Ben Adcock

Generalized Sampling and Infinite-Dimensional Compressed Sensing

We introduce and analyze a framework and corresponding method for compressed sensing in infinite dimensions. This extends the existing theory from finite-dimensional vector spaces to the case of separable Hilbert spaces. We explain why such a new theory is necessary by demonstrating that existing finite-dimensional techniques are ill suited for solving a number of key problems. This work stems from recent developments in generalized sampling theorems for classical (Nyquist rate) sampling that allows for reconstructions in arbitrary bases. A conclusion of this paper is that one can extend these ideas to allow for significant subsampling of sparse or compressible signals. Central to this work is the introduction of two novel concepts in sampling theory, the stable sampling rate and the balancing property, which specify how to appropriately discretize an infinite-dimensional problem.

A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases

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.

Beyond Consistent Reconstructions: Optimality and Sharp Bounds for Generalized Sampling, and Application to the Uniform Resampling Problem

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 basis or frame. Unlike more common approaches for this problem, such as the consistent reconstruction technique of Eldar and others, it leads to numerical methods possessing both guaranteed stability and accuracy. The purpose of this paper is twofold. First, we give a complete and formal analysis of generalized sampling, the main result of which being the derivation of new, sharp bounds for the accuracy and stability of this approach. Such bounds improve upon those given previously and result in a necessary and sufficient condition, the stable sampling rate, which guarantees a priori a good reconstruction. Second, we address the topic of optimality. Under some assumptions, we show that generalized sampling is an optimal, stable method. Correspondingly...

On the resolution power of Fourier extensions for oscillatory functions

Functions that are smooth but non-periodic on a certain interval possess Fourier series that lack uniform convergence and suffer from the Gibbs phenomenon. However, they can be represented accurately by a Fourier series that is periodic on a larger interval. This is commonly called a Fourier extension. When constructed in a particular manner, Fourier extensions share many of the same features of a standard Fourier series. In particular, one can compute Fourier extensions which converge spectrally fast whenever the function is smooth, and geometrically fast if the function is analytic, much the same as the Fourier series of a smooth/analytic and periodic function. With this in mind, the purpose of this paper is to describe, analyse and explain the observation that Fourier extensions, much like classical Fourier series, also have excellent resolution properties for representing oscillatory functions. The resolution power, or required number of degrees of freedom per wavelength, depends on a user-controlled parameter and, as we show, it varies between 2 and @p. The former value is optimal and is achieved by classical Fourier series for periodic functions, for example. The latter value is the resolution power of algebraic polynomial approximations. Thus, Fourier extensions with an appropriate choice of parameter are eminently suitable for problems with moderate to high degrees of oscillation.

Generalized Sampling: Stable Reconstructions, Inverse Problems and Compressed Sensing over the Continuum

Abstract 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 sampling to inverse and ill-posed problems. And third, the combination of generalized sampling with sparse recovery techniques. This final contribution leads to a theory and set of methods for infinite-dimensional compressed sensing, or as we shall also refer to it, compressed sensing over the continuum.

A Stability Barrier for Reconstructions from Fourier Samples

We prove that any stable method for resolving the Gibbs phenomenon---that is, recovering high-order accuracy from the first

On Stable Reconstructions from Nonuniform Fourier Measurements

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Generalized sampling: extension to frames and inverse and ill-posed problems

Fourier coefficients of an analytic and nonperiodic function---can converge at best root-exponentially fast in

Linear Stable Sampling Rate: Optimality of 2D Wavelet Reconstructions from Fourier Measurements

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Efficient Compressed Sensing SENSE pMRI Reconstruction With Joint Sparsity Promotion

. Any method with faster convergence must also be unstable, and in particular, exponential convergence implies exponential ill-conditioning. This result is analogous to a recent theorem of Platte, Trefethen, and Kuijlaars concerning recovery from pointwise function values on an equispaced