XXXV, Part B3. Istanbul 2004
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol .
original curve as
(10) enlargement of a |
indicated by 'o' are
the computed hom
Note that equation (9) shows that in fact two homography od (v,— X)
matrices provide sufficient set of linear equations for the ar RE
fundamental matrix. xp 12
Odi c x, X)
3.3 Finding the homography matrix using ICP
Oyp Ly
In this section we describe procedure to find the homography
matrices of free form planar curves using ICP algorithm. This
time the ICP algorithm is carried out in image plane. The
chosen image is the first (left) image of the photogrammetric
model. Every curve in the right image that was digitized and
matched to curve from the left image is transformed to the left
image using initial homography matrix. The initial component
where: L,, is the distance between points 1,2 of the closest
segment from left image curve.
Now for the derivatives of equation (8) for hl..h8 we rewrite
equation (8) as:
of the homography can be computed using regular assumption Nx
for initial relative orientation parameter and applying equation Xp = D
(6). One possible value set for initial components of H is (11)
simply: Ny
; 1.5 ypzci-
Ho = {1,0,0,0,1,0,0,0,1}. After we determine the initial i D
component of the homography matrix we transform the curve and:
from right image plane to the left image plane and compute the
closest points between the two corresponds curves. The
registration is computed using least square adjustment and = EH
Levenberg-Marquardt (Fitzgibbon 2001) algorithm is tested as Oh, ONy Oh oD Oh,
well. ap _ oyp ONy | Oyp OD
The proposed algorithm is sketched as follows: utu UE. I
Oxp _ Oxp ONy Oxp OD
(12)
Oh ONy Oh, OD Oh, Fi
e find the corresponding curves in both images
e set the initial homography matrix to SO
H={1,0,0,0,1,0,0,0,1}.
e Transform the curve from right image to left Gd Mou mf ms
image using the initial matrix. dem . xr e
e Find the closest points between the curves. Oh, L, D "
e Compute the design matrix A and the error Qd. ioc ri Ja s ot
vector e for the current components of the Tou n RT
homography matrix. 2 12 =
ed Y; y, di LA
e Compute the update vector x for the current
components. ;
e — Update the homography matrix H=H + x : rr
e Continue to transfer the curve and update the Od | X =X 1, (13)
homography matrix until maximum distance oh, Ls D Fiet
between closest points smaller than tolerance. ; e
od be er
Compared to the algorithm presented in the previous section, oh, La D » m EN S we can
the current procedure reduces the dimension of the problem ad x f ge using initial |
from 3D to 2D. In addition, here, only one transformation from EEE AS)
right image plane to left image plane is required. Oh, Lis D 1)
; ; Od v,~y, —f Nx KK ANY |
3.4 Design matrix for registration zn EET EX 100
eh, Le D» Lio p:
The error is actually the distance computed between the closest od yay, uen Fox 3 amer fo Ny m
points obtained. The optimal situation is that all the distances oh, = Tae Cp L ug d
are zero. The design matrix is computed by the taking 8 12 12 60
derivatives of the distance function with respect to any
component of the homography matrix. The distance is where: xr, yr are the right image coordinates of the specific of\
computed between every point that was transformed from right point. xl \ me
to left image and the closest point to it from the correspond % ies
curve. When the curve is a free form curve then the distance is 3.5 Experiments el NC
actually computed between the point and the closest segment to N
the point. The segment from the left image curve is for now The proposed procedure has been tested on synthetic data anl x.
considered as fixed. showing good converging between the curves as shown in E
The derivative of equation (2) for xp and yp (the transformed figure 6. The initial homography matrix component for this n. tsi
experiment was H-11,0,90/f,0,1,0,0,0,17 where f is the focal i 20 4
point from right image) is:
10
length and the value 90 is given to keep the model scale close to
the image scale. Figure 4 describes the synthetic images of the
curves. For example, one of the curves is transformed from right
image to the left one using initial values for the homography
matrix. After two iteration of ICP algorithm we get close to the
Figure 5. Cui
