Final Report Summary - SFSASDA (Spline-like function spaces with applications to scattered data approximations)
We developed mathematical methods for scattered data approximation that can be used to implement efficient algorithms in signal processing and related areas. We considered the problem in spaces generated by translates of compressed or dilated copies of a function (or finitely many functions) along irregularly spaced points. Both the functional analytic side and the computational aspects of the problem have been covered.
After studying the behaviour of 'smooth warping' functions, we found a transformation rule that provides a method for constructing Riesz bases and frames for the proposed spaces by choosing proper affine transformations. The systems of affine transformations of a function have a simple structure and hence there are suitable for numerical implementations. Therefore the proposed technique has a wide range of applications like approximation of geophysical data, wireless communication and medical imaging.
Algorithms for constructing this kind of Riesz bases have been implemented in Matlab for several 'mother' functions (for example splines) and different warping functions. All the experiments showed that the affinely obtained systems and the warped systems are very similar.
The concept of frame is an important background for sampling theory and signal processing. Different from bases, frame decompositions are redundant. This property is advantageous in de-noising, error robustness or sparsity. In acoustics, a typical example of frames of translates are filterbanks. For example the phase vocoder corresponds to such a filterbank with regular shifts, which is often used in signal processing applications like time stretching. Introducing irregular shifts gives rise to a generalisation of this analysis/synthesis system.
We studied properties of a set of irregular translates of a function in L^2(R^d). This was achieved by looking at a set of exponentials restricted to a subset E of R^d with frequencies in a countable set. The results were obtained by analysing which properties of this set of exponentials are preserved when multiplied by the Fourier transform of a function in L^2(E). Using density results due to Beurling, we proved the existence and gave ways to construct frames by irregular translates.
Finite frames arise in many applications, where we usually work in finite dimensional spaces. Dual frames are an essential tool when we want to reconstruct a function. We considered a finite dimensional Hilbert space and introduced the concept of 'mixed frame potential', which generalises the notion of the Benedetto-Fickus frame potential and measures the biorthogonality of two systems of vectors. We characterised the minimisers of this new potential on a restricted domain.
We obtained necessary and sufficient conditions on a real sequence {c_m}_{m=1,¿,N} in order to have a generalised dual pair of frames {f_m}_{m=1,¿,N},{g_m}_{m=1,¿,N} such that =c_m. Moreover we found that the elements of this generalised dual pair of frames can be constructed with any desired norm.
It has been worked on the infinite dimensional version of the Bourgain-Tzafriri restricted invertibility theorem. Results for the wavelet case have been obtained using a result about interpolating sets.
After studying the behaviour of 'smooth warping' functions, we found a transformation rule that provides a method for constructing Riesz bases and frames for the proposed spaces by choosing proper affine transformations. The systems of affine transformations of a function have a simple structure and hence there are suitable for numerical implementations. Therefore the proposed technique has a wide range of applications like approximation of geophysical data, wireless communication and medical imaging.
Algorithms for constructing this kind of Riesz bases have been implemented in Matlab for several 'mother' functions (for example splines) and different warping functions. All the experiments showed that the affinely obtained systems and the warped systems are very similar.
The concept of frame is an important background for sampling theory and signal processing. Different from bases, frame decompositions are redundant. This property is advantageous in de-noising, error robustness or sparsity. In acoustics, a typical example of frames of translates are filterbanks. For example the phase vocoder corresponds to such a filterbank with regular shifts, which is often used in signal processing applications like time stretching. Introducing irregular shifts gives rise to a generalisation of this analysis/synthesis system.
We studied properties of a set of irregular translates of a function in L^2(R^d). This was achieved by looking at a set of exponentials restricted to a subset E of R^d with frequencies in a countable set. The results were obtained by analysing which properties of this set of exponentials are preserved when multiplied by the Fourier transform of a function in L^2(E). Using density results due to Beurling, we proved the existence and gave ways to construct frames by irregular translates.
Finite frames arise in many applications, where we usually work in finite dimensional spaces. Dual frames are an essential tool when we want to reconstruct a function. We considered a finite dimensional Hilbert space and introduced the concept of 'mixed frame potential', which generalises the notion of the Benedetto-Fickus frame potential and measures the biorthogonality of two systems of vectors. We characterised the minimisers of this new potential on a restricted domain.
We obtained necessary and sufficient conditions on a real sequence {c_m}_{m=1,¿,N} in order to have a generalised dual pair of frames {f_m}_{m=1,¿,N},{g_m}_{m=1,¿,N} such that =c_m. Moreover we found that the elements of this generalised dual pair of frames can be constructed with any desired norm.
It has been worked on the infinite dimensional version of the Bourgain-Tzafriri restricted invertibility theorem. Results for the wavelet case have been obtained using a result about interpolating sets.