Boris Rubin

Inversion formulas for the spherical Radon transform and the generalized cosine transform

The k-dimensional totally geodesic Radon transform on the unit sphere Sn and the corresponding cosine transform can be regarded as members of the analytic family of intertwining fractional integrals R?f(?)=?n,k(?)?Snf(x)sind(x,?)?+k?ndx,d(x,?) being the geodesic distance between x?Sn and the k-geodesic ?. We develop a unified approach to the inversion of R?f for all ??0,1?k?n?1,n?2. The cases of smooth f and f?Lp are considered. A series of new inversion formulas is obtained. The convolution?backprojection method is developed.

Reconstruction of functions from their integrals overk-planes

Thek-plane Radon transform assigns to a functionsf(x) on ℝn the collection of integralsf(τ)=∫τf over allk-dimensional planesτ. We give a systematic treatment of two inversion methods for this transform, namely, the method of Riesz potentials, and the method of spherical means. We develop new analytic tools which allow to invertf(τ) under minimal assumptions forf. It is assumed thatfεLp, 1≤p<n/k, orf is a continuous function with minimal rate of decay at infinity. In the framework of the first method, our approach employs intertwining fractional integrals associated to thek-plane transform. Following the second method, we extend the original formula of Radon for continuous functions on ℝ2 tofεLp(ℝn) and all 1≤k<n. New integral formulae and estimates, generalizing those of Fuglede and Solmon, are obtained.

Inversion and characterization of the hemispherical transform

Explicit inversion formulas are obtained for the hemispherical transform(FΜ)(x) = Μ{y ∃Sn :x. y ≥ 0},x ∃Sn, whereSn is thendimensional unit sphere in ℝn+1,n ≥ 2, and Μ is a finite Borel measure onSn. If Μ is absolutely continuous with respect to Lebesgue measuredy onSn, i.e.,dΜ(y) =f(y)dy, we write(F f)(x) = ∫x.y> 0f(y)dy and consider the following cases: (a)f ∃C∞(Sn); (b)f ∃ Lp(Sn), 1 ≤ p < ∞; and (c)f ∃C(Sn). In the case (a), our inversion formulas involve a certain polynomial of the Laplace-Beltrami operator. In the remaining cases, the relevant wavelet transforms are employed. The range ofF is characterized and the action in the scale of Sobolev spacesLpγ (Sn) is studied. For zonalf ∃ L1(S2), the hemispherical transformF f was inverted explicitly by P. Funk (1916); we reproduce his argument in higher dimensions.

Fractional integrals and wavelet transforms associated with Blaschke-Levy representations on the sphere

AbstractA family of the spherical fractional integrals

Helgason-Marchaud inversion formulas for Radon transforms

T^\alpha f = \gamma _{n,\alpha } \int {_{\Sigma _n } } \left| {xy} \right|^{\alpha - 1} f(y)dy

Funk, Cosine, and Sine Transforms on Stiefel and Grassmann Manifolds

on the unit sphere Σn in ℝn+1 is investigated. This family includes the spherical Radon transform (α = 0) and the Blaschke-Levy representation (α>1). Explicit inversion formulas and a characterization ofTαƒ are obtained for ƒ belonging to the spacesC∞,C, Lp and for the case when ƒ is replaced by a finite Borel measure. All admissiblen ≥ 2,α ε ℂ, andp are considered. As a tool we use spherical wavelet transforms associated withTα. Wavelet type representations are obtained forTα ƒ, ƒ εLp, in the case Reα ≤ 0, provided thatTα is a linear bounded operator inLp.

The inversion of fractional integrals on a sphere

The generalized Calderón reproducing formula involving “wavelet measure” is established for functions f ∈ Lp(ℝn). The special choice of the wavelet measure in the reproducing formula gives rise to the continuous decomposition of f into wavelets, and enables one to obtain inversion formulae for generalized windowed X-ray transforms, the Radon transform, and k-plane transforms. The admissibility conditions for the wavelet measure μ are presented in terms of μ itself and in terms of the Fourier transform of μ.

METHOD OF MEAN VALUE OPERATORS FOR RADON TRANSFORMS IN THE SPACE OF MATRICES

Let X be either the hyperbolic space H n or the unit sphere S n , and let Ξ be the set of all k-dimensional totally geodesic submanifolds of X, 1 ≤ k < n - 1. For x ∈ X and ξ ∈ Ξ, the totally geodesic Radon transform f(x) → f(ξ) is studied. By averaging f(ξ) over all ξ at a distance θ from x, and applying Riemann-Liouville fractional differentiation in 0, S. Helgason has recovered f(x). We show that in the hyperbolic case this method blows up if f does not decrease sufficiently fast. The situation can be saved if one employs Marchauds fractional derivatives instead of the Riemann-Liouville ones. New inversion formulas for f(ξ); f ∈ L p (X), are obtained.