Otsu's Threshold

Implement Otsu's method to find the optimal binarization threshold that maximizes between-class variance.

Given a list of pixel intensities (values 0-255), find the threshold $t$ that best separates the pixels into two classes (foreground and background) by maximizing the between-class variance $\sigma_B^2$.

Algorithm:

For each candidate threshold $t$ from 0 to 255:

1. Class probabilities:

   • $w_0(t) = \sum_{i=0}^{t} p(i)$ (background weight)
   • $w_1(t) = \sum_{i=t+1}^{255} p(i)$ (foreground weight)

2. Class means:

   • $\mu_0(t) = \frac{\sum_{i=0}^{t} i \cdot p(i)}{w_0(t)}$
   • $\mu_1(t) = \frac{\sum_{i=t+1}^{255} i \cdot p(i)}{w_1(t)}$

3. Between-class variance: $\sigma_B^2(t) = w_0(t) \cdot w_1(t) \cdot (\mu_0(t) - \mu_1(t))^2$

4. The optimal threshold maximizes $\sigma_B^2(t)$.

where $p(i)$ is the probability of intensity $i$ (histogram normalized by total pixel count).

Return the threshold value (integer) that maximizes $\sigma_B^2$.