Optimal Compressive Imaging of Fourier Data

Gitta Kutyniok, Wang-Q Lim
2018 SIAM Journal of Imaging Sciences  
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
more » ... ames 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. where Ω c J = Ω 1 ∪ Ω 2 ∪ Ω 3 with Ω 1 = {n ∈ Ω c J : |n 1 | > 2 J(1+ρ) }, Ω 2 = {n ∈ Ω c J : |n 2 | > 2 J(1+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}.
doi:10.1137/16m1098541 fatcat:v5ne3q25hjhz5hgyrytc7omyte