From Fig. 2, it can be seen that sets of type $O(i,j)$, inside the three directional subbands, at scale 2 may be found significant until the third sorting pass, when $Thc=7$. Also, the first significant sets of type $L(i,j)$ in the horizontal, vertical, and diagonal subbands may be found significant until the fifth, fourth, and sixth passes, respectively. Notice that the encoding process is blind beyond the current threshold. That is, no information of the subbands is available to the process. Hence, all the descendants in a tree must be tested for significance, even though they belong to subbands with a threshold less than that of the current threshold, thus performing unnecessary comparison operations during a sorting pass. According to Fig. 2, for an image size of $M\xd7N$, the maximum number of unnecessary comparison operations $C$ in the first sorting pass can be approximated by Eq. (2): Display Formula
$C=M\xd7N\u2211l=0L\u22121122(L\u2212l)\u2211k=13Il,k,$(2)
Display Formula$Il,k={1,Thc>Thl,k0,otherwise,$(3)
where $Thl,k$ is the threshold of the subband at scale $l$ and direction $k$. $Il,k$ indicates subbands with a threshold less than $Thc$. However, as the number of passes increase, according to the bit rate requirements, Eq. (3) becomes Display Formula$Il,k={Thmax\u2212Thl,k,Thc>Thl,k0,otherwise.$(4)