Analysis vs. synthesis sparsity for $α$-shearlets

Abstract

There are two notions of sparsity associated to a frame Ψ = (ψ i ) i∈I : Analysis sparsity of f means that the analysis coefficients ( f, ψ i ) i∈I are sparse, while synthesis sparsity means that we can write f = i∈I c i ψ i with sparse synthesis coefficients (c i ) i∈I . Here, sparsity of a sequence c = (c i ) i∈I means c ∈ ℓ p (I) for a given p < 2. We show that both notions of sparsity coincide if Ψ = SH (ϕ, ψ; δ) = (ψ i ) i∈I is a discrete (cone-adapted) shearlet frame with sufficiently nice generators ϕ, ψ and sufficiently small sampling density δ > 0. The required 'niceness' of ϕ, ψ is explicitly quantified in terms of Fourier-decay and vanishing moment conditions. In addition to ℓ p -sparsity, we even allow weighted ℓ p -spaces ℓ p w s as a sparsity measure, with weights of the form w s = 2 js (j,ℓ,δ,k) where j encodes the scale of the corresponding shearlet elements.More precisely, we show that the shearlet smoothness spaces S p,q s R 2 introduced by Labate et al. simultaneously characterize analysis and synthesis sparsity with respect to a shearlet frame, in the sense that-for suitable ϕ, ψ, δ-the following are equivalent:As an application, we prove that shearlets yield (almost) optimal approximation rates for the class of cartoonlike functions:, where f N is a linear combination of N shearlets. This might appear to be a well-known statement, but an inspection of the existing proofs reveals that these only establish analysis sparsity of cartoon-like functions, which implies f, where g N is a linear combination of N elements of the dual frame Ψ to the shearlet frame Ψ. This is not completely satisfying, since only limited knowledge about the structure and properties of Ψ is available.In addition to classical shearlets, we also consider more general α-shearlet systems. For these, the parabolic scaling is replaced by α-parabolic scaling. The resulting systems range from ridgelet-like systems (for α = 0) over classical shearlets (α = 1 2 ) to wavelet-like systems (α = 1). In this more general case, the shearlet smoothness spaces S p,q s R 2 have to be replaced by the α-shearlet smoothness spaces S p,q α,s R 2 . We completely characterize the existence of embeddings between these spaces for different values of α. This allows us to decide whether sparsity with respect to α 1 -shearlets implies sparsity with respect to α 2 -shearlets, even for α 1 = α 2 .

Structured, compactly supported Banach frame decompositions of decomposition spaces

“…As we will see in Chapters 7 and 8, this is indeed possible for Besov spaces and α-modulation spaces. In addition, in the companion paper [86] we verify that the theory also applies to shearlet smoothness spaces. Furthermore, it turns out that in each of these cases one can find compactly supported prototype functions γ which fulfill the relevant criteria.…”

“…Probably, this intuition is what originally lead Labate et al to the introduction of the shearlet smoothness spaces [64], although they did not have the machinery to rigorously prove that the usual discrete, cone-adapted shearlet systems indeed yield Banach frames and atomic decompositions for the shearlet smoothness spaces. Using the theory developed here, we show in the companion paper [86] that this is indeed the case. Furthermore, [86] employs the results developed here to show that suitable discrete, cone-adapted shearlet systems achieve an almost optimal approximation rate for the class of cartoonlike functions.…”

“…Using the theory developed here, we show in the companion paper [86] that this is indeed the case. Furthermore, [86] employs the results developed here to show that suitable discrete, cone-adapted shearlet systems achieve an almost optimal approximation rate for the class of cartoonlike functions. At a first glance, this might appear to be a well-known statement, but a closer inspection of the classical results about approximation of cartoon-like functions by shearlets (see e.g.…”

“…Thus, in this aspect, our approach improves upon (generalized) coorbit theory. As shown in the companion paper [86] (see in particular [86, Theorems 4.2 and 4.3]), the theory developed here is in particular able to handle discrete cone-adapted shearlet frames. For this case, our theory indeed shows that analysis sparsity is equivalent to synthesis sparsity.…”

“…In stark contrast, the theory developed in the present paper is able to handle Besov spaces (see Chapter 8), as well as shearlet smoothness spaces (see the companion paper [86]).…”