Constructive frames-based realizations in time-frequency analysis

The general aim of the project is to advance the field of computational time- frequency analysis through new and nearly optimal constructive approximations that could be implemented on a computer. The project will develop a unifying discretization recipe to guarantee convergence for the sequential or parallel algorithms designed for either spline-type frames, Gabor frames or wavelet frames. The mathematical framework to control the approximations are the function spaces like the Wiener amalgam spaces and especially the Segal algebra S_0(G) as a tool for analyzing the atoms. The new realizable and optimal computational procedures will be employed to analyze experimental data, most of the times corrupted by noise and occasionally almost completely wrong. As for concrete applications, we will use the newly developed realizable constructive procedures in mechanical processing for detecting damages of assets and also in medical image processing.

In this project we have studied mathematical methods to improve the representation and the analysis of functions, signals and images in spline-type spaces. These spaces are a generalization of the standard spaces invariant to shifts. Alongside with the mathematical concepts, we have developed the new techniques from the perspective of their realization on a computer. Following our first goal, we have created a unifying framework for constructive realizable algorithms based on spline-type spaces for Gabor and wavelet frames. The procedure for obtaining multiresolution wavelet-like systems in the framework of spline-type spaces considers only the generators that prominently intersect the spectrum of the signal to be approximated. The system constructed in this way can analyse jointly three different features of a signal namely time, scale and frequency. By reformulating Gabor systems as modulated spline-type systems, a speed boost to the computation of such systems is achieved. Also, we have considered new computational aspects to improve the approximation of HilbertSchmidt operators via generalized Gabor multipliers. One aspect is to consider the approximation of the symbol of a HilbertSchmidt operator as an L2 projection in the spline-type space associated to a Gabor multiplier. The second goal of the project was orientated towards the stability of these new algorithms. While exploring the reformulation of Gabor systems through multi-window spline-type spaces, we have considered the continuous dependence between deformations of a spline-type system through modulations and the approximation error. Then the concept of stability for shift-invariant spaces, interpreted as boundedness from below of the synthesis operator, was explored for general spline-type spaces. Inspired by wavelets on local fields, but using only minimal theory of locally compact Abelian groups, we also explored the stability for multilevel analysis. Our last approach to stability was driven by its definition in the case of numerical solutions of partial differential equations. Initially coined to express the growth of rounding error, in the Lax- Richtmyer theory it has been reformulated as an intrinsic property of the discretization scheme, independent of the particular initial value of the problem. The final goal of the project was related to the application of the new algorithms in engineering and medical problems. We have successfully applied our techniques for the detection of the mandibular canal in panoramic radiographies with a Gabor-Hough algorithm, for the Gabor filtering of 3D medical images and for detecting proteine coding regions using multi-scale splines constructions. Also, we proposed a deep learning exploration of the structural health condition of cantilever beams based on time-frequency extended signatures. Besides these expected results, some complementary results for nD image analysis using topological invariants were obtained.