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Determination of Range Conditions for General Projection Pair Operators

Richard Huber, Rolf Clackdoyle, Laurent Desbat

TL;DR

The paper develops Projection Pair Range Conditions (PPRCs) to characterize the range of two-projection operators in the plane, formalizing a kernel condition that links projection geometry, weights, and Jacobians to projection kernels $(V_1,V_2)$. This framework recovers known range conditions for parallel and fanbeam geometries and extends to mixed parallel-fanbeam configurations, deriving explicit kernels in favorable cases and clarifying when singular kernels necessitate principal-value interpretations. Crucially, the authors prove that no PPRC exists for the exponential fanbeam transform, implying a dense range and that any data pair $(g_1,g_2)$ can be approximated in $L^2$ by projections of smooth $f$; this contrasts with classical projections and has implications for data consistency testing and parameter identification. A numerical demonstration corroborates the density result, showing how an appropriate smooth $f$ can reproduce given data to near machine precision in a discretized setting, highlighting both theoretical and practical aspects of range behavior in tomographic models.

Abstract

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.

Determination of Range Conditions for General Projection Pair Operators

TL;DR

The paper develops Projection Pair Range Conditions (PPRCs) to characterize the range of two-projection operators in the plane, formalizing a kernel condition that links projection geometry, weights, and Jacobians to projection kernels . This framework recovers known range conditions for parallel and fanbeam geometries and extends to mixed parallel-fanbeam configurations, deriving explicit kernels in favorable cases and clarifying when singular kernels necessitate principal-value interpretations. Crucially, the authors prove that no PPRC exists for the exponential fanbeam transform, implying a dense range and that any data pair can be approximated in by projections of smooth ; this contrasts with classical projections and has implications for data consistency testing and parameter identification. A numerical demonstration corroborates the density result, showing how an appropriate smooth can reproduce given data to near machine precision in a discretized setting, highlighting both theoretical and practical aspects of range behavior in tomographic models.

Abstract

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' . 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 -projections operator . The most well-known example of an -projections operator is the restriction of the Radon transform to finitely many projection angles. Characterizations of the range of -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 in the plane, i.e., operators formed by two projections () applied to functions in . We find that the set of annihilators to 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 sense by projections of smooth functions. We also illustrate the use of this theory on a mixed parallel-fanbeam projection pair operator.

Paper Structure

This paper contains 10 sections, 8 theorems, 55 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Theoretical Investigations of General Projection Pair Operators
  3. Preliminaries and single projection operators
  4. The identification of projection pair range conditions
  5. Known Range Conditions as PPRCs
  6. Classical projections as single projection operators
  7. PPRCs for mixed parallel-fanbeam projection geometries
  8. Pairwise Range Conditions for the Exponential Fanbeam Transform
  9. No PPRC for $\mathop{\mathrm{\mathcal{E}}}\nolimits^{\lambda_1,\lambda_2}$
  10. Numerical approximation of inconceivable data

Key Result

Lemma 3

Let $\mathop{\mathrm{\mathfrak{p}}}\nolimits$ be a single projection operator. Then $\mathop{\mathrm{\mathfrak{p}}}\nolimits$ maps from $\mathcal{C}^\infty_c(\mathop{\mathrm{\Omega}}\nolimits)$ into $L^\infty_c(\mathop{\mathrm{R}}\nolimits)$, and is linear. Moreover, there is a constant $c>0$ such t

Figures (9)

