Generalized tomographic maps

TL;DR

This paper explores new generalizations of tomographic transforms for quadratic surfaces, analyzing elliptic, hyperbolic, and hybrid types, and establishing inverse maps for these generalized tomograms.

Contribution

It introduces novel methods for quadratic surface tomography, including deformation-based and shift-based approaches, expanding the theoretical framework of tomographic transforms.

Findings

Inverse tomographic maps can be consistently defined for all generalized cases.

Two distinct methods for defining tomographic sections are proposed.

The generalized transforms include deformations of hyperplanes and quadrics.

Abstract

We introduce several possible generalizations of tomography for quadratic surfaces. We analyze different types of elliptic, hyperbolic and hybrid tomograms. In all cases it is possible to consistently define the inverse tomographic map. We find two different ways of introducing tomographic sections. The first method operates by deformations of the standard Radon transform. The second method proceeds by shifting a given quadric pattern. The most general tomographic transformation can be defined in terms of marginals over surfaces generated by deformations of complete families of hyperplanes or quadrics. We discuss practical and conceptual perspectives and possible applications.