Wavelets and subband coding
Multiresolution analysis for surfaces of arbitrary topological type
ACM Transactions on Graphics (TOG)
Curves and surfaces for CAGD: a practical guide
Curves and surfaces for CAGD: a practical guide
EURASIP Journal on Applied Signal Processing
Single-knot wavelets for non-uniform B-splines
Computer Aided Geometric Design
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This paper investigates the mathematical background of multiresolution analysis in the specific context where the signal is represented by irregularly sampled data at known locations. The study is related to the construction of nested piecewise polynomial multiresolution spaces represented by their corresponding orthonormal bases. Using simple spline basis orthonormalization procedures involves the construction of a large family of orthonormal spline scaling bases defined on consecutive bounded intervals. However, if no more additional conditions than those coming from multiresolution are imposed on each bounded interval, the orthonormal basis is represented by a set of discontinuous scaling functions. The spline wavelet basis also has the same problem. Moreover, the dimension of the corresponding wavelet basis increases with the spline degree. An appropriate orthonormalization procedure of the basic spline space basis, whatever the degree of the spline, allows us to (i) provide continuous scaling and wavelet functions, (ii) reduce the number of wavelets to only one, and (iii) reduce the complexity of the filter bank. Examples of the multiresolution implementations illustrate that the main important features of the traditional multiresolution are also satisfied.