[Paper Review] On the determination of a function from its cone transform with fixed central axis

Sunghwan Moon|arXiv (Cornell University)|Mar 26, 2015

Statistical and numerical algorithms15 references3 citations

TL;DR

This paper studies the n-dimensional cone transform with a fixed central axis, deriving two inversion formulas, stability estimates, and uniqueness and reconstruction results for limited data. It establishes that the cone transform is an isometry and provides explicit inversion methods for reconstructing functions from their cone integrals, with applications in Compton camera imaging and SPECT.

ABSTRACT

A Radon-type transform called a cone transform that assigns to a given function its integral over various sets of cones has arisen in the last decade in the context of the study of Compton cameras used in Single Photon Emission Computed Tomography. Here, we study the cone transform for which the central axis of the cones of integration is fixed. We present many of its properties, such as two inversion formulas, a stability estimate, and uniqueness and reconstruction for a local data problem.

Motivation & Objective

• To investigate the mathematical properties of the cone transform with a fixed central axis in n-dimensional space.
• To establish inversion formulas for reconstructing a function from its cone transform data.
• To analyze the stability and uniqueness of reconstruction under limited data conditions.
• To derive range conditions for the cone transform in odd dimensions.
• To provide a theoretical foundation for Compton camera imaging in Single Photon Emission Computed Tomography (SPECT).

Proposed method

• Formalizes the n-dimensional cone transform using integration over cones with fixed central axis and vertex on a plane.
• Derives two inversion formulas using Fourier analysis and radial symmetry properties of the transform.
• Applies the Fourier slice theorem analog to relate the cone transform to the Fourier transform of the function.
• Uses projection operators and inverse Fourier transforms to handle limited data reconstruction.
• Introduces Sobolev space estimates and proves that the cone transform is an isometry under appropriate function spaces.
• Applies the support theorem for the Radon transform to establish uniqueness in the limited data problem.

Experimental results

Research questions

• RQ1 Can a function be uniquely reconstructed from its cone transform when only partial data (limited |v|) is available?
• RQ2 What are the inversion formulas for the cone transform with a fixed central axis in n dimensions?
• RQ3 How stable is the reconstruction process under small perturbations in the cone transform data?
• RQ4 What are the range conditions for the cone transform in odd-dimensional spaces?
• RQ5 What is the relationship between the cone transform and the Radon transform in terms of isometry and inversion?

Key findings

• Two inversion formulas are derived for the cone transform with a fixed central axis, enabling exact reconstruction of the original function from its cone integrals.
• The cone transform is shown to be an isometry on appropriate function spaces, ensuring stability in reconstruction.
• A stability estimate is established in Sobolev spaces, quantifying the error in reconstruction under data perturbations.
• Uniqueness and reconstruction are proven for the local data problem, where cone transform data is available only for |v| in (a,b).
• The range of the cone transform in odd dimensions is characterized using the support theorem and Fourier analysis.
• The limited data cone transform is reconstructed via projection operators and inverse Fourier transforms, with a precise relation to the Riesz potential and the full transform.