Chapter Summary

Chapter Summary

Key Points

• 1.

  Signed distance functions represent 3D geometry as a continuous scalar field, with the surface at the zero level set $\{f(\mathbf{p}) = 0\}$. The Eikonal equation $\|\nabla f\| = 1$ characterises valid SDFs and enables efficient ray-surface intersection via sphere tracing.

• 2.

  Neural SDFs (DeepSDF) parameterise the distance function as an MLP with positional encoding, providing a compact, resolution-free representation that bypasses the $O(N^3)$ memory cost of voxel grids --- critical for mmWave imaging where wavelength-scale discretisation is otherwise prohibitive.

• 3.

  GeRaF reconstructs 3D geometry from mmWave radar by fitting a neural SDF to matched-filter power images. The lensless radar imaging model is bridged to neural rendering via a differentiable MF power rendering equation, trained end-to-end with Eikonal regularisation.

• 4.

  Occupancy networks provide a simpler, binary alternative to SDFs: they classify each 3D point as inside or outside an object. Easier to train (binary cross-entropy, no Eikonal loss), but they lack distance information and surface normals, limiting them to coarse scene understanding tasks like blockage prediction.

• 5.

  Eikonal regularisation enforces valid SDF geometry without ground-truth distance labels, enabling self-supervised SDF learning from raw radar measurements. Gropp et al. showed that the Eikonal loss plus surface-point constraints suffice to recover the signed distance function.

• 6.

  Joint geometry-material estimation from multi-view RF data recovers not only where surfaces are but what they are made of: reflectivity, roughness, and permittivity. The angular dependence of the Fresnel reflection coefficient provides the material-discriminating information that multi-view acquisitions exploit.