Isometry property and inversion of the Radon transform over a family of paraboloids

초록

Integral geometry problems involve finding a desired function from its integrals on a surface. These problems are closely intertwined with the generalized Radon transform, and obtaining an inversion formula for it is pivotal in solving integral geometry problems. The applications of integral geometry span various fields, including tomography, radar, and radiology. Particularly noteworthy is the recovery of a function from integrals over a parabola, which holds significance in reflection seismology. In our study, we concentrate on the transform that maps a real-valued smooth function with compact support to integrals over the paraboloid. This transform, along with its dual, can be expressed as convolutions of kernels and given functions, and we have derived inversion formulas based on their isometric properties.

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