Neural skeleton: Implicit neural representation away from the surface

Mattéo Clémot, Julie Digne

10.1016/j.cag.2023.06.012

overview

Contribution

TV-Loss

L_{TV} = \int_{R^{3}} ∥ \nabla ∥ \nabla u ∥ (p) ∥ d p

L_{TV} = \int_{R^{3} - \partial Ω} ∥ \nabla ∥ \nabla u ∥ (p) ∥ d p

Coverage Axis Method

Input: a set of sketetal candicate points with their medial sphere redius and a set of surface points.
Object: Find the small subset of skeletal set that the balls can cover all the surface points
$\begin{aligned} min & ∥ v ∥_{2} \\ s.t. & D v \geq 1 \end{aligned}$
$v_{i} = {0, 1}^{M}$
$D \in {0, 1}^{N \times M}, D_{i j} = 1_{| | p_{i} - s_{j} | | - r_{j} \leq δ}$

How to obtain sketetal points and surface points with INR

Uniform sampling surface points with NeuralPull Method
- $p \leftarrow p - \frac{\nabla u (p)}{∥ \nabla u (p) ∥^{2}} u (p)$
- To uniformize it,
  - $p \leftarrow p - α r$
  - $r_{i} = \sum_{p_{j} \in N_{K} (p_{i})} w (p_{i}, p_{j}) Π_{T} (\frac{p_{j} - p_{i}}{‖ p_{j} - p_{i} ‖})$
  - $w (p_{i}, p_{j}) = \exp (- \frac{{‖ p_{i} - p_{j} ‖}^{2}}{σ}) with σ = 16 D / N .$
Sketetal Points , make the simple observation that if we cast a ray from a surface point in the opposite direction to the gradient, then this ray cross the medial axis before hitting the surface again. In addition, at the medial axis crossing, the SDF should be minumum(assuming the SDF to be positive outside and negative inside the object)

例子