之graythresh函數（OTSU算法）

graythresh函數是使用OTSU（大津）算法返回輸出最佳閾值的算法。graythresh函數的代碼詳解如下：

function [level em] = graythresh(I)
%GRAYTHRESH Global image threshold using Otsu's method.
%   LEVEL = GRAYTHRESH(I) computes a global threshold (LEVEL) that can be
%   used to convert an intensity image to a binary image with IM2BW. LEVEL
%   is a normalized intensity value that lies in the range [0, 1].
%   GRAYTHRESH uses Otsu's method, which chooses the threshold to minimize
%   the intraclass variance of the thresholded black and white pixels.
%
%   [LEVEL EM] = GRAYTHRESH(I) returns effectiveness metric, EM, as the
%   second output argument. It indicates the effectiveness of thresholding
%   of the input image and it is in the range [0, 1]. The lower bound is
%   attainable only by images having a single gray level, and the upper
%   bound is attainable only by two-valued images.
%
%   Class Support
%   -------------
%   The input image I can be uint8, uint16, int16, single, or double, and it
%   must be nonsparse.  LEVEL and EM are double scalars. 
%
%   Example
%   -------
%       I = imread('coins.png');
%       level = graythresh(I);
%       BW = im2bw(I,level);
%       figure, imshow(BW)
%
%   See also IM2BW.

%   Copyright 1993-2009 The MathWorks, Inc.
%   $Revision: 1.9.4.10 $  $Date: 2009/11/09 16:24:24 $

% Reference:
% N. Otsu, "A Threshold Selection Method from Gray-Level Histograms,"
% IEEE Transactions on Systems, Man, and Cybernetics, vol. 9, no. 1,
% pp. 62-66, 1979.

% One input argument required.
iptchecknargin(1,1,nargin,mfilename);
iptcheckinput(I,{'uint8','uint16','double','single','int16'},{'nonsparse'}, ...
              mfilename,'I',1);

if ~isempty(I)
  % Convert all N-D arrays into a single column.  Convert to uint8 for
  % fastest histogram computation.
  I = im2uint8(I(:));%將輸入轉換爲無符號8位整型
  num_bins = 256;%設置灰度級爲256級
  counts = imhist(I,num_bins);
  %imhist(I,n)其中，I爲灰度的輸入圖像，n爲指定的灰度級數目，缺省值爲256；
  %imhist(X,map)就算和顯示索引色圖像X的直方圖，map爲調色板。
  %[counts,x]=imhist()
  %用stem(x,counts)同樣可以顯示直方圖。counts和x分別爲返回直方圖數據向量和相應的彩色向量。
  
  % Variables names are chosen to be similar to the formulas in
  % the Otsu paper.
  p = counts / sum(counts);%每一個灰度級下的像素點的個數除以總的像素點得到該灰度級發生的概率
  omega = cumsum(p);%求累加和。按行累加，即概率分佈
  mu = cumsum(p .* (1:num_bins)');%灰度級的累加和

  %PS：跟香農公式計算的開頭差不多，只是不排序
  mu_t = mu(end);%整個圖像的平均灰度即mu的最後一個值
  %%求類間方差
  sigma_b_squared = (mu_t * omega - mu).^2 ./ (omega .* (1 - omega));

  % Find the location of the maximum value of sigma_b_squared.
  % The maximum may extend over several bins, so average together the
  % locations.  If maxval is NaN, meaning that sigma_b_squared is all NaN,
  % then return 0.
  maxval = max(sigma_b_squared);
  isfinite_maxval = isfinite(maxval);
  if isfinite_maxval
    idx = mean(find(sigma_b_squared == maxval));
    % Normalize the threshold to the range [0, 1].
    level = (idx - 1) / (num_bins - 1);
  else
    level = 0.0;
  end
else
  level = 0.0;
  isfinite_maxval = false;
end

% compute the effectiveness metric
if nargout > 1
  if isfinite_maxval
    em = maxval/(sum(p.*((1:num_bins).^2)') - mu_t^2);
  else
    em = 0;
  end
end