# Otsu Thresholding : Why 'minimum within class variance' gives the optimum threshold?

I was trying to understand the Otsu thresholding algorithm in image processing. For that purpose I found a useful link. I got the algorithm flow, but a fundamental doubt arises. Why 'minimum within class variance' ${\sigma_W}^2$ (notation given in the link) gives the optimum threshold? Please can someone clear my doubt?

Otsu threshold assumes a bi-modal (e.g. two-class) histogram. Nobuyuki Otsu shows that in such a case, minimizing the within-class variance is the same as maximizing between-class variance. To separate the bi-modal histogram, that is actually what you need to do. It would correspond to maximizing the distance between the clusters, which would give you the optimum point of separation.

Please check the paper for further details:

http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=04310076

Nobuyuki Otsu (1979). "A threshold selection method from gray-level histograms". IEEE Trans. Sys., Man., Cyber. 9 (1): 62–66

• Perfect answer although yet to see the paper. Any way that's the matter of seeing a proof right. 'minimizing the within-class variance is the same as maximizing between-class variance'- I think I have read something like this when reading about PCA. – dexterdev Apr 19 '14 at 8:16