Tomographic projectors#
The discretized reconstruction problem of CT can be stated as retrieving the 3D object \(\mathbf x \in \mathbb{R}^{N_x N_y N_z}\) discretized on a voxel grid by solving the linear inverse problem
where the forward projection \(\mathbf A \colon \mathbf x \mapsto \mathbf y\) represents the X-ray transform, and \(\mathbf y \in \mathbb{R}^{N_\theta N_u N_v}\) denotes the measured projections at \(N_\theta\) angles of an \((N_u, N_v)\)-sized detector. The adjoint, \(\mathbf A^T \colon \mathbf y \mapsto \mathbf x\), is the backprojection. In practice, \(\mathbf y\) is often not the direct quantity measured by a detector in an X-ray setup, but is instead obtained after several preprocessing steps of the raw measurement data (e.g., using a \(\log\)-transform to invert Beer-Lambert’s law[1]).
Several discretization strategies can be considered to construct \(\mathbf A\) as well as its transpose \(\mathbf A^T\)[2][3]. Each of the \(N_u N_vN_\theta\) rows of \(\mathbf A\), c.q.columns of \(\mathbf A^T\), discretizes a single line integral associated with the X-ray transform. That is, it chooses interpolation weights to approximate the integral
which describes the straight line from a point source \(s_\theta\) to a detector pixel midpoint \(d_{\theta, u, v}\) through the volume \(x\). For 3D projectors, ASTRA estimates the line integral above with the Joseph kernel[4]. In this ray-driven approach, each integration point takes a trilinearly interpolated value from neighboring points on the voxel grid. Importantly, the line is modeled such that it precisely arrives at a detector pixel’s midpoint, and, hence, no re-interpolation is needed in the sinogram. Conversely, during a voxel-driven backprojection, a bilinear interpolation at each angle of the sinogram sums up to the voxel’s value[5]. In this case, all lines of backprojection go precisely through the voxel’s center, and now re-interpolation is avoided in the volume. The reader is referred to Hansen et al.[2] (Chapter 9) for interpolation formulae.
Due to the difference in forward and backward lines, the 3D conebeam FP and BP projectors in ASTRA are unmatched, i.e., the backprojector is not the exact transpose of the forward projector. The approach, however, is advantageous for an implementation on GPUs. All parallel threads (discussed in Section ref{sec:kernel}) are independent of each other, which avoids potential race conditions, i.e., when two threads would write to the same memory simultaneously[6]. On the other hand, unmatched projectors lead to nonconvergence in iterative algorithms due to a nonsymmetry of the iteration matrix[7]. In the presence of noise, this does not always pose a problem[8].