Dynamical sampling: Time–space trade-off

Aldroubi, Akram; Davis, Joan M.; Krishtal, Ilya A.

doi:10.1016/j.acha.2012.09.002

Cited by 79 publications

(65 citation statements)

References 16 publications

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“…Theorems 2.1 and 2.4 parallel the finite dimensional results we obtained in [3]. For example, one can use more complicated choices for Ω ⊂ {1, .…”

Section: Introductionmentioning

confidence: 48%

“…For example, in Theorem 2.4, the dynamical samples without the samples in the extra sampling set Ω still form a uniqueness set (the operator A has a trivial kernel). The latter was not the case in [3]. …”

Section: Introductionmentioning

confidence: 82%

“…. , mn − 1} in Theorem 2.4, and the admissible choices are determined by the same equivalence relations as in the finite dimensional case (see [3] for more details). Moreover, some of the the methods we use for obtaining stability results in Section 3 are the same as in [3].…”

Section: Introductionmentioning

confidence: 99%

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Exact Reconstruction of Signals in Evolutionary Systems Via Spatiotemporal Trade-off

Aldroubi

Davis

Krishtal

2014

J Fourier Anal Appl

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Abstract. We consider the problem of spatiotemporal sampling in which an initial state f of an evolution process f t = A t f is to be recovered from a combined set of coarse samples from varying time levels {t 1 , . . . , t N }. This new way of sampling, which we call dynamical sampling, differs from standard sampling since at any fixed time t i there are not enough samples to recover the function f or the state f ti . Although dynamical sampling is an inverse problem, it differs from the typical inverse problems in which f is to be recovered from A T f for a single time T . In this paper, we consider signals that are modeled by 2 (Z) or a shift invariant space V ⊂ L 2 (R).

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“…Theorems 2.1 and 2.4 parallel the finite dimensional results we obtained in [3]. For example, one can use more complicated choices for Ω ⊂ {1, .…”

Section: Introductionmentioning

confidence: 48%

Section: Introductionmentioning

confidence: 82%

See 1 more Smart Citation

Exact Reconstruction of Signals in Evolutionary Systems Via Spatiotemporal Trade-off

Aldroubi

Davis

Krishtal

2014

J Fourier Anal Appl

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“…We assign a finite nonnegative integer l i to each i ∈ Ω. Under what conditions on Ω and l i is the sequence Problem 1.2 is motivated by the spatiotemporal sampling and reconstruction problem arising in spatially invariant evolution systems [1,2,3,6,7,8,21,23,26,27]. Let f ∈ ℓ 2 (I) be an unknown vector that is evolving under the iterated actions of a convolution operator A, such that at time instance t = n it evolves to be A n f .…”

Section: Problem 12mentioning

confidence: 99%

“…These works studied the spatiotemporal sampling and reconstruction problem in the continuous diffusion field f (x, t) = A t f (x), where f (x) = f (x, 0) is the initial signal and A t is the time varying Gaussian convolution kernel determined by the diffusion rule. In [6,7], Aldroubi and his collaborators develop the mathematical framework of Dynamical Sampling to study the spatiotemporal sampling and reconstruction problem in discrete spatially invariant evolution processes, which can be viewed as a discrete version of diffusion-like processes. Our results can be viewed as an extension of [6,7] to the irregular setting and the algebraic characterization given in [6] can be viewed as a special case of our characterization for the unions of periodic constructions.…”

Section: Problem 12mentioning

confidence: 99%

Universal Spatiotemporal Sampling Sets for Discrete Spatially Invariant Evolutionary Systems

Tang

2017

IEEE Trans. Inform. Theory

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Abstract. Let (I, +) be a finite abelian group and A be a circular convolution operator on ℓ 2 (I). The problem under consideration is how to construct minimal Ω ⊂ I and li such that Y = {ei, Aei, · · · , A l i ei : i ∈ Ω} is a frame for ℓ 2 (I), where {ei : i ∈ I} is the canonical basis of ℓ 2 (I). This problem is motivated by the spatiotemporal sampling problem in discrete spatially invariant evolution systems. We will show that the cardinality of Ω should be at least equal to the largest geometric multiplicity of eigenvalues of A, and we consider the universal spatiotemporal sampling sets (Ω, li) for convolution operators A with eigenvalues subject to the same largest geometric multiplicity. We will give an algebraic characterization for such sampling sets and show how this problem is linked with sparse signal processing theory and polynomial interpolation theory. IntroductionLet (I, +) be a finite abelian group, we denote by ℓ 2 (I) the space of square summable complex-valued functions defined on I with the inner product given byWe denote by {e i : i ∈ I} the canonical basis of ℓ 2 (I), where e i is the characteristic function of {i} ⊂ I.Definition 1.1. The operator A on ℓ 2 (I) is called a circular convolution operator if there exists a complex-valued function a defined on I such thatThe function a is said to be the convolution kernel of A.2010 Mathematics Subject Classification. Primary 94A20, 94A12, 42C15, 15A29.

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Time-Variant System Approximation via Later-Time Samples

Aceska

Kim

2021

Springer Proceedings in Mathematics &Amp; Statistics

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Dynamical sampling: Time–space trade-off

Cited by 79 publications

References 16 publications

Exact Reconstruction of Signals in Evolutionary Systems Via Spatiotemporal Trade-off

Exact Reconstruction of Signals in Evolutionary Systems Via Spatiotemporal Trade-off

Universal Spatiotemporal Sampling Sets for Discrete Spatially Invariant Evolutionary Systems

Time-Variant System Approximation via Later-Time Samples

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