16K and 8K vertices, resp.).[]{data-label="fig:ring_stab"}](pic/unique_quiv.png){width="12cm"}

In case of multiple rings, $\tilde{p}$ is selected only on the biggest ring $ring_{N_r}$; for each concentric ring, the starting point is the point $p_i$, which is the closest one to $\tilde{p}$. Generally the number of elements $p_i \in \mathcal{R}$ varies from one ring to another, because of the increasing radius of the sphere and the irregularity of the tessellation, see Figure \[fig:ring\_construction\](b). To have the same number of elements on every ring, we sample $C$ with $P$ points, where $P$ is a fixed number, called the *spatial resolution*. The results of this sampling is $S$, a set of equidistant samples of $C$, $s_j$ with $j=1, \ldots, P$. In details, the equidistant re-sampling is performed as follows:

-   we set the expected distance $\delta r$ between two successive points in $S$ as $\delta r= \dfrac{2\pi R}{P}$;

-   we set $s_0=\tilde{p}$ and extract the points $s_j$ on $C$ such that $$|s_{j-1}-s_j|\approx \delta r, \qquad j=\{1, \ldots, P\}.$$

The value $h(s_j)$ is linearly approximated from the values the function $h$ assumes on the extrema of the corresponding segment in $C$.

### Choice of the ring radii

With the edgeLBP we are interested to code local variations on 