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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part Bl. Istanbul 2004
that the nature of the terrain being imaged can be expected to
have virtually no impact upon the metric performance of sensor
orientation based on bias-compensated RPCs. The practical
achievement of sub-pixel ground point determination is also
demonstrated for base-level (most economical) IKONOS Geo
and QuickBird Basic stereo imagery products.
2. BIAS-COMPENSATED RPC BUNDLE
ADJUSTMENT
2.1 The adjustment model
The RPC model provides a direct mapping from 3D object
space coordinates (usually offset normalised latitude, longitude
and height) to 2D image coordinates (usually offset normalised
line and sample values). Here we give only a cursory account of
this model in the form that provides bias-compensation. For a
more comprehensive account of standard rational function
models, as applied to HRSI, the reader is referred to Tao & Hu
(2002), Di et al. (2003) and Grodecki & Dial (2003). For the
present discussion we present the model in the form
F (UVMW)
Dx A ERE iiS
0 i : F,(U,V,W)
F (UN W) (D)
SB BIB s-—————
0 i 2 EUV)
where / and s are line and sample coordinates, and F; are third-
order polynomial functions of object space coordinates U, V
and W. The A4; and B; terms describe image shift and drift effects
and they provide the 'bias-compensation'. Within this model
there are three logical choices of ‘additional parameter’ (AP)
sets to effect the bias correction:
1) Ag Ay, ... By, which describe an affine transformation.
il) Ag, Aj, Bp, Bj, which model shift and drift.
iii) Ay, Bo, which effect an image coordinate translation only.
The solution of the APs in Eq. I can be carried out via a multi-
image bundle adjustment, as developed by Fraser & Hanley
(2003) and Grodecki & Dial (2003). The model of Eq. 1 has
also been referred to as the adjustable RPC model