ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision“, Graz, 2002
0.5 (Goudriaan, 1988). Therefore, the probability of an
individual LIDAR pulse reflecting from a specific cell z; in the
grid (and not reflecting from the cells i-1, i-2, ..., 0 that it has
already passed through) will be given by:
gy, | x)= ([Tleswt-t/coss - LAD, (8,2) 42,)]) (5)
k=0
x[1-exp(-1/cos6, - LAD, -G(0,,z,) dz,)]
This function defines a probability density for LIDAR reflection
y», anywhere along a three-dimensional pulse vector f£. In
addition, in our model it is assumed that the probability of a
laser pulse reflecting if it penetrates to within 6 meters of the
terrain (DTM) elevation is 1. In addition, foliage reflectance is
assumed to be constant.
The likelihood function is then given by the following
expression, which represents the joint probability of the LIDAR
data:
iex / cos0- LAD, 66.24) (6)
0
(yx) - [] I
"| x[1-exp(-1/cos@- LAD, -G(0,z,) -dz,)]
The object configuration X that maximizes this function will
represent the maximum likelihood estimate (MLE) of the true
object configuration x. However, given that MLE does not
penalize large numbers of overlapping objects, it is likely that
the MLE will be overly sensitive to the data and therefore will
not represent a realistic forest object configuration. Through a
Bayesian approach, prior knowledge relating to tree
distributions and interactions can be incorporated into the
model through the specification of the prior model, leading to
more accurate estimates of the true object configuration.
3.3.3 Forest object processes: In the Bayesian object
recognition approach, the underlying prior distribution, and the
resulting posterior probability distribution of the true object
configuration given the observed image data, usually takes the
form of a spatial point process, a stochastic geometric model
for an irregular, random pattern of points. These models allow
for inference to be carried out relating to the spatial position of
individual objects as well as the attributes of these individuals.
These models also allow inter-object interaction, as well as
possible global properties of a distribution of objects to be
incorporated into the spatial model (Ripley, 1991).
If we define the environment E(A) of a set A to be the set of
neighbors of points in 4, a point process is a Markov process if
the conditional distribution on 4 given the rest of the process
depends only on the process in E(4). One of the most common
Markov point process models is the pair-wise interaction model,
which has the form:
px) =afT]ex,x) (7)
i<j
where n(x) is the number of points in x and g(x;x;) is an
interaction function (Ripley, 1981). This model therefore places
a constant multiplicative penalty on each pair of interacting
points. This type of model can be used to represent varying radii
of inhibition surrounding biological phenomenon, and therefore
can provide a useful model for forest object processes where
trees exhibit pair-wise interactions.
A marked point process is a point process with a characteristic
(mark) attached to each point in the process. Therefore a
marked point process on RY is a random sequence
x =[s,;m, |where the points § constitute an (unmarked)
point process in A^ and the m are the marks corresponding to
each location s.
In this model, s denotes the location of a tree, while m
represents a vector of object attributes including height, crown
width, crown height, and crown curvature.
Given a probability distribution for the marks, v(m), the prior
model, representing the Markov object process, takes the
following form, where forest object interactions depend upon
the individual tree attributes (marks):
px) a p"? Tv, JI gx) (8)
i«j
In our model, two crowns were considered to be overlapping if
the ratio of the distance between the center of the crowns and
the sum of the crown radii was less than 0.75. The mark
distribution was a multivariate normal distribution, with
parameters determined from stand observations.
3.3.4 Simulation-based posterior inference for the MAP
forest object configuration: In Bayesian analysis, all
inferences are based upon the posterior distribution:
p(x] y)e<l(y|x)p(x). The typical objective of Bayesian
object recognition is to estimate the true configuration of
objects x, given the observed data y. In particular, the maximum
a posteriori (MAP) estimate, representing the mode of the
posterior distribution, is of primary interest in the context of
object recognition.
Within our model formulation, the posterior distribution is also
a Markov object process. Due to the complex nature of the
posterior distribution in this case, posterior inference was
conducted via Markov chain Monte Carlo (MCMC) simulation.
In MCMC, one constructs a Markov chain with an equilibrium
distribution converging to the target distribution (the posterior
distribution in the case of Bayesian inference). Ideally, this
Markov chain should be constructed so as to efficiently move
throughout the set of possible configurations, while maintaining
the correct equilibrium distribution.
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