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ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision“, Graz, 2002
In this case, proposed moves for the Markov chain include 1)
change of object parameters, 2) birth of an object, 3) death of an
object, 4) merging of two objects, and 5) splitting a single
object into two objects. Due to the change of dimension when
proposing to add, delete, merge, or split objects in the
configuration, the Metropolis-Hastings-Green reversible jump
MCMC algorithm was used to maintain the correct equilibrium
distribution (Green, 1995; Hurn and Syversveen, 1998). The
change of parameters is achieved by drawing a sample from a
multivariate normal distribution centered on the parameter
vector for a selected object. The birth of an object is carried out
by drawing a sample from the multivariate normal mark
distribution for tree objects.
3.3.5 MAP estimation via simulated annealing: Finding the
object configuration that maximizes the posterior probability
(i.e. MAP estimate) is essentially a combinatorial optimization
problem. Simulated annealing, an optimization technique with
its origins in statistical mechanics, provides a means of arriving
at the global optimum for a given system through a process of
first “melting” the system at a high temperature, then gradually
lowering the temperature until the system freezes at the optimal
configuration (Kirkpatrick et al, 1983). In the context of
Bayesian object recognition, it has been shown that samples
obtained, via MCMC, from the tempered posterior distribution
[p(x] »L 7 will converge to the MAP solution as 7 — 0 (van
Lieshout, 1994). In our algorithm, an annealing schedule of
FH. = A-H, was used, where H is the temperature at iteration
n+1
n, and X is the cooling rate.
4. EXAMPLE
The algorithm was run on a 0.21 ha area within a lightly thinned
unit of the Capitol State Forest (see Figure 3). The interaction
parameter used in the prior distribution was set to e”°, which
places a moderately heavy penalty on severely overlapping tree
crowns. The intensity parameter for the object process, 3, was
also set to e? .
For this example, 133900 iterations of the
MCMC algorithm were run with the cooling rate, ^, set to
0.999975, and initial temperature (H) of 20. The MCMC
algorithm started with zero objects.
Figure 3. Location of example area (delineated in white) within
area of Capitol State Forest, WA.
The locations, heights, and tree crown dimensions
corresponding to the MAP estimate of the true object
configuration within this area are shown in Figure 4,
superimposed on the three-dimensional scatter plot of the
LIDAR data and terrain model. A two-dimensional
representation of the MAP estimate and LIDAR data is shown
in Figure 5.
Three-dimensional perspective view of MAP
estimate of tree locations and crown dimensions
superimposed on LIDAR data. LIDAR pulse
footprints are drawn to scale and are color-coded by
elevation.
Figure 4.
1.72740%10°
1.72730x107
172710x105 —
Figure 5. Planimetric view of MAP estimate of crown
configuration (black circles) superimposed on
LIDAR data. LIDAR pulse footprints are drawn to
scale and are color-coded by elevation.
The estimates obtained from the object recognition algorithm
were compared to photogrammetric measurements of crown
locations made from large-scale (1:7000) aerial photography
within a 0.21 ha area (see Figure 6).
