Broadly speaking, my group’s work concerns the reconstruction of objects (images, signals, functions, etc) from data. Applications range from image and signal processing to high-dimensional approximation and the numerical solution of PDEs. I am particularly interested in the development, application and analysis of sparse recovery techniques in these areas. I also have ongoing interests in high-order methods in approximation and numerical PDEs, image and signal processing, uncertainty quantification, sampling theory, computational harmonic analysis, mathematics of information and data science.

Listed below are some short summaries of my group’s areas of research. For links to the papers cited, please see my Publications page.

## Current Areas of Research

**Local structure in compressed sensing**

Standard compressed sensing (CS) is based on global principles: sparsity and incoherence. However, in many problems where CS is applied, e.g. imaging, these principles are poorly suited.

Read more!This work was recently featured in SIAM News.

**Relevant Papers**

- B. Adcock, A. C. Hansen, C. Poon and B. Roman,
*Breaking the coherence barrier: a new theory for compressed sensing*. - B. Adcock, A. C. Hansen, and B. Roman,
*The quest for optimal sampling: computationally efficient, structure-exploiting measurements for compressed sensing.* - B. Roman, B. Adcock and A. C. Hansen,
*On asymptotic structure in compressed sensing*.

**Infinite-dimensional compressed sensing**

Standard compressed sensing (CS) concerns vectors and matrices. However, many real-life signals and images are analog/continuous-time, and therefore better modelled as functions in function spaces acted on by linear operators. Applying discrete CS techniques to continuous problems can is plagued by poor reconstructions and issues such as the inverse crime. This raises the question of whether or not one can extend CS techniques and theory to infinite dimensions.

Read more!**Relevant papers**

- B. Adcock and A. C. Hansen,
*Generalized sampling and infinite-dimensional compressed sensing.* - B. Adcock, A. C. Hansen, B. Roman and G. Teschke,
*Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum.*

**High-dimensional approximation via compressed sensing**

Many applications, including uncertainty quantification and parametric modelling, call for the accurate approximation of smooth, high-dimensional functions. Due to phenomena such as the curse of dimensionality, this is often a challenging task. Compressed sensing offers a way to ameliorate (or even circumvent) this issue by computing sparse approximations in terms of tensor products of orthogonal polynomials.

Read more!**Relevant papers**

- B. Adcock,
*Infinite-dimensional compressed sensing and function interpolation.* - B. Adcock,
*Infinite-dimensional l1 minimization and function approximation from pointwise data.* - B. Adcock, S. Brugiapaglia and C. Webster,
*Compressed sensing approaches for polynomial approximation of high-dimensional functions*.

**Sparse regularization in medical imaging**

Sparse regularization techniques have the potential to significantly enhance reconstruction quality and/or reduce scan time in medical imaging. Our work in this areas seeks to further enhance these techniques in applications such as parallel MRI through the development of fast algorithms that exploit additional structure beyond sparsity.

Read more!**Relevant papers**

- I. Y. Chun, B. Adcock and T. Talavage,
*Efficient compressed sensing SENSE pMRI reconstruction with joint sparsity promotion*. - I. Y. Chun, B. Adcock and T. Talavage,
*Non-Convex Compressed Sensing CT Reconstruction Based on Tensor Discrete Fourier Slice Theorem*.

**Frames and numerical approximation**

Orthogonal bases are ubiquitous in numerical approximation. However, there are many problems where finding a ‘good’ (e.g. rapidly-convergent or computationally-efficient) orthonormal basis is difficult or impossible. One such example is finding global high-order approximations on irregular domains. However, in many of these cases one can easily construct frames with the desired properties. Frames are standard tools in image and signal processing and sampling theory. But they are far less well known in numerical approximation.

Read more!**Relevant papers**

- B. Adcock and D. Huybrechs,
*Frames and numerical approximation.* - B. Adcock, D. Huybrechs and J. Martin-Vaquero,
*On the numerical stability of Fourier extensions*.

