Convergence of Ray- and Pixel-Driven Discretization Frameworks in the Strong Operator Topology

TL;DR

This work analyzes the discretization of the 2D parallel-beam Radon transform in tomography by viewing ray-driven forward and pixel-driven backprojection as convolutional finite-rank operators. It proves convergence in the strong operator topology as the discretization steps vanish under balanced resolutions, providing a rigorous justification for rd–pd* discretization schemes. The authors extend the theory to limited- and sparse-angle settings, and complement the results with numerical experiments (e.g., FORBILD phantom) that validate the convergence and reveal angle-dependent behavior in the pixel-driven approach. The findings illuminate when unconstrained discretizations approximate the continuous Radon problem well and suggest practical guidance for choosing discretization parameters in tomographic reconstruction.

Abstract

Tomography is a central tool in medical applications, allowing doctors to investigate patients' interior features. The Radon transform (in two dimensions) is commonly used to model the measurement process in parallel-beam CT. Suitable discretization of the Radon transform and its adjoint (called the backprojection) is crucial. The most commonly used discretization approach combines what we refer to as the ray-driven Radon transform with what we refer to as the pixel-driven backprojection, as anecdotal reports describe these as showing the best approximation performance. However, there is little rigorous understanding of induced approximation errors. These methods involve three discretization parameters: the spatial-, detector-, and angular resolutions. Most commonly, balanced resolutions are used, i.e., the same (or similar) spatial- and detector resolutions are employed. We present an interpretation of ray- and pixel-driven discretizations as `convolutional methods', a special class of finite-rank operators. This allows for a structured analysis that can explain observed behavior. In particular, we prove convergence in the strong operator topology of the ray-driven Radon transform and the pixel-driven backprojection under balanced resolutions, thus theoretically justifying this approach. In particular, with high enough resolutions one can approximate the Radon transform arbitrarily well. Numerical experiments corroborate these theoretical findings.

Figures (14)

• Figure 1: On the left, an illustration of the used geometry with a straight line $L_{\phi,s}$ in direction $\vartheta_\phi^\perp$ with normal distance to the center $s$ (which also corresponds to the detector offset). On the right, an illustration of the backprojection, where for fixed $x$, we integrate values along a sine-shaped trajectory in the sinogram domain. This trajectory corresponds to all lines $L_{\phi,s}$ passing through $x$.
• Figure 2: On the left, the spatial domain $\mathop{\mathrm{\Omega}}\nolimits$ (in fact, the larger domain $[-1,1]^2$) is divided into pixels $X_{ij}$ with width $\delta_x\times \delta_x$. On the right, the discretization of the sinogram domain $\mathop{\mathrm{\mathcal{S}}}\nolimits$ into pixels $\Phi_q\times S_p$ with pixel centers $(\phi_q,s_p)$ and with width $|\Phi_q|\times \delta_s$ is shown.
• Figure 3: Depiction of the ray-driven weight function $t\mapsto \delta_x^2\mathop{\mathrm{\omega^\mathrm{rd}_{\delta_\mathit{x}}}}\nolimits(\phi,t)$ for fixed $\phi\in \{0^\circ,20^\circ,45^\circ\}$ in the first three plots. For fixed $\phi$, these are trapezoid functions (like the $20^\circ$ case), whose incline, height, and width depend on $\phi$ and $\delta_x$. In the extreme case $\phi=45^\circ$, the function turns into a hat function, while for $\phi=0^\circ$ it turns into a piecewise constant function (note the values for $\pm \frac{\delta_x}{2}$). On the right, in the last plot, the pixel-driven weight function $t\mapsto \delta_s^2\mathop{\mathrm{\omega^\mathrm{pd}_{\delta_\mathit{s}}}}\nolimits(t)$ (a hat function) independent of $\phi$ is shown. Note the difference in scales between the ray-driven and pixel-driven functions.
• Figure 4: Illustration of the ray-driven forward (left) and the pixel-driven backprojection (right). The ray-driven method splits integration along a straight line into the sum of values on pixels times their intersection length (colored segments). The pixel-driven backprojection approximates the angular integral \ref{['equ_def_backprojection']} (along the violet curve $x\cdot\vartheta_\phi$) by a finite sum (Riemann sum) of angular evaluations $x\cdot\vartheta_q$ (the cyan crosses), whose values are approximated via linear interpolation in the detector dimension (using the neighboring orange pixel centers).
• Figure 5: Depiction of the discrete $4096\times4096$ pixel FORBILD head phantom placed in the $[-12.5,12.5]^2$ square in a) and the corresponding $4096\times1800$ analytical Radon transform in b).

Theorems & Definitions (22)

• Definition 2.1: Sinogram domain
• Remark 2.2
• Definition 2.3: $L^2$ spaces
• Definition 2.4: Radon transform
• Definition 2.5: Weight functions
• Lemma 2.6: Closed form of the intersection length
• Definition 2.7: Convolutional discretizations
• Remark 2.8
• Theorem 3.1: Convergence in the strong operator topology
• Remark 3.2