Refinable vector splines and multi-wavelets with shortest matrix filters

Ranirina, Dinna (2018-03)

Thesis (PhD)--Stellenbosch University, 2018.

Thesis

ENGLISH ABSTRACT : A widely used class of basis functions in signal analysis is obtained from the dilation and integer shifts of a given (compactly supported) wavelet ψ : R → R, by means of which a (scalar) signal can be decomposed into its low frequency and high frequency components. Whereas initially much attention was devoted to orthogonal wavelet decomposition techniques (see for example [1] and [2]), the recent book [3] introduced a more general approach to wavelet construction in which orthogonality is not a requirement and which yielded signi cant advantages in some application areas. An interesting extension is to consider instead, with the view to the decomposition of a vector-valued signal, as presented for the orthogonal case in, for example, [4], a multi-wavelet Ψ : R → R ν . The main focus of this study is to extend the methods in [3], in order to characterize, by means of matrix Laurent polynomial identity systems, a class of multi-wavelets based on general (not necessarily orthogonal) space decomposition. As main building blocks are used re nable vector functions , together with their corresponding matrix re nement sequences. Three di erent classes of re nable vector splines are analysed, with particular focus also on their integer-shift linear independence and stability properties, before explicitly constructing their corresponding spline multi-wavelets. The low-pass and high-pass decomposition matrix lter sequences thus obtained are the shortest possible for the given re nable vector spline, and the spline multi-wavelet is of minimal support for these optimal matrix lters. Moreover, our approach yields explicit formulations for the re nable vector splines, as well as for their corresponding spline multi-wavelets and matrix lter sequences. Computationally e cient algorithms are developed, and examples are calculated, with accompanying illustrating graphs.

AFRIKAANSE OPSOMMING : 'n Wydgebruikte klas van basisfunksies in seinanalise word verkry uit die dilasie en heeltalskuiwe van 'n gegewe ( kompak-ondersteunde) gol e ψ : R → R, deur middel waarvan 'n (skalaar-) sein in lae en hoë frekwensie komponente ontbind kan word. Waar daar aanvanklik baie aandag bestee is aan ortogonale gol eontbindingstegnieke ( sien byvoorbeeld [1] en [2]), het die onlangse boek [3] 'n meer algemene benadering bekendgestel waarin ortogonaliteit nie 'n vereiste is nie, en wat beduidende voordele in sommige toepassingsgebiede opgelewer het. 'n Interessante uitbreiding is om instede te beskou, met die oog op die ontbinding van 'n vektorsein, soos aangebied vir die ortogonale geval in, byvoorbeeld, [4], 'n multi-gol e Ψ : R → R ν . Die hoo okus van hierdie studie is om die metodes van [3] uit te brei, met die doel om, deur middel van matriks-Laurentpolinome, 'n klas multigol es gebaseer op algemene ( nie noodwendig ortogonale) ruimte-dekomposisie te karakteriseer. As hoofboustene word gebruik verfynbare vektorfunksies, tesame met ooreenkomstige matriks-verfyningsrye. Drie verskillende klasse verfynbare vektorlatfunksies word ge-analiseer, met spesi eke aandag ook op hulle heeltalskuif lineêre onafhanklikheiden stabiliteitseienskappe, voordat hulle ooreenkomstige latfunksie multi-gol es eksplisiet gekonstrueer word. Die lae-deurgang en hoëdeurgang ontbindings matriks lterrye wat sodoende verkry word is die kortste moontlik vir die gegewe verfynbare vektorlatfunksie, en die latfunksie multi-gol e is van minimale steun vir hierdie optimale matriks lters. Ons benadering lewer boonop eksplisiete formulerings vir die verfynbare vektorlatfunksies, asook vir hulle ooreenkomstige latfunksie multi-gol es en matriks lterrye. Berekeningsdoeltre ende algoritmes word ontwikkel, en voorbeelde word uitgewerk, met bygaande illustrerende gra eke.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/103739
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