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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