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A simple range characterization for spherical mean transform in even dimensions

Divyansh Agrawal, Gaik Ambartsoumian, Venkateswaran P. Krishnan, Nisha Singhal

TL;DR

This work provides a complete range characterization of the spherical mean transform in even dimensions by deriving symmetry-based, nonlocal integral conditions for the data's spherical-harmonic coefficients. The radial case is established first and then lifted to the general case via a spherical-harmonics decomposition; crucial to the arguments are two new Bessel-function identities (an integral cross-product and a Nicholson-type formula) and an elliptic-integral equality used in the necessity part. The results yield a constructive, simpler criterion for data to lie in the SMT range, with direct implications for data completion and reconstruction in tomography, as well as new standalone contributions to the theory of special functions. Collectively, the paper advances both the mathematical understanding of SMT range descriptions and practical approaches to handling incomplete or noisy SMT data in imaging applications.

Abstract

The paper presents a new and simple range characterization for the spherical mean transform of functions supported in the unit ball in even dimensions. It complements the previous work of the same authors, where they solved an analogous problem in odd dimensions. The range description in even dimensions consists of symmetry relations, using a special kind of elliptic integrals involving the coefficients of the spherical harmonics expansion of the function in the range of the transform. The article also introduces a pair of original identities involving normalized Bessel functions of the first and the second kind. The first result is an integral cross-product identity for Bessel functions of integer order, complementing a similar relation for Bessel functions of half-integer order obtained in the aforementioned work of the same authors. The second result is a new Nicholson-type identity. Both of these relations can be considered as important standalone results in the theory of special functions. Finally, as part of the proof of one of the theorems, the authors derive an interesting equality involving elliptic integrals, which may be of independent interest.

A simple range characterization for spherical mean transform in even dimensions

TL;DR

This work provides a complete range characterization of the spherical mean transform in even dimensions by deriving symmetry-based, nonlocal integral conditions for the data's spherical-harmonic coefficients. The radial case is established first and then lifted to the general case via a spherical-harmonics decomposition; crucial to the arguments are two new Bessel-function identities (an integral cross-product and a Nicholson-type formula) and an elliptic-integral equality used in the necessity part. The results yield a constructive, simpler criterion for data to lie in the SMT range, with direct implications for data completion and reconstruction in tomography, as well as new standalone contributions to the theory of special functions. Collectively, the paper advances both the mathematical understanding of SMT range descriptions and practical approaches to handling incomplete or noisy SMT data in imaging applications.

Abstract

The paper presents a new and simple range characterization for the spherical mean transform of functions supported in the unit ball in even dimensions. It complements the previous work of the same authors, where they solved an analogous problem in odd dimensions. The range description in even dimensions consists of symmetry relations, using a special kind of elliptic integrals involving the coefficients of the spherical harmonics expansion of the function in the range of the transform. The article also introduces a pair of original identities involving normalized Bessel functions of the first and the second kind. The first result is an integral cross-product identity for Bessel functions of integer order, complementing a similar relation for Bessel functions of half-integer order obtained in the aforementioned work of the same authors. The second result is a new Nicholson-type identity. Both of these relations can be considered as important standalone results in the theory of special functions. Finally, as part of the proof of one of the theorems, the authors derive an interesting equality involving elliptic integrals, which may be of independent interest.
Paper Structure (15 sections, 15 theorems, 160 equations)

This paper contains 15 sections, 15 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Spherical mean transform
  5. Spherical harmonics
  6. Faà di Bruno formula for $D$ derivatives
  7. Bessel functions
  8. Range characterization
  9. Proof of main results
  10. An identity involving elliptic integrals; proof of Theorem \ref{['Elliptic_Integrals']}
  11. Proof of necessity in Theorem \ref{['range']}
  12. A new Nicholson-type cross product formula for Bessel functions; proof of Theorem \ref{['Nicholson-type formula']}
  13. A cross product identity for Bessel functions; Proof of Theorem \ref{['CP-Identity']}
  14. Proof of sufficiency in Theorem \ref{['range']}
  15. Proof of Theorem \ref{['Thm1.4']}

Key Result

Theorem 2.1

Let $\mathbb{B}$ denote the unit ball in $\mathbb{R}^n$ for even $n \geq 2$ and $\alpha=\frac{n-2}{2}$. A function $g \in C_c^\infty ((0,2))$ is representable as $g = \mathcal{R} f$ for a radial function $f \in C_c^\infty(\mathbb{B})$ if and only if $h(t) \coloneqq t^{n-2} g(t)$ satisfies for each $

Theorems & Definitions (26)

  • Theorem 2.1: Range characterization - radial case
  • Theorem 2.2: Range characterization - general case
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1: Funk-Hecke
  • Lemma 3.2: Faá di Bruno
  • Lemma 3.3
  • Theorem 3.4
  • Lemma 4.1
  • ...and 16 more