelets is accomplished using fractal interpolation functions and splines. These give rise to multiwavelets. In the latter case piecewise polynomial wavelets are exhibited using an intertwining multiresolution analysis.
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