.. (iv) 
at surface 
across the 
given by 
a planar 
.. (V) 
Z in the X 
define the 
ad in the 
| now be 
.. (vi)a 
.. (Vi)b 
derived 
are 
jns (vi)a 
function 
.. (Vii) 
1 in terms 
e surface 
Z, (0Z/0X) 
el of the 
th the left 
Jhbouring 
FAy,), its 
(p; Yn*^ 
del. Thus 
jhbouring 
... (viii) 
yield :- 
ps 
.. (ix) 
   
where the superscript ? indicates a priori estimate and 
dxp, dyp, dAxm and dAyg are the corrections to the a 
priori value. Estimates of xm" and ym can be obtained 
from a suitable method, such as feature detection or even 
manual selection, and estimates of Axp? and AyR® are 
computed using provisional values of Xo, Yo, Zo, 
(oZ/oX), (0Z/oY) and the planar surface model. The terms 
olp/axp are olR/0YR the gradients of the intensities in the 
x and y directions across the right image. 
Looking at eqn (ix), it should be noted that corrections 
dxp and dypg are the terms that are sought in the 
solution. In order to use the surface model in the 
matching process, a relationship for the terms dAxR and 
dAyR is needed. Consider the term AxR, which can be 
expressed as :- 
AXR = (OxR/0X)AX + (OxR/0Y)AY + (OxR/0Z)AZ e (X) 
Substituting eqns (vi)a, (vi)b and (vii) into egn (x) and 
replacing (oZ/0X) and (oZ/oY) by G and H respectively 
will give :- 
422. 3Y AX zx, OY | … (x) 
Equation (xi) expresses AxR in terms of the known shifts 
AXL. , AyL on the left image, the partial derivatives of the 
collinearity equations (calculated using provisional values 
of xn, yn. and computed Xo, Yo, Zo) and the terms G, 
H as obtained from the planar surface model. As such, 
only G and H are not known and to be solved, hence :- 
dAxg = 2*8 qq, 9*8 qu. ... (xii) 
2G oH 
The partial derivatives (dAxR/0G) and (odAxr/0H) are 
obtainable from eqn(xi), which are :- 
  
JAXg | dxg OX oxg OX i 
X e BL : 
2G E e az | (diga 
0AXR OXR oY OXR oY Yir 
bus pov .. (xiii)b 
oH Jm ex e Sz ay, | X 
Similarly, the relationships for Aym and dAyg are :- 
BE QX ay, AY oy 
ah „Na X 1 
oZ OX 0Z dy, 
Da 4 OR OV oY m 
RTT Ty, ju 
557 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
while, the partial derivatives are given by :- 
= fe es À — ÁÀ s 
oG 3 C eL (je 
RE an ays on 5 
oH 3$ TE HY pa 
Partial derivatives (OxR/0Xa), (0yR/0Xo), (OxR/0Yo), 
(0yR/0Yo), (OXR/0Zo) and (oyr/0Zo) can be obtained from 
the collinearity equation and their form is well 
documented in most photogrammetric books. 
Substituting eqns(xiii) and (xv) into the eqn (ix), the 
linearised observation equation would be :- 
Ip (X_+AXL,YL+AYL) + nGc-Ax| y Ay, ) = 
Ip (xR%+axR® yR®+AyR®) 
E al 
On. d Rd 
E «De v 
+ 
[lg OAXR . Olr OAYR 
Pau MR LOIR. dG 
Hoe 36 Nu 36 
i 
+ SR 20 La … (xvi) 
  
OXR oH YR oH 
Equation (xvi) is non-linear in dxR, dyR, dG and dH, thus 
an iterative least squares solution is needed to solved for 
the corrections. 
2.2 Computational Steps 
The steps needed in evaluating the coefficients can be 
summarised as follows :- 
(a) Select a window of n x n pixels in the left image 
and let the coordinates of the central point be 
(XL, yL). The initial estimate of corresponding 
position of the central point (xm, yR) is then 
obtained. 
(b) Using these coordinates and the relative 
orientation — parameters, the corresponding 
(provisional) object coordinates, Xo, Yo, Zo, are 
computed. 
(c) Determine the shift Ax; and Ay, so as to represent 
the position of a neighbouring point with respect 
to (xj, yL). This is followed by computing the 
partial derivatives for the planar surface model 
and subsequently AX, AY and AZ, as shown in 
eqns (vi)a, (vi)b and (vii), are evaluated at this 
position. 
(d) With the information obtained in (c) the values of 
AXR and Aym are calculated using eqns (xi) and 
(xiv), thus yielding the corresponding coordinates 
of xL',yL' on the right image, i.e. xn, yR'. 
(e) The values of xm, ym' obtained in (d) would 
enable the computation of the partial derivatives 
OXR/0X, dyR/0X, dyYR/0X, dyR/0Y, OXR/0Z and