International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
dp
(df)
(dR),
M
(df)
(dR)
: ; i Tee
The least-squares solution of system (5), minimizing e Ze le ;
is given by
dE
d$
with A » 4, A,]
- -(a (az, 8^ y AJ. 4^ (xz B" y" M" (6)
and
> TA pa 49 2 d
&--X,B(BE,B ) (M +4 a£ a$| ) (7)
Since the original system (3) is not linear, the solution (6)
requires an iterative approach ultimately yielding the best (in
the least-squares sense) estimates for the orientation parameters.
As shown, it rigorously combines prior information (from any
positioning system, e.g., GPS/INS) and geometric relations
between parametric curves in object space and their realization
in SAR image.
4. GENERALIZATION TO FREE-FORM CURVES
In this section we formalize the orientation determination
problem when some elongated objects, represented as free-form
curves are known in object space and can be extracted in the
image. A 3D free-form curve r,, is represented by a sequence
of vertices 1 = (y; . The set of vertices V induces an ordered set
^ 1* J J
of line segments 7- di where segment iJ.) connects the two
vertices |y, and (y,,, (Figure 3).
VN-1
V2 € e © 0-9
e
l4 . © In-1 VN
V4 e e @
Figure 3: Representation of a 3-D free-form curve.
R . . ^
Let Q-zYw, GEI ¥,=0 ) be a partial projection of -
rzi
represented by a disjoint set of W components {,,}- each
comprising a connected set of n,, 2-D pixels in SAR image
wih 1x jxn,. As before, we assume that there is no point-to-
point correspondence between features in object space and their
(partial) projections in SAR image.
Then, the problem is to come up with the parameters that would
describe the relationships between object and image features in
the best (in least-squares sense) way.
First, we select a subset A Qof image pixels that belong to
the projected control feature. Subsequently, given V and A,
together with approximated auxiliary trajectory, we follow a
similar procedure employed for parametric curves, but now
using piecewise linear features in object space. Hence, in each
iteration a temporary association between every pixel in A and
some line segment {/;} with a corresponding segment
parameter t, is established. Note, that the correspondence
between a given pixel location and its associated line segment is
dynamic, and may change from iteration to iteration.
The proposed orientation method with free-form curves is based
on the parametric formalism introduced in the previous section.
There are some important differences, however. As has been
already mentioned the parametric model has been developed for
space curves having first order continuous derivatives. Clearly,
this is not the case for free-form curves with singularities at the
vertices of r f At these singular locations none of the equations
of system (5) that require object space derivatives cag) be
formed. Hence, it is important to discuss how to address these
singular cases when encountered. A simple way to circumvent
this problem is not to estimate the curve parameter / at the
vertices. In this case, the closest point on the corresponding line
segment will be kept fixed, that is, the degree of freedom to
move along the otherwise unique tangent direction is removed.
This solution is plausible in situations where the object space
curve consists of relatively long segments, thus reducing the
chance for the closest point to coincide with a vertex. For the
opposite case, with many short vertices it is recommended to
approximate the set of vertices in the neighborhood of the
closest point by an analytical curve, e.g. cubic spline, to
eliminate singularities. This strategy will allow us to employ the
parametric model developed in the previous section without any
change.
5. SUMMARY AND FUTURE WORK
This paper has reported several preliminary results from an
ongoing R&D research project on registration of airborne and
space-borne SAR images employing feature-based
photogrammetry techniques. In particular, ERS-2 and
RADARSAT space sensors along with some airborne SAR
systems will be studied in the near future. For each particular
sensor, the proposed mathematical model will be examined with
respect to the quantity of linear features available, their shape as
well as their distribution (spatial configuration) within the SAR
image. ;
Apart from being an elegant solution to the problem of
accurately identifying control feature in SAR images, our
proposed stochastic model should yield more accurate estimates
for instantaneous position and velocity vectors, being a crucial
factor particularly for SAR air-borne platforms often being
subject to unstable atmospheric conditions yielding non-smooth
variations in their navigation parameters. Obtaining the same
performance with traditional methods would either require
using highly accurate on-board positioning and navigation
equipment or alternatively an extremely dense distribution of
GCPs across the entire image - practically impossible
requirement for typical SAR mages.
While by no means, the work done so far and reported herein
may be considered complete, this paper has provided the
motivation for employing FBP techniques for orienting and
subsequently registering SAR imagery and by doing that has
paved the way for a new paradigm in SAR processing.
International Arc
Ue
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