In this paper we study the general reconstruction of a compactly supported function from its Fourier coefficients using compactly supported shearlet systems. We assume that only finitely many Fourier samples of the function are accessible and based on this finite collection of measurements an approximation is sought in a finite dimensional shearlet reconstruction space. We analyse this sampling and reconstruction process by a recently introduced method called generalized sampling. In particular by studying the stable sampling rate of generalized sampling we then show stable recovery of the signal is possible using an almost linear rate. Furthermore, we compare the result to the previously obtained rates for wavelets.
@cite_6: Abstract In this paper we study the problem of computing wavelet coefficients of compactly supported functions from their Fourier samples. For this, we use the recently introduced framework of generalized sampling. Our first result demonstrates that using generalized sampling one obtains a stable and accurate reconstruction, provided the number of Fourier samples grows linearly in the number of wavelet coefficients recovered. For the class of Daubechies wavelets we derive the exact constant of proportionality. Our second result concerns the optimality of generalized sampling for this problem. Under some mild assumptions we show that generalized sampling cannot be outperformed in terms of approximation quality by more than a constant factor. Moreover, for the class of so-called perfect methods, any attempt to lower the sampling ratio below a certain critical threshold necessarily results in exponential ill-conditioning. Thus generalized sampling provides a nearly-optimal solution to this problem.
@cite_11: In this paper we analyze two-dimensional wavelet reconstructions from Fourier samples within the framework of generalized sampling. For this, we consider both separable compactly-supported wavelets and boundary wavelets. We prove that the number of samples that must be acquired to ensure a stable and accurate reconstruction scales linearly with the number of reconstructing wavelet functions. We also provide numerical experiments that corroborate our theoretical results.
The use of GS to study the reconstruction problem from Fourier measurements has already been investigated for wavelets in 1D @cite_6 and 2D @cite_11 . The authors of the respective works showed, that for dyadic multiresolution analysis (MRA) wavelets the stable sampling rate is linear, meaning up to a constant one has to sample as many Fourier coefficients as one wants to reconstruct wavelet coefficients. Moreover, it was shown in @cite_6 that a defiance of this rate leads to unstable approximations.