Implement Hierarchical SLIM (H-SLIM) for Large-Scale Mesh Parameterization

[csparker247]

The current implementation of ABF++ followed by LSCM reconstruction faces significant performance and memory bottlenecks when processing high-resolution heritage science meshes (exceeding 1M faces). LSCM requires a direct global solve that scales poorly in terms of RAM and computation time, making real-time or interactive workflows difficult for dense scans. Furthermore, the current pipeline is strictly conformal, offering limited flexibility for other distortion metrics like area preservation or ARAP (As-Rigid-As-Possible).

Proposed Solution: Hierarchical SLIM (H-SLIM)

Integrate Scalable Locally Injective Mappings (SLIM) within a hierarchical framework. SLIM uses a Reweighted Least Squares (RWLS) approach that is more memory-efficient than LSCM and guarantees local injectivity (no triangle flips), provided it has a valid initialization.

By leveraging a coarse-to-fine hierarchy, we can use the existing ABF++ logic for a robust "global" start and use SLIM for high-performance "local" refinement on the full-resolution mesh.

Technical Workflow

1. Coarse Initialization:
   • Generate a decimated proxy mesh (e.g., 10k–50k faces). Possible use ACVD ([Feature] Implement ACVD #62) and/or QEM.
   • Apply the existing ABF++ $\rightarrow$ LSCM pipeline to generate a valid, flip-free initial $(u, v)$ embedding.
2. Prolongation:
   • Interpolate the coarse $(u, v)$ coordinates onto the original high-resolution mesh (1M+ faces).
3. Iterative Refinement (SLIM):
   • Run a sequence of SLIM iterations on the fine mesh to minimize a chosen energy function $E$.
   • Utilize Preconditioned Conjugate Gradient (PCG) for the global step to maintain a low memory footprint.
   • Energy Minimization: $$E(u, v) = \sum_{f \in F} A_f \cdot e(\mathbf{J}_f)$$
     where $e(\mathbf{J}_f)$ is the energy density (e.g., Symmetric Dirichlet or ARAP) based on the Jacobian of the mapping for each face $f$.

Key Advantages

• Scalability: Moves from a heavy direct solve to an iterative approach that can be early-exited once visual convergence is met.
• Metric Flexibility: Allows "plug-and-play" energy metrics. We can prioritize Symmetric Dirichlet to prevent extreme compression/stretching or ARAP to preserve the physical dimensions of heritage items.
• Robustness: ABF++ provides a high-quality "seed" that ensures the SLIM solver starts in a valid, locally injective state.
• GPU Potential: The "Local" phase of the SLIM optimization is embarrassingly parallel and prime for GPU acceleration.

Task Breakdown

• Implement a Reweighted Least Squares (RWLS) solver loop.
• Add support for multiple energy densities (ARAP, Symmetric Dirichlet, MIPS).
• Integrate the prolongation operator to transfer UVs across the mesh hierarchy.
• (Optional) Implement a matrix-free PCG solver to further reduce memory overhead.

Relationships

• The coarse mesh creation depends on the mesh coarsening implementation provided in feat: Hierarchical LSCM (HLSCM) - cascadic multigrid UV parameterization #44 and [Feature] Implement ACVD #62

References

Rabinovich, Michael, et al. "Scalable locally injective mappings." ACM Transactions on Graphics (TOG) 36.4 (2017): 1. [PDF]