Orthogonal Function Series Expansions and the Null Space of the Radon Transform

Abstract

The range of the Radon transform for compactly supported functions is spanned by products of Gegenbauer polynomials and spherical harmonics. The inverse transform of those basis functions is given for arbitrary dimensions and arbitrary Gegenbauer polynomials. The resulting inversion formula is applied to study the “ghosts”, i.e. the functions in the null space of the transform for finitely many projections. They are characterized by their series expansions and results concerning the optimal choice of directions for the data acquisition are deduced. The application to nuclear magnetic resonance zeugmatography is discussed.