In computer vision and image processing, Otsu's method is used to automatically perform
histogram shape-based image thresholding, 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 (e.g. foreground and background)
then calculates the optimum threshold separating those two classes so that their combined spread (intra-class variance) is minimal.

Let the pixels of given picture be represented in L gray levels [1,2,…L]. The number of pixels with level i is denoted by n_i and the total number of pixels by  N = n₁ + n₂ + …+ n_L.
In order to simplify the discussion, the gray-level histogram is normalized and regarded as a probability distribution:

Now suppose that we dichotomize the pixel into two classes C₀ and C₁ (background and objects) by a threshold at level k: C₀ denotes pixels with levels [0, … k] and C₁ denotes pixels with levels [k+1, … ,L]. Then the probabilities of class occurrence and the class mean levels, respectively, are given by

The optimal threshold k* is

where

Reference:

1. Nobuyuki Otsu, A threshold selection method from gray-level histograms, IEEE, January 1979
2. https://en.wikipedia.org/wiki/Otsu's_method

void OtsuBinarization(Image* input_image, Image* output_image)
{
	int max_x = input_image->get_width();
	int max_y = input_image->get_height();
	const int L = 256;
	float hist[L] = { 0.0F };

	//calculate grayscale histogram
	for (int x = 0; x < max_x; ++x)
		for (int y = 0; y < max_y; ++y)
		{
			Pixel cur;
			cur = input_image->get_pixel(x, y);
			int graylevel = max(0.0, min(255.0, 0.299*cur.R + 0.587*cur.G + 0.114*cur.B));
			hist[graylevel] += 1;
		}

	int N = max_x*max_y;

	//normalize histogram
	for (int i = 0; i < L; ++i)
		hist[i] /= N;


		float ut = 0;
		for (int i = 0; i < L; ++i)
			ut += i*hist[i];

		int max_k = 0;
		int max_sigma_k_ = 0;
		for (int k = 0; k < L; ++k)
		{
			float wk = 0;
				for (int i = 0; i <= k; ++i)
					wk += hist[i];
				float uk = 0;
				for (int i = 0; i <= k; ++i)
					uk += i*hist[i];

			float sigma_k = 0;
			if (wk != 0 && wk != 1)
				sigma_k = ((ut*wk - uk)*(ut*wk - uk)) / (wk*(1 - wk));

			if (sigma_k > max_sigma_k_)
			{
				max_k = k;
				max_sigma_k_ = sigma_k;
			}
		}

	for (int x = 0; x < max_x; ++x)
		for (int y = 0; y < max_y; ++y)
		{
			Pixel cur;
			cur = input_image->get_pixel(x,y);
			int graylevel = max(0.0, min(255.0, 0.299*cur.R + 0.587*cur.G + 0.114*cur.B));
			if (graylevel < max_k)
				output_image->set_pixel(x,y, RGB(0, 0, 0));
			else
				output_image->set_pixel(x,y, RGB(255, 255, 255));
		}
}

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