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We introduce a generalized framework for sampling and reconstruction in separable Hilbert spaces. Specifically, we establish that it is always possible to stably reconstruct a vector in an arbitrary Riesz basis from sufficiently many of its samples in any other Riesz basis. This framework can be viewed as an extension of the well-known consistent… (More)

- Ben Adcock, Anders C. Hansen
- Foundations of Computational Mathematics
- 2016

We introduce and analyze an abstract framework, and corresponding method, for compressed sensing in infinite dimensions. This extends the existing theory from signals in finite-dimensional vectors spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary, and demonstrate that existing finite-dimensional techniques are… (More)

- Ben Adcock, Anders C. Hansen, Clarice Poon, Bogdan Roman
- ArXiv
- 2013

We introduce a mathematical framework that bridges a substantial gap between compressed sensing theory and its current use in real-world applications. Although completely general, one of the principal applications for our framework is the Magnetic Resonance Imaging (MRI) problem. Our theory provides a comprehensive explanation for the abundance of numerical… (More)

This paper presents a framework for compressed sensing that bridges a gap between existing theory and the current use of compressed sensing in many real-world applications. In doing so, it also introduces a new sampling method that yields substantially improved recovery over existing techniques. In many applications of compressed sensing, including medical… (More)

- ANDERS C. HANSEN
- 2010

We show that it is possible to compute spectra and pseudospectra of linear operators on separable Hilbert spaces given their matrix elements. The core in the theory is pseudospectral analysis and in particular the n-pseudospectrum and the residual pseudospecrum. We also introduce a new classification tool for spectral problems, namely, the Solvability… (More)

We introduce a simple and efficient method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we establish, provided the dimension of the reconstruction space is chosen suitably in relation to the… (More)

- Ben Adcock, Anders C. Hansen, Clarice Poon
- SIAM J. Math. Analysis
- 2013

Generalized sampling is a recently developed linear framework for sampling and reconstruction in separable Hilbert spaces. It allows one to recover any element in any finite-dimensional subspace given finitely many of its samples with respect to an arbitrary frame. Unlike more common approaches for this problem, such as the consistent reconstruction… (More)

- Bogdan Roman, Anders C. Hansen, Ben Adcock
- ArXiv
- 2014

This paper demonstrates how new principles of compressed sensing, namely asymptotic incoherence, asymptotic sparsity and multilevel sampling, can be utilised to better understand underlying phenomena in practical compressed sensing and improve results in real-world applications. The contribution of the paper is fourfold: First, it explains how the sampling… (More)

- Alexander Bastounis, Anders C. Hansen
- 2015 International Conference on Sampling Theory…
- 2015

Compressed sensing (CS) is one of the great successes of computational mathematics in the past decade. There are a collection of tools which aim to mathematically describe compressed sensing when the sampling pattern is taken in a random or deterministic way. Unfortunately, there are many practical applications where the well studied concepts of uniform… (More)

- Ben Adcock, Anders C. Hansen, Bogdan Roman, Gerd Teschke
- ArXiv
- 2013

The purpose of this paper is to report on recent approaches to reconstruction problems based on analog, or in other words, infinite-dimensional, image and signal models. We describe three main contributions to this problem. First, linear reconstructions from sampled measurements via so-called generalized sampling (GS). Second, the extension of generalized… (More)