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Reconstruction of a function defined on a sphere using the Funk transform

Rafik Aramyan

Abstract

It is known that the Funk transform (the Funk-Radon transform) is invertible in the class of even (symmetric) continuous functions defined on the unit 2-sphere S^2. In this article, for the reconstruction of f from C(S^2) (can be non-even), an additional condition (to reconstruct an odd function) is found, and the injectivity of the so-called two data Funk transform is considered. An iterative inversion formula of the transform is presented. Such inversions have theoretical significance in convexity theory, integral geometry and spherical tomography. Also, the Funk-Radon transform is used in Diffusion-weighted magnetic resonance imaging.

Reconstruction of a function defined on a sphere using the Funk transform

Abstract

It is known that the Funk transform (the Funk-Radon transform) is invertible in the class of even (symmetric) continuous functions defined on the unit 2-sphere S^2. In this article, for the reconstruction of f from C(S^2) (can be non-even), an additional condition (to reconstruct an odd function) is found, and the injectivity of the so-called two data Funk transform is considered. An iterative inversion formula of the transform is presented. Such inversions have theoretical significance in convexity theory, integral geometry and spherical tomography. Also, the Funk-Radon transform is used in Diffusion-weighted magnetic resonance imaging.

Paper Structure

This paper contains 9 sections, 7 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Inversion formula in terms of the Fourier coefficients
  3. The derivation of the Fourier coefficients
  4. Averaging the inversion formula
  5. A representation for the Fourier coefficients
  6. The Inversion formula
  7. The weighted Funk transforms with respect to $\Omega$
  8. Computational Implementation
  9. Conclusion

Key Result

Theorem 1

Let $f\in{ \cal C}^{1}(\mathbf S^2)$ (can be non even) and $\Omega\in\mathbf S^2$. The value of $f(\Omega)$ can be recovered using the two data Funk transform ($\Omega\in\mathbf S^2$ is taken for the north Pole).

Figures (2)

  • Figure 1: An illustration to (\ref{['4.1']}).
  • Figure 2: An illustration to Theorem 1.3.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Lemma 1
  • Theorem 4
  • Theorem 5