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_36: The authors describe two different applications of wavelet transforms in magnetic resonance imaging (MRI). In each case, the use of wavelets offers tangible benefits over the traditional Fourier-transform-based imaging. These stem from the simultaneous time and frequency localization properties of the wavelets. The first application utilizes the localization properties of wavelets to acquire single T sub 2 weighted images in times which are short relative to current methods. An interesting combinatorial scheduling problem arises from the multiscale structure inherent in the formulation of this procedure. Its solution is sketched and applied to the imaging process. The second example is a technique for potentially 'instant' imaging. Image acquisition is modeled as an inverse wavelet transform of the spin density in the region in question. Signal acquisition takes place in as few as one or two echoes after the excitation by RF pulses designed to excite a moving band of spins in a slice. These sweeping RF pulses excite strips of spins which rephase at different times during acquisition. They replace phase-encoding gradients; no fast pulsed gradients are required. >
@cite_34: A theoretical comparison of the wavelet and Fourier encoding methods is made with respect to resolution, sensitivity to artifact, and signal-to-noise ratios (SNR's). A general mathematical description is developed in which magnetic resonance (MR) image encoding is represented by a projection of the function representing the "MR signal density" onto an approximation subspace of the finite energy functions. Characteristics of the subspace are used to define a "generalized" point-spread function for space-variant systems. Using the formal model of MR image encoding it is shown that wavelet encoding approaches the resolution limit defined by Fourier encoding. Artifact is treated according to whether or not the source of the variation in the measured data is stationary. Nonstationary imperfections perturb the projection operation and result in encoding method-dependent effects which can be modeled by a distortion matrix suitable for treating shift variant systems. A framework is developed in which to derive expressions for SNR's applicable to a general class of MR encoding methods.
To enhance the quality of the Fourier inversion of the sampled data, great effort has been done in applications to improve the reconstruction process by changing the reconstruction basis or adapting the sampling process. For example in @cite_36 the authors introduced which allows the user to directly sample wavelet coefficients instead of Fourier coefficients by adapting the acquisition device. However, as for now this method also has several shortcomings such as low signal-to-noise ratio @cite_34 . On the other hand, the proposed method GS can be understood as a post processing method that does not require to change the acquisition device since the sparsifying transform will be incorporated in the reconstruction process and not in the sampling process.