Project overview

• Distance on meshes*
  • Reproducing Kernel Hilbert Spaces
  • Embedding of shapes into space of currents
  • Currents metric
• Code (Implementation in Jax)
• Experiments
  • Translation, scaling, rotation
  • Refinement of mesh 

*Stanley Durrleman. Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution. Human-Computer Interaction [cs.HC]. Université Nice Sophia Antipolis, 2010. English. fftel-00631382f

Currents metric on shapes

• A solution to the (point)-correspondence issue
• Convergence of 'mesh distance' to the initial continuous representation as the grids become finer
• Works for piecewise smooth curves and surfaces

Let $\tilde{S}$ be a piecewise smooth curve or surface, and let $n(x)$ denote the tangent or normal at $x\in \tilde{S}$.

The shape $S$ can be characterized by


for test vector fields $w\in W$. 

$S(w) = \int_S w(x)^t n(x)d\lambda(x)$

Reproducing Kernel  Hilbert Space

Choose $W$ as a Reproducing Kernel Hilbert Space with the Gaussian Kernel

This is an infinite dimensional Hilbert space of functions satisfying that $x\mapsto f(x)$ is cont. for all $f\in W$.

$K(x,y) = -\exp(-||x-y||^2) / (2\sigma^2) )$.

Reproducing Kernel  Hilbert (dual) Space

Distance of shapes

The Currents metric of two shapes is

$d(S,S') = || S-S' ||_{W^*} = (\langle S-S', S-S' \rangle)^{1/2}$

If $S = \sum_i \delta^{\alpha_i}_{x_i}$ and $S' = \sum_i \delta_{y_i}^{\beta_i}$ then 

$d(S,S') = \sum_{i,j} \alpha_i^t K(x_i,x_j) \alpha_j + \sum_{i,j} \beta_i^tK(y_i,y_j) \beta_j$

$-2 \sum_{i,j} \alpha_i^tK(x_i,y_i)\beta_j$

Influence of kernel

$K(x,y) = -\exp(-||x-y||^2) / (2\sigma^2) )$.

Gaussian Kernel:

The Code

The Meshes

Experiments

Translation

Scaling

Rotation

Sphere from cube

Mesh refinement

Our Questions

• Should the normals be scaled by area weights? 
• What are good use cases?
• What $\sigma$-values are appropriate? 

Thank you for listening!