Optimisation in the construction of multidimensional wavelets

Wavelet transforms overcome the limitation of Fourier analysis in accessing localised time-frequency contents of a signal and in dealing with non-stationary signals. After the celebrated construction of Daubechies' compactly supported, smooth and orthonormal wavelets with multiresolution structure, a plethora of research works focused on higher-dimensional extensions and constructions of wavelets with properties tailored to particular applications. Recently, the construction of smooth, compactly supported, orthonormal and complex-valued multidimensional wavelets has been formulated as feasibility problems that are subsequently solved by employing projection algorithms. In another timeline, the development of Clifford-Fourier transforms has laid down the foundation for generalising the classical Fourier and wavelet analyses to provide the basic theory required for the construction of quaternion-valued wavelets with compact support, prescribed regularity, orthonormality and multiresolution structure. In this thesis, we present an optimisation approach to wavelet architecture that hinges on the Zak transform to formulate the construction as a minimisation problem. Such an approach allows for construction of scaling functions and wavelets on the line with symmetry and cardinality properties. We also extend the feasibility approach to wavelet construction by adding symmetry and cardinality in the design criteria. Solutions admit compactly supported, smooth, orthogonal wavelets on the line and on the plane with symmetry and cardinality properties. In the context of solving feasibility problems, we introduce a reformulation technique that converts many-set feasibility problems into their equivalent two-set problems. In dealing with convergence analyses of constraint-reduced variants of projection algorithms, we generalise a classical result which guarantees that the composition of two projectors onto subspaces is a projector onto their intersection. We also consider heuristic approaches to solving feasibility problems by setting up two-stage global-then-local search methods - pairing up the Douglas-Rachford algorithm with recently introduced centering methods. Finally, we present a feasibility problem formulation for wavelet construction yielding novel examples of smooth, compactly supported, orthonormal and quaternion-valued wavelets on the plane.