img=Import["http://i.stack.imgur.com/myVKd.png"]

pos = ImageValuePositions[img, 1];
newPos = Nest[Function[pts, Mean@*Rest /@ Nearest[pts, pts, 3]], pos, 50];
ReplaceImageValue[ConstantImage[0, ImageDimensions[img]], newPos -> 1]

Graphics[Line /@ Nearest[newPos, newPos, 3]]