Sampling Conditions for the Circular Radon Transform.

Area of Science:

• Applied Mathematics
• Image Reconstruction
• Medical Imaging

Background:

• Function recovery from circular or spherical mean values underpins modern imaging like computed tomography.
• Significant advancements exist for continuous data in circular Radon transform problems.
• Practical imaging relies on discrete sampling, posing a challenge for existing inversion formulas.

Purpose of the Study:

• To address the challenge of recovering functions from a discrete set of circular mean values.
• To apply Shannon sampling theory to determine necessary angular and radial sampling conditions.
• To enable accurate function recovery in practical computed tomography scenarios.

Main Methods:

• Utilized Shannon sampling theory to analyze discrete circular mean value data.
• Derived specific sampling conditions for both angular and radial data acquisition.
• Focused on the inversion of the circular Radon transform for b0-bandlimited functions.

Main Results:

• Established sharp sampling conditions for discrete circular mean value data.
• Demonstrated that any essentially b0-bandlimited function can be recovered.
• Quantified the required number of angular and radial samples for accurate reconstruction.

Conclusions:

• The derived sampling conditions provide a theoretical basis for discrete data acquisition in tomographic imaging.
• This work bridges the gap between theoretical inversion formulas and practical, discrete data collection.
• Enables robust function recovery in applications like photoacoustic and ultrasound computed tomography.