Covariant tomography of fields

This paper develops 'covariant tomography', a local framework for solving Inverse Boundary Value Problems (IBVP) for parallel transport equation on star-shaped domains. By integrating geometric decomposition with specific interior extensions - radial, heat equation, or harmonic - the method reconstructs currents and gauge potentials from boundary data. The choice of extension directly dictates the regularity of the recovered interior fields. A primary contribution is the 'tower' algorithm, which reduces higher-order systems, such as Maxwell equations, to a sequence of coupled first-order equations. We establish a formal solvability criterion (Theorem \ref{Th_TowerTheorem}), proving that higher-order IBVPs are solvable if and only if this tower is sequentially solvable. The framework is validated through low-dimensional examples and electromagnetic potential reconstruction in $\mathbb{R}^{3}$.