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 2 1 1 Ap ri l 2020 i-1 2 2 i -2 i-1 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 13

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