Implementation of Steiner Point of Fuzzy Set

Algorithm 1

Implementations of algorithm.

if 𝐽 is a color image, convert it into the gray

𝑔_min∶=minhistogram(𝐽)

𝑔_max∶=maxhistogram(𝐽)

𝐽₀∶=(𝐽−𝑔_min)/(𝑔_max−𝑔_min)

define 𝑒 as a large number to record the image difference, for example, 𝑒∶=10⁶

define 𝑑 as the 𝛼-cut step with size in [0,1], for example, 𝑑=0.05

for 𝛼 from 0 to 1 step 𝑑

      𝐽₁∶=(𝐽₀>𝛼)  // 𝛼-cut image

     compute the convex hull of the object in image 𝐽₁, that is,

          𝐾:= convexhull(𝐽₁)

     calculate the Steiner point of the object:

          𝑠_𝑥,𝑠_𝑦:= CalculateSteinerPoint(𝐾)

          //Here, (𝑠_𝑥,𝑠_𝑦) is the 2-dimension Steiner point

     computing image distance between two neighbor gray:

          𝑑𝐽₁,𝐽₀∶=|𝐽₁−𝐽₀|

     record the Steiner point corresponding to the minimum varying image gray:

     if 𝑑(𝐽₁,𝐽₀)<𝑒 then

          𝑒∶=𝑑(𝐽₁,𝐽₀)

          𝑠_𝑥0,𝑠_𝑦0∶=𝑠_𝑥,𝑠_𝑦  //Record the approximate Steiner point

     end if

end for

Function CalculateSteinerpoint(𝐾)

denote 𝑞_𝑖∈𝐾 (𝑖=0,…,𝑛)

for 𝑖 from 0 to 𝑛 step  1

     compute the external angle 𝜃_𝑖 of convex polygon at 𝑞_𝑖

     (𝑠_𝑥,𝑠_𝑦)∶=(𝑠_𝑥,𝑠_𝑦)+𝑞_𝑖𝜃_𝑖

end for