each pixel $i\in I$, a *ring* of pixels around $i$ (called $ring_i$) is considered (see Figure \[fig:LBP\_ring\]) and a 8-digit binary array $str_i$ defined as follow: $$str_i(j)=\Bigl\{  \begin{array}{ll}   1 & if \quad h(i)<h(i_j)\\
                                        0 & otherwise           \\
                    \end{array}$$ where $i_j$ is the $j-th$ pixel of the ring around $i$, sorted clockwise and starting from the top-left pixel.

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   ![In (a) the $ring$ of the pixel $i$ is shown; while (b) and (c) show two examples of concentric $ring$s.[]{data-label="fig:LBP_ring"}](pic/multiring_1.jpg "fig:"){width="3.5cm"}   ![In (a) the $ring$ of the pixel $i$ is shown; while (b) and (c) show two examples of concentric $ring$s.[]{data-label="fig:LBP_ring"}](pic/multiring_2.jpg "fig:"){width="3.5cm"}   ![In (a) the $ring$ of the pixel $i$ is shown; while (b) and (c) show two examples of concentric $ring$s.[]{data-label="fig:LBP_ring"}](pic/multiring_3.jpg "fig:"){width="3.5cm"}
                                                                                          (a)                                                                                                                                                                                  (b)                                                                                                                                                                                  (c)
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The LBP operator of a pixel $i$ is defined by: $$LBP(i)=\sum_j str_i(j)\alpha(j),$$ where $\alpha$ is a weight function. Throughout this paper we consider $\alpha_1(j)=1, \forall j$. Notice that in this case, the $LBP(i)$ value is independent of the ordering of $ring_i$. Finally, the LBP descriptor of the pattern in $I$ is defined as the histogram of the values $LBP(i)$.

The LBP operator was extended to multiple rings around each pixel in 