Name
Abstract | ||
|---|---|---|
In this paper, an approach is introduced based on differential operators to construct wavelet-like basis functions. Given a differential operator L with rational transfer function, elementary building blocks are obtained that are shifted replicates of the Green's function of L. It is shown that these can be used to specify a sequence of embedded spline spaces that admit a hierarchical exponential B-spline representation. The corresponding B-splines are entirely specified by their poles and zeros; they are compactly supported, have an explicit analytical form, and generate multiresolution Riesz bases. Moreover, they satisfy generalized refinement equations with a scale-dependent filter and lead to a representation that is dense in L2. This allows us to specify a corresponding family of semi-orthogonal exponential spline wavelets, which provides a major extension of earlier polynomial spline constructions. These wavelets are completely characterized, and it is proven that they satisfy the following remarkable properties: 1) they are orthogonal across scales and generate Riesz bases at each resolution level; 2) they yield unconditional bases of L2-either compactly supported (B-spline-type) or with exponential decay (orthogonal or dual-type); 3) they have N vanishing exponential moments, where N is the order of the differential operator; 4) they behave like multiresolution versions of the operator L from which they are derived; and 5) their order of approximation is (N-M), where N and M give the number of poles and zeros, respectively. Last but not least, the new wavelet-like decompositions are as computationally efficient as the classical ones. They are computed using an adapted version of Mallat's filter bank algorithm, where the filters depend on the decomposition level. |
Year | DOI | Venue |
|---|---|---|
2006 | 10.1109/TSP.2006.870544 | IEEE Transactions on Signal Processing |
Keywords | Field | DocType |
hierarchical exponential b-spline representation,differential equation,riesz base,new wavelet-like base,exponential decay,splines,multiresolution analysis,exponential moment,green's functions,operator l,corresponding b-splines,polynomial spline construction,index terms—continuous-time signal processing,differential operator,l2-either compactly,multireso- lution approximation,differential operators,embedded spline space,wavelets.,green function,signal processing,wavelets,polynomials,transfer functions,indexing terms,differential equations,transfer function,wavelet transforms,wavelet analysis,spline,satisfiability,poles and zeros,filter bank | B-spline,Spline (mathematics),Mathematical optimization,Exponential function,Exponential decay,Multiresolution analysis,Differential operator,Operator (computer programming),Rational function,Mathematics | Journal |
Volume | Issue | ISSN |
54 | 4 | 1053-587X |
Citations | PageRank | References |
18 | 1.32 | 10 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| I. Khalidov | 1 | 18 | 1.32 |
| Unser, M. | 2 | 3438 | 442.40 |