Automatic regularization parameter choice for tomography using a double model approach

Technical Summary: Automatic Regularization Parameter Choice for Tomography Using a Double Model Approach

1. Problem Statement

In X-ray Computed Tomography (CT), reconstructing an image from projection data is an ill-posed inverse problem. When data is incomplete (e.g., sparse sampling), the reconstruction process is highly sensitive to noise and discretization errors. To stabilize the solution, variational regularization is used, which introduces a regularization parameter ($\alpha$) to balance data fidelity against prior assumptions (like smoothness or sparsity).

The core challenge is the automatic selection of $\alpha$. Traditional methods have significant drawbacks:

• L-curve and Cross-validation: Often rely on unstable feature detection or computationally expensive grid searches.
• Discrepancy Principle: Requires precise knowledge of the noise level ($\eta$), which is rarely available in real-world scenarios.
• Data-driven/Deep Learning: Require massive training datasets and lack interpretability.

2. Methodology

The authors propose a novel paradigm: treating regularization parameter selection as a closed-loop feedback control problem rather than a static estimation problem.

A. The Double Model Strategy:
Instead of comparing a reconstruction to a "ground truth" (which is unavailable), the method compares two different reconstructions of the same data. The problem is solved on two geometrically distinct computational grids (the primary grid $A$ and a rotated grid $A_\theta$). Because discretization errors are grid-dependent, an under-regularized solution will produce uncorrelated artifacts on the two grids, leading to low similarity.

B. Feedback Control Loop:

1. Sensing: The system calculates the Structural Similarity Index (SSIM) between the primary reconstruction ($x_1$) and the rotated/re-aligned secondary reconstruction ($\tilde{x}_2$). This SSIM serves as the feedback signal $S(\alpha)$.
2. Actuation: A Proportional (P) Controller operates in the logarithmic domain of $\alpha$. It dynamically adjusts the regularization strength based on the error between the measured similarity and a user-defined target similarity ($S_{ref}$).
3. Monotonicity Assumption: The method relies on the fact that as $\alpha$ increases, noise is suppressed and the two reconstructions converge toward a common signal, causing $S(\alpha)$ to increase monotonically (until the point of extreme over-smoothing).

C. Algorithm:
The controller starts with a very small $\alpha$ and iteratively updates it using the rule:
$$\log_{10}(\alpha_{k+1}) = \log_{10}(\alpha_k) + K_p \cdot (S_{ref} - S_k)$$
The process terminates once the similarity stays within a tolerance band of the target for a set number of steps.

3. Key Contributions

• Control-Theoretic Framework: Reframes parameter tuning as a "reference tracking" task, moving away from the search for a single "mathematically optimal" $\alpha$.
• Discretization as Information: Turns discretization error—usually viewed as a nuisance—into a measurable signal for stability.
• User-Centric Design: By allowing the user to set $S_{ref}$, the method allows for a transparent trade-off between noise suppression (high similarity) and structural detail (lower similarity).
• Solver Agnosticism: The framework works regardless of whether the regularizer is Total Variation (TV) or Tikhonov ($L_2$).

4. Results

The method was tested on two complex datasets: Walnut (smooth gradients) and Pine Cone (high-frequency structures).

• Convergence & Stability: The controller successfully drove the system to the target $S_{ref}$ across all tests, demonstrating robustness to different regularizers (TV and Tikhonov) and different geometric complexities.
• Comparative Analysis: When compared to the L-curve and Discrepancy Principle, the proposed controller was found to be less "conservative." While the classical methods produced very smooth images with extremely high similarity, they often suppressed fine structural details. The controller, by targeting a specific (often lower) similarity level, preserved significantly more high-frequency detail (e.g., the internal septa of the walnut).
• Pareto Efficiency: The controller operates near the "knee" of the consistency-vs-detail curve, providing a better balance of information recovery than traditional heuristics.

5. Significance

This work introduces a highly practical engineering approach to inverse problems. By shifting the goal from finding the best $\alpha$ to regulating a specific quality criterion, the method provides a way to automate CT reconstruction that is intuitive for clinicians or technicians. It offers a principled way to navigate the fundamental trade-off between image stability and structural detail without requiring ground-truth data or prior noise estimates.