function [IDX,sep] = otsu(I,n)
%OTSU Global image thresholding/segmentation using Otsu's method.
% IDX = OTSU(I,N) segments the image I into N classes by means of Otsu's
% N-thresholding method. OTSU returns an array IDX containing the cluster
% indices (from 1 to N) of each point. Zero values are assigned to
% non-finite (NaN or Inf) pixels.
%
% IDX = OTSU(I) uses two classes (N=2, default value).
%
% [IDX,sep] = OTSU(...) also returns the value (sep) of the separability
% criterion within the range [0 1]. Zero is obtained only with data
% having less than N values, whereas one (optimal value) is obtained only
% with N-valued arrays.
%
% Notes:
% -----
% It should be noticed that the thresholds generally become less credible
% as the number of classes (N) to be separated increases (see Otsu's
% paper for more details).
%
% If I is an RGB image, a Karhunen-Loeve transform is first performed on
% the three R,G,B channels. The segmentation is then carried out on the
% image component that contains most of the energy.
%
% Example:
% -------
% load clown
% subplot(221)
% X = ind2rgb(X,map);
% imshow(X)
% title('Original','FontWeight','bold')
% for n = 2:4
% IDX = otsu(X,n);
% subplot(2,2,n)
% imagesc(IDX), axis image off
% title(['n = ' int2str(n)],'FontWeight','bold')
% end
% colormap(gray)
%
% Reference:
% ---------
% Otsu N, <a href="matlab:web('http://dx.doi.org/doi:10.1109/TSMC.1979.4310076')">A Threshold Selection Method from Gray-Level Histograms</a>,
% IEEE Trans. Syst. Man Cybern. 9:62-66;1979
%
% See also GRAYTHRESH, IM2BW
%
% -- Damien Garcia -- 2007/08, revised 2010/03
% Visit my <a
% href="matlab:web('http://www.biomecardio.com/matlab/otsu.html')">website</a> for more details about OTSU
error(nargchk(1,2,nargin))
% Check if is the input is an RGB image
isRGB = isrgb(I);
assert(isRGB | ndims(I)==2,...
'The input must be a 2-D array or an RGB image.')
%% Checking n (number of classes)
if nargin==1
n = 2;
elseif n==1;
IDX = NaN(size(I));
sep = 0;
return
elseif n~=abs(round(n)) || n==0
error('MATLAB:otsu:WrongNValue',...
'n must be a strictly positive integer!')
elseif n>255
n = 255;
warning('MATLAB:otsu:TooHighN',...
'n is too high. n value has been changed to 255.')
end
I = single(I);
%% Perform a KLT if isRGB, and keep the component of highest energy
if isRGB
sizI = size(I);
I = reshape(I,[],3);
[V,D] = eig(cov(I));
[tmp,c] = max(diag(D));
I = reshape(I*V(:,c),sizI(1:2)); % component with the highest energy
end
%% Convert to 256 levels
I = I-min(I(:));
I = round(I/max(I(:))*255);
%% Probability distribution
unI = sort(unique(I));
nbins = min(length(unI),256);
if nbins==n
IDX = ones(size(I));
for i = 1:n, IDX(I==unI(i)) = i; end
sep = 1;
return
elseif nbins<n
IDX = NaN(size(I));
sep = 0;
return
elseif nbins<256
[histo,pixval] = hist(I(:),unI);
else
[histo,pixval] = hist(I(:),256);
end
P = histo/sum(histo);
clear unI
%% Zeroth- and first-order cumulative moments
w = cumsum(P);
mu = cumsum((1:nbins).*P);
%% Maximal sigmaB^2 and Segmented image
if n==2
sigma2B =...
(mu(end)*w(2:end-1)-mu(2:end-1)).^2./w(2:end-1)./(1-w(2:end-1));
[maxsig,k] = max(sigma2B);
% segmented image
IDX = ones(size(I));
IDX(I>pixval(k+1)) = 2;
% separability criterion
sep = maxsig/sum(((1:nbins)-mu(end)).^2.*P);
elseif n==3
w0 = w;
w2 = fliplr(cumsum(fliplr(P)));
[w0,w2] = ndgrid(w0,w2);
mu0 = mu./w;
mu2 = fliplr(cumsum(fliplr((1:nbins).*P))./cumsum(fliplr(P)));
[mu0,mu2] = ndgrid(mu0,mu2);
w1 = 1-w0-w2;
w1(w1<=0) = NaN;
sigma2B =...
w0.*(mu0-mu(end)).^2 + w2.*(mu2-mu(end)).^2 +...
(w0.*(mu0-mu(end)) + w2.*(mu2-mu(end))).^2./w1;
sigma2B(isnan(sigma2B)) = 0; % zeroing if k1 >= k2
[maxsig,k] = max(sigma2B(:));
[k1,k2] = ind2sub([nbins nbins],k);
% segmented image
IDX = ones(size(I))*3;
IDX(I<=pixval(k1)) = 1;
IDX(I>pixval(k1) & I<=pixval(k2)) = 2;
% separability criterion
sep = maxsig/sum(((1:nbins)-mu(end)).^2.*P);
else
k0 = linspace(0,1,n+1); k0 = k0(2:n);
[k,y] = fminsearch(@sig_func,k0,optimset('TolX',1));
k = round(k*(nbins-1)+1);
% segmented image
IDX = ones(size(I))*n;
IDX(I<=pixval(k(1))) = 1;
for i = 1:n-2
IDX(I>pixval(k(i)) & I<=pixval(k(i+1))) = i+1;
end
% separability criterion
sep = 1-y;
end
IDX(~isfinite(I)) = 0;
%% Function to be minimized if n>=4
function y = sig_func(k)
muT = sum((1:nbins).*P);
sigma2T = sum(((1:nbins)-muT).^2.*P);
k = round(k*(nbins-1)+1);
k = sort(k);
if any(k<1 | k>nbins), y = 1; return, end
k = [0 k nbins];
sigma2B = 0;
for j = 1:n
wj = sum(P(k(j)+1:k(j+1)));
if wj==0, y = 1; return, end
muj = sum((k(j)+1:k(j+1)).*P(k(j)+1:k(j+1)))/wj;
sigma2B = sigma2B + wj*(muj-muT)^2;
end
y = 1-sigma2B/sigma2T; % within the range [0 1]
end
end
function isRGB = isrgb(A)
% --- Do we have an RGB image?
% RGB images can be only uint8, uint16, single, or double
isRGB = ndims(A)==3 && (isfloat(A) || isa(A,'uint8') || isa(A,'uint16'));
% ---- Adapted from the obsolete function ISRGB ----
if isRGB && isfloat(A)
% At first, just test a small chunk to get a possible quick negative
mm = size(A,1);
nn = size(A,2);
chunk = A(1:min(mm,10),1:min(nn,10),:);
isRGB = (min(chunk(:))>=0 && max(chunk(:))<=1);
% If the chunk is an RGB image, test the whole image
if isRGB, isRGB = (min(A(:))>=0 && max(A(:))<=1); end
end
end
