Otsu's method

An example image thresholded using Otsu's algorithm

Original image

In computer vision and image processing, Otsu's method is used to automatically perform histogram shape-based image thresholding,^[1] or, the reduction of a graylevel image to a binary image. The algorithm assumes that the image to be thresholded contains two classes of pixels or bi-modal histogram (e.g. foreground and background) then calculates the optimum threshold separating those two classes so that their combined spread (intra-class variance) is minimal.^[2] The extension of the original method to multi-level thresholding is referred to as the Multi Otsu method^[3]. Otsu's method is named after Nobuyuki Otsu (大津展之, Ōtsu Nobuyuki^?).

Method

In Otsu's method we exhaustively search for the threshold that minimizes the intra-class variance, defined as a weighted sum of variances of the two classes:

$\sigma^2_w(t)=\omega_1(t)\sigma^2_1(t)+\omega_2(t)\sigma^2_2(t)$

Weights ω_i are the probabilities of the two classes separated by a threshold t and $\sigma^2_ i$ variances of these classes.

Otsu shows that minimizing the intra-class variance is the same as maximizing inter-class variance:^[2]

$\sigma^2_b(t)=\sigma^2-\sigma^2_w(t)=\omega_1(t)\omega_2(t)\left[\mu_1(t)-\mu_2(t)\right]^2$

which is expressed in terms of class probabilities ω_i and class means μ_i.

The class probability ω₁(t) is computed from the histogram as t:

$\omega_1(t)=\Sigma_0^t p(i)$

while the class mean μ₁(t) is:

$\mu_1(t)=\Sigma_0^t p(i)*x(i)$

where x(i) is the value at the center of the ith histogram bin. Similarly, you can compute ω₂(t) and μ_t on the right-hand side of the histogram for bins greater than t.

The class probabilities and class means can be computed iteratively. This idea yields an effective algorithm.

Algorithm

1. Compute histogram and probabilities of each intensity level
2. Set up initial ω_i(0) and μ_i(0)
3. Step through all possible thresholds $t = 1 \ldots$ maximum intensity
   1. Update ω_i and μ_i
   2. Compute $\sigma^2_b(t)$
4. Desired threshold corresponds to the maximum $\sigma^2_b(t)$

References

1. ^ M. Sezgin and B. Sankur (2003). "Survey over image thresholding techniques and quantitative performance evaluation". Journal of Electronic Imaging 13 (1): 146–165. doi:10.1117/1.1631315. 
2. ^ ^{a b} Nobuyuki Otsu (1979). "A threshold selection method from gray-level histograms". IEEE Trans. Sys., Man., Cyber. 9 (1): 62–66. doi:10.1109/TSMC.1979.4310076. 
3. ^ Ping-Sung Liao and Tse-Sheng Chen and Pau-Choo Chung (2001). "A Fast Algorithm for Multilevel Thresholding". J. Inf. Sci. Eng. 17 (5): 713–727. 

External links

• Lecture notes on thresholding - covers the Otsu method.
• A plugin for ImageJ using Otsu's method to do the threshold.
• A full explanation of Otsu's method with a worked example and Java implementation.
• Implementation of Otsu's method in ITK
• Otsu Thresholding in C# A straightforward C# implementation with explanation.