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Exact inversion of an integral transform arising in Compton camera imaging.

Fatma Terzioglu¹

¹University of Chicago, Department of Statistics, Chicago, Illinois, United States.

Journal of Medical Imaging (Bellingham, Wash.)

|March 25, 2020

Summary

This study presents an exact inversion method for the 3D Compton transform, crucial for Compton camera imaging. The technique converts Compton data to Radon projections for function recovery, enhancing imaging applications.

Keywords:

Compton camera Compton transform cone transform exact inversion image reconstruction

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Area of Science:

Medical Imaging
Industrial Imaging
Homeland Security Imaging
Astronomy

Background:

The Compton transform is integral to passive gamma-ray source detection using Compton cameras.
Compton cameras have diverse applications in medical, industrial, security, and astronomical imaging.

Purpose of the Study:

To develop an exact inversion formula for the three-dimensional (3-D) Compton transform.
To enable accurate reconstruction of functions from their integrals over conical surfaces.

Main Methods:

Formulated a generalized identity linking the Compton and Radon transforms.
Devised a two-step method: first, recover Radon transform from Compton data, then invert the Radon transform.
The method is detector geometry-independent, provided an admissibility condition is met.

Main Results:

An exact inversion formula for the 3-D Compton transform was derived.
A two-step inversion technique was successfully developed and validated.
Numerical simulations confirmed the stability of the inversion algorithm.

Conclusions:

An exact inversion method for the 3-D Compton transform is now available.
The proposed technique offers a stable and versatile approach for Compton imaging reconstruction.
This advancement holds significant potential for improving imaging capabilities in various fields.