For $i$’th point, there can be two different conditions: totally belonging to one cluster or belonging to several clusters at same time. When the normalized distance between this point and the $k$’th cluster center is small enough, this point shall totally belong to the $k$’th class as shown by Eqs. (13) and (14): Display Formula
$when\u2009\u2009Min[dn(xi,cj)]<\u2212pf,(j=1,\u2026,M),$(13)
Display Formula$U(xi,cj)={1,dn(xi,ck)\u2264dn(xi,cl)\u2009(l=1,\u2026,M,l\u2260k)0,otherwise,$(14)
where $U(xi,cj)$ represents the degree of the fuzzy membership with which the $i$’th point belongs to the $j$’th class and $pf$ is a parameter introduced in the fuzzy function design to adjust the range of acceptance of fuzzy operation. Otherwise, when the distances between $i$’th point and every cluster center are all larger than $\u2212pf$, the point shall belong to several clusters and a judgment shall be made to choose the essential clusters as shown by Eqs. (15) and (16): Display Formula$when\u2200\u2009\u2009dn(xi,cj)>\u2212pf,(j=1,\u2026,n),$(15)
Display Formula$U(xi,cj)={0,dn(xi,cj)>pf(pf\u2212Min[dn(xi,cj),pf]2\u2211k=1M(pf\u2212Min[dn(xi,ck),pf]2,otherwise,$(16)
where $Min[dn(xi,cj),pf]$ means the minimum value of $dn(xi,cj)$ and $pf$, which can ensure the pixel far from the cluster center has no effect on the membership calculation. And $U(xi,cj)=0$$dn(xi,cj)>pf$ means the distance between $i$’th point and $j$’th cluster is too large so that the membership of $i$’th point to $j$’th cluster shall be set to 0.