# Cardinal spline wavelet decomposition based on quasi-interpolation and local projection

Thesis (MSc (Mathematics))--University of Stellenbosch, 2009.

Thesis

Wavelet decomposition techniques have grown over the last two decades into a powerful tool in signal analysis. Similarly, spline functions have enjoyed a sustained high popularity in the approximation of data. In this thesis, we study the cardinal B-spline wavelet construction procedure based on quasiinterpolation and local linear projection, before specialising to the cubic B-spline on a bounded interval. First, we present some fundamental results on cardinal B-splines, which are piecewise polynomials with uniformly spaced breakpoints at the dyadic points Z/2r, for r ∈ Z. We start our wavelet decomposition method with a quasi-interpolation operator Qm,r mapping, for every integer r, real-valued functions on R into Sr m where Sr m is the space of cardinal splines of order m, such that the polynomial reproduction property Qm,rp = p, p ∈ m−1, r ∈ Z is satisfied. We then give the explicit construction of Qm,r. We next introduce, in Chapter 3, a local linear projection operator sequence {Pm,r : r ∈ Z}, with Pm,r : Sr+1 m → Sr m , r ∈ Z, in terms of a Laurent polynomial m solution of minimally length which satisfies a certain Bezout identity based on the refinement mask symbol Am, which we give explicitly. With such a linear projection operator sequence, we define, in Chapter 4, the error space sequence Wr m = {f − Pm,rf : f ∈ Sr+1 m }. We then show by solving a certain Bezout identity that there exists a finitely supported function m ∈ S1 m such that, for every r ∈ Z, the integer shift sequence { m(2 · −j)} spans the linear space Wr m . According to our definition, we then call m the mth order cardinal B-spline wavelet. The wavelet decomposition algorithm based on the quasi-interpolation operator Qm,r, the local linear projection operator Pm,r, and the wavelet m, is then based on finite sequences, and is shown to possess, for a given signal f, the essential property of yielding relatively small wavelet coefficients in regions where the support interval of m(2r · −j) overlaps with a Cm-smooth region of f. Finally, in Chapter 5, we explicitly construct minimally supported cubic B-spline wavelets on a bounded interval [0, n]. We also develop a corresponding explicit decomposition algorithm for a signal f on a bounded interval. ii Throughout Chapters 2 to 5, numerical examples are provided to graphically illustrate the theoretical results.