On minimal trajectories for mobile sampling of bandlimited fields

Gröchenig, Karlheinz; Romero, José Luis; Unnikrishnan, Jayakrishnan; Vetterli, Martin

doi:10.1016/j.acha.2014.11.002

Cited by 32 publications

(31 citation statements)

References 25 publications

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“…Finally, by Plancherel, we have G s * f, e n =Ĝ s (n) ·f (n), which we indeed can compute knowing the action of the chosen filterĜ s (n) as well asf (n). Applying now our choice for the sampling sets ∆ J,s from Subsection 3.3, the linear system of equations (13) becomes…”

Section: 5mentioning

confidence: 99%

Optimal Compressive Imaging of Fourier Data

Kutyniok¹,

Lim²

2018

SIAM J. Imaging Sci.

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Applications such as Magnetic Resonance Tomography acquire imaging data by point samples of their Fourier transform. This raises the question of balancing the efficiency of the sampling strategies with the approximation accuracy of an associated reconstruction procedure. In this paper, we introduce a novel sampling-reconstruction scheme based on a random anisotropic sampling pattern and a compressed sensing type reconstruction strategy with a variant of dualizable shearlet frames as sparsifying representation system. For this scheme, we prove asymptotic optimality in an approximation theoretic sense for cartoon-like functions as a model class for the imaging data. Finally, we present numerical experiments showing the superiority of our scheme over other approaches.where1+ρ) and |n 1 | ≤ 2 J(1+ρ/2) }, and Ω 3 = {n ∈ Ω c J : |n 2 | > 2 J(1+ρ) and |n 1 | > 2 J(1+ρ/2) }. Next, for each λ = (j, s, m, p) ∈Λ J,s , the decay conditions in Definition 2.3 imply n∈Ω c J G. KUTYNIOK AND W.-Q LIM and in the case j 0 < J/4, we set Λ 0 J,s = {λ = (j, s, m, p) ∈Λ J,s : j = −1, j 0 , . . . , J 4 − 1} ∪ {λ = (j, s, m, p) ∈Λ J,s : int(supp(ψ ♯ λ )) ∩ Γ ∩ int(Q j,ℓ ) = ∅ and |k j,s (ℓ)| ≤ 2 J −j 4 for Q j,ℓ ∈ Q j , j = J 4 , . . . , J}. ♯(Λ 0 J,s ) J j=j 0 {ℓ:|k j,s (ℓ)|≤2 J −j 4 ,Q j,ℓ ∈Q j } 2 2Jρ (1 + |k j,s (ℓ)|)

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Section: 5mentioning

confidence: 99%

Optimal Compressive Imaging of Fourier Data

Kutyniok¹,

Lim²

2018

SIAM J. Imaging Sci.

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“…for some C 1 ,C 2 > 0 absolute constants. A related problem for band-limited signals in R 2 (i.e., the Paley Wiener spaces PW σ ) with time varying sampling locations corresponding to trajectories but timeindependent function can be found in [16].…”

Section: The Dynamical Sampling Problemmentioning

confidence: 99%

Dynamical Sampling and Systems from Iterative Actions of Operators

Aldroubi

Petrosyan

2017

Applied and Numerical Harmonic Analysis

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In this chapter, we review some of the recent developments and prove new results concerning frames and Bessel systems generated by iterations of the form {A n g : g ∈ G, n = 0, 1, 2, . . . }, where A is a bounded linear operator on a separable complex Hilbert space H and G is a countable set of vectors in H . The system of iterations mentioned above was motivated from the so called dynamical sampling problem. In dynamical sampling, an unknown function f and its future states A n f are coarsely sampled at each time level n, 0 ≤ n < L, where A is an evolution operator that drives the system. The goal is to recover f from these space-time samples.

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“…For us, the motivation to study the iterative actions of normal operators comes from sampling theory and related topics [2,9,21,29,31,5,6,18,37,38,16,15]. Specifically, the motivation derives from the so called dynamical sampling problem [3,4,13,19,25]: Let the initial state of a system be given by a vector f in a Hilbert space H and assume that the initial state is evolving under the action of a bounded operator A ∈ B(H) to the states f n = Af n−1 , f 0 = f.…”

Section: Introductionmentioning

confidence: 99%

Iterative actions of normal operators

Aldroubi

Cabrelli

Çakmak

et al. 2017

Journal of Functional Analysis

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Let A be a normal operator in a Hilbert space H, and let G ⊂ H be a countable set of vectors. We investigate the relations between A, G and L that make the system of iterations {A n g : g ∈ G, 0 ≤ n < L(g)} complete, Bessel, a basis, or a frame for H. The problem is motivated by the dynamical sampling problem and is connected to several topics in functional analysis, including, frame theory and spectral theory. It also has relations to topics in applied harmonic analysis including, wavelet theory and time-frequency analysis.2010 Mathematics Subject Classification. 46N99, 42C15, 94O20.

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On minimal trajectories for mobile sampling of bandlimited fields

Cited by 32 publications

References 25 publications

Optimal Compressive Imaging of Fourier Data

Optimal Compressive Imaging of Fourier Data

Dynamical Sampling and Systems from Iterative Actions of Operators

Iterative actions of normal operators

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