The full transformation semigroup TnTn is defined to consist of all functions from Xn={1,…,n}Xn={1,…,n} itself, under the operation composition. In \cite{JMH1}, for any αα in TnTn, Howie and denoted collapse by c(α)=⋃t∈\im(α){tα−1:|tα−1|≥2}c(α)=⋃t∈\im(α){tα−1:|tα−1|≥2}. Let OnOn be order-preserving transformations CnCn decreasing on XnXn=under its natural order, respectively. E(On)E(On) set ide...

We introduce order-k α-hulls and αshapes — generalizations of α-hulls and α-shapes. Being also a generalization of k-hull (known in statistics as “k-depth contour”), order-k α-hull provides a link between shape reconstruction and statistical depth. As a generalization of α-hull, order-k α-hull gives a robust shape estimation by ignoring locally up to k outliers in a point set. Order-k α-shape p...