Image reconstruction from radially incomplete spherical Radon data
This work advances theoretical understanding and practical reconstruction for imaging modalities using spherical Radon transforms, such as medical, radar, and sonar imaging.
The paper proves unique inversion of the spherical Radon transform from radially incomplete data, providing reconstruction formulas for images inside, outside, or on both sides of the data acquisition sphere, and demonstrates an accurate and efficient computational algorithm on numerical examples.
We study inversion of the spherical Radon transform with centers on a sphere (the data acquisition set). Such inversions are essential in various image reconstruction problems arising in medical, radar and sonar imaging. In the case of radially incomplete data, we show that the spherical Radon transform can be uniquely inverted recovering the image function in spherical shells. Our result is valid when the support of the image function is inside the data acquisition sphere, outside that sphere, as well as on both sides of the sphere. Furthermore, in addition to the uniqueness result our method of proof provides reconstruction formulas for all those cases. We present a robust computational algorithm based on our inversion formula and demonstrate its accuracy and efficiency on several numerical examples.