## Past Areas of Research

**Generalized Sampling**

The classical result in sampling theory, the Nyquist-Shannon Sampling Theorem, states that a bandlimited signal can be recovered from countably-many equispaced samples taken at or above a certain critical rate (the Nyquist rate). However, in practice one never has access to infinitely-many samples. This raises a different question: how well can one stably recover a given signal from finitely-many such samples?

Read more!**Relevant papers**

- B. Adcock and A. C. Hansen,
*A generalized sampling theorem for stable reconstructions in arbitrary bases*. - B. Adcock, A. C. Hansen and C. Poon,
*Beyond consistent reconstructions: optimality and sharp bounds for generalized sampling, and application to the uniform resampling problem*. - B. Adcock, A. C. Hansen, E. Herrholz and G. Teschke,
*Generalized sampling: extensions to frames and inverse and ill-posed problems*. - B. Adcock, A. C. Hansen and C. Poon,
*On optimal wavelet reconstructions from Fourier samples: linearity and universality of the stable sampling rate*. - B. Adcock, M. Gataric and A. C. Hansen,
*On stable reconstructions from nonuniform Fourier measurements*. - B. Adcock, M. Gataric and A. C. Hansen,
*Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples*. - B. Adcock, M. Gataric and A. C. Hansen,
*Density theorems for nonuniform sampling of bandlimited functions using derivatives or bunched measurements*.

**High-order function approximation using variable transforms**

Variable transforms are useful in a variety of numerical computations to increase accuracy, efficiency and/or stability. Our work in this area aims to develop new variable transform techniques for a range of practical approximation problems.

Read more!**Relevant papers**

- B. Adcock and R. Platte,
*A mapped polynomial method for high-accuracy approximations on arbitrary grids.* - B. Adcock and M. Richardson,
*New exponential variable transform methods for functions with endpoint singularities.* - B. Adcock, M. Richardson and J. Martin-Vaquero,
*Resolution-optimal exponential and double-exponential transform methods for functions with endpoint singularities.*

**High-order function approximation with non-standard Fourier series**

In a number of different applications, such as spectral methods for PDEs, scattered data approximation, etc, one seeks to recover a smooth function to high accuracy from its values on a given set of points. Our work in this area involves the use of certain non-standard Fourier series for such reconstructions.

Read more!**Relevant papers**

- B. Adcock, D. Huybrechs and J. Martin-Vaquero,
*On the numerical stability of Fourier extensions.* - B. Adcock and J. Ruan,
*Parameter selection and numerical approximation properties of Fourier extensions from fixed data.* - B. Adcock and D. Huybrechs,
*On the resolution power of Fourier extensions for oscillatory functions.* - B. Adcock, A. Iserles and S. P. Nørsett,
*From high oscillation to rapid approximation II: Expansions in Birkhoff series.* - B. Adcock,
*On the convergence of expansions in polyharmonic eigenfunctions.* - B. Adcock,
*Multivariate modified Fourier series and application to boundary value problems.*

**Resolution of the Gibbs phenomenon**

The Gibbs phenomenon occurs when a piecewise smooth function is expanded as a Fourier series. Characteristic features are oscillations near the discontinuities and lack of uniform convergence of the expansion. Resolution of the Gibbs phenomenon refers to postprocessing the Fourier coefficients to obtain higher orders of convergence, and has applications to both image processing and the numerical solution of PDEs.

Read more!**Relevant papers**

- B. Adcock, A. C. Hansen and A. Shadrin,
*A stability barrier for reconstructions from Fourier samples*. - B. Adcock and A. C. Hansen,
*Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon*. - B. Adcock and A. C. Hansen,
*Generalized sampling and the stable and accurate reconstruction of piecewise analytic functions from their Fourier coefficients*. - B. Adcock,
*Gibbs phenomenon and its removal for a class of orthogonal expansions*. - B. Adcock,
*Convergence acceleration of modified Fourier series in one or more dimensions*.