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function [p, r, nc, G, xy] = find_components (A,sorted)
%FIND_COMPONENTS finds all connected components of an image.
% Two pixels in an image are in the same connected component if and only if
% they are adjacent and have the same value. "Adjacent" in this context means
% north/south or east/west, not diagonally. That is, A(2,3) is adjacent to
% A(2,2), A(2,4), A(1,3), and A(3,3) only, and is not adjacent to A(3,4).
%
% Let [p r] = find_components(A) where A is m-by-n. The result is a permutation
% p and component boundaries r. p is a permutation of 1:m*n, and refers to the
% linear indexing of A. That is, A(i,j) is refered to as A(i+j*m) in the list
% p. The kth connected component consists of A (p (r(k) : r(k+1)-1)).
% The number of connected components is nc = length (r) - 1. The components
% are ordered by the smallest linear index in each component (assuming you have
% MATLAB 7.5 or later, which uses DMPERM from CSparse; otherwise the ordering
% is not defined).
%
% With a single output argument, c = find_components (A) just returns a list
% of the nodes in the largest component (the component with the largest value
% if ties, and if there are 2 components still tied, return the one containing
% the smallest node index).
%
% The are no restrictions on the image A except that it must be a 2D matrix,
% and the operator "==" must be defined on its entries. If sorting of the
% components by size is requested or if the largest component is requested,
% double(A) must be computable.
%
% Usage:
% c = find_components (A) ; % just return nodes in the largest component
% [p r nc G xy] = find_components (A) ; % sorted by least node number
% [p r nc G xy] = find_components (A, 1) ; % sorted by size, ties by value
% find_components (A) ; % just plot the graph
%
% Example:
%
% A = [ 1 2 2 3
% 1 1 2 3
% 0 0 1 2
% 0 1 3 3 ]
% [p r nc] = find_components (A,1)
% [m n] = size (A) ;
% for k = 1:nc
% a = A (p (r (k))) ;
% fprintf ('\ncomponent %d, size %d, value %d\n', k, r(k+1)-r(k), a) ;
% C = nan (m,n) ;
% C (p (r (k) : r (k+1)-1)) = a ;
% fprintf ('A = \n') ; disp (A) ;
% fprintf ('the component = \n') ; disp (C) ;
% input (': ', 's') ;
% end
%
% The optional outputs G and xy give a graph representation of the problem
% which can be viewed with gplot(G,xy).
%
% See also LARGEST_COMPONENT, FIND_COMPONENTS_EXAMPLE, DMPERM, GPLOT
% Copyright 2008, Tim Davis, University of Florida
%-------------------------------------------------------------------------------
% number the nodes
%-------------------------------------------------------------------------------
% K is a matrix containing the linear index into A. For the example above,
%
% K = [ 1 5 9 13
% 2 6 10 14
% 3 7 11 15
% 4 8 12 16 ]
%
% The "nodes" of A are simply their linear indices. For example, A(2,3)
% is called node 10.
[m n] = size (A) ;
N = m*n ; % N = number of nodes in A
K = reshape (1:N, m, n) ;
%-------------------------------------------------------------------------------
% look to the east
%-------------------------------------------------------------------------------
% If A(i,j) == A(i,j+1), then East(i,j) is set to K(i,j+1). That is, East(i,j)
% gives the node number of the Eastern neighbor of the node A(i,j).
%
% In this example, East = [
% 0 9 0 0
% 6 0 0 0
% 7 0 0 0
% 0 0 16 0 ]
%
% because (for example) node 5 has an Eastern neighbor, node 9 (A(5) == A(9)).
East = [(K (:,2:n) .* (A (:,1:n-1) == A (:,2:n))) zeros(m,1)] ;
% E gives the node numbers of all nodes with Eastern neighbors. For example,
% E = [2 3 5 12]' ;
E = find (East) ;
%-------------------------------------------------------------------------------
% look to the south
%-------------------------------------------------------------------------------
% If A(i,j) == A(i+1,j), then South(i,j) is set to K(i+1,j). That is,
% South(i,j) gives the node number of the Southern neighbor of the node A(i,j).
