Reconstruction of functions from their integrals overk-planes

Rubin, Boris

doi:10.1007/bf02772213

Cited by 44 publications

(45 citation statements)

References 24 publications

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“…After pioneering papers by P. Funk and J. Radon, it is known [3,4,7,8,9] that f can be explicitly reconstructed from ϕ = Rf by applying differentiation of order k/2 (if k is odd this differentiation is fractional and interpreted in a suitable sense) to R * r ϕ in the r 2 -variable. However, it was recently proved by Helgason [4,p.…”

Section: G(x S) Sgn(s−t) Ds G(x S)mentioning

confidence: 99%

A generalization of the Mader-Helgason inversion formulas for Radon transforms

Antipov

Rubin

2012

Trans. Amer. Math. Soc.

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Abstract. In 1927, Philomena Mader derived elegant inversion formulas for the hyperplane Radon transform on R n . These formulas differ from the original ones by Radon and seem to be forgotten. We generalize Mader's formulas to totally geodesic Radon transforms in any dimension on arbitrary constant curvature space. Another new interesting inversion formula for the k-plane transform was presented in the recent book "Integral geometry and Radon transform" by S. Helgason. We extend this formula to arbitrary constant curvature space. The paper combines tools of integral geometry and complex analysis.

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Section: G(x S) Sgn(s−t) Ds G(x S)mentioning

confidence: 99%

A generalization of the Mader-Helgason inversion formulas for Radon transforms

Antipov

Rubin

2012

Trans. Amer. Math. Soc.

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“…We refer to [7,12,22,24,27,28,31] for the basic properties and applications of Riesz potentials. In this article we restrict to real α ∈ (0, n).…”

Section: )mentioning

confidence: 99%

“…More information can be found in [5,7,24]. Let G n,k and G n,k be the affine Grassmann manifold of all non-oriented k-dimensional planes τ in R n and the ordinary Grassmann manifold of k-dimensional linear subspaces ζ of R n , respectively.…”

Section: Inversion Of K-plane Transformsmentioning

confidence: 99%

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On some inversion formulas for Riesz potentials and k-plane transforms

Rubin

2011

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“…The latter result applies well beyond the Schwartz class, to all continuous functions [H,Thm. 6.2] (or even functions in L q loc , with 1 ≤ q ≤ n/k [R,Thm. 4.1]), with O(r −n−ε ) decay.…”

Section: X-rays Of Forms and Projections Of Currents 159mentioning

confidence: 99%

X-rays of forms and projections of currents

Solomon

2011

Trans. Amer. Math. Soc.

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Abstract. We study a new Radon-like transform that averages projected pforms in R n over affine (n − k)-spaces. We then prove an explicit inversion formula for our transform on the space of rapidly-decaying smooth p-forms. Our transform differs from the one in the work by Gelfand, Graev and Shapiro (1969). Moreover, if it can be extended to a somewhat larger space of p-forms, our inversion formula will allow the synthesis of any rapidly-decaying smooth p-form on R n as a (continuous) superposition of pullbacks from p-forms on k-dimensional subspaces. In turn, such a synthesis implies an explicit formula (which we derive) for reconstructing compactly supported currents in R n (e.g., compact oriented k-dimensional subvarieties) from their oriented projections onto k-planes.

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Reconstruction of functions from their integrals overk-planes

Cited by 44 publications

References 24 publications

A generalization of the Mader-Helgason inversion formulas for Radon transforms

A generalization of the Mader-Helgason inversion formulas for Radon transforms

On some inversion formulas for Riesz potentials and k-plane transforms

X-rays of forms and projections of currents

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