Non rigid diffusion, last experiments

Emery Pierson, Lei Li, Maks Ovsjanikov

Small recap

Let an input shape \(\mathcal{S}\), we wish to find the correspondance between this shape and a template shape \(\mathcal{T}\).

We do it using functional map. A functional map is a linear map between the function space \(\mathcal{S} \to \mathbb{R}\) and \(\mathcal{T} \to \mathbb{R}\).

Let \(\Phi_\mathcal{S} \in \mathbb{R}^{n \times n_S}\), \(\Phi_\mathcal{T} \in \mathbb{R}^{n \times n_T}\), the Laplace eigenvectors of both shapes. The functional map can be represented as a matrix \(C\) taking coefficient in \(S\) basis to the ones in \(T\) basis.

Functional map work by matching features in the eigenbasis, but often fails because of underlying symmetry

Suppose now we have a shape diffusion model (here it is in PCA coefficient space). Can we regularize those results with diffusion models?

Proposed approach

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Transfer to template connectivity

We transfer the coordinates by convert the matrix \(C_{S \to T}\) to a point to point map \(\Pi_{S \to T}\).

\[ F(X_S, C) = \Pi_{S \to T}^C X_S, \] where \[ \Pi_{S \to T}^C (p, q) = \frac{\exp(-\delta_{pq}*\alpha)}{\sum_q {\exp(-\delta_{pq}*\alpha)}}, \]

with \(\delta_{pq} = ||\Phi_T(q) - C\Phi_S(p)||²\)

Last week results

Remarks of last week

Train a pose + shape model -> changed

What happens with a ground-truth fmap?

Other experiments with ground-truth

Add some usual functional maps loss (orthogonality/others)

First experiment: replace SDS with ground-truth

Given our transfer function \(F(X_S, C)\) we suppose we know \(X_{S \to T}^{\text{gt}} = F(X_S, C_{\text{gt}})\). Starting from a noisy \(C\), can we recover \(C_{\text{gt}}\) from \(X_{S \to T}^{\text{gt}}\), by minimizing the loss below ?

\[ || F(X_S, C) - X_{S \to T}^{\text{gt}} || ² \]

No. But if we add a new term to the loss we want to minimize. Given the transferred vertices \(X_{\mathcal{S} \to \mathcal{T}}\), we compute the point correspondance in the spatial coordinates (+normals, to avoid erros when parts are gluing to each other). We then extract a functional map based on this correspondance, \(C_{\text{spatial}}\), and minimize:

\[ \mathcal{L}_{proper} = || C - C_{\text{spatial}}||^2 \]

With this supplementary loss, the functional map converges.