linear least
the perpen-
es in all the
58. In fact, it
le to find the
;, in order to
lel edge and
is unknown
presentative
:d 2D model
(1)
id z axis. In
1€ 3D model
ged into the
(3)
el within the
lel in image
ixels. Figure
four images
conditions:
is computed
/hich satisfy
e estimation
1 (1) has the
g procedure
ects will not
e 2 indicates
uncertainty
ints into the
lected edge-
(4)
Babak Ameri
pres . / Ne i NS t
/ ult. / M s .
/ i 4 V ; ‘
i à ^B Y A NN. \ : Y
/ : ‘ FS: i xX [s ^
Dh \ Se vi Ferry ur: e
/ / N a A \ / 1 f* 2 / | f£ pt
x E 7 / / / # in
f E = A x we
Se - / eT \ * X
pe b d Eu
Figure 2: Selected edge-pixels during the first iteration of the estimation process
£x NN P
n /
PS = |
= ^ = \ z > Md J N
~ \ "M Dee, 2 A 55 X
; NS , 7 : , , To T
: : / S 7 / / T.
d E d i
; = e :
| =
Figure 3: Selected edge-pixels during the last iteration of the estimation process
In this formulation, the orthogonal distance e; represents an added error parameter, which acts as a cost function and
should be minimized during the estimation (see figure 4).
p; (X, y;)
+ Pend
€;
Figure 4: Regression of a 2D image edge to the representative edge-pixels
Linearization of the equation (4) with respect to its parameters (d (r,3)» O(r,j)) Tesults in the following formulation:
Of(r j) Of(r ) : ;
LAO jy + A Ady jy — li — ei(zi "9, yi2) ©)
96)9—09 » (7,3) Ödja=a?_ en : 3
where
L = dj) - zime(0) sin 00. + yim cos 0t, iy
For every selected edge-pixel (aime, y;"9) of each 2D edge model €(j,r)» Within every image I,., an equation of type (5)
is inserted into the system of equations. The total system of equations can be written in matrix form as:
2 p-1
Alinear UE cm linear =e 3 TOP linear” (6)
The linear, is the observation vector containing the orthogonal distance between the candidate pixels and their respec-
tive 2D model edge in image space. x is the vector of unknowns consisting of the correction of the edge parameters
(A0,,j, Ad,,;), Atinear is the associated design matrix including derivatives of the observation equations with respect to
the unknowns. The matrix Pjinear; iS the corresponding weight matrix which is introduced as a diagonal matrix and is
determined based on the normalized gradient magnitude of each candidate pixel, and e is a error vector with the following
statistical assumptions:
E(e)=0 E(e*e) = oëP.!
linear”
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 27
