Edge-Aligned Beam Placement in Scanning Probe Tomography via Reconstruction-Free Sequential Design of Experiments

In X-ray tomography, adopting sequential experimental design methods that strategically select informative measurements to maximize information gain while minimizing exposure time and cost is crucial for improving reconstruction quality. This work proposes a novel adaptive beam selection method that identifies edge-aligned measurements directly from the sinogram, bypassing intermediate image reconstructions to enhance computational efficiency and reduce susceptibility to reconstruction errors.

The gist

Our method selects the next set of measurement beams by maximizing an acquisition function that balances exploration and exploitation over the domain of all possible measurements, improving reconstruction quality while reducing measurement redundancy.

Sequential Design of Experiments Framework

The procedure follows a sequential design-of-experiments framework where measurements are acquired in batches, with each batch informed by data from prior measurements. The process is summarized in Algorithm 1:

1. Input an acquisition function Ψ.

2. Initialization: Set the initial set of measurements Φ0 and set k ← 0.

3. For k = 0 to N do: Collect the measurements Yk(Φk) = [y1, y2,..., ynk]⊤ where yi = Rf ∀ui ∈ Φk.

4. Formulate optimal design of experiment problem: U∗k = arg max u1,.,unr Ψ(U, Φk, Yk(Φk)).

5. Update Φk+1 = Φk ∪ U∗k.

6. End for N iterations to obtain the final set of measurements.

Acquisition Function Design

The acquisition function Ψ is constructed by combining two well-established heuristics: projections intersecting regions of high image gradient (edge alignment) and measurements obtained along geometrically similar beams being redundant. To promote edge-aligned beams, a zero-mean Gaussian Process (GP) prior is imposed on the sinogram: y(u) ∼ GP(0, K(u, u′)), where the covariance function K is chosen as the Matern 3/2 kernel to capture smooth variations across the sinogram. The interpolated sinogram value at a measurement beam u∗ can be obtained using the posterior mean of the GP (Equation 2).

The acquisition function is then defined as the product of two terms:

1. The confidence interval width, denoted as Ωˆk(u), which promotes exploration by targeting regions of high uncertainty. This width is computed from the GP posterior covariance: σˆk(u) = K(u, u) − K(u, Φk)⊤K(Φk, Φk) + σ2 I−1 K(u, Φk). The width is then calculated as omegaˆk(u) = 2Z((1 + β)/2)pσˆk(u), where Z(·) denotes the inverse cumulative distribution function of the standard normal distribution.

2. The edge-mapping term, EˆYˆk; α, which provides a finite-difference approximation of the gradient in the edge-mapping function. This is defined as EˆYˆk; α = 1 − exp (−α q ∆y squared k,θ + ∆y squared k,r), where ∆θ and ∆r represent perturbations between consecutive angles and scanning positions, respectively.

The optimal batch U∗k is determined by solving u∗k = arg max u∈Φ ψ(u) = arg max u∈Φ omegaˆk(u)EˆYˆk; α. A minimum distance threshold δψ is applied post-processing to reduce online computational complexity, ensuring that sequentially selected beams are not clustered.

Numerical Experiments and Results

The effectiveness of the framework was evaluated using five representative samples (Shepp–Logan phantom, Skewed ellipse set, Torso, Biological sample, and Siemens Star inspired sample). The reconstruction performance is quantified by the root mean square error (RMSE) between the reconstruction using Simultaneous Algebraic Reconstruction Technique [10] and ground-truth samples.

The results demonstrated that Algorithm 1 achieves improved reconstruction quality with the same number of measurements as baseline strategies, particularly in early iterations with few measurement beams. The behavior showed that identifying edges in the sinogram effectively corresponds to positioning measurement beams along the sample boundaries, leading to a concentration of measurements around salient features. Furthermore, an additional benefit derived from the GP prior is that its predictive mean acts as a filter for measurement noise, maintaining higher accuracy even when noise levels reach 5%. The proposed algorithm outperforms the baseline across all samples, especially in early iterations.

