A Novel Interpretation of the Radon Transform's Ray- and Pixel-Driven Discretizations under Balanced Resolutions

TL;DR

This paper addresses discretization of the Radon transform in planar CT under balanced resolutions and unifies ray-driven forward projections with pixel-driven backprojections through a convolutional discretization framework. It introduces weight-based operators $\mathcal{R}_\omega$ using $\omega^{\mathrm{rd}}_\delta$ and $\omega^{\mathrm{pd}}_\delta$, and proves strong-operator convergence results: for discretization sequences with $\frac{\delta_s}{\delta_x}$ bounded the ray-driven forward approximation converges to the true Radon transform, and under $\frac{\delta_s}{\delta_x}\to 0$ the ray-driven backprojection as well as the pixel-driven backprojection converge appropriately; these findings are complemented by numerical simulations highlighting the role of weight functions and angular discretization. The results provide theoretical support for the widespread use of rd-pd* approaches at balanced resolutions and point toward potential extensions to other tomography geometries and to unmatched operators. The work thus offers a rigorous foundation for discretization choices in CT and guides future investigations into operator-norm convergence in regimes where $\delta_s/\delta_x\to 0$.

Abstract

Tomographic investigations are a central tool in medical applications, allowing doctors to image the interior of patients. The corresponding measurement process is commonly modeled by the Radon transform. In practice, the solution of the tomographic problem requires discretization of the Radon transform and its adjoint (called the backprojection). There are various discretization schemes; often structured around three discretization parameters: spatial-, detector-, and angular resolutions. The most widespread approach uses the ray-driven Radon transform and the pixel-driven backprojection in a balanced resolution setting, i.e., the spatial resolution roughly equals the detector resolution. The use of these particular discretization approaches is based on anecdotal reports of their approximation performance, but there is little rigorous analysis of these methods' approximation errors. This paper presents a novel interpretation of ray-driven and pixel-driven methods as convolutional discretizations, illustrating that from an abstract perspective these methods are similar. Moreover, we announce statements concerning the convergence of the ray-driven Radon transform and the pixel-driven backprojection under balanced resolutions. Our considerations are supported by numerical experiments highlighting aspects of the discussed methods.

Figures (6)

• Figure 1: On the left is the geometry of the Radon transform; in the middle is the discretization of the spatial domain $\mathop{\mathrm{\Omega}}\nolimits$ into pixels $X_{ij}$ with width $\delta_x\times \delta_x$; on the right is the discretization of the sinogram domain $\mathop{\mathrm{\mathcal{S}}}\nolimits$ into pixels $\Phi_q\times S_p$ of width $|\Phi_q|\times \delta_s$.
• Figure 2: Depiction of the ray-driven weight function $t\mapsto \delta_x^2\mathop{\mathrm{\omega^\mathrm{rd}_{\delta}}}\nolimits(\phi,t)$ for fixed $\phi\in \{0^\circ,30^\circ,45^\circ\}$. On the right, the depiction of the pixel-driven weight $t\mapsto \delta_s^2\mathop{\mathrm{\omega^\mathrm{pd}_{\delta}}}\nolimits(t)$.
• Figure 3: Illustration of the ray-driven Radon transform (left) and pixel-driven backprojection (right). In the former, integration along a straight line is split into the sum of integrals along the intersections with pixels. The pixel-driven backprojection discretizes the angular integral \ref{['equ_def_backprojection']} (along the blue curve $(x\cdot\vartheta_\phi,\phi)$) by a finite sum of angular evaluations $(x\cdot\vartheta_q,\phi_q)$ (red dots), with linear interpolation in the detector dimension (using the points represented by the green dots).
• Figure 4: Illustration of the error incurred by ray-driven backprojections for Example 1. In the first row, a balanced resolutions setting with fixed $N_\phi=90$ but increasing $N_x=N_s$ is shown. The amplitude of the errors and the relative $L^2$ error, do not reduce with finer spatial and detector resolutions. In d) and e), we increase $N_\phi$ to $N_\phi=180$ and $N_\phi=360$, yielding slight improvements. Lastly, we depict the error when reducing the spatial resolution to $N_x=1000$ while keeping $N_s=4000$ detector pixels, leaving the balanced resolution setting and significantly reducing the error.
• Figure 5: Illustration of errors for Example 2, depicting the approximation errors of ray-driven backprojection with $\hat{q}$ such that $\phi_{\hat{q}}=\frac{\pi}{4}$. Under balanced resolutions in a) and b), these errors are very significant and do not reduce with an increased number of angles. In c), we leave the balanced resolution setting, reducing errors significantly.

Theorems & Definitions (5)

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