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 imaging, the standard principles of incoherence and sparsity are lacking. Whilst compressed sensing is often used successfully in such applications, it is done largely without mathematical explanation. The framework introduced in this paper provides such a justification. It does so by replacing these standard principles with three more general concepts: asymptotic sparsity, asymptotic incoherence and multilevel random subsampling. Moreover, not only does this work provide such a theoretical justification, it explains several key phenomena witnessed in practice. In particular, and unlike the standard theory, this work demonstrates the dependence of optimal sampling strategies on both the incoherence structure of the sampling operator and on the structure of the signal to be recovered. Another key consequence of this framework is the introduction of a new structured sampling method that exploits these phenomena to achieve significant improvements over current state-of-the-art techniques.2010 Mathematics Subject Classification: 94A20, 94A08 (primary); 42C40, 65R32, 92C55 (secondary)

Deep learning, due to its unprecedented success in tasks such as image classification, has emerged as a new tool in image reconstruction with potential to change the field. In this paper, we demonstrate a crucial phenomenon: Deep learning typically yields unstable methods for image reconstruction. The instabilities usually occur in several forms: 1) Certain tiny, almost undetectable perturbations, both in the image and sampling domain, may result in severe artefacts in the reconstruction; 2) a small structural change, for example, a tumor, may not be captured in the reconstructed image; and 3) (a counterintuitive type of instability) more samples may yield poorer performance. Our stability test with algorithms and easy-to-use software detects the instability phenomena. The test is aimed at researchers, to test their networks for instabilities, and for government agencies, such as the Food and Drug Administration (FDA), to secure safe use of deep learning methods.

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 ill-suited for solving a number of important problems.This work stems from recent developments in generalized sampling theorems for classical (Nyquist rate) sampling that allows for reconstructions in arbitrary bases. The main conclusion of this paper is that one can extend these ideas to allow for significant subsampling of sparse or compressible signals. The key to these developments is the introduction of two new concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize the fundamentally infinitedimensional reconstruction problem.

In a previous paper [4] we described the numerical properties of function approximation using frames, i.e. complete systems that are generally redundant but provide infinite representations with coefficients of bounded norm. Frames offer enormous flexibility compared to bases. We showed that, in spite of extreme ill-conditioning, a regularized projection onto a finite truncated frame can provide accuracy up to order √ ǫ, where ǫ is an arbitrarily small threshold. Here, we generalize the setting in two ways. First, we assume information or samples from f from a wide class of linear operators acting on f , rather than inner products with the frame elements. Second, we allow oversampling, leading to least-squares approximations. The first property enables the analysis of fully discrete approximations based, for instance, on function values only. We show that the second property, oversampling, crucially leads to much improved accuracy on the order of ǫ rather than √ ǫ. Overall, we demonstrate that numerical function approximation using truncated frames leads to highly accurate approximations in spite of having to solve ill-conditioned systems of equations. Once the approximations start to converge, i.e. once sufficiently many degrees of freedom are used, any function f can be approximated to within order ǫ with coefficients of small norm.

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 reconstruction technique (Eldar et al). However, whilst the latter imposes stringent assumptions on the reconstruction basis, and may in practice be unstable, our framework allows for recovery in any (Riesz) basis in a manner that is completely stable.Whilst the classical Shannon Sampling Theorem is a special case of our theorem, this framework allows us to exploit additional information about the approximated vector (or, in this case, function), for example sparsity or regularity, to design a reconstruction basis that is better suited. Examples are presented illustrating this procedure.