Evaluating Bresenham-based Binary versus Siddon-based Weighted Ray-Tracing Algorithms for Sparse-View 3D Reconstruction of Coronary Vasculature

Technical Summary: Evaluating Bresenham-based Binary versus Siddon-based Weighted Ray-Tracing Algorithms for Sparse-View 3D Reconstruction of Coronary Vasculature

Problem Statement
Interventional C-arm imaging is a standard tool for guiding minimally invasive cardiac procedures, offering reduced radiation exposure compared to computed tomography (CT) by utilizing a limited number of projections. However, reconstructing three-dimensional (3D) data from such sparse views (typically around 9 projections) presents a severely ill-posed inverse problem. Standard analytical methods like Filtered Back Projection (FBP) fail under these conditions due to angular undersampling, resulting in severe aliasing and streak artifacts. Furthermore, clinical low-dose conditions introduce high levels of Poisson quantum noise. The core challenge addressed in this work is the geometric inaccuracy introduced by the mathematical approximation of the X-ray path through a discrete voxel grid. Specifically, the study investigates whether the choice of forward projector—binary (Bresenham) versus weighted (Siddon)—fundamentally limits the ability to recover accurate vascular morphology, particularly in the presence of noise and sparse sampling.

Methodology
The study employs a digital twin validation framework using three patient-specific coronary models derived from the Vascular Model Repository. These models serve as ground truth for an in silico simulation of the C-arm imaging process.

1. Simulation Setup:

   • Data Generation: High-resolution 3D vascular meshes are forward-projected to generate 9 sparse-view 2D projections spanning a 180° arc.
   • Noise Modeling: To simulate clinical low-dose conditions, Poisson noise is added to the projections, corresponding to an incident intensity of $5 \times 10^3$ photons per pixel.
   • Ground Truth: Crucially, the ground-truth projections are generated using a high-fidelity weighted (Siddon) model to avoid "inverse crime" (testing an algorithm with the same model used to generate data).

2. Reconstruction Framework:

   • Solver: The reconstruction utilizes the Simultaneous Iterative Reconstruction Technique (SIRT) combined with Total Variation (TV) minimization to address the ill-posed nature of the problem and suppress noise.
   • Optimization: The problem is formulated as a regularized convex optimization: $\hat{x} = \arg\min_{x \ge 0} (||Ax - b||^2 + \lambda R(x))$, where $A$ is the system matrix (projector), $b$ is the projection data, and $R(x)$ is the TV regularization term.
   • Variables: The primary independent variable is the system matrix $A$, implemented in two distinct ways:
     • Binary Approach (Bresenham): Uses integer arithmetic to select voxels based on proximity to the ray center. Voxels are assigned a weight of 1 if intersected and 0 otherwise, creating a "staircase" effect.
     • Weighted Approach (Siddon): Calculates the exact geometric length of the ray segment within each voxel. Voxels are assigned continuous fractional weights based on these intersection lengths, providing sub-voxel accuracy.

3. Evaluation Metrics:

   • Geometric Overlap: Dice Similarity Coefficient (DSC) and Jaccard Index (IoU) calculated on binarized vessel masks.
   • Volume Preservation: Volume Ratio (VR) comparing reconstructed volume to ground truth.
   • Intensity Fidelity: Root Mean Square Error (RMSE) calculated on continuous grayscale volumes prior to binarization.

Key Contributions
The primary contribution of this work is the isolation and quantification of the geometric errors inherent to binary ray-tracing algorithms in sparse-view reconstruction. The study demonstrates that:

• Fundamental Limitation of Binary Methods: The Bresenham algorithm introduces an irreducible baseline error. By treating continuous rays as blocky chains of voxels, it artificially dilates vessel boundaries. This morphological dilation cannot be corrected by downstream regularization (TV) because the error is embedded in the system matrix $A$ itself.
• Necessity of Weighted Projection: The Siddon algorithm, by accounting for partial volume effects through exact intersection lengths, acts as an intrinsic spatial anti-aliasing filter. This allows the iterative solver to converge on a solution that preserves the true morphological boundaries of the vasculature.
• Interplay with Regularization: The study highlights that while TV regularization is effective at suppressing noise, its ability to preserve fine anatomical details is contingent upon the accuracy of the underlying forward projector.

Results
Quantitative and visual evaluations across three distinct coronary models under simulated low-dose noise conditions yielded the following results:

• Binary (Bresenham) Performance:

  • Exhibited significant geometric failure, characterized by artificial vessel thickening and volume inflation.
  • Volume Ratio: Mean VR of $1.95 \pm 0.47$ (nearly doubling the vessel volume).
  • Overlap: Mean Dice coefficient of $0.61 \pm 0.05$ and Jaccard index of $0.44 \pm 0.05$.
  • Intensity: High RMSE ($0.08 \pm 0.04$), indicating poor recovery of physical attenuation values.
  • Visuals: Reconstructions showed merged adjacent branches and fragmented distal vessels.

• Weighted (Siddon) Performance:

  • Demonstrated robustness to noise and superior geometric fidelity.
  • Volume Ratio: Mean VR of $1.05 \pm 0.11$ (minimal overestimation).
  • Overlap: Mean Dice coefficient of $0.80 \pm 0.06$ and Jaccard index of $0.67 \pm 0.09$.
  • Intensity: RMSE reduced by over 50% to $0.03 \pm 0.03$.
  • Visuals: Successfully preserved tubular morphology and distal branch connectivity, with effective noise suppression.

Significance and Claims
The paper claims that accurate 3D reconstruction of coronary vasculature from sparse-view, low-dose C-arm imaging is feasible, but strictly contingent upon the use of suitable backprojection algorithms. The authors assert that the fundamental geometric accuracy of the system matrix $A$ is a prerequisite for any successful reconstruction, regardless of the sophistication of the regularization or solver used.

Specifically, the study argues that:

1. Binary approximations are insufficient: The "staircase" artifacts introduced by binary ray-tracers lead to morphological dilation that renders the reconstructed models unsuitable for quantitative clinical applications (e.g., Fractional Flow Reserve estimation), which are highly sensitive to vessel radius deviations.
2. Weighted projection is mandatory for high fidelity: To recover distal vessel connectivity and mitigate volumetric dilation, the forward projector must utilize weighted, sub-voxel accurate modeling (Siddon).
3. Implications for AI: The findings suggest that deep learning models trained on synthetic data generated with coarse binary projectors may inherently learn flawed representations of X-ray physics, encoding geometric errors that persist even when processing real clinical data.

The study concludes that while the weighted method significantly improves reconstruction quality, it does not completely resolve the reconstruction of the smallest vascular structures, and future work is required to address motion artifacts and further refine regularization for thin tubular structures.