Abstract:
This dissertation sets a novel approach to analyze second generation wavelet schemes by providing
a basis function and decomposition method. Moreover, the representation of gray–scale images
with normal multiresolution approximation in less smooth spaces, such as Besov spaces, Bα
p,q(Ω),
1 ≤ p, q ≤ ∞ for 0 < α < 1, where Ω be a Lipschitz domain in Rd
, d ≥ 1, and d is odd.
The assumption is that a normal multiresolution approximation is parameterized in a regular
interval, and then with the Lagrangian interpolation formula a basis function is constructed by
using Hardy’s multiquadric function. The basis function is shift–invariant and, generates a space
Sj = span{ϕ(2jx − k) : for all x ∈ R and for all k ∈ Z} for j ∈ N0. Approximation properties
of this setting is explored in Sobolev spaces.
Since the above basis function does not satisfy the requirements of a compact support; it is
resort to consider the second divided difference of the basis function. Thus, the wavelet transform
on the real line is defined on the basis of quasi–interpolating basis function. In addition, the
local properties of the function are also studied; for instance, the case of pointwise convergence.
As such, the above stated basis function is generalized to multivariate setting in a bounded
simply connected domain Ω ⊂ Rd
, d ≥ 1, with the localization concept of multiquadric functions
and 1–unisolvence property. Thus, the characterization of the Besov spaces, Bα
p,q(Ω), in terms of
vertical offset coefficients of functions with respect to these bases. As a consequence, it is seen
that Horizon images with 0 < α < 1 are characterized by the coefficients with respect to these
normal wavelet basis functions.
As an application of the multiquadric basis function, an efficient image compression scheme,
called Normal Multiresolution Triangulation Interpolation scheme, is presented in this dissertation.
