* INTRODUCTION

This is a very simple implementation of a variant of KSP applied to
multi-target tracking dubbed "Multi-Tracked Paths" (MTP).

It works with negative edge length and stops when it can not find any
path of negative total length, instead of fixing the total number of
paths to a constant K.

* INSTALLATION

This source code should compile with any C++ compiler. Just execute

  make
  ./mtp_example

It will create a synthetic dummy example, save its description in
tracker.dat, and print the optimal detected trajectories.

If you now execute

  ./mtp tracker.dat

It will load the tracker.dat example, run the detection, save the
detected trajectories in result.trj, and the underlying graph with
occupied edges in graph.dot. You can produce a pdf from the latter
with the dot command from graphviz:

  dot < graph.dot -T pdf -o graph.pdf

* IMPLEMENTATION

The two main classes are MTPGraph and Tracker.

The MTPGraph class stores a directed acyclic graph (DAG), with a
length for each edge -- which can be negative -- and can compute the
family of paths in this graph that minimizes the sum of edge lengths.

This means that it will iteratively add paths as long as it can find
some with negative length. If there are no such path, it will compute
no path at all. Note that the solution it finds is globally
optimal. Note that the procedure is similar to that of KSP, in the
sense that the family it computes eventually is globally optimal, even
if the procedure is iterative.

The Tracker class allows

 (1) to define a spatial topology composed of

     - a number of locations
     - the allowed motions between them (i.e. a Boolean flag for each
       pair of locations)
     - the entrances (a Boolean flag for each location)
     - the exits (a Boolean flag for each location)

 (2) to define a number of time steps

 (3) to set for every location and time a detection score, which
 should be equal to log(P(Y = 1 | X)/P(Y = 0 | X)) where Y stands for
 the location occupancy and X for the observations.

From this setting, it computes the best set of disjoint trajectories
consistent with the topology, which maximizes the overall detection
score (i.e. the sum of the detection scores of the nodes visited by
the trajectories)

The Tracker class uses the MTPGraph. From the definition of the
spatial topology, it builds a graph with one source, one sink, and two
nodes per location and time. This structure ensures the trajectories
computed by the tracker to be node-disjoint by forcing the paths
computed by the MTPGraph to be edge-disjoint. The edges from the
source or to the sink, or between these pairs, are of length zero, and
the edge between the two nodes of such a pair has a length equal to
the opposite of the detection score.

The file mtp.cc gives a very simple usage example of the Tracker
class.

--
François Fleuret
August 2012