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Suppose we have L gray scale layers.  t* is the layer over which all layers are considered Black. and the rest is considered white. So if we consider Co for white and C1 black then,  Co = {0,1,2,….t} and C1={t+1,t+2,…L-1}.

To determine the threshold value we need to find the t=t* for which the ratio of class variance and total variance is maximum. Selection_002

where sigmaB is the class variance for black pixels and sigmaT is total variance.

Selection_003 . here both W0 and W1 are functions of t. Selection_004  .  and W is a constant. Pi is the probability of ith gray level .  which makes Selection_005 . the mew values are found from following equations ,  Selection_006. here UT is constant whereas Ut and U0 and U1 are functions of t.

sigmaT is determined using Selection_007.

thats all the equations we need. A java Implementation of it can be seen here.

Source:

CHARACTER RECOGNITION SYSTEMS
A Guide for Students and Practioners

By MOHAMED CHERIET et all

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