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element of A in (1). The two dimensional projective transformation relates the points on one plane to their
correspondingly projected positions on another plane.
84 4X - 242 Y - 843 2 a54X + a99Ÿ + 293
= (2)
8434X + azoY * 843
x Y!
T 844X + ano Y + 833
The CT scan surview however introduces a singular transformation (figure 4) for to distinct lines in the plane there
corresponds one, and only one line, the line on the CT scan surview. Such a transformation cannot be uniquely
inverted for to every picture point there corresponds an infinity of space points.
CT surview image line - x
ES
Y |
|
| 2.
|
|
|
Space ray
| *
Space points
| Space lines
i
[e LO
Figure 4: Illustration of singular transformation
In figure 5 consider the one-dimensional space - i.e. the line E. Line E represents one of the horizontal sections of a
surview. On line E, P has a coordinate x in one dimension. One can interpret X, Y as rectangular coordinates in two
dimensional space, i.e. X, Y in the CT scan slice system, and in this space one can choose a line Y=k parallel to the X
line as the line E. If one joins the point P on E to the origin O by a straight line, then for points on this line X/Y are
constant. If one lets Y=k=1 say, in general one can write:
—=Xx (3)
Accordingly the introduction of homogeneous coordinates signifies the representation of the line E into a space pencil
of rays with the origin O as centre, of which E is a section; i.e. the homogenous coordinates of a point are the space
coordinates of the points of the projecting ray of that point.
Y
AN n i > x LineE
vx] .
vL > X
0
Figure 5: lllustrating homogeneous coordinates
The equations (2) are in terms of X, Y. but for the one-dimensional coordinates x and using the homogenous
coordinates one can rewrite equation (3):
xeu AM Min De ie Alt 7 aY ta (4)
Y' 44X*842Y 843 894X-*8505Y 893 894X*855Y tao
5. TREATING THE AP AND LAT SURVIEWS AS A TWO DIMENSIONAL PROJECTIVE TRANSFORMATION
As the geometry of the central projection pertaining to the surviews only holds true in the XY plane of the CT scan slice
system, the AP and LAT images can be viewed as a whole series of two dimensional projections "stacked together
along the Z axis" (figure 6 ).
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences”, Zurich, March 22-24 1995