Fourier-based Inversion of Partial X-ray Transforms in n Dimensions

Abstract

We present two theorems describing analytic left-inverses of partial X-ray transforms. The first theorem concerns X-ray data collected with an arbitrary distribution of parallel projections; it contains a convolution-backprojection formula and a backprojection-convolution formula for recovering the transformed volume, provided the data is sufficient. The second theorem concerns X-ray data collected with a cone-beam; it contains a backprojection-convolution formula for recovering the transformed volume, provided the data is amenable to this method (for example: (n-1)-dimensional source loci that `surround' the reconstruction support; detectors of finite size are supported). These theorems are the outcome of a modestly general and rigorous investigation undertaken into the existence of backprojection-convolution methods in n-dimensional space. Necessary and sufficient conditions on the experiment geometry are established for the existence of such methods, as are the particular error metrics minimised by backprojection-convolution methods and the uniqueness of those minimum-error solutions. A major practical outcome of this work is the production of the first known exact inversion methods for cone-beam geometries where the X-ray source point loci are multidimensional, such as (in 3D) a cylinder or a sphere of X-ray source positions. A separate article describes a practical computer implementation for the case of a cylinder in 3D.