A Theory for Sampling Signals From a Union of Subspaces

Lu, Yue M.; Minh, N.

doi:10.1109/tsp.2007.914346

Cited by 243 publications

(284 citation statements)

References 33 publications

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“…Combining this theorem with the fact that almost all linear maps have the URP gives a similar result to the one we derive below, but for the special case of mappings and allows us to reduce the required observation dimension by one and therefore extends the results from [17] and [31]. In particular, we can use a slight variation of the proof used by Sauer et al…”

Section: B the Case M ≥ K Maxsupporting

confidence: 58%

“…In [17] a very similar result was derived for a countably infinite union of subspaces. The difference to our result is that for the finite union of subspaces, our theorem states that almost all maps have the desired property, whilst in [17] it was shown that the set of maps with the property is dense, which is a slightly weaker statement (though derived for a more general model) as density of a set does not imply anything about the measure of the set.…”

Section: B the Case M ≥ K Maxmentioning

confidence: 56%

“…This model is a special case of the union of subspaces model introduced in [17] with the difference that we will restrict the discussion to mixtures of finitely many finite dimensional subspaces, i.e we assume L < ∞.…”

Section: B Unions Of Linear Subspacesmentioning

confidence: 99%

“…In this paper we follow a similar approach to that in [17] and rely heavily on the difference between any two vectors x i − x j , both from A. The vectors x i − x j lie in a union of subspaces S i,j and k max gives the largest dimensions of any of these subspaces.…”

Section: B Unions Of Linear Subspacesmentioning

confidence: 99%

“…This one to one property of Φ for elements of A is clearly an important aspect in sampling as it specifies a 'minimal' requirement that allows us to sample a signal without loss of information. Whilst this property has previously been studied in [17], our first main contribution is to show that under appropriate conditions almost all linear maps are one to one. For the mixture model with finitely many finite dimensional subspaces, our results are therefore stronger than those derived in [17], which only states that the set of one to one maps is a dense subset of all linear maps.…”

Section: Contributionmentioning

confidence: 99%

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Sampling Theorems for Signals From the Union of Finite-Dimensional Linear Subspaces

Blumensath¹,

Davies²

2009

IEEE Trans. Inform. Theory

246

361

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Compressed sensing is an emerging signal acquisition technique that enables signals to be sampled well below the Nyquist rate, given that the signal has a sparse representation in an orthonormal basis. In fact, sparsity in an orthonormal basis is only one possible signal model that allows for sampling strategies below the Nyquist rate. In this paper we consider a more general signal model and assume signals that live on or close to the union of linear subspaces of low dimension. We present sampling theorems for this model that are in the same spirit as the Nyquist-Shannon sampling theorem in that they connect the number of required samples to certain model parameters.Contrary to the Nyquist-Shannon sampling theorem, which gives a necessary and sufficient condition for the number of required samples as well as a simple linear algorithm for signal reconstruction, the model studied here is more complex. We therefore concentrate on two aspects of the signal model, the existence of one to one maps to lower dimensional observation spaces and the smoothness of the inverse map. We show that almost all linear maps are one to one when the observation space is at least of the same dimension as the largest dimension of the convex hull of the union of any two subspaces in the model. However, we also show that in order for the inverse map to have certain smoothness properties such as a given finite Lipschitz constant, the required observation dimension necessarily depends logarithmically This is a corrected version of the papers in which a few small errors have been corrected. Importantly, the dependence on δ in Theorem 3.3 and its corollaries has been corrected. VERSION: DECEMBER 3, 2009 1 on the number of subspaces in the signal model. In other words, whilst unique linear sampling schemes require a small number of samples depending only on the dimension of the subspaces involved, in order to have stable sampling methods, the number of samples depends necessarily logarithmically on the number of subspaces in the model. These results are then applied to two examples, the standard compressed sensing signal model in which the signal has a sparse representation in an orthonormal basis and to a sparse signal model with additional tree structure.

