Rank matrix cascade algorithm, hermite interpolation

[摘要] ENGLISH ABSTRACT: (Math symbols have changed) Wavelet and subdivision techniques have developed, over the last two decades, intopowerful mathematical tools, for example in signal analysis and geometric modelling.Both wavelet and subdivision analysis are based on the concept of a matrix–refinablefunction, i.e. a finitely supported matrix function which is self-replicating in the sensethat it can be expressed as a linear combination of the integer shifts of its own dilationwith factor 2:F = TAF = åk∈ZF(2 ・ −k)Ak.The coefficients Ak, k ∈ Z of d × d matrices, of this linear combination constitute theso-called matrix- mask sequence. Wavelets are in fact constructed as a specific linearcombination of the integer shifts of the 2-dilation of a matrix- refinable function cf. [2;9], whereas the convergence of the associated matrix- subdivision schemec0 = c, cr+1 = SAcr, r ∈ Z+,SA : c = (ck : k ∈ Z) 7→ SAc = åℓ∈ZAk−2ℓ cℓ : k ∈ Z!,subject to the necessary condition thatrank := dim\ǫ∈{0,1}ny ∈ Rd : Qǫy = yo > 0, Qǫ := åj∈ZAǫ+2j, ǫ ∈ {0, 1},( cf. [26]) , implies the existence of a finitely supported matrix- function which is refinablewith respect to the mask coefficients defining the refinement equation and thesubdivision scheme.Throughout this thesis, we investigate in time–domain for a given matrix mask sequence,the related issues of the existence of a matrix–refinable function and the convergenceof the corresponding matrix– cascade algorithm, and finally we apply someresults to the particular research area of Hermite interpolatory subdivision schemes.The dissertation is organized as follows:In order to provide a certain flexibility or freedom over the project, we establishedin Chapter 1 the equivalence relation between the matrix cascade algorithm and thematrix subdivision scheme, subject to a well defined class of initial iterates.Despite the general noncommutativity of matrices, we make use in the full rank caseQǫ = I, ǫ ∈ {0, 1}, of a symbol factorization, to develop in Chapter 2 some usefultools, yielding a convergence result which comes as close to the scalar case as possible:we obtained a concrete sufficient condition on the mask sequence based on the matrixversion of the generating function introduced in [3, page 22] for existence and convergence.Whilst the conjecture on nonnegative masks was confirmed in 2005 by Zhou [29],our result on scalar case provided a progress for general mask sequences. We thenapplied to obtain a new one-parameter family of refinable functions which includesthe cardinal splines as a special case, as well as corresponding convergent subdivisionschemes.With the view to broaden the class of convergent matrix-masks, we replaced in chapter3 the full rank condition by the rank one condition Qǫu = u, ǫ ∈ {0, 1}, u :=(1, . . . , 1)T, then improved the paper by Dubuc and Merrien [13] by using the theoryof rank subdivision schemes by Micchelli and Sauer [25; 26], and end up this improvementwith a generalization of [13, Theorem 13, p.8] in to the context of rank subdivisionschemes.In Chapter 4, we translated the concrete convergence criteria of the general theory fromTheorem 3.2, based on the r-norming factor introduced in [13, Definition 6, p.6], intothe context of rank, factorization and spectral radius (cf. [26]), and presented a carefulanalysis of the relationship between the two concepts. We then proceed with generalizationsand improvements: we classified the matrix cascade algorithms in term ofrank = 1, 2, . . . , d, and provided a complete characterization of each class with the useof a more general r−norming factor namely τ(r)-norming factor. On the other hand,we presented numerical methods to determine, if possible, the convergence of eachclass of matrix cascade algorithms.In both the scalar and matrix cases above, we also obtained explicitly the geometricconstant appearing in the estimate for the geometric convergence of thematrix-cascadealgorithm iterates to the matrix- refinable function. This same geometric convergencerate therefore also holds true for the corresponding matrix–cascade algorithm.Finally, in Chapter 5, we apply the theory and algorithms developed in Chapter 4 tothe particular research area of Hermite interpolatory subdivision schemes: we provideda new convergence criterium, and end up with new convergence ranges of theparameters' values of the famous Hermite interpolatory subdivision scheme with twoparameters, due to Merrien [23].

[发布机构] Stellenbosch University