In computer vision and image processing, Otsu's method, named after Nobuyuki Otsu (大津展之, Ōtsu Nobuyuki), is used to perform automatic image thresholding. In the simplest form, the algorithm returns a single intensity threshold that separate pixels into two classes – foreground and background. This threshold is determined by minimizing intra-class intensity variance, or equivalently, by maximizing inter-class variance. Otsu's method is a one-dimensional discrete analogue of Fisher's discriminant analysis, is related to Jenks optimization method, and is equivalent to a globally optimal k-means performed on the intensity histogram. The extension to multi-level thresholding was described in the original paper, and computationally efficient implementations have since been proposed.

Let, ${\displaystyle H}$ be the normalised histogram of the pixels in an image (s.t. it becomes the probability distribution of pixel intensities) with ${\displaystyle L}$ bins. There are two classes of this histogram: ${\displaystyle C_{0}}$ for background pixels, and ${\displaystyle C_{1}}$ for foreground pixels. The primary disciminator of pixels (to assort them into classes) is the threshold ${\textstyle t}$. ${\displaystyle C_{0}}$ includes pixels from ${\displaystyle 0}$ to ${\displaystyle (t-1)}$, and ${\displaystyle C_{1}}$ includes from ${\displaystyle t}$ to ${\displaystyle (L-1)}$.

The algorithm is then global search for an optimal threshold ${\displaystyle t^{*}}$ such that intra-class variance (variance of pixels intensities in ${\displaystyle C_{0}}$ or ${\displaystyle C_{1}}$) is minimised.

Let, ${\displaystyle \omega _{0}}$ denote the cumulative probability of ${\displaystyle C_{0}}$, and ${\displaystyle \omega _{1}}$denote of ${\displaystyle C_{1}}$.$${\displaystyle {\begin{aligned}\omega _{0}(t)&=\sum _{i=0}^{t-1}P(i),\\\omega _{1}(t)&=\sum _{i=t}^{L-1}P(i).\end{aligned}}}$$For a classes ${\displaystyle C_{0}}$ and ${\displaystyle C_{1}}$, the conditional probability of selecting the ${\displaystyle i}$-th pixel in those classes is ${\displaystyle P(i/C_{0})}$ and ${\displaystyle P(i/C_{1})}$ respectively.

Now, let ${\displaystyle \mu _{0}(t)}$ and ${\displaystyle \mu _{1}(t)}$ be the mean (pixel intensity) of ${\displaystyle C_{0}}$ and ${\displaystyle C_{1}}$ respectively.

${\displaystyle {\begin{aligned}\mu _{0}(t)&=\sum _{i=0}^{t-1}iP(i/C_{0})=\sum _{i=0}^{t-1}{\frac {iP(i)}{\omega _{0}(t)}}={\frac {\sum _{i=0}^{t-1}iP(i)}{\omega _{0}(t)}}.\end{aligned}}}$

Similarly,

${\displaystyle \mu _{1}(t)={\frac {\sum _{i=t}^{L-1}iP(i)}{\omega _{1}(t)}}.}$