3.  The step $2$ is repeated $\forall e \in L$, until the list is empty.

To achieve a multi-ring representation, for any vertex $v \in V$ we consider $N_r$ rings, $\{ring_1^v, \ldots, ring_{N_r}^v\}$. Let $S^v_{l}$ be the surface portion of $T$ that contains $v$ and has the $ring^v_k$ as its boundary, $l=1\ldots N_r-1$, then the relation $S^v_{l} \subset S^v_{l+1}$ holds for each $l$. When extending the edgeLBP evaluation to multiple rings, the algorithm takes advantage of the nested nature of the rings and extracts $S^v_l$ with respect to increasing values of the radius $R$. In general, the sphere-surface intersection can produce multiple, closed curves that bound either a multiple connected or a dis-connected portion of the surface, as detailed in [@blowing_bubbles]. Using a region growing approach, we dynamically consider only the $S^v_l$ components.Therefore, $S^v_l$ is always a connected region that contains $v$; however, it can become multiply connected. If all the $N_r$ components of $S^v_l$ are simply connected and all the $N_r$ rings do not intersect the surface boundary (if any), the $v$ is considered an *admissible* vertex for the edgeLBP, otherwise it is *non-admissible*.

### Ring re-ordering and sampling

Each $ring$ is represented as the piecewise, linear curve $C$ 