ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision“, Graz, 2002 
  
laser radar data used. The intent is to speed up the algorithm 
and reduce the size of memory needed. 
Since the data is very dense, more than 10 points per square 
meter, the data set is re sampled into a regular grid. This is 
not such a good idea when dealing with more sparse data 
sets. The re sampling of the data is done in the simplest way 
possible, using the lowest point in each mesh. This is 
preferable instead of make any interpolation between near 
points. In a forest area a single grid mesh often contains both 
measurements belonging to the trees and to the ground, in 
such a case it is better to use the lowest measured point rather 
than some mean between the ground and the tree points. Grid 
points with no data are left blank, no interpolation is done 
between neighbour points. 
From this re sampling of the data a two dimensional height 
image is received, (x, y). The active shape model is 
represented by a height matrix, v(x > y) , With the same size 
as I. 
E from (1) adds stiffness to the model In this 
implementation Z;, is a function of the first derivative of 
the model, giving it elasticity but not rigidity. The result is a 
model able to form sharp edges, from this the model is more 
to be liken with a net of rubber bands rather than splines. 
Wanting the model to bend down steep slopes but still stretch 
out flat under buildings and other objects, the function E;nt 
is non linear. 
x+1 y 
Ei (x,y)=C Y_ Y |arctan(v(x, y) -v(m,n)) (2) 
mz-x-lnzy-l 
This has of course its minimum when the model forms a 
plane where v(x, y)sv. C is the elasticity constant and 
define how hard the model is to form. By adjusting this 
constant C it is possible to control how rough the model will 
turn out. 
Ej, from (1) is a function of the distance between / and v. 
Ej, (x, y) 2 -Ae 4C) 0» = 
This function makes the model attach to / , having the 
minimum energy when v=/. The constants 4 and a controls 
the attraction strength and attraction length of the model. 
This function attract the model with a strong force but on a 
short range, i.e. either the model is unaffected by / in a point 
or v is glued to it. In Figure 3 the principle of the behaviour 
of the model due to this is shown. The strong attraction force 
makes the model attach to the ground points once the model 
is close enough. Points belonging to measurements in the 
tree, leave the model unaffected. The internal elasticity of the 
model produces forces too weak to pull the model away from 
already attached nodes but are strong enough to stretch the 
surface where it is not influenced of the image attraction. 
A- 116 
; Measured terrain points, 
8 TM 
+ ;opEye “ data. 
   
    
The optimized 
active contour. 
Figure 3. A section of the model. The measured points from 
the tree is too far from the model to attract it, the elasticity of 
the model streches it between the ground points. 
The minimization of (1) is done in an iterative process where 
E(v) decreases by updating the nodes of the model surface in 
small steps until a local minimum is found. In each node of 
the model the new Z value that minimizes the energy 
function is calculated. When the model then is updated, these 
values no longer minimize the energy function because the 
value of the energy function in a point is dependent on the 
neighbour nodes. In the next iterations new Z values are 
calculated until the process converge towards the final 
solution. To speed up this process, the iterations are stopped 
when the maximum step size of the model is less than 5 cm. 
The start value of v is a plane under the lowest point in /. To 
speed up the algorithm , an external component of the energy 
function is added initially 
E a (X V) = —Gv(x, y) (4) 
This rises the model guiding it close towards the ground 
enough to let the function Æ,, attract the model. When a 
local minimum of the energy function is found, £,,, causes 
the model to bubble up in areas where the model has not 
been attracted to the height image. This is obvious under 
large objects, as buildings, where the model is forming a 
cushion instead of being stretched out. The simple solution of 
this problem is to turn the external component off and run 
the algorithm again. This only takes a few iterations since the 
model now is close to the solution of (1).