Nonlocal inversion formulas in real integral geometry

1978 ◽  
Vol 11 (3) ◽  
pp. 173-179 ◽  
Author(s):  
I. M. Gel'fand ◽  
S. G. Gindikin
Keyword(s):  
Integral Geometry ◽  
Inversion Formulas

The vertical slice transform on the unit sphere

2019 ◽  
Vol 22 (4) ◽  
pp. 899-917 ◽  
Author(s):  
Boris Rubin
Keyword(s):  
Unit Sphere ◽  
Radon Transform ◽  
Integral Geometry ◽  
Singular Value ◽  
Hypersingular Integrals ◽  
Inversion Formulas ◽  
Spherical Mean ◽  
Singular Value Decompositions ◽  
Vertical Slice ◽  
Class Of Functions

Abstract The vertical slice transform in spherical integral geometry takes a function on the unit sphere Sn to integrals of that function over spherical slices parallel to the last coordinate axis. This transform was investigated for n = 2 in connection with inverse problems of spherical tomography. The present article gives a survey of some methods which were originally developed for the Radon transform, hypersingular integrals, and the spherical mean Radon-like transforms, and can be adapted to obtain new inversion formulas and singular value decompositions for the vertical slice transform in the general case n ≥ 2 for a large class of functions.

Crofton's function and inversion formulas in real integral geometry

1991 ◽  
Vol 25 (1) ◽  
pp. 1-5 ◽  
Author(s):  
I. M. Gel'fand ◽  
M. I. Graev
Keyword(s):  
Integral Geometry ◽  
Inversion Formulas ◽  
And Inversion

Erdélyi–Kober fractional integrals and radon transforms for mutually orthogonal affine planes

2020 ◽  
Vol 23 (4) ◽  
pp. 967-979
Author(s):  
Boris Rubin ◽  
Yingzhan Wang
Keyword(s):  
Fractional Integrals ◽  
Radon Transforms ◽  
Affine Planes ◽  
Inversion Formulas ◽  
Radial Functions ◽  
Compactly Supported ◽  
Radial Case ◽  
Compactly Supported Functions

AbstractWe apply Erdélyi–Kober fractional integrals to the study of Radon type transforms that take functions on the Grassmannian of j-dimensional affine planes in ℝn to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. We obtain explicit inversion formulas for these transforms in the class of radial functions under minimal assumptions for all admissible dimensions. The general (not necessarily radial) case, but for j + k = n − 1, n odd, was studied by S. Helgason [8] and F. Gonzalez [4, 5] on smooth compactly supported functions.

scholarly journals Lagrange Inversion and Combinatorial Species with Uncountable Color Palette

2021 ◽  
Author(s):  
Sabine Jansen ◽  
Tobias Kuna ◽  
Dimitrios Tsagkarogiannis
Keyword(s):  
Lagrange Inversion ◽  
Inversion Formulas ◽  
Color Palette

AbstractWe prove a multivariate Lagrange-Good formula for functionals of uncountably many variables and investigate its relation with inversion formulas using trees. We clarify the cancellations that take place between the two aforementioned formulas and draw connections with similar approaches in a range of applications.

scholarly journals Integral-geometry morphological image analysis

2001 ◽  
Vol 347 (6) ◽  
pp. 461-538 ◽  
Author(s):  
K. Michielsen ◽  
H. De Raedt
Keyword(s):  
Image Analysis ◽  
Integral Geometry ◽  
Morphological Image Analysis ◽  
Morphological Image

An integral geometry underdetermined problem for a family of curves

2015 ◽  
Vol 56 (2) ◽  
pp. 217-230
Author(s):  
D. S. Anikonov ◽  
D. S. Konovalova
Keyword(s):  
Integral Geometry ◽  
Family Of Curves
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