Given the string T = abracadabra$:

0123456789AB
abracadabra$

C is the array mapping every symbol to the number of symbols in T that preceed
it in alphabetical order:

    C
$   0
a   1
b   6
c   8
d   9
r  10

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

Computing LF from BWT only requires a simple computation.
The column SA is NOT necessary and it does not need to be stored.

   BWT  SA               SA      LF[i]= C[c] + Rank_c(BWT,i) where c=BWT[i]
0   a   $                B          1 =    1 + 0
1   r   a$               A          A =    A + 0
2   d   abra$            7          9 =    9 + 0  i.e.     SA[9]=6=dabra$
3   $   abracadabra$     0          0 =    0 + 0  preceeds SA[2]=7= abra$
4   r   acadabra$        3          B =    A + 1
5   c   adabra$          5          8 =    8 + 0
6   a   bra$             8          2 =    1 + 1
7   a   bracadabra$      1          3 =    1 + 2
8   a   cadabra$         4          4 =    1 + 3
9   a   dabra$           6          5 =    1 + 4
A   b   ra$              9          6 =    6 + 0
B   b   racadabra$       2          7 =    6 + 1

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

Let's rebuild T from BWT. We start from i s.t. BWT[i]=$ and follow the LF chain,
rebuilding T from last to front.

  BWT[i]   i    LF[i]
    $      3    0
    a      0    1
    r      1    A
    b      A    6
    a      6    2
    d      2    9
    a      9    5
    c      5    8
    a      8    4
    r      4    B
    b      B    7
    a      7    3
    $      3

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

The PSI function is also the result of an easy computation.

    C
$   0
a   1
b   6
c   8
d   9
r  10

Select_c(S,i) returns the index j such that S[j]=c and it is the i-th occurrence
of c (0-based: the first occurrence is the 0-th).

   BWT  SA               SA      PSI[i]= Select_c(BWT,r-C[c])
                                 where i \in [C[c],C[c+1])
0   a   $                B          3 = Select_$(BWT,0-0)    0 \in [0,1) => c=$
1   r   a$               A            =
2   d   abra$            7            =
3   $   abracadabra$     0            =
4   r   acadabra$        3          8 = Select_a(BWT,4-1)    4 \in [1,6) => c=a
5   c   adabra$          5            =    i.e.     SA[4]=3=acadabra$
6   a   bra$             8            =    preceeds SA[8]=4=cadabra$
7   a   bracadabra$      1            =
8   a   cadabra$         4            =
9   a   dabra$           6            =
A   b   ra$              9            =
B   b   racadabra$       2            =