The Radon transform intertwines wavelets and shearlets

Bartolucci, Francesca; Mari, Filippo De; Vito, Ernesto De; Odone, Francesca

doi:10.1016/j.acha.2017.12.005

Cited by 21 publications

(49 citation statements)

References 31 publications

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“…This version of the Radon transform is proved to be particularly well-adapted to the structure of the classical shearlet transform, see [5] and [11]. We recall that the key idea in shearlet analysis is to construct a family of analyzing functions…”

Section: Introductionmentioning

confidence: 99%

“…where the integral converges in the weak sense. In [5] we have shown that the classical shearlet transform can be realised by applying first the horizontal (affine) Radon…”

Section: Introductionmentioning

confidence: 99%

“…In formula (2), P C f is reconstructed via the classical shearlet transform and P C v f via the vertical shearlet transform and this allows to restrict the shearing parameter s over a compact interval. In this paper, applying the "shearlets on the cone" construction to our results presented in [5], we obtain for any f ∈ L 1 (R 2 ) ∩ L 2 (R 2 ) a reconstruction formula of the form (2), i.e. where both the scale parameter a and the shearing parameter s range over compact intervals, and where the coefficients depend on f only through its Radon transform.…”

mentioning

confidence: 97%

“…The Radon transform of a signal f is a function on the affine projective space P 1 × R = {Γ | Γ line of R 2 } whose value at a line is the integral of f along that line. We label lines in the plane by pairs (v, t) ∈ R 2 as x + vy = t and we define the horizontal (affine) Radon transform Rf :This version of the Radon transform is proved to be particularly well-adapted to the structure of the classical shearlet transform, see [5] and [11]. We recall that the key idea in shearlet analysis is to construct a family of analyzing functionsby translating, shearing and dilating a fixed initial function ψ ∈ L 2 (R 2 ), called mother shearlet.…”

mentioning

confidence: 99%

“…If ψ satisfies the admissible condition (4) we can recover any signal f ∈ L 2 (R 2 ) from its shearlet transform through the reconstruction formulawhere the integral converges in the weak sense. In [5] we have shown that the classical shearlet transform can be realised by applying first the horizontal (affine) Radon…”

mentioning

confidence: 99%

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Cone-Adapted Shearlets and Radon Transforms

Bartolucci

Mari

Vito

2020

Applied and Numerical Harmonic Analysis

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We show that the cone-adapted shearlet coefficients can be computed by means of the limited angle horizontal and vertical (affine) Radon transforms and the onedimensional wavelet transform. This yields formulas that open new perspectives for the inversion of the Radon transform. IntroductionThe inversion of the Radon transform is a classical ill-posed inverse problem and consists in reconstructing an unknown signal f on R 2 from its line integrals [14]. The Radon transform of a signal f is a function on the affine projective space P 1 × R = {Γ | Γ line of R 2 } whose value at a line is the integral of f along that line. We label lines in the plane by pairs (v, t) ∈ R 2 as x + vy = t and we define the horizontal (affine) Radon transform Rf :This version of the Radon transform is proved to be particularly well-adapted to the structure of the classical shearlet transform, see [5] and [11]. We recall that the key idea in shearlet analysis is to construct a family of analyzing functionsby translating, shearing and dilating a fixed initial function ψ ∈ L 2 (R 2 ), called mother shearlet. Once we have this family of analyzing functions, we define the shearlet transform of any f ∈ L 2 (R 2 ) by S ψ f (b, s, a) = f, S b,s,a ψ . If ψ satisfies the admissible condition (4) we can recover any signal f ∈ L 2 (R 2 ) from its shearlet transform through the reconstruction formulawhere the integral converges in the weak sense. In [5] we have shown that the classical shearlet transform can be realised by applying first the horizontal (affine) Radon

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Section: Introductionmentioning

confidence: 99%

“…where the integral converges in the weak sense. In [5] we have shown that the classical shearlet transform can be realised by applying first the horizontal (affine) Radon…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 97%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Cone-Adapted Shearlets and Radon Transforms

Bartolucci

Mari

Vito

2020

Applied and Numerical Harmonic Analysis

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Continuity properties of the shearlet transform and the shearlet synthesis operator on the Lizorkin type spaces

Bartolucci

Pilipović

Teofanov

2022

Mathematische Nachrichten

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We develop a distributional framework for the shearlet transform Sψ0.16em:0.16emS0true(R2true)0.16em→0.16emscriptS(S)${\mathcal {S}}_{\psi }\,{:}\, {\mathcal {S}}_0\big (\mathbb {R}^2\big )\,{\rightarrow }\,{\mathcal {S}}(\mathbb {S})$ and the shearlet synthesis operator Sψt-0.16em:-0.16emscriptS(S)-0.16em→-0.16emS0true(R2true)${\mathcal {S}}^t_{\psi }\!: \!{\mathcal {S}}(\mathbb {S})\!\rightarrow \!{\mathcal {S}}_0\big (\mathbb {R}^2\big )$, where S0true(R2true)${\mathcal {S}}_0\big (\mathbb {R}^2\big )$ is the Lizorkin test function space and scriptSfalse(double-struckSfalse)${\mathcal {S}}(\mathbb {S})$ is the space of highly localized test functions on the standard shearlet group S$\mathbb {S}$. These spaces and their duals S0′(double-struckR2),0.16emS′(S)$\mathcal {S}_0^\prime {\big(\mathbb {R}^2\big)},\, \mathcal {S}^\prime (\mathbb {S})$ are called Lizorkin type spaces of test functions and distributions. We analyze the continuity properties of these transforms when the admissible vector ψ belongs to S0true(R2true)${\mathcal {S}}_0\big (\mathbb {R}^2\big )$. Then, we define the shearlet transform and the shearlet synthesis operator of Lizorkin type distributions as transpose mappings of the shearlet synthesis operator and the shearlet transform, respectively. They yield continuous mappings from S0′(double-struckR2)$\mathcal {S}_0^\prime {\big(\mathbb {R}^2\big)}$ to S′(S)$\mathcal {S}^\prime (\mathbb {S})$ and from S′(S)$\mathcal {S}^\prime (\mathbb {S})$ to S0′true(R2true)$\mathcal {S}_0^\prime \big (\mathbb {R}^2\big )$. Furthermore, we show the consistency of our definition with the shearlet transform defined by direct evaluation of a distribution on the shearlets. The same can be done for the shearlet synthesis operator. Finally, we give a reconstruction formula for Lizorkin type distributions, from which follows that the action of such generalized functions can be written as an absolutely convergent integral over the standard shearlet group.

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