Normal Multiresolution Approximation of Piecewise Smooth Images

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.