Last experiments

Emery Pierson, Lei Li, Maks Ovsjanikov

Last suggestions

Instead of working by optimizing SDS Loss directly, the experiment consists of searching whereas we can use the transferred vertices to recover the ground truth functional map.

Let $\mathcal{S}$ the input shapes, $ the template shape, $C$ the functional map matrix. We suppose we are given a way to transfer the coordinates of $S$, $X_\mathcal{S}$, to $X_{S \to T}$ using $C$. The transfer function is denoted $F(X_S, C)$.

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 ?

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

Experiments

Now, there a 3 different ways of transferring the data:

“Matrix transfer”: $F(X_S, C) = \phi_T C \phi_S^T X_S$ (transferring in Laplacian basis directly)
“Distance transfer”: $F(X_S, C) = \Pi_{S \to T}^C X_S$, where $\Pi_{S \to T}^C (p, q) = Softmax(\delta_{pq})$, where $\delta_{pq} = ||\Phi_T(q) - C\Phi_S(p)||²$
“Product transfer” $F(X_S, C) = \Pi_{S \to T}^C X_S$ where now $\delta_{pq} = \frac{\langle \Phi_T(q), C\Phi_S(p) \rangle}{|| \Phi_T(q)|| || C\Phi_S(p)||}$