  • Figure 1: On the top, an illustration of a generic bijection $\gamma$ between the domain $\mathop{\mathrm{\Omega}}\nolimits$ in gray on the left, and $\mathop{\mathrm{\mathcal{M}}}\nolimits$ on the right. The colored straight lines on the right (associated with fixed values for $\mathop{\mathrm{r}}\nolimits$) are transformed by $\gamma$ into curves covering $\mathop{\mathrm{\Omega}}\nolimits$. The dashed black lines illustrate $\gamma$ for fixed $\mathop{\mathrm{t}}\nolimits$. In particular, $\mathop{\mathrm{T}}\nolimits(\mathop{\mathrm{r}}\nolimits)$ is the intersection of the colored line representing a fixed $\mathop{\mathrm{r}}\nolimits$ with $\mathop{\mathrm{\mathcal{M}}}\nolimits$, and is depicted for the orange $\mathop{\mathrm{r}}\nolimits$. In the bottom row, analogous illustrations for the parallel beam geometry setting (on the left), and for a fanbeam geometry setting (on the right) are shown.
  • Figure 2: Geometric illustration of Assumption \ref{['Assumption_curve_independent']} for two fanbeam projections as described in Remark \ref{['Remark_Assumption_violated_Fanbeam']}. It shows in orange the domain $\mathop{\mathrm{\Omega}}\nolimits$, and in red and blue the corresponding straight lines (curves), solid insight $\mathop{\mathrm{\Omega}}\nolimits$, and dashed outside. On the left, the fanbeam vertex points are on the left, so that the line connecting them (in teal) is outside $\mathop{\mathrm{\Omega}}\nolimits$, while on the right, the connecting line is inside $\mathop{\mathrm{\Omega}}\nolimits$. In particular, for the $\mathop{\mathrm{r}}\nolimits_1$ and $\mathop{\mathrm{r}}\nolimits_2$ forming said intersection line, there are infinitely many intersection points.
  • Figure 3: Illustration of Lemma \ref{['lemma_connected_separation']} as described in Remark \ref{['remark_assumption_curves_independent']} for parallel beam projections with directions $\vartheta_1=(1,0)^T$ and $\vartheta_2=\frac{1}{\sqrt 2} (1,1)^T$ (associated with angles $0^\circ$ and $45^\circ$). On the left, an illustration of the intersection point of the thick lines $\gamma_1(\mathop{\mathrm{r}}\nolimits_1,\cdot)$ and $\gamma_2(\mathop{\mathrm{r}}\nolimits_2,\cdot)$ being $X(\mathop{\mathrm{r}}\nolimits_1,\mathop{\mathrm{r}}\nolimits_2)$. In the center and on the right, an illustration of typical functions $\mathop{\mathrm{h}}\nolimits_1(\mathop{\mathrm{\rho}}\nolimits_1(x))$ and $\mathop{\mathrm{h}}\nolimits_2(\mathop{\mathrm{\rho}}\nolimits_2(x))$.
  • Figure 4: Illustration of the maps $\gamma_{\theta}^\text{par}$ (see \ref{['equ_definition_curve_parallel']}) on the left and $\gamma_{\lambda}^\text{fan}$ (see \ref{['equ_definition_curve_fanbeam']}) on the right related to the parallel beam and fanbeam projection curves, respectively. The dashed lines represent the curves induced by various $\mathop{\mathrm{r}}\nolimits \in \mathop{\mathrm{R}}\nolimits$ (on the left, each $\mathop{\mathrm{r}}\nolimits\in \mathop{\mathrm{R}}\nolimits$ represents a shift of the ray, while on the right, each $\mathop{\mathrm{r}}\nolimits$ is a different angle for the ray) and fixed $\theta$ or $\lambda$, while the orange $\mathop{\mathrm{t}}\nolimits$ and teal $\mathop{\mathrm{r}}\nolimits$ are the coordinates associated with the $x=\gamma(\mathop{\mathrm{r}}\nolimits,\mathop{\mathrm{t}}\nolimits)$ represented by the red dot.
  • Figure 5: Illustration of the assumption on the geometry in Section \ref{['section_appendix_different_projection_geometries']}, with blue parallel lines with angle $\theta$, and red fanbeam lines originating in $\lambda$. On the left, the assumption is satisfied, while on the right, it is violated as the teal line emitted from $\lambda$ coincides with a parallel line, or (equivalently) the parallel lines passing through $\mathop{\mathrm{\Omega}}\nolimits$ do hit $\lambda$.
  • ...and 4 more figures

Theorems & Definitions (27)

  • Definition 2
  • Lemma 3
  • proof
  • Definition 5
  • Definition 6
  • Remark 7
  • Lemma 9
  • proof
  • Remark 10
  • Lemma 11
  • ...and 17 more