Wolfgang Förstner
from which two contraints can be derived
Ar) X=0 B(y)Xz0 . wih A(r)-2z3-1zw3-wl Bíy') z2-—y'3z w2 -—'3
The planes A and B pass through the image point and the projection centre and span the image ray as ANB =
(u'3 — w'1) n (w'2 — v'3) — -w'(u'20 3 &- v 30 1 9 w'102) uingg C0 D = -DNC,VC, D, the factor
—w' X 0 has no effect. A similar derivation of observation equations and constraints can be performed for observed lines.
4.2 ORIENTATION OF ONE IMAGE
The projection matrix P can easily be determined, if n > 6 corresponding points x; and X; are given. Its row vectors can
be collected in the 12-vector u! — (17,27, 37) leading to
1
zGGXO-(0X)V f o-X] 05 3X] x i m
( y (3-X;)— (2X;) /” of —X! yx] 2 zo a" eno YES he
* An estimate for u is the adequately normalized eigenvector ü corresponding to the smallest eigenvalue of the matrix
C — (C,). leading to an estimated projection matrix
—
P-(123)' - (Dia) - (KRB| - KRX,)
This requires at least n > 6 image points to be given. It can be partitioned into a left 3 x 3-matrix D and a right 3 x 1-vektor
d from which the parameters of the exterior and the interior orientation can be directly computed in four steps:
s eT XU spr
1 X.--D'ad 2. KK =DD' using a Choleski partitioning 3. R=K D 4. K-K/Ks;
The normalization of the calibration matrix is necessary only if one wants to interprete its entries. Due to the generality of
the model this procedure is much simpler than the one given in (BOPP & KRAUS 1978). It only works in case the points
are not coplanar and do not sit on an algebraic curve of third order (FAUGERAS 1993).
A similar estimation procedure can be developed for observed lines, leading to P from which the rows of P can be
determined by joining the corresponding principle rays, e. g. 1-2^3-3n1A1n2.
5 TWO IMAGES
We model the geometry of two images using the projection matrices
P,-(12,3)! -KiRi(IO PP» (4,5,6!' - K&R;(I| - T)
thus putting the origin of the object coordinate system into the first projection centre. We will also use the reduced image
coordinates
ke! = RK 'x’ ky! = BG: x!
which represent the intersections of the image rays with two normalized cameras looking downwards, having horizontal
image planes, and principle distance c; = ca, — 1.
We do not need to model the projection of 3D lines as they do not contribute to the orientation of an image pair.
5.1 EPIPLOAR LINES AND COPLANARITY
For each object point the two projecting lines L' and L" need to intersect which can be expressed in two ways as a function
of the rows in the projection matrices (cf. Fig. 1)
L'nL"- (w2n3-4v'3n14w'1n2)n(u'5n6--v"604-4 w'An5) - (A, (x')NB1(y"))N(A2(2")NB2(y")) = 0
This coplanarity condition is linear in all image coordinates and can be expressed as
T |2.3;5,6| |2,3;6,4| 12,
x Fx"z with F= |3,1;5,6| |3,1;6,4| 3,
|1.2;,5.6| |1,2,6,4| I1
1 ,
300 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
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