Linear Artificial Tomography (LAT)

• Linear Artificial Tomography (LAT) is a tomographic imaging method that uses straight-line, parallel translations to gather truncated projection data for ROI reconstruction.
• It employs BPF-type algorithms such as MP-BPF and MZ-BPF, which combine differentiation, weighted backprojection, and finite-support inverse Hilbert transforms.
• The approach minimizes artifacts through tailored redundancy weighting and smooth tapering, enabling robust, cost-effective imaging even with limited data.

Linear Artificial Tomography (LAT) encompasses tomographic imaging systems in which the locus of source–detector pairs is restricted to a straight line or a limited set of straight-line segments, in contrast to the traditional circular or spiral trajectories. Recent research on parallel translational computed tomography (PTCT) provides an explicit framework for LAT, demonstrating that high-fidelity region-of-interest (ROI) images can be reconstructed even from truncated data by leveraging novel backprojection–filtration (BPF) algorithms with finite-support Hilbert inversion and carefully designed redundancy weighting (Wu et al., 2016).

1. System Architecture and Data Acquisition Model

LAT, as instantiated in PTCT, operates with a fixed 2D Cartesian frame $(x,y)$, where the object is compactly supported and centered near the origin. The X-ray source and detector translate synchronously along parallel, but oppositely directed, straight lines, typically at heights $+h$ (source) and $-d$ (detector) relative to the origin. The source trajectory is parameterized as

$$\Phi(\lambda) = ( \lambda\cos\psi - h\sin\psi, \, \lambda\sin\psi + h\cos\psi ), \;\; \lambda \in [\lambda_{\min}, \lambda_{\max}]$$

where $\psi$ denotes the scan orientation.

The data acquisition in fan-beam geometry is modeled by

$$p(\lambda, s) = \int_{-\infty}^{\infty} f\bigl( \Phi(\lambda) + t\,\mathbf{e}(\lambda,s) \bigr)\,dt$$

where $\mathbf{e}(\lambda,s)$ is the ray direction from the source at $\Phi(\lambda)$ to detector coordinate $s$. Measurements are sampled as $p_{j,k} = p(\lambda_j, s_k)$ with $j = 1, \ldots, J$, $k = 1, \ldots, K$. In typical LAT/PTCT deployments, only the central subset of detector coordinates is sampled, leading to data truncation relative to the full object support.

2. BPF-Type Algorithmic Framework: MP-BPF and MZ-BPF

To address severe truncation artifacts in PTCT, two BPF-type algorithms were introduced: MP-BPF and MZ-BPF. The BPF approach proceeds in three main stages for each L-PI-line (an Editor's term for a straight chord in object space covered by the scan geometry):

1. Differentiation: The measured projection data is differentiated with respect to detector coordinate:

   $\frac{\partial}{\partial s} p(\lambda, s)$

2. Weighted Backprojection: For each line segment, a 1D Hilbert image is constructed as

   $$b_\ell(r) = \int_{\Lambda(r)} w(r,\lambda) \frac{\partial}{\partial s} p\left(\lambda, s(r,\lambda)\right) d\lambda$$

where $\Lambda(r)$ is the set of source positions for which the L-PI-line through $r$ is fully visible, and $w(r, \lambda)$ is a redundancy weight.

1. Finite-support Inverse Hilbert Transformation: Recovery of the local function values is performed by inverse Hilbert transform along each L-PI-line:

   $$f(r) = \int_{X_1(r)}^{X_2(r)} h_{\rm inv}(X, r) \, b_\ell(X) \, dX$$

Alternatively, the differentiation and backprojection can be folded into a linear-filtration step with principal value Hilbert kernel

$h(s) = \frac{1}{\pi s}$

applied via

$$g(\lambda, s) = \int_{-\infty}^\infty h(s-\sigma) p(\lambda, \sigma) d\sigma$$

and

$$f(r) = \int_{\Lambda(r)} w(r, \lambda) g(\lambda, s(r, \lambda)) d\lambda$$

3. Redundancy Weights and Multi-Linear Trajectories

In multi-translation configurations (2T, 3T, ...), the same L-PI-line can be measured multiple times due to overlapping coverage. Two redundancy weight functions are proposed:

• MP Weight: Defined as $w_{\rm MP}(r, \lambda) = 1 / M(r)$, where $M(r)$ counts how many times point $r$ is covered.
• MZ Weight: To avoid discontinuities at scan endpoints, the MZ weight employs a taper:

  $$w_{\rm MZ}(r, \lambda) = \frac{s(\lambda)}{\sum_{\lambda' \in \Lambda(r)} s(\lambda')}$$

with $s(\lambda)$ a tapering window (typically a small-cosine window vanishing at segment ends).

