In: Paparoditis N., Pierrot-Deseilligny M.. Mallet C.. Tournaire O. (Eds). IAPRS. Vol. XXXVIII. Part 3A - Saint-Mandé, France. September 1-3. 2010 
121 
comparable result with CHM-based approach. However, the 
main defect of the method is that the point distribution in the 
tree crown is not only related to tree species, but can be also 
affected by many factors such as scan rate, scan direction, scan 
angle, scan pattern and shadow effects from neighbourhood 
trees. 
Under such a situation, there is a reasonable demand in the 
perspective of computer vision that a more generalized way for 
single trees detection from remote sensing data to be developed. 
Perrin et al. (2005) employed marked point process to extract 
tree crown from aerial photo of a plantation. This method 
considered the image as a realization of a marked point process 
and searched for the best configuration in a completely 
stochastic way. The method produced good result from the 
image. However, as the nature of the method ignored any priori 
information can be extracted from the image, the process of 
searching for global optimum takes a long time to converge and 
its computing cost is expensive. Descombes and Pechersky 
(2006) presented a three state Markov Random Field model to 
detect tree crown from aerial image and define the problem as a 
segmentation problem. Although the approach defines a 
template of tree crown and works on a local mask, it actually 
calculates on the pixel level. As well, this model does make use 
of any knowledge can be extracted from the image. 
3. ORGANIZATION OF THE PAPER 
The presented work places the detection of individual trees in a 
stochastic framework using a novel Markov Random Field 
model, but bring it to high level by addressing the difficulties of 
individual tree detection problem in terms of problem 
representation and objective function. In our approach, trees are 
modeled as objects in the image by extracting priori information 
from ALS image. Then a Markov Random Field model is 
defined on the objects. The configuration is updated towards the 
global optimum at which point we get the detected trees. The 
method is particularly interesting for following reasons: 
i. it extracts priori features from ALS data and models 
trees as objects, so the sites in the Markov Random 
Field model are objects, not pixels; 
ii. it allows to introduce a priori knowledge of objects, as 
well as consider likelihood between the represented 
object and image, which is the property and advantage 
offered by a Markov Random Field model; 
iii. it greatly reduces the searching space by modelling 
objects and their relationships at high level and makes 
the computation much less heavy. 
To present our method, we first introduce how we represent 
trees as objects in the ALS data and how the neighbourhood 
system is defined. Then we propose the energy formulation 
based on a data term which measures how features extracted 
from the data support the object as an individual tree, and a 
contextual term which take into consideration some interactions 
between neighbouring objects. The model updating and 
optimization are followed at the third place. Finally, 
experimental results of the method on ALS data acquired from a 
coniferous forest are presented. 
4. THE PROPOSED MODEL 
We introduce in this section about how the proposed Markov 
Field Model is built in three parts: problem representation, 
objective function design and model optimization. 
4.1 Markov Random Field 
Markov Random Field (MRF) was first introduced into 
computer vision community by Geman and Geman (1984). The 
appeal of MRF theory is that it provides a systemic framework 
to encode contextual constrains into the priori probability and 
MRF based approaches has been successfully applied to 
modeling both low level and high level vision problems (Li, 
1994). 
In a probabilistic frame, a random field X is said to be a Markov 
Random Field, if the value of random variable X s G X only 
depends on its local environment through a neighborhood 
system ¥ defined as (Li, 1994): 
s£V(s) 
Vr G S\{s}, s G V(r) »r6 V(s) 
In the measurable space (fl,T, P), MRF model can be described 
by the probability law P(X = x) the event x to be a realization 
of X as: 
Vx G f1, Vs G S, P(A r s = x s |Aj. = x r ,r G 5\{s}) (2) 
= P(X S = x s \X r — x r , r G ¥ s ) 
The Bayesian model is then used to solve the inverse problem 
of how to retrieve the best configuration x given the 
observations D. By Bayesian law, which relates the a priori and 
conditional probability, the a posteriori probability can be 
written as: 
P(X\D) = °c P(D\X)P(X) (3) 
The problem is then converted into maximizing the a posteriori 
probability (MAP) problem: 
x MAP = arg max xen P(X\D) (4) 
Which is equivalent to 
*map = arg (- l °d( P ( D \ X )) ~ logifPW) (5) 
According to the Hammersley-Clifford theorem (Besag, 1974), 
a MRF field and a Gibbs field are equivalent. The a priori 
probability of X can there written as: 
P(X = x) = Z c _1 x e" u ‘W (6) 
Where Z c = * s normalization constant and U c the 
priori energy, or contextual energy as referred to in 4.2. 
Let the conditional probability be expressed also in the 
exponential form: 
P(D\X = x) = Z d ~ l x e" u *W (7) 
Where Z d = Yj X ea e ~ Ud ^ ' s normalization constant and U d the 
likelihood energy, or data energy as referred to in 4.2.