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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004
vector f, -Q, - Q, can be calculated, which corresponds to
the rock glacier flow between these two epochs.
The combined system offers the possibility to introduce
additional constraints between the object points Q, and Q;.
Constraints controlling direction and/or length of the flow
vector can therefore be directly integrated into the matching
process. One implementation of such a flow constraint is
described in Section 2.2.3.
2.2 Modifications of the standard MPCM algorithm
In this Section the standard MPCM algorithm (using central
perspective images) will be adapted for direct matching in
POPs. First, the well-known collinearity equations are replaced
by a modified function, which describes the projection of an
arbitrary point Q in object space into a POP (next Section).
Furthermore, a linearized form of this function will be derived,
which allows geometric constraints to be formulated in the same
way as in the standard MPCM (Section 2.2.2).
2.2.1 Modified collinearity equations
The projection ray p; is defined by the projection center C; and a
given object point Q. Intersecting this ray with the rough DTM
yields the intersection point D'. This point must be projected
into the ground plane to obtain the corresponding point P'; in
the image POP; (see Figure 2).
Instead of the strict, iterative DTM intersection a more coarse
method will be used: At an approximate position (D;) on the
rough DTM the normal vector n; is calculated, defining the
tangential plane in this point. Intersecting the projection ray
with this plane gives only an approximate solution, but has two
advantages:
* Direct and fast computation of the intersection point.
* The projection of an object point into a POP can be
given in closed formulas.
Using this method, the calculation of the intersection point is
straightforward:
À, _(P,)-C,)n; (1)
p;:n;
D,-C,-A,p, with
where p; - Q-C,... projection ray.
The origin of the image coordinate system of the POP is located
at position X? ^" in the object coordinate system (x-axis parallel
to the X-axis, y-axis opposite to the Y-axis, pixel size 4X and
AY). Then Equation (2) can be used to transform the planimetric
X-and-Y-cootdinates of D'; into the image coordinates Xd of
the projected point P", :
= t€ (X - ror = x
= (pr, X, yx eG; (2)
; Y POP ^
Yo = -(p uc Ys Jay EG
895
Equation (2) permits the computing of the projected point in the
POP associated with C; from a given object point Q. This
function thus represents the modified collinearity equation and
will be called G; and Gj, respectively, for the x- and y-
component of the projected point.
In the MPCM adjustment the linearized form of (2) is needed,
which relates small changes in the object point coordinates to a
translation of the projected point in the POP:
dx. = 8er at + Sc dr + ac: : dz
er cox oY 0Z (3)
¥ J G 3 N77 x , S
din TEX eO. SU ou
2 OX oY 0Z
In order to derive the partial derivatives of functions G;* and
G/, a linearization of Equation (1) with respect to the object
point Q has to be done:
dD = dA, pP; + À, : dp; = À, . dp; An) (4)
(n, : P; )
dX
where dp, =dQ =| dY |... translation of the object point.
dz
From Equation (4) the first and second component D'* and D'/'
can be extracted and inserted into the linearized form of
Equation (2). Replacing the term dp; by dQ and sorting for
parameters dX. dY and dZ, respectively, gives the partial
derivatives of the modified collinearity equations, listed
explicitly in Equation (5):
0G; c À, fx n? : s ; 06; = À 7H : p?
9XO7 AXAC (np BY SAX n.p.)
HE st 0 e (ME A. am
02 ALL hap) ox. AY (n.- p.)
SS 9G, À, nf. p!
LN M.
ày - AYl(n.-p.) 97 AY! (n,-p,)
2.2.2 Epipolar constraint for POPs
Epipolar geometry can also be used to constrain matching in
POPs. The situation with POPs is more complex, however,
because of the DTM involved. In general, the epipolar line in
the POP will no longer be a straight line but a general curve.
This can be seen by intersecting the epipolar plane with the
rough DTM. The projection of this intersection curve into the
ground plane yields an "epipolar curve" in the POP. The
tangent line at a certain point on this curve therefore
corresponds to the epipolar line of a perspective image.