%
% In this example, South = [
% 2 0 10 14
% 0 0 0 0
% 4 0 0 0
% 0 0 0 0 ]
%
% because (for example) node 4 has a Southern neighbor, node 4 (A(3) == A(4)).
% Then S gives the node numbers of all nodes with Southern neighbors.
South = [(K (2:m,:) .* (A (1:m-1,:) == A (2:m,:))) ; zeros(1,n)] ;
S = find (South) ;
%-------------------------------------------------------------------------------
% create the graph G
%-------------------------------------------------------------------------------
% The graph G is N-by-N for an image A of size m-by-n with N=m*n nodes.
% There is an edge between two nodes if they are neighbors (that is, if the
% two nodes are adjacent and have the same value). A diagonal is added to
% the graph so that DMPERM knows that the matrix G does not need to first be
% permuted to reveal a maximum matching ... that step is skipped.
%
% Ignoring the diagonal entries, in this example, G =
%
% (2,1) 1
% (1,2) 1
% (6,2) 1
% (4,3) 1
% (7,3) 1
% (3,4) 1
% (9,5) 1
% (2,6) 1
% (3,7) 1
% (5,9) 1
% (10,9) 1
% (9,10) 1
% (16,12) 1
% (14,13) 1
% (13,14) 1
% (12,16) 1
%
% If drawn as a 4-by-4 mesh, with edges between neighbors, G looks like this:
%
% 1 5 - 9 13
% | | |
% 2 - 6 10 14
%
% 3 - 7 11 15
% |
% 4 8 12 - 16
%
% If the nodes are labeled according the value of A, the graph looks like this:
%
% 1 2 - 2 3
% | | |
% 1 - 1 2 3
%
% 0 - 0 1 15
% |
% 0 1 3 - 3
G = sparse ([K(E) ; K(S)], [East(E) ; South(S)], 1, N, N) ;
clear K East E South S % free up some space in case the problem is large
G = G + G' + speye (N) ;
%-------------------------------------------------------------------------------
% find the connected components of G
%-------------------------------------------------------------------------------
% Note that p==q and r==s, since the matrix G is square with zero-free diagonal.
% nc gives the number of connected components.
[p q r s] = dmperm (G) ; %#ok (s is unused, present for comments)
nc = length (r) - 1 ;
%-------------------------------------------------------------------------------
% sort the components by size, if requested (the rest is all optional)
%-------------------------------------------------------------------------------
if (nargin < 2)
% the default is not to sort the components
sorted = 0 ;
end
if (nargout == 1)
% largest component is requested, so we must sort
sorted = 1 ;
end
if (sorted)
[ignore i] = sortrows ([diff(r)' double(A(p(r(1:end-1)))')], [-1 -2]) ;
if (nargout == 1)
% just return the largest component
c = i (1) ;
p = p (r (c) : r (c + 1) - 1) ;
else
% sort all the components (this can be costly)
p2 = zeros (1,N) ;
r2 = zeros (1,nc+1) ;
k2 = 0 ;
for k = 1:nc
% get the nodes and the size of the kth largest component, c
c = i (k) ;
nodes = p (r (c) : r (c + 1) - 1) ;
csize = length (nodes) ;
% place the nodes in the new output permutation
p2 (k2+1 : k2+csize) = nodes ;
r2 (k) = k2+1 ;
k2 = k2 + csize ;
end
r2 (nc+1) = N+1 ;
p = p2 ;
r = r2 ;
end
end
%-------------------------------------------------------------------------------
% return the XY coordinates, if requested or if the graph is to be plotted
%-------------------------------------------------------------------------------
if (nargout >= 5 || nargout == 0)
x = repmat (1:n, n, 1) ;
y = repmat (m:-1:1, 1, m) ;
xy = [x(:) y(:)] ;
end
%-------------------------------------------------------------------------------
% plot the graph if no outputs requested
%-------------------------------------------------------------------------------
if (nargout == 0)
if (N < 100)
gplot (G, xy, 'o-') ;
else
gplot (G, xy) ;
end
axis ([0 n+1 0 m+1]) ;
title (sprintf ('%d connected components', nc)) ;
end