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Hungarian based particle linking

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Find best links between two sets of points based on their euclidean distance, using Hungarian algo

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The file munkres.m from Yi Cao, available at FEX submission #20652 is REQUIRED by this function.

HUNGARIANLINKER link two lists of points based on the hungarian algorithm.

target_indices = HUNGARIANLINKER(source, target) finds for each point in
'source' the closest point in 'target'. These 2 inputs must be arrays
with one point per row, and have their cartesian coordinates in each
column (1D, 2D, 3D, ...). Source to target assignment is based on the
famous hungarian algorithm using its excellent implementation by the
excellent Yi Cao. The two arrays might not have the same number of
points.

The indices of the 'target' points are returned in an array
'target_indices', so that each row in 'source' matches the corresponding
row in 'target(target_indices, :)'.

The linking is exclusive: one source point is linked to at most one
target point, and conversely. The linking is globally optimal: the sum of
the square distance is minimized, contrary to the naive nearest neighbor
approach.

target_indices = HUNGARIANLINKER(source, target, max_distance) adds
a condition on distance. If the nearest neighbor is found to be at a
distance larger than the given 'max_distance', they are not linked, and
the 'target_indices' receive the value -1 for this source point. The same
happens if all target points are exhausted.

[ target_indices target_distances ] = HUNGARIANLINKER(source, target)
additionaly return the distance to the matched target point. Un-matched
source points have a distance value set to NaN.

[ target_indices target_distances unmatched_targets ] =
HUNGARIANLINKER(source, target)
additionaly return the indices of the points in 'target' that have not
been linked.

[ target_indices target_distances unmatched_targets total_cost ] =
HUNGARIANLINKER(source, target)
additionaly return the globally optimized value of the square distance
sum.

The matching algorithm used here is one of the best available and ensures
that the resulting assignment is a optimum. However the price to pay is
an increased complexity. The cost for the naive nearest neighbor approach
roughly scales as O(p^2) where p is the number of source points. The
munkres implementation of the hungarian algorithm by Yi Cao is in O(p^3),
and is the best so far.

EXAMPLE:

n_points = 20;
source = 10 * rand(n_points, 2);
target = source + rand(n_points, 2);
target_indices = hungarianlinker(source, target);
colors = hsv(n_points);
figure
hold on
for i = 1 :n_points
plot(source(i,1), source(i,2), 'o', 'Color', colors(i,:))
plot(target(target_indices(i),1), target(target_indices(i),2), 's', ...
'Color', colors(i,:))
plot( [ source(i,1) target(target_indices(i),1) ] , ...
[ source(i,2) target(target_indices(i),2) ], ...
'Color', colors(i,:))
end


Jean-Yves Tinevez <jeanyves.tinevez@gmail.com>.
However all credits should go to Yi Cao, which did the hard job of
implementing the Munkres algorithm; this file is merely a wrapper for it.

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