Scale-invariant projection optimization in tomographic volumetric additive manufacturing

1. Problem Statement

Tomographic Volumetric Additive Manufacturing (TVAM) is a 3D printing technique that cures an entire volume of resin simultaneously using tomographically delivered light patterns, rather than layer-by-layer.

• The Core Challenge: The inverse problem in TVAM involves calculating projection patterns that generate a specific material response (e.g., polymerization degree) within a target volume while strictly suppressing unintended curing (spillage) in the surrounding liquid.
• Limitations of Existing Methods:
  • Current approaches often rely on absolute dose scaling, requiring manual threshold searches or heuristic weighting to balance target fidelity against out-of-bounds exposure.
  • They struggle to decouple the shape of the projection from the absolute physical scale of the dose.
  • As forward models become more physically realistic (accounting for blur, diffusion, and scattering), computational complexity increases, and the optimization often loses interpretability regarding process constraints.
  • Existing metrics (like Process Window) are often insufficient for evaluating high-resolution grayscale targets or spatially varying dose distributions.

2. Methodology: Scale-Invariant Projection Optimization (SiPO)

The authors propose SiPO, a framework that decouples projection shape optimization from absolute dose scaling. The methodology proceeds in four stages:

A. Domain Definition and Forward Model

• Object Space: Divided into three regions:
  1. Gelation Region ($\Omega_{gel}$): The intended target.
  2. Band Region ($\Omega_{band}$): A boundary layer where spillage is penalized (crucial for controlling artifacts).
  3. Exterior Region ($\Omega_{ext}$): Unaffected background.
• Projection Space: Divided into active rays (contributing to the target) and masked rays (excluded).
• Forward Operator: The dose accumulation is modeled as a linear operator $\mathbf{A}^T = \mathbf{K}\mathbf{P}^T$, where $\mathbf{P}^T$ is the backprojection and $\mathbf{K}$ is an effective Point Spread Function (PSF) operator accounting for blur and diffusion.

B. Response-to-Dose Mapping

To handle the non-linear relationship between optical dose and material response (e.g., polymerization conversion), the framework uses an inverse material response function ($\mathcal{M}^{-1}$).

• The target material response $\mathbf{m}_T$ is pre-processed into an equivalent target dose field $\mathbf{f}_T$.
• This allows the core optimization to remain a linear problem regarding the decision variables (projection patterns), while non-linearities are confined to pre- and post-processing.

C. Scale-Invariant Metrics and Formulation

Instead of optimizing absolute dose, SiPO optimizes normalized metrics that are invariant to global scaling:

1. Dose-to-Target Value Ratio (DTVR): Measures spatial uniformity within the target.
2. Dose Spillage Ratio (DSR): Measures the worst-case out-of-bounds exposure relative to the minimum target dose.

• Mathematical Transformation: The objective is formulated as a Linear-Fractional Program (LFP). Using the Charnes–Cooper transformation, this LFP is converted into a standard Linear Program (LP).
• Solver: To handle large-scale problems (millions of voxels/beamlets) without constructing dense matrices, the authors use a Matrix-Free Primal-Dual Hybrid Gradient (PDHG) solver implemented in PyTorch.

D. Practical Process Control Cases

The framework offers two deterministic formulations for specific industrial needs:

• Case 1 (Minimize Spillage): Strictly enforces material tolerances (lower/upper bounds) within the target and minimizes dose spillage in the band region. This maximizes process separation.
• Case 2 (Maximize Conformity): Enforces a hard inhibition threshold (e.g., oxygen inhibition) in the band region and minimizes the deviation from the target pattern within the gelation region. This maximizes target fidelity.

3. Key Contributions

1. Unified Scale-Invariant Framework: A mathematical structure that treats binary and grayscale targets uniformly by separating geometric shape optimization from physical scaling.
2. Band Region as a Design Variable: The introduction of the "band region" ($\Omega_{band}$) is not just for bookkeeping but acts as a controllable variable to allocate "sacrificial spillage," allowing the optimizer to manage where artifacts occur.
3. Deterministic Process Control: The derivation of specific LP formulations (Case 1 and Case 2) that replace heuristic weight tuning with hard physical constraints (tolerances or inhibition thresholds), providing clear diagnostic feedback (feasibility vs. infeasibility).
4. Scalability: The use of matrix-free PDHG solvers enables the optimization of high-resolution 3D problems that would be computationally prohibitive for standard commercial LP solvers.

4. Results

The framework was validated using 2D benchmarks (UC Berkeley logo) and a 3D physical model (Thinker target):

• Binary Targets:
  • The General Formulation achieved near-perfect target conformity (DTVR $\approx$ 1.0) but with lower separation between target and background.
  • Case 1 aggressively suppressed spillage (highest Process Separation Ratio, PSR) but resulted in wider target dose variations.
  • Case 2 offered a compromise, maintaining strong separation while recovering significant target fidelity.
• Grayscale Targets: The framework successfully handled spatially varying dose prescriptions. The trade-off between conformity and separation remained consistent with the binary results.
• 3D Blur-Aware Models: When tested with a 3D forward model including diffusion-like blur (Gaussian PSF):
  • The framework remained effective, though the feasible process window was compressed due to physical blur.
  • The "band region" definition was found to be less critical in blur-dominated scenarios, as the PSF naturally smooths boundary leakage.
• Band Region Sensitivity: Removing the explicit band region had minimal impact on separation for simple geometries but degraded target conformity in Case 2, confirming the band's role in allocating optimization freedom.

5. Significance

• Bridging Theory and Practice: SiPO connects rigorous inverse planning (common in radiation therapy) with physics-aware TVAM process design, making optimization results directly interpretable in terms of manufacturing tolerances.
• Robustness: By decoupling shape from scale, the method is robust against variations in resin properties and exposure times, as the physical scale is calibrated after the geometric optimization.
• Scalability: The matrix-free approach makes high-fidelity, 3D volumetric printing optimization computationally feasible, paving the way for complex applications in bioprinting, aerospace, and functional materials.
• Strategic Workflow: The paper proposes a two-stage design strategy: first, use Case 1 to determine the physically feasible process bounds; second, use Case 2 to maximize fidelity within those bounds, avoiding computationally expensive parameter sweeps.