Broadly speaking, my group investigates the recovery of objects (images, signals, functions, etc) from data. We pursue applications in areas such as image and signal processing, Uncertainty Quantification, and the numerical solution of PDEs. We are particularly interested in the development, analysis and implementation of sparse regularization techniques for these problems and the associated compressed sensing (CS) theory. Our recent interests include the application of machine learning techniques – in particular, neural networks and deep learning – to problems in computational science. We also have ongoing interests in high-dimensional approximation, image and signal processing, high-order methods in approximation and numerical PDEs, computational harmonic analysis and sampling theory.

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

## Current Areas of Research

**Compressive imaging and local structure**

Image reconstruction has been one of the key beneficiaries of sparse regularization and CS. Standard CS is based on global principles: sparsity and incoherence. However, these principles do not adequately capture the structure inherent to most imaging problems. In such problems, it turns out that local structure plays a crucial role. Leveraging this additional structure – through the design of optimized sampling strategies and recovery algorithms – we obtain significant gains in reconstruction quality.

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In a series of papers we introduced a new mathematical framework for CS based on broader and more realistic local principles. This theory not only explains why CS works so well in imaging applications, it also opens the door for substantially improved approaches in a range of practical applications through the design of optimized sampling strategies. Applications include single-pixel/lensless imaging, MRI, NMR, radio interferometry, fluorescence microscopy and beyond. Practical implementations of these ideas have been carried out in [3], as well as by Siemens (Wang et al, 2014); a major manufacturer of MR scanners.

This work was recently featured on the cover of 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*. - C. Li and B. Adcock,
*Compressed sensing with local structure: uniform recovery guarantees for the sparsity in levels class.*

**High-dimensional approximation and applications in computational science**

The challenge of approximating a high-dimensional function from data arises in many areas of computational science. This is often a key task in Uncertainty Quantification (UQ), for example, and/or the numerical solution of parametric PDEs. This problem is rendered difficult by the well-known curse of dimensionality. Classical approaches, such as projection, interpolation and least-squares, are known to have significant drawbacks in such scenarios. Conversely, in the last five years, CS has begun to emerge as a promising way to ameliorate (or even circumvent) this issue.

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My group’s work in this direction has been twofold. First, we have developed a set of weighted l1 regularization strategies for computing sparse polynomial approximations of smooth, high-dimensional functions. As shown in [1,2], these techniques provably overcome the curse of dimensionality (up to a logarithmic factor in the dimension) and are optimal, in the sense that they achieve the same asymptotic sample complexities as oracle least-squares estimators. Second, we have sought to address the inherent infinite-dimensionality of the approximation problem by proposing an infinite-dimensional CS framework [3]. Ongoing work includes the development of optimal sampling strategies, extending this framework to more complicated approximation problems, including derivative sampling [4] and problems on irregular domains [5], investigating and developing new optimization algorithms to handle corrupted data [6,7], and pursuing applications in UQ.

This work was recently featured by SFU Science.

**Relevant papers**

- B. Adcock,
*Infinite-dimensional compressed sensing and function interpolation.* - B. Adcock, S. Brugiapaglia and C. Webster,
*Compressed sensing approaches for polynomial approximation of high-dimensional functions*. - B. Adcock,
*Infinite-dimensional l1 minimization and function approximation from pointwise data.* - B. Adcock and Y. Sui,
*Compressive Hermite interpolation: sparse, high-dimensional approximation from gradient-augmented measurements*. - B. Adcock and D. Huybrechs,
*Approximating smooth, multivariate functions on irregular domains.* - B. Adcock, A. Bao, J. Jakeman and A. Narayan,
*Compressed sensing with sparse corruptions: Fault-tolerant sparse collocation approximations.* - B. Adcock, A. Bao and S. Brugiapaglia.
*Correcting for unknown errors in sparse high-dimensional function approximation.*

**Relevant talks**

- Compressed sensing and high-dimensional approximation: progress and challenges, iTWIST 2018, Marseille, France, November 2018.
- Polynomial approximation of high-dimensional functions: from regular to irregular domains. Workshop on Surrogate models for UQ in complex systems, Isaac Newton Institute, Cambridge, UK, February 2018.

