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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
7 
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Figure 2. Imaging geometry and rectification plane. The side 
view (left) and top view (right). 
Based on the relative orientation between the image plane and 
the rectification plane and camera widening angles, locations of 
the image corner co-ordinates were calculated on the rectified 
image plane. These co-ordinates and traditional least minimum 
square adjustment were utilized to determine eight parameters 
of projective transformation from image plane to plane (Mikhail 
et al, 2001): 
mi at ER (1) 
ex+fy+1 ext ue 
Co 
e (2) 
A - 0 Aixe0cki0l^l rx 
Y 3n. —rvuh 0 — 0 —] P» iy 
A 
us f 
Sı 
82 | 
where X, y = image coordinates, 
X, Y = co-ordinates in rectification plane and 
€9,€,,€5, fy fi J25 8182 7 
unknown parameters. 
After rectification process objects were geometrically correct in 
rectification plane. As the relative orientation of image plane 
and rectification plane were constant for all images, same 
transformation parameters were utilized for all image 
rectifications. 
  
  
Figure 3. Original image (left). distortion free image (middle) 
and rectified image (right). 
3.3 The tree crown dimension measurements 
The parameters for individual trees, such as tree height and 
crown height, were extracted from the images. Reference scale 
bar height and crown width to left and right from trunk in 0.5 m 
elevation intervals were also extracted. The origin for height 
measurements was tree base. The pixel units were transformed 
to metric units based on the known distance in the image, 
reference scale bar, for enabling comparison to other data 
sources. 
  
  
Figure 4. The tree height and crown widths were measured from 
first and second image. 
3.4 The determination of trunk tilting 
In remote sensing aided forest inventory tree trunks are typically 
assumed to be straight. In this study, trunk tilting was estimated. 
The tree top shift dx, that is deviation from vertical line initiated 
from the tree base, in horizontal direction was determined in 
both images. The trunk was represented as a vector (Figure 5). 
Vealtbtck (3) 
where 7 i and k are unit vectors in direction of co-ordinate 
system axes, a and b are unknown parameters and c is vertical 
component of trunk vector. A vertical trunk vector is simplified 
Pack (4) 
where c is the height of the tree. Unknown parameters a and 5 
were solved from equation pair: 
a=aa+ab 
; (5) 
b = bb— ba 
where a and bb are projections of shifts dx/ and dx2 to co- 
ordinate system axis: 
Ix dxl 
meta bbs ee (6) 
cos( B — / 2) cos(@X) 
ab and ba are calculated from angles between image planes and 
parameters a and 5 (Figure 5): 
ab - b-tan(B —z/2) (7) 
ba = a -tan(a)