SHARC: Reference point driven Spherical Harmonic Representation for Complex Shapes

1. Problem Statement

The paper addresses the challenge of representing complex 3D shapes in a manner that is both compact (low storage/bandwidth) and capable of preserving fine geometric details.

• Limitations of Explicit Representations: Triangle meshes and point clouds scale linearly with detail, making them expensive to transmit, edit, or use in optimization tasks (e.g., neural rendering).
• Limitations of Neural Implicit Representations: While methods like Neural SDFs offer high fidelity and infinite resolution, they suffer from high inference latency, large parameter overhead, and a "black box" nature where decoding is computationally expensive.
• Limitations of Existing Skeletal/SH Methods:
  • Medial Axis Transforms (MAT): Often produce overly complex structures or fail to capture high-frequency details due to reliance on simple primitives (unions of spheres).
  • Previous SH approaches (e.g., MASH): Rely on "multi-view" surface patching with view cones, requiring a large number of anchor points and expensive iterative optimization to resolve geometric details, particularly at patch boundaries.

2. Methodology: SHARC Framework

SHARC (Spherical Harmonic Anchored Reference Collection) synthesizes arbitrary, genus-agnostic shapes by encoding the distance field from a set of optimally placed internal reference points to the surface using Spherical Harmonics (SH).

The pipeline consists of three main stages:

A. Candidate Generation

Instead of constructing a complex Voronoi graph or Medial Axis (which is sensitive to boundary noise), SHARC samples random points within the object's bounding box and uses a ray-casting parity check (Even-Odd Rule) to filter for points strictly inside the volume. This creates a dense pool of potential reference points.

B. Visibility-Aware Reference Point Selection

The algorithm selects a sparse subset of reference points ($C$) from the candidate pool using a greedy selection loop. The selection maximizes a composite score function $H(c)$ that balances three criteria:

1. Centrality ($E_{cen}$): Favors points deep inside the object (approximating the Medial Axis), defined as the inverse of the distance to the nearest surface point.
2. Uniformity ($E_{uni}$): Penalizes clustering, encouraging new anchors to be far from already selected ones to ensure spatial coverage.
3. Visibility & Proximity Coverage ($E_{cov}$): The core innovation. It calculates how well a candidate point $c$ "sees" the surface.
   • Visibility: A surface point is counted only if the ray from $c$ to the surface does not intersect the geometry elsewhere.
   • Proximity: A surface point is counted only if it is within a configurable distance threshold ($\tau_{prox}$) from $c$.
   • Goal: This ensures that the SH representation for a specific point only encodes the local, visible geometry, preventing the model from trying to fit distant, occluded surfaces with a single low-bandwidth function.

C. SH Distance Field Encoding

For each selected reference point $c$:

1. Ray-casting: The algorithm casts rays in all directions (using a HEALPix grid to avoid polar distortion) to measure the radial distance to the surface.
2. Transformation: The resulting radial distance field $r(\theta, \phi)$ is converted into Spherical Harmonic coefficients ($a_{\ell}^m$) using the Fast Spherical Harmonic Transform (FSHT).
3. Smoothing: To mitigate ringing artifacts (Gibbs phenomenon) caused by discontinuities, a Lanczos filter is applied to the spectral coefficients before reconstruction.

D. Reconstruction

To recover the 3D shape:

1. Inverse Transform: The smoothed SH coefficients are evaluated to generate a dense raw point cloud.
2. Local Consistency Filtering: Since SH fields are omnidirectional, a point generated by a distant reference point might be noisy. The method filters the cloud by retaining a point only if its generating reference point is among the $k$-nearest neighbors to that point. This resolves competition between overlapping fields.
3. Normal Orientation & Meshing: Normals are computed via PCA and oriented using the vector from the reference point to the surface point (ensuring outward direction). Finally, a Screened Poisson Surface Reconstruction generates the watertight mesh.

3. Key Contributions

1. Execution Time: SHARC achieves significantly faster encoding and reconstruction times compared to state-of-the-art (SOTA) methods, including neural fields and other skeletal approaches.
2. Complexity vs. Accuracy Balance: The method achieves a superior trade-off between the number of primitives (reference points + SH coefficients) and reconstruction quality. It uses far fewer primitives than competitors while maintaining higher fidelity.
3. Proximity-Based Reconstruction: Unlike methods that blend local patches, SHARC uses a proximity-based filtering rule to resolve overlaps. By prioritizing the nearest reference point for any given surface area, it ensures sharper, more accurate reconstructions without the need for complex iterative optimization.
4. Genus-Agnostic: The framework handles shapes with arbitrary topologies (holes, complex genus) without requiring the shape to be star-shaped or decomposed into simple patches manually.

4. Experimental Results

The authors evaluated SHARC on four datasets: Princeton Shape Benchmark (PSB), Stanford 3D Scanning Repository, Thingi10k, and ShapeNet.

• Quantitative Performance:

  • Accuracy: SHARC consistently achieved the lowest Chamfer Distance (dCD) and Hausdorff Distance (HD) across all datasets. For example, on the Thingi10k dataset, SHARC achieved a dCD of 0.39% compared to MASH's 0.52% and Raw MAT's 2.23%.
  • Compactness: SHARC used drastically fewer primitives. On average, it used ~45–120 reference points, whereas Raw MAT required ~30k–80k spheres, and MASH required 200–400 anchors.
  • Storage Efficiency: For a 10M-face "Thai Statue" (181 MB), SHARC compressed it to 1.49 MB (121x reduction) with minimal error, whereas MASH required significantly more time to reach similar accuracy.

• Qualitative Performance:

  • Visual comparisons showed SHARC preserving high-frequency details (e.g., dragon scales, facial features) that were smoothed out or lost in MAT and MASH reconstructions.
  • It successfully handled complex topologies where other methods failed to establish connectivity or produced artifacts.

• Efficiency:

  • SHARC demonstrated near-constant runtime scaling as the number of primitives increased, whereas competitors like MASH showed steep linear increases in computation time.

5. Significance and Future Work

• Significance: SHARC provides a lightweight, interpretable, and highly efficient alternative to neural implicit representations and traditional skeletal methods. It bridges the gap between the compactness of explicit primitives and the high-fidelity detail of implicit fields. Its ability to encode complex shapes with very few parameters makes it ideal for transmission, storage, and real-time applications.
• Future Directions:
  • Adapting SHARC as a shape descriptor for recognition tasks.
  • Extending the framework to encode surface textures (RGB, radiance) for a unified UV-free representation.
  • Making the reference point selection process differentiable to allow for end-to-end learning of optimal anchor positions.

In conclusion, SHARC represents a significant advancement in 3D geometry processing by leveraging optimally placed internal reference points and spherical harmonics to achieve high-fidelity, compact, and fast shape reconstruction.