Convergence of ray- and pixel-driven discretization frameworks in the strong operator topology

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 $ \Omega $ (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 $ \mathcal{S} $ 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 \omega^\mathrm{rd}_{\delta_\mathit{x}}(\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 \omega^\mathrm{pd}_{\delta_\mathit{s}}(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 (4) (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)

Figure 6.  Illustration of the pointwise absolute difference between the analytical Radon transform of the FORBILD phantom and the ray-driven projection (on the left) or the pixel-driven projection (on the right). For both, the balanced situation $ N_x = N_s = 4096 $ and $ N_\phi = 1800 $ is used

Figure 7.  Illustration of the projections for $ \phi = \frac{3}{4}\pi = 135^\circ $ on the left, and for $ \phi = 135.1^\circ $ on the right. Even though the angles only differ by a tenth of a degree, the pixel-driven method reveals significant errors in the former setting that have all but disappeared in the latter one

Figure 8.  Illustration of the relative $ L^2 $ error of the individual projections (i.e., for variable $ \phi $) $ E_{\phi_q}(g, \mathcal{R}^\mathrm{rd}_{\delta} f) $ and $ E_{\phi_q}(g, \mathcal{R}^\mathrm{pd}_{\delta} f) $. The extreme cases $ \frac{\pi}{4} $ and $ \frac{3\pi}{4} $ are far beyond the shown scale (by a factor of 10), but there are also other projections whose errors far exceed the average errors (around $ 5\ 10^{-4} $)

Figure 9.  Illustration of the relative $ L^2 $ errors for fixed $ N_\phi = 360 $ and increasing balanced resolutions $ N_x = N_s $ between $ 256 $ and $ 8192 $ in 32 steps. On the top, the errors $ E(g, \mathcal{R}_\delta f) $ for the entire sinogram are depicted, while the figure below shows the relative error of the worst single projection $ E_\text{max}(g, \mathcal{R}_\delta f) $. As can be seen, the pixel-driven error appears stagnant

Figure 10.  Development of relative ray-driven backprojection errors for fixed balanced resolution $ N_x = N_s = 2000 $ and increasing $ N_\phi $. The behavior is roughly $ \sqrt{\delta_\phi} $; see the red crosses for comparison

Figure 11.  Illustration of the error $ E( \mathcal{R}^*g, \mathcal{R}^\mathrm{rd^*}_{\delta}g) $ for the fixed $ N_\phi = 90 $ and increasing $ N_x $ and three different ratios $ \frac{\delta_s}{\delta_x} = \frac{N_x}{N_s} $

Figure 12.  The first row shows the pointwise absolute difference $ | \mathcal{R}^\mathrm{rd^*}_{\delta}g- \mathcal{R}^*g| $ for increasing balanced resolutions $ \frac{\delta_s}{\delta_x} = 1 $. The second line shows the same differences for $ \frac{\delta_s}{\delta_x} = \frac{1}{2} $. In both cases $ N_\phi = 90 $ remains fixed. Note the different colormaps between the first and second row

Figure 13.  The evolution of relative $ L^2 $ errors $ E( \mathcal{R}^*g, \mathcal{R}^\mathrm{rd^*}_{\delta}g) $ in the ray-driven backprojection with increasing $ N_s $ while $ N_\phi = 90 $ is fixed and $ N_x = 1000 $ or $ N_x = 2000 $

Figure 14.  Illustration supporting the proof of Lemma 2.6. In a), we depict the different cases passing through a square $ Z $ for fixed $ \phi $ (here $ 105^\circ $), where the teal lines describe the case 4 $ |s| = \underline s(\phi) $ in which the corners are precisely hit, the brown line $ |s_1|< \underline s(\phi) $ (case 1) and the red line $ |s_2| \in [\underline s(\phi), \overline s(\phi)[ $ (case 2). Moreover, the blue line is representative of case 5 (hitting exactly one corner), while the violet line representing case 3 does not hit $ Z $, and the dashed magenta line is representative of case 6 with the line intersecting with one side of $ Z $. Figures b) and c) detail the geometry of case 1 and case 2, depicting relevant right triangles.