A Note on the Sonar Transform and Related Radon Transforms

2022/06/12 by Rubin, Boris
#FOS: Mathematics #Functional Analysis (math.FA) #Primary 44A12 #Secondary 47G10

paper · doi:10.48550/arxiv.2206.05854

Abstract

The sonar transform in geometric tomography maps functions on the Euclidean half-space to integrals of those functions over hemispheres centered on the boundary hyperplane. We obtain sharp L^p-L^q estimates for this transform and new explicit inversion formulas under minimal assumptions for functions. The main results follow from intriguing connection between the sonar transform, the Radon transform over paraboloids, and the transversal Radon transform, which integrates functions over hyperplanes, meeting the last coordinate axis.

Related

• Operator identities relating sonar and Radon transforms in Euclidean space
• Explicit inversion formulas for the spherical mean Radon transform
• Ellipsoidal and hyperbolic Radon transforms; microlocal properties and injectivity
• Reconstruction of a function from its two spherical Radon transforms with the centers on a plane
• Surface of revolution Radon transforms with centers on generalized surfaces in ℝⁿ