Conclusion

This work presents a novel adaptive beam selection framework for scanning probe tomography that leverages edge information directly from the sinogram, eliminating the need for intermediate image reconstructions. By modeling the sinogram as a GP and designing an acquisition function that balances uncertainty with edge alignment, the proposed method efficiently identifies informative beams and reduces measurement redundancy. The results underscore the method’s potential benefits and motivate future advances in adaptive scanning hardware.

As a fastidious and diligent researcher, I have analyzed the proposed method—the Adaptive Beam Selection for Efficient Scanning Probe Tomography—and identified several high-impact areas where this framework, when adapted to AI systems (specifically in medical imaging, material science, and complex simulation), can yield significant improvements.

Here are the specific improvements and what the resulting AI system can achieve:

The core innovation is replacing computationally expensive image reconstructions with a Bayesian Gaussian Process (GP) prior on the sinogram domain, combined with an acquisition function that balances exploration (uncertainty width) and exploitation (edge-alignment heuristic). This allows for adaptive, sequential measurement selection.

Here are the specific improvements applicable to AI systems:

1. The framework can be adapted from X-ray CT reconstruction to other complex imaging modalities where data acquisition is costly or time-consuming (e.g., high-resolution microscopy, neutron scattering).

2. The method can replace traditional, fixed experimental designs in machine learning training loops or material characterization workflows with an adaptive sampling strategy that maximizes information gain per unit of computational budget.

3. The integration of the GP prior allows the system to inherently model and filter measurement noise during the design phase, leading to more robust downstream inference.

Specific Improvements for AI Systems:

1. In reinforcement learning (RL) agents tasked with exploration in a high-dimensional state space (e.g., navigating complex physical environments or optimizing chemical reaction pathways), the acquisition function can be adapted to select the next action (measurement/probe) that maximizes the predicted change in the posterior belief, rather than relying solely on sampled rewards.

2. In generative models for molecular design or material synthesis, where experimental data is sparse and expensive to acquire, this method can be used as a sequential design protocol:

pinpoint regions in the high-dimensional parameter space (the sinogram domain) that are most likely to reveal critical structural features (edges) based on current model uncertainty. The system then selects the next experiment specifically targeting those regions, drastically reducing the required number of costly physical experiments needed to constrain a complex generative model.

3. In Bayesian Neural Networks (BNNs) used for uncertainty quantification in deep learning models, the acquisition function can be used to guide targeted data collection or hyperparameter tuning. Instead of randomly sampling data points, the system selects inputs that maximize the expected reduction in model variance (i.e., inputs where the GP uncertainty is high and edge-alignment suggests structural importance).

What these improved AI systems can do:

1. In materials science and drug discovery, these systems can perform Active Experimentation. They can intelligently design the next set of physical measurements required to validate or refine a complex simulation/AI model of a material's internal structure, achieving high-fidelity structural understanding with significantly fewer total experimental runs than traditional exhaustive testing.

2. In complex simulations (e.g., fluid dynamics, climate modeling), the system can adapt its probe locations (the beams) to focus on areas exhibiting the highest predicted gradient changes or phase transitions, allowing the AI to rapidly discover critical physical phenomena within a computationally feasible budget.

3. In high-stakes medical image analysis (where acquiring full 3D scans is prohibitive), this framework could be used for adaptive data acquisition during a scan, selecting only those projection angles and positions that maximize the diagnostic information gain for specific pathological features, leading to faster diagnosis with lower radiation/exposure doses.

In X-ray scanning probe tomography, reconstruction quality generally improves with larger numbers of projections. However, additional projections increase experiment costs, acquisition time, and the radiation dose imparted to the sample. One mitigation to these trade-offs is to adopt a sequential design of experiments, in which each subsequent measurement is determined as a function of previously acquired data in order to maximize information gain. In scanning probe tomography, a widely used heuristic to maximize information is to align beams with the edges of the sample. A key challenge, however, is that the true sample is unknown, so identifying edge-aligned beams typically requires reconstructing the sample based on available measurements. This work proposes a novel sequential design method that identifies edge-aligned measurements directly from the sinogram, bypassing any reconstruction, thereby improving computational efficiency and reducing the experimental design's susceptibility to reconstruction errors. Our method dynamically selects the next set of measurement beams by maximizing an acquisition function that balances exploration and exploitation over the domain of all possible measurements, improving reconstruction quality while reducing measurement redundancy.