** Numerical approximation with redundancy**

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 approximation systems on irregular domains, another is constructing approximation systems that incorporate a finite number of ‘feature functions’ (e.g. singularities or oscillations). However, in many of these cases one can easily redundant systems of functions with the desired properties. Redundant systems (dictionaries) are standard tools in image and signal processing and sampling theory. But they are far less well understood in numerical analysis and approximation.

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My group and collaborators are developing the theory, implementation and application of numerical methods based on redundant systems for a range of different problems. These include efficient approximation and PDE solvers on irregular domains in two and more dimensions, numerical methods for treating singularities and beyond. Our work has typically assumed the underlying system is a so-called frame. The basic theory of numerical frame approximation was introduced in [1,2], based on earlier work on so-called Fourier extensions [3]. Applications to multivariate approximation on irregular domains are considered in [4].

**Relevant papers**

- B. Adcock and D. Huybrechs,
*Frames and numerical approximation.* - B. Adcock and D. Huybechs,
*Frames and numerical approximation II: generalized sampling.* - B. Adcock, D. Huybrechs and J. Martin-Vaquero,
*On the numerical stability of Fourier extensions*. - B. Adcock and D. Huybrechs,
*Approximating smooth, multivariate functions on irregular domains.*

**Sparse regularization and parallel acquisition: medical imaging and beyond**

Sparse regularization techniques have the potential to significantly enhance reconstruction quality and/or reduce acquisition time in many applications. Typically, one considers a scenario of single sensor acquiring measurements of an object. However, many practical acquisition systems involve multiple sensors acting in parallel. This is the case, for instance, in parallel MRI, where multiple coils simultaneously acquire Fourier data of the image.

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Our work in this area seeks to further enhance sparse regularization for parallel acquisition systems through the development of fast algorithms that exploit additional structure beyond sparsity. In [1] we introduced a new regularization which promotes joint sparsity between the measured coil images, and a fast algorithm for its implementation. This approach improves reconstruction accuracy over the current state-of-the-art techniques. Ongoing work involves the further development of this model and the development of new CS theory [3,4] to optimize its performance in applications.

**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*. - I. Y. Chun and B. Adcock,
*Compressed sensing and parallel acquisition.* - I. Y. Chun and B. Adcock,
*Uniform recovery from subgaussian multi-sensor measurements.*

## Past Areas of Research

**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 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.

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In [1] we introduced a framework for CS in Hilbert spaces, based on the ideas of generalized sampling (see below). Whilst finite-dimensional CS is a special case of this more general theory, applying this framework directly to the underlying continuous model allows one to avoid the aforementioned issues and thereby obtain higher-fidelity reconstructions. Practical validation of these ideas have been carried out in applications such as electron microscopy and Helium Atom Scattering [3].

**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.* - B. Roman, B. Adcock and A. C. Hansen,
*On asymptotic structure in compressed sensing*.

**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?

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In [1] we introduced a new sampling framework to address this issue, known as generalized sampling (GS). An important concept in GS, introduced in [2], is the stable sampling rate (SSR), which specifies the relationship between the number of samples and the degrees of freedom in the reconstruction. Linearity of the SSR for wavelet spaces was shown in [4]. We also extended the original GS framework to the case of nonuniform samples [5,6], ill-posed problems [3] and sampling with derivatives [7]. Open problems include further generalizations to other function spaces and to other types of sampling and reconstruction systems.

**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.

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First, we developed new parametrized mapping techniques for the efficient approximation of functions with endpoint singularities [2,3]. Second, we used certain parametrized conformal mappings to better approximate functions from scattered data [1]. Key aspects of this work are the analysis of stability and convergence of the various approximations uniformly in the mapping parameters, and the understanding of how such techniques yield high accuracy without necessarily converging in a classical sense.

**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.

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An attractive means to do this is to use a Fourier series on a larger domain, known as the Fourier extension technique. In [1,2,3] we provided analysis for the convergence and stability of Fourier extensions and in particular, established their benefits for approximating oscillatory functions. Another approach is to use expansions in various eigenfunctions [4,5,6], which are particularly well suited in higher dimensions.

**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.

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In [1] we proved a stability barrier for this problem, and in [2,3] several approaches for attaining this barrier were proposed. The Gibbs phenomenon in related eigenfunction expansions was also studied in [4,5].

**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*.