Balancing MultiWavelets (about balancing order and other discrete-time properties of multiwavelets)
Jérôme Lebrun · 2022
Multiwavelets are a generalization of wavelets where one allows the multiresolution analysis to be generated by a finite number of scaling functions instead of only one. This approach allows to overcome some limitations in the design of filter banks preventing the construction of non-trivial, orthonormal bases from compactly supported, symmetric wavelets.However, in order to obtain multifilter banks that are appropriate for processing scalar signals, new conditions, coined ''balancing'', have to be imposed in the design of multiwavelets. A thorough study of the discrete-time properties of multifilter banks shows that the balancing conditions are closely related to the important issue of the preservation of discrete-time polynomial signals. Balancing is then proved to be equivalent to a very natural factorization of the lowpass synthesis refinement mask and to some Strang-Fix conditions on a time-varying scalar subdivision operator. Connections are also made with the usual properties of the associated multiresolution analysis (approximation power, moments of the scaling functions, superfunction theory). MultiCoiflets, the natural generalization of Coiflets, come out as a special case of balanced multiwavelets.With the help of computational algebraic geometry, in particular Groebner bases methods, the design of several families of orthonormal balanced multiwavelets and multiCoiflets with compact support, symmetries and minimal-length multifilters is detailed. A new concept of discrete-time balanced smoothness is introduced to measure the influence of ergodic properties (zeros at preperiodic points of invariant cycles) on the smoothness of the iterated multifilter bank. Finally, the description of a significance tree image coder based on balanced multiwavelets concludes this thesis dissertation.