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Section: B the Case M ≥ K Maxsupporting

confidence: 58%

Section: B the Case M ≥ K Maxmentioning

confidence: 56%

Section: B Unions Of Linear Subspacesmentioning

confidence: 99%

Section: B Unions Of Linear Subspacesmentioning

confidence: 99%

Section: Contributionmentioning

confidence: 99%

See 3 more Smart Citations

Sampling Theorems for Signals From the Union of Finite-Dimensional Linear Subspaces

Blumensath¹,

Davies²

2009

IEEE Trans. Inform. Theory

246

361

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show abstract

On the convergence properties of sampling Durrmeyer‐type operators in Orlicz spaces

Costarelli

Piconi

Vinti

2022

Mathematische Nachrichten

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Here, we provide a unifying treatment of the convergence of a general form of sampling-type operators, given by the so-called sampling Durrmeyer-type series. The main result consists of the study of a modular convergence theorem in the general setting of Orlicz spaces 𝐿 𝜑 (ℝ). From the latter theorem, the convergence in 𝐿 𝑝 (ℝ), in 𝐿 𝛼 log 𝛽 𝐿, and in the exponential spaces can be obtained as particular cases. For the completeness of the theory, we provide a pointwise and uniform convergence theorem on ℝ, in case of bounded continuous and bounded uniformly continuous functions; in this context, we also furnish a quantitative estimate for the order of approximation, using the modulus of continuity of the function to be approximated. Finally, applications and examples with graphical representations are given for several sampling series with special kernels.

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Removal of nuisance signals from limited and sparse ¹H MRSI data using a union‐of‐subspaces model

Lam

Johnson

et al. 2015

Magnetic Resonance in Med

103

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Purpose To remove nuisance signals (e.g., water and lipid signals) for 1H MRSI data collected from the brain with limited and/or sparse (k, t)-space coverage. Methods A union-of-subspace model is proposed for removing nuisance signals. The model exploits the partial separability of both the nuisance signals and the metabolite signal, and decomposes an MRSI dataset into several sets of generalized voxels that share the same spectral distributions. This model enables the estimation of the nuisance signals from an MRSI dataset that has limited and/or sparse (k, t)-space coverage. Results The proposed method has been evaluated using in vivo MRSI data. For conventional CSI data with limited k-space coverage, the proposed method produced “lipid-free” spectra without lipid suppression during data acquisition at 130 ms echo time. For sparse (k, t)-space data acquired with conventional pulses for water and lipid suppression, the proposed method was also able to remove the remaining water and lipid signals with negligible residuals. Conclusions Nuisance signals in 1H MRSI data reside in low-dimensional subspaces. This property can be utilized for estimation and removal of nuisance signals from 1H MRSI data even when they have limited and/or sparse coverage of (k, t)-space. The proposed method should prove useful especially for accelerated high-resolution 1H MRSI of the brain.

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A Theory for Sampling Signals From a Union of Subspaces

Cited by 243 publications

References 33 publications

Sampling Theorems for Signals From the Union of Finite-Dimensional Linear Subspaces

Sampling Theorems for Signals From the Union of Finite-Dimensional Linear Subspaces

On the convergence properties of sampling Durrmeyer‐type operators in Orlicz spaces

Removal of nuisance signals from limited and sparse ¹H MRSI data using a union‐of‐subspaces model

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A Theory for Sampling Signals From a Union of Subspaces

Cited by 243 publications

References 33 publications

Sampling Theorems for Signals From the Union of Finite-Dimensional Linear Subspaces

Sampling Theorems for Signals From the Union of Finite-Dimensional Linear Subspaces

On the convergence properties of sampling Durrmeyer‐type operators in Orlicz spaces

Removal of nuisance signals from limited and sparse 1H MRSI data using a union‐of‐subspaces model

Contact Info

Product

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Removal of nuisance signals from limited and sparse ¹H MRSI data using a union‐of‐subspaces model