Systematic Analysis of Penalty-Optimised Illumination Design for Tomographic Volumetric Additive Manufacturing via the Extendable Framework TVAM AID Using the Core Imaging Library

1. Problem Statement

Tomographic Volumetric Additive Manufacturing (TVAM) is a manufacturing technique that creates complex 3D objects by simultaneously solidifying a photopolymer resin using light projected from multiple angles. Unlike layer-based 3D printing, TVAM cures the entire volume at once, offering significant speed advantages.

The core challenge is Illumination Plan Design: determining the light intensity (sinogram) for each projection angle such that the accumulated light energy dose ($f$) solidifies the "in-part" regions ($\Omega_{in}$) while keeping "out-of-part" regions ($\Omega_{out}$) liquid.

• The Constraint: Light cannot be blocked selectively; shining light on a target voxel also illuminates voxels behind and in front of it.
• The Physics: Solidification occurs only when the accumulated dose exceeds a threshold $\tau$.
• The Optimization Gap: Traditional reconstruction methods (like Filtered Back-Projection) often yield negative values, which are physically impossible for light sources. Iterative solvers are required to enforce non-negativity, but existing methods often struggle to balance two competing goals:
  1. Process Window (PW): Maximizing the gap between the minimum dose in solid regions and the maximum dose in liquid regions to ensure robustness.
  2. In-Part Dose Range (IPDR): Minimizing the variation of dose within the solid regions to ensure uniform curing times and prevent sedimentation or optical distortion.

2. Methodology

The authors propose TVAM AID (Adaptive Illumination Design), a reproducible framework built on the open-source Core Imaging Library (CIL). The problem is formulated as a constrained least-squares optimization:

$$g^* = \arg\min_{g \ge 0} \|A^T g - t\|_2^2$$

Where $g$ is the illumination plan (sinogram), $A^T$ is the back-projection operator, and $t$ is the target dose profile. The novelty lies in how the objective function (penalty) is constructed for different voxel types ($\Omega_{in}$, $\Omega_{out}$, and external regions).

The paper systematically compares four penalty function strategies:

1. L2-norm (L2N): Uses standard squared error penalties shifted to specific thresholds ($\tau_{lower}$ for out-of-part, $\tau_{upper}$ for in-part).
   • Pros: Produces uniform dose distributions.
   • Cons: Strictly penalizes any deviation, often resulting in a small Process Window.
2. One-Sided Penalties (OSP): Removes penalties for "good" deviations (e.g., no penalty if out-of-part dose is $<\tau_{lower}$ or in-part dose is $>\tau_{upper}$).
   • Pros: Maximizes the Process Window.
   • Cons: Leads to high IPDR (non-uniform curing) because high doses in the in-part region are not penalized.
3. OSP with High Dose Penalty (OSPW, $w > 0$): Adds a penalty for in-part doses exceeding $\tau_{upper} + w$.
   • Pros: Attempts to balance PW and IPDR.
   • Cons: The parameter $w$ is often fixed in literature (e.g., to normalize to 1), limiting flexibility.
4. OSPW ($w = 0.0$) - The Proposed Optimal: A hybrid approach using One-Sided Penalties for out-of-part voxels (to maximize PW) and L2-norm penalties for in-part voxels (to minimize IPDR).
   • Innovation: This approach treats the thresholds ($\tau_{lower}, \tau_{upper}$) as design parameters rather than fixed constants, allowing for a systematic parameter sweep to find optimal values.

Metrics Used:

• Process Window (PW): $min(f_{in}) - max(f_{out})$. (Higher is better).
• In-Part Dose Range (IPDR): $max(f_{in}) - min(f_{in})$. (Lower is better).
• Voxel Error Rate (VER): The ratio of out-of-part voxels exceeding the minimum in-part dose. (Zero is ideal).
• Robustness: The authors introduce a method to exclude outlier voxels (e.g., top/bottom 0.05%) from metric calculations to prevent single erroneous voxels from skewing results.

3. Key Contributions

1. Critical Analysis of Penalty Functions: The paper rigorously demonstrates that a hybrid penalty strategy (OSPW with $w=0$) outperforms pure L2-norm or pure One-Sided approaches by simultaneously maximizing the Process Window and minimizing the In-Part Dose Range.
2. Thresholds as Design Parameters: Instead of treating thresholds as fixed, the authors perform a systematic parameter sweep (325 combinations) to characterize how $\tau_{lower}$ and $\tau_{upper}$ affect printing metrics. They identify optimal ranges (e.g., $\tau_{lower} \approx 0.72, \tau_{upper} \approx 0.96$) and recommend robust default values.
3. Extensible Framework (TVAM AID): By leveraging CIL, the framework lowers the barrier to entry for TVAM research. It allows researchers to easily swap penalty functions, add regularization, or apply weights without rebuilding the core solver.
4. 3D Validation: The study validates the approach on complex 3D geometries (including a Gyroid structure), confirming that the 2D slice-based optimization holds true for volumetric printing.

4. Results

• Performance Comparison: The OSPW ($w=0.0$) approach consistently outperformed the state-of-the-art OSMO (Object Space Model Optimisation) algorithm across all tested geometries (DTU logo, simple disk, resolution test, and 3D Gyroid).
  • IPDR: OSPW achieved significantly lower IPDR (more uniform curing) than OSMO.
  • PW: OSPW achieved higher Process Windows, even when OSMO was tuned with its own "optimal" thresholds.
• Threshold Sensitivity: The study revealed that while optimal thresholds vary slightly by geometry, the default values ($\tau_{lower}=0.70, \tau_{upper}=0.90$) provide near-optimal performance for most cases, simplifying practical implementation.
• Outlier Robustness: The study confirmed that standard metrics are highly susceptible to outliers. Removing just 0.05% of voxels from the calculation stabilized the metrics, particularly for the OSPW method, revealing that the "worst" slices were often driven by single erroneous voxels rather than systemic failure.
• Flexibility: For the 3D Gyroid, relaxing the penalty to OSPW ($w=0.1$) further improved the Process Window without significantly degrading IPDR, demonstrating the framework's ability to adapt to specific geometric challenges.

5. Significance

This work provides a theoretical and practical foundation for next-generation TVAM systems.

• Quality Improvement: By minimizing IPDR, the method reduces the time window where solid and liquid coexist, mitigating issues like sedimentation and optical distortion during printing.
• Robustness: By maximizing PW, the printing process becomes more tolerant to variations in resin chemistry and exposure time.
• Reproducibility: The use of CIL ensures that these optimization strategies are transparent, reproducible, and easily extensible to include advanced features like regularization or grayscale printing (functional materials).
• Paradigm Shift: The paper moves TVAM illumination design from "trial-and-error" threshold selection to a systematic, data-driven approach where thresholds are optimized parameters.

In conclusion, the authors demonstrate that a carefully designed hybrid penalty function, implemented within a flexible open-source framework, can significantly enhance the quality, speed, and reliability of volumetric additive manufacturing.