Practical computations discretize the integrals; reconstructions are implemented with efficient 1D FFTs for Hilbert convolution and numeric quadrature for the finite inverse Hilbert step.

4. Artifact Suppression, ROI Reconstruction, and Stability

Truncation artifacts arise in single-sweep (1T) scans with limited angular spans, as some object voxels are not seen over the complete PI-fan—the region in the frequency domain rendered missing leads to ill-posedness and streak artifacts in standard FBP reconstructions.

BPF-type methods recover the ROI exactly, provided each L-PI-line passing through the ROI is fully measured by at least one scan segment. When employing well-designed redundancy weights and finite-support inverse Hilbert operators, truncation artifacts are entirely eliminated within the ROI. The formal error bound for BPF-type region-of-interest reconstruction is

$$\| \Delta f \|_{L^2({\rm ROI})} = O(\log(1/\delta))$$

for projection errors $\|\Delta p\| \le \delta$, indicating mild logarithmic ill-posedness as opposed to the much more severe behavior observed in filtered backprojection with incomplete data (Wu et al., 2016).

5. Numerical Performance and Comparative Evaluation

Numerical studies in 256×256 Shepp–Logan and abdominal CT phantoms were conducted to benchmark the BPF-type algorithms. Each translation mode was tested with both nontruncated and truncated detector data.

Key quantitative metrics include root mean squared error (RMSE):

Mode	Data Type	FBP RMSE	MZ-BPF RMSE	MP-BPF RMSE
1T	Nontruncated	0.1301	0.1253	0.1280
2T	Nontruncated	0.0301	0.0305	0.0322
3T	Nontruncated	0.0199	0.0201	0.0208
1T	Truncated ROI	221.55 HU	182.45 HU	162.10 HU
2T	Truncated ROI	547.28 HU	22.25 HU	23.40 HU
3T	Truncated ROI	511.34 HU	19.65 HU	19.96 HU

With nontruncated data, all algorithms yield high-fidelity reconstructions for the full support. For truncated data, only the BPF-type methods achieve artifact-free, ROI-accurate recovery, whereas FBP exhibits severe degradation (rim and chord artifacts).

6. Generalizations and Implications for LAT

PTCT instantiates a broad class of LAT systems wherein the data acquisition locus is one or a small number of straight-line segments; the same analytic strategy generalizes to nonparallel, fan-arranged, or piecewise linear scan geometries by appropriately defining L-PI-lines, redundancy weights, and coverage conditions.

The principled use of finite-support inverse Hilbert transforms along coverage lines, together with smoothly tapered redundancy weighting, extends the classical BPF method to the linear scan domain. Furthermore, partial extensions to cone-beam LAT systems are conceivable, replacing L-PI-lines with virtual L-PI-sheets and generalizing finite Hilbert inversion to higher dimensions. This suggests substantial flexibility for realizing low-complexity hardware while enabling exact or near-exact ROI reconstructions (Wu et al., 2016).

7. Design Principles and Future Directions

Several guidelines emerge for future LAT system design:

• Every ROI voxel must be covered by at least one straight-line scan (or segment), ensuring the interior support condition for stable ROI recovery.
• A smooth tapering window (MZ style) should be applied at translation endpoints to suppress limited-angle streaking artifacts.
• The analytic framework of finite inverse Hilbert transform and redundancy weighting provides the foundation for systematic artifact suppression and accuracy in truncated-data reconstructions.

A plausible implication is that these principles, when implemented in hardware with linear scan geometries, can deliver robust, cost-effective tomographic imaging in situations where rotary acquisition is impractical or cost-prohibitive. LAT is thus positioned as a foundation for a new class of “low-end” yet high-quality tomographic systems (Wu et al., 2016).