# OTSU Thresholding
- an adaptive thresholding way for binarization in image processing. By going through all possible threshold values (from 0 to 255), it can find the optimal threshold value of input image. [[see more](https://medium.com/@hbyacademic/otsu-thresholding-4337710dc519#baf7)]

## Preliminaries
- Given an image, threshold $t$ separates all the pixels $P_i$ into $C_1$ and $C_2$, where $C_1 = \{P_i \mid P_i \gt t \}$ and $C_2 = \{P_i \mid P_i \leq t \}$
- $W_1$ and $W_2$ are the probabilities of the two classes separated by $t$, where $W_1+W_2=1$
- $V_T$ : total variance
- $V_b$ : between-class variance
- $V_w$ : within-class variance
- $V_T = V_b + V_w$
- $V_1$ : $C_1$ variance
- $V_2$ : $C_2$ variance
- $V_w = W_1 \times V_1 + W_2 \times V_2$
- $\mu_T$ : total mean
- $\mu_1$ : $C_1$ mean
- $\mu_2$ : $C_2$ mean
- $W_1 \times \mu_1 + W_2 \times \mu_2 ＝ \mu_T$
- $N$ : total pixel count
- $\sum\limits_{i=0}^N{P_i} = N \times \mu_T$

## $V_T$ and $V_w$ Expansion
- $V_t \\
=\frac{\sum\limits_{i=1}^N{(P_i - \mu_t)^2}}{N} = \frac{\sum\limits_{i=0}^N{({P_i}^2 - 2 \times P_i \times \mu_t + {\mu_t}^2)}}{N}\\
=\frac{(\sum\limits_{i=0}^N{P_i}^2) - {N \times \mu_t}^2}{N} =\frac{\sum\limits_{i=0}^N{P_i}^2}{N} - {\mu_t}^2$

- $V_w \\
= W_1 \times V_1 + W_2 \times V_2 \\
= W_1 \times (\frac{\sum\limits_{P_i \in C_1}{(P_i - \mu_1)^2}}{N \times W_1}) + W_2 \times (\frac{\sum\limits_{P_i \in C_2}{(P_i - \mu_2)^2}}{N \times W_2}) \\
= \frac{\sum\limits_{P_i \in C_1}{(P_i - \mu_1)^2}}{N} + \frac{\sum\limits_{P_i \in C_2}{(P_i - \mu_2)^2}}{N} \\
= \frac{(\sum\limits_{P_i \in C_1}{P_i}^2) - N \times W_1 \times {\mu_1}^2}{N} + \frac{(\sum\limits_{P_i \in C_2}{P_i}^2) - N \times W_2 \times {\mu_2}^2}{N} \\
= \frac{\sum\limits_{P_i \in C_1}{P_i}^2}{N} - W_1 \times {\mu_1}^2 + \frac{\sum\limits_{P_i \in C_2}{P_i}^2}{N} - W_2 \times {\mu_2}^2$

## Vb = ?

- $V_b \\
= V_t - V_w \\
= (\frac{\sum\limits_{i=0}^N{P_i}^2}{N} - {\mu_t}^2) - (\frac{\sum\limits_{P_i \in C_1}{P_i}^2}{N} - W_1 \times {\mu_1}^2 + \frac{\sum\limits_{P_i \in C_2}{P_i}^2}{N} - W_2 \times {\mu_2}^2) \\
= W_1 \times {\mu_1}^2 + W_2 \times {\mu_2}^2 {\color{red}{-{\mu_t}^2}} \\
= W_1 \times {\mu_1}^2 + W_2 \times {\mu_2}^2 - {\color{red}{2 \times {\mu_t}^2 + {\mu_t}^2}} \\
{\color{orange}{\because \mu_t = W_1 \times \mu_1 + W_2 \times \mu_2}} \\
{\color{orange}{\because W_1 + W_2 = 1}} \\
= W_1 \times {\mu_1}^2 + W_2 \times {\mu_2}^2 - 2 \times (W_1 \times \mu_1 + W_2 \times \mu_2) \times {\mu_t} + (W_1 + W_2) \times {\mu_t}^2 \\
= W_1 \times ({\mu_1}^2 - 2 \times {\mu_1} \times {\mu_t} + {\mu_t}^2) + W_2 \times ({\mu_2}^2 - 2 \times {\mu_2} \times {\mu_t} + {\mu_t}^2) \\
= {\color{magenta}{W_1 \times (\mu_1 - \mu_t)^2 + W_2 \times (\mu_2 - \mu_t)^2}} \\
{\color{orange}{\because \mu_t = W_1 \times \mu_1 + W_2 \times \mu_2}} \\
= W_1 \times [(1 - W_1) \times \mu_1 - W_1 \times \mu_2]^2 + W_2 \times [W_1 \times \mu_1 - (1 - W_2) \times \mu_2]^2 \\
{\color{orange}{\because W_1 + W_2 = 1}} \\
= W_1 \times [(W_2) \times \mu_1 - W_1 \times \mu_2]^2 + W_2 \times [W_1 \times \mu_1 - (W_1) \times \mu_2]^2 \\
= W_1 \times W_2 \times (\mu_1 - \mu_2)^2 \times (W_2 + W_1) \\
{\color{orange}{\because W_1 + W_2 = 1}} \\
= {\color{blue}{W_1 \times W_2 \times (\mu_1 - \mu_2)^2}}$

- $V_b \\
= {\color{magenta}{W_1 \times (\mu_1 - \mu_t)^2 + W_2 \times (\mu_2 - \mu_t)^2}} \\
= {\color{blue}{W_1 \times W_2 \times (\mu_1 - \mu_2)^2}}$