Determination of range conditions for general projection pair operators

Abstract In the context of tomographic imaging, a projection is a set of line integrals over a collection of curves. This work discusses the range of tomographic ‘projection pair operators’ P consisting of two such projections. We restrict our investigations to a planar setting (operators that act on functions in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> ). Range conditions for P can be associated with the set of annihilators of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi>Rg</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . We find that this set of annihilators contains at most one dimension, and the corresponding range conditions can be explicitly determined by what we refer to as ‘kernel conditions’. With this tool, we show that under certain assumptions, the exponential fanbeam projection pair operator has dense range.