Determination of Range Conditions for General Projection Pair Operators

Tomographic techniques are vital in modern medicine, allowing doctors to observe patients' interior features. Individual steps in the measurement process are modeled by `single projection operators' $p$. These are line integral operators over a collection of curves that covers the regions of interest. Then, the entire measurement process can be understood as a finite collection of such single projections, and thus be modeled by an $N$-projections operator $P=(p_1,\dots,p_N)$. The most well-known example of an $N$-projections operator is the restriction of the Radon transform to finitely many projection angles. Characterizations of the range of $N$-projections operators are of intrinsic mathematical interest and can also help in practical applications such as geometric calibration, motion detection, or model parameter identification. In this work, we investigate the range of projection pair operators $\mathcal{P}$ in the plane, i.e., operators formed by two projections ($N=2$) applied to functions in $\mathbb{R}^2$. We find that the set of annihilators to $\mathrm{rg}(\mathcal{P})$ that are regular distributions contains at most one dimension and a range condition can be explicitly determined by what we refer to as `kernel conditions'. With this tool, we examine the exponential fanbeam transform for which no range conditions were known, finding that no (regular) range condition exists, and therefore, arbitrary data can be approximated in an $L^2$ sense by projections of smooth functions. We also illustrate the use of this theory on a mixed parallel-fanbeam projection pair operator.