IEEE Systems, Man and Cybernetics Magazine - April 2020 - 13

Algorithm S5. Dark Block Extraction [54].
Input : D - n # n dissimilarity matrix
				 a -  minimum cluster size as a fraction of n
Output: k -  the number of dark blocks
	 1	Transform D to Dl with d ij = 1- e -d /v, where v is a
 scale parameter obtained by applying Otsu's
-algorithm [184] on D
	2	Apply VAT to Dl to generate RDI I (1)
	 3	Threshold I (1) using Otsu's algorithm [184] to generate
 binary image I (2)
	 4	Filter I (2) using morphological operator with directional
 "line" structuring elements of size l 1 = an to obtain
filtered binary image I (3)
	 5	Perform distance transformation on I (3) to obtain a
 new grayscale image I (4) and scale the pixel values
to [0, 1]
	 6	Project pixel values of I (4) onto the main diagonal
 axis of the image to form a projection signal H (1)
	 7	Smooth H (1) using simple average filter h of length
 l 2 = 2an to obtain the filtered signal H (2)
	 8	Compute first-order derivative of H (2) to obtain
 signal H (3)
	 9	Find peak position p i (positive-to-negative zero-crossing
 points) and valley position (negative-to-positive zerocrossing points) in H (3)
	10	Select major peaks by removing minor peaks and
 valleys using a filter size of length l 3 = 2an
	11	 k =  number of major peaks
ij

	16	 for t ! 1 to kl do
n #|S t|
m
nt = c
	17		
N
	18		 Draw n t unique random indices Su t from S t
	19	 end
kl
	20	 Su = ' Su t ; D n = dist " x Su , x Su ,
t=1

	21	 Apply VAT to D n returning D )n, P  and d
	22	 Apply iVAT to D )n returning D l)n

Algorithm S7. Clustering Using
Scalable iVAT (clusiVAT) [79]-[81].
Input : X = " x 1, x 2,f, x N , - N  p -dimensional data points
				 k l -  overestimate of actual number of clusters
				 n -  approximating sample size
Output: D l)n - n # n iVAT reordered dissimilarity matrix of D n
				 u : X " " 1, 2,f, k , -  cluster membership
	1	Apply siVAT on X returning D l)n, Su, P, d
	2	Choose the number of clusters k using siVAT image
	3	t = argmax d i
1# i # k

	 4	Form the aligned partition:
	5	u ) = " t 1 : t 2 - t 1 :f: t k - t k - 1 ,
	6	u Su = u P) ; 1# i # k
	 7	for xt ! Xt = X - X Su do
	 8		 j = argmin " dist " x st , x i ,,
Pi

i

i ! Su

	 9		 u st = u j
	10	end (nearest prototype rule)

Algorithm S6. Asymmetric iVAT [13].
Input : X = " x 1, x 2,f, x N , - N  p -dimensional data points
				 k l -  overestimate of actual number of clusters
				 n -  approximating sample size
Output: D l)n - n # n iVAT reordered dissimilarity matrix
  of D n
				 Su -  indices of samples in D n
				 P -  VAT reordering indices of D n
				 d -  ordering of MST cut magnitudes
	 1	Select the indices m of kl distinguished objects
	2	m 1 = 1
	3	y = " dist " x 1, x 1 ,,f, dist " x 1, x N ,,
	 4	for t ! 2 to kl do
	 5		 y = ^min " y 1, dist " x m , x 1 ,,f,
	 6		 min " y N , dist " x m , x N ,,h
	 7		 m t = argmax {y j}
t-1

t-1

	 8	end

1#j#N

	 9	Group objects in X = " x 1, x 2,f, x N , with their
-nearest distinguished objects
	10	 S 1 = S 2 = f = S k l = 4
	11	 for t ! 1 to N do
l = argmin " dist " x m , x t ,,
	12		
j

1#j#k

Sl = Sl , " t ,
	13		
	14	 end

Algorithm S8. InsertPosition.
Input : Pn -  VAT reordering indices of D )n
			
d n -  MST cut magnitude order of D )n
			Fn -  MST connection indices of D )n
V = " v 1,v 2,f,v n , Distance of x n+1 from
			
			 " x 1, x 2,f, x N ,
Output: i -  insertion position of x n+1
				Pn + 1 -  initialization of VAT reordering indices of D )n +1
				d n + 1 -  initialization of MST cut magnitude order
 of D )n +1
				Fn + 1 -  initialization of MST connection indices
 of D )n +1
	1	Y = VP = " v P ,v P ,f,v P ,
	2	i = n + 1
	3	 j = argmin (V )
n

1

	

n

	 4	for t ! 1 to n - 1 do
	 5		 if min  ^" Y1,Y2,f,Yt ,h 1 d n then
	 6			 i = t + 1
j = argmin ^" Y1,Y2,f,Yt ,h
	 7			
	 8			 break
	 9		 end
	10	 end
t

	11	 Pn + 1 = " Pn , Pn ,f, Pn , n + 1 ,
	12	 d n + 1 = " d n , d n ,f, d n , min^Y1,Y2,f,Yi -1 h,
	13	 Fn + 1 = " Fn , Fn ,f, Fn , j ,
1

	15	Randomly select data near each distinguished
  object to form D n

2

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IEEE Systems, Man and Cybernetics Magazine - April 2020

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