0123456789A
missisippi$

c C[c]  number of characters in missisippi$ that come before c
$   0
i   1
m   5
p   6
s   8

Rank_c(BWT,i) is the number of cs in BWT that preceed the c at
position i

i BWT SA                        LF[i] = C[c] + Rank_c(BWT,i)
0  i   A $              0 $        1  =    1 + 0
1  p   9 i$             1 i        6  =    6 + 0
2  s   6 ippi$                     8  =    8 + 0
3  s   4 isippi$                   9  =    8 + 1
4  m   1 issisippi$                5  =    5 + 0
5  $   0 missisippi$    5 m        0  =    0 + 0
6  p   8 pi$            6 p        7  =    6 + 1
7  i   7 ppi$                      2  =    1 + 1
8  i   5 sippi$         8 s        3  =    1 + 2
9  s   3 sisippi$                  A  =    8 + 2
A  i   2 ssisippi$                 4  =    1 + 3

How to reconstruct the initial string from the BWT only?

BWT[i]  i  LF[i]
$       5  0
i       0  1
p       1  6
p       6  7
i       7  2
s       2  8
i       8  3
s       3  9
s       9  A
i       A  4
m       4  5
        5

LF PSI

i   SA[i]      LF[i]  SA[LF[i]]    PSI[i]  SA[PSI[i]]
3   4 isippi$   9     3 sisippi$    8      5 sippi$

=========================================

c C[c]  number of characters in missisippi$ that come before c
$   0
i   1
m   5
p   6
s   8

Select_c(BWT,n)=j if BWT[j]=c and it is the n-th occurrence of c in
BWT (where the first one is the 0-th ones)

                               let c s.t. i \in [C[c],C[c+1])
i BWT SA                       PSI[i] = Select_c(BWT,i-C[c])
0  i   A $              0 $         5 = Select_$(BWT,0-0) c=$ i\in[0,1)
1  p   9 i$             1 i         0 = Select_i(BWT,1-1) c=i i\in[1,5)
2  s   6 ippi$                      7 = Select_i(BWT,2-1) c=9 i\in[1,5)
3  s   4 isippi$                    8 = Select_i(BWT,3-1) c=i i\in[1,5)
4  m   1 issisippi$                 A = Select_i(BWT,4-1) c=i i\in[1,5)
5  $   0 missisippi$    5 m         4 = Select_m(BWT,5-5) c=m i\in[5,6)
6  p   8 pi$            6 p         1 = Select_p(BWT,6-6) c=p i\in[6,8)
7  i   7 ppi$                       6 = Select_p(BWT,7-6) c=p i\in[6,8)
8  i   5 sippi$         8 s         2 = Select_s(BWT,8-8) c=s i\in[8,B)
9  s   3 sisippi$                   3 = Select_s(BWT,9-8) c=s i\in[8,B)
A  i   2 ssisippi$                  9 = Select_s(BWT,A-8) c=s i\in[8,B)

How to rebuild the initial message using PSI?

BWT[i]  i  PSI[i]
$       5  4
m       4  A
i       A  9
s       9  3
s       3  8
i       8  2
s       2  7
i       7  6
p       6  1
p       1  0
i       0  5
$       5