i.e., $\sigma_B^2(t) = p_0(t) p_1(t) \left[ {\mu_0(t) - \mu_1(t)} \right]^2$, from the discriminant analysis literature as its objective function and determines the optimal threshold $t^*$ as the gray level that maximizes $\sigma_B^2(t)$, i.e., $t^* = \mathop {{\mathop{\rm argmax}\nolimits} }\limits_{t \in \left\{0,1,\dotsc,L - 1 \right\}} \sigma_B^2(t)$. Between-class variance can be viewed as a measure of class separability or histogram bimodality. It can be efficiently calculated using: $$\sigma_B^2(t) = \frac{{\left[ {\mu_T p_0(t) - \mu(t)} \right]^2 }}{{p_0(t) p_1(t)}}$$

It should be noted that the efficiency of Otsu’s method can be attributed to the fact that it operates on histogrammed pixel gray values, which are non-negative integers. Var-Part and PCA-Part, on the other hand, operate on the projections of the points on the partitioning axis, which are often fractional. This problem can be circumvented by linearly scaling the projection values to the limits of the histogram, i.e., $0$ and $L-1$. Let $y_i$ be the projection of a point ${{\bf x}}_i$ on the partitioning axis. $y_i$ can be mapped to histogram bin $b$ given by: $$b = \left\lfloor {\frac{{L\left( {y_i - \mathop {\min }\limits_j y_j } \right)}}{{\mathop {\max }\limits_j y_j - \mathop {\min}\limits_j y_j }}} \right\rfloor$$ where $\lfloor z \rfloor$ is the floor function which 