Compressed Sensing in Hilbert Spaces

Traonmilin, Yann; Puy, Gilles; Gribonval, Rémi; Davies, Mike E.

doi:10.1007/978-3-319-69802-1_12

Cited by 42 publications

(2 citation statements)

References 39 publications

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“…Most of the studies in this field are confined to the case where both B and E are finite dimensional [9,13,17,19]. In the last few years, some efforts have been provided to get a better understanding of (1) and (2) where B and E are sequence spaces [2,3,29,28]. Finally, a different route, which will be followed in this paper, is the case where E = M, the space of Radon measures on a continuous domain.…”

Section: Introductionmentioning

confidence: 99%

Exact solutions of infinite dimensional total-variation regularized problems

Flinth

Weiss

2018

Information and Inference: A Journal of the IMA

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We study the solutions of infinite dimensional linear inverse problems over Banach spaces. The regularizer is defined as the total variation of a linear mapping of the function to recover, while the data fitting term is a near arbitrary function. The first contribution describes the solution's structure: we show that under mild assumptions, there always exists an m-sparse solution, where m is the number of linear measurements of the signal. Our second contribution is about the computation of the solution. While most existing works first discretize the problem, we show that exact solutions of the infinite dimensional problem can be obtained by solving one or two consecutive finite dimensional convex programs depending on the measurement functions structures. These results extend recent advances in the understanding of total-variation regularized inverse problems.

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Section: Introductionmentioning

confidence: 99%

Exact solutions of infinite dimensional total-variation regularized problems

Flinth

Weiss

2018

Information and Inference: A Journal of the IMA

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“…In the 90s, it has been recognized that 1 regularizations are much better for providing sparse and interpretable reconstructions, leading to the LASSO [4] and the basis pursuit [5] and then inspiring the field of compressed sensing [6][7][8][9][10]. These ideas have been initially developed in finite dimension, and have been extended to infinite-dimensional settings by several authors [11][12][13][14][15][16][17].…”

Section: Comparison With Previous Work and Motivationmentioning

confidence: 99%

TV-based reconstruction of periodic functions

Fageot

Simeoni

2020

Inverse Problems

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We introduce a general framework for the reconstruction of periodic multivariate functions from finitely many and possibly noisy linear measurements. The reconstruction task is formulated as a penalized convex optimization problem, taking the form of a sum between a convex data fidelity functional and a sparsity-promoting total variation based penalty involving a suitable spline-admissible regularizing operator L. In this context, we establish a periodic representer theorem, showing that the extreme-point solutions are periodic L-splines with less knots than the number of measurements. The main results are specified for the broadest classes of measurement functionals, spline-admissible operators, and convex data fidelity functionals. We exemplify our results for various regularization operators and measurement types (e.g., spatial sampling, Fourier sampling, or square-integrable functions). We also consider the reconstruction of both univariate and multivariate periodic functions.

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Compressive Imaging: Structure, Sampling, Learning

Adcock

Hansen

2021

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Images are everywhere. Whether on a social media page, a doctor's desk to aid diagnosis or a scientist's computer screen to help study a chemical, physical or biological process, the modern world is awash with digital images.All images are produced by acquiring measurements using a physical device. The list of different acquisition devices is long and varied. It includes simple devices found in most homes, such as digital cameras; specialist medical imaging equipment found in hospitals, such as a Magnetic Resonance Imaging (MRI) or X-ray Computed Tomography (CT) scanner; or scientific devices, such as electron microscopes, found in laboratories.The concern of this book is the task of image reconstruction. This is the algorithmic process of converting the raw data (the measurements) into the final image seen by the end user. The overarching aim of image reconstruction is to achieve the four following, and competing, objectives:Objective #1 (accuracy): to produce the highest-quality images.Accuracy is of course paramount. High-quality images are desirable in virtually all applications. However, in direct competition with this is:Objective #2 (sampling): to use as few measurements as possible.Acquiring more measurements usually comes at a cost. It could mean an additional outlay of time, power, monetary expense or risk, depending on the application at hand. Reducing the number of measurements is often the primary goal of image reconstruction. For example, in MRI, taking more measurements involves a longer scan time, which can be unpleasant and challenging for the patient -especially in paediatric MRI. It also makes the measurements acquired more susceptible to corruptions due, for instance, to patient motion. In X-ray CT, the number of measurements loosely corresponds to the amount of radiation to which the patient is exposed. Acquiring fewer measurements per scan opens the door for more frequent scans, which in turn allows for more effective treatment monitoring.Objective #3 (stability): to ensure that errors in the measurements or in the numerical computation do not significantly impact the quality of the recovered image.All imaging systems introduce error in the measurements, due to noise, corruptions or modelling assumptions. There are also round-off errors in the numerical computations performed by image reconstruction algorithms. It is vital that reconstruction algorithmsIn Fourier imaging the (noiseless) measurements y ∈ C m correspond to samplesis an important example of Fourier imaging. Key Point #10. Current deep learning strategies for compressive imaging are prone to instabilities, with certain small perturbations in the measurements leading to severe image artefacts. This arises because of the training procedure, which can cause the resulting network to over-perform. 1.7.4 Stable and Accurate Neural Networks for Compressive Imaging This situation is rather unsatisfactory. While neural networks have produced stunning results in various imaging tasks, their use in compressive imaging appears hampered 1.10...

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Compressed Sensing in Hilbert Spaces

Cited by 42 publications

References 39 publications

Exact solutions of infinite dimensional total-variation regularized problems

Exact solutions of infinite dimensional total-variation regularized problems

TV-based reconstruction of periodic functions

Compressive Imaging: Structure, Sampling, Learning

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