Vector-driven: A new projection and backprojection algorithm based on vector mapping

Abstract

X-ray imaging is a technique used to visualize the internal structures of objects, primarily employed in medical diagnostics. X-rays, which are special electromagnetic waves with wavelengths ranging from 10 nanometers to 10 picometers, penetrate objects and are absorbed by varying internal densities. This differential absorption allows detectors to capture and measure the X-rays' intensities, creating images that reveal internal structures. A major advantage of X-ray imaging is the ability to observe internal structures without invasive procedures. However, radiation exposure poses significant risks, especially in medical applications, requiring a careful balance between radiation dose and image quality. This study focuses on the medical applications of X-ray imaging, particularly the need for multiple projections to achieve 3D reconstructions from 2D projections. The trade-off between image quality and radiation exposure requires careful parameter optimization. Image reconstruction transforms 2D projections into 3D images, a process involving forward projection (3D to 2D transformation) and backprojection (2D to 3D transformation). The body is modeled as an array of attenuation coefficients, which absorb X-rays to varying degrees. The reconstruction process involves calculating line integrals on the image to fill detector cells, storing projections in a sinogram for further processing. Two main domains are used: the image domain (representing the reconstructed image) and the sinogram domain (used for filtering and analyzing projection quality). Projection and backprojection are critical operations in image reconstruction, with three main approaches: pixel-driven (PD), ray-driven (RD), and distance-driven (DD). Each approach models the geometrical structure of X-ray imaging differently. PD calculates X-ray beams passing through pixel centers to detector cells, RD attaches X-rays to detector cell centers and evaluates contributions along the X-ray trace, and DD maps pixel and detector cell boundaries onto a common axis to calculate overlapping regions. DD is the state-of-the-art for projection and backprojection. However, DD algorithm faces problems like index mismatching and non-rectangular overlapping when the detector is tilted. These issues significantly limit DD's applicability, necessitating new solutions. This study proposes a new algorithm called vector-driven to address the tilted detector problem, non-rectangular overlapping, and index matching issues in DD. The algorithm involves three stages: mapping, redefining boundaries, and calculating overlapping regions. The first stage projects voxels directly onto the detector plane, eliminating distortions caused by mapping onto a common axis. The second stage interpolates non-rectangular voxel distributions onto a regular grid, simplifying overlap calculations. The third stage uses standard overlap calculation methods after redefining boundaries. This approach enhances robustness against geometrical limitations, making the reconstruction system more flexible and suitable for custom designs. The proposed algorithm also facilitates parallelization, improving computational efficiency. To examine the proposed model, we developed a tomographic imaging toolbox using Python 3.8.0 to simulate X-ray imaging. The system is organized into modules following object-oriented programming principles, categorized by fundamental stages of tomographic imaging. The setup includes components like the beam source, detector, data object, projector, backprojector, and reconstructor. Each module corresponds to real or imaginary entities linked by attributes and methods. Several experiments compared the efficiency of the proposed method against DD and branchless distance-driven (BDD). The proposed vector-driven algorithm proved robustness against rotations, capturing projections without causing artifacts. However, DD and BDD exhibited significant distortions, particularly with detector rotations around y and z axis. The VD algorithm's robustness makes it more compatible with flexible digital breast tomosynthesis (DBT) applications. Timing is a less critical criterion compared to accuracy, but important for handling high-resolution images and extensive computations required by AI applications. The proposed algorithm, while not the fastest, performed acceptably and can be further optimized. The proposed vector-driven algorithm outperformed DD and BDD in handling rotations and geometrical distortions, making it a robust and feasible solution for projection and backprojection operations. Future improvements could include parallelization, different interpolators, and new filtering topologies to enhance performance and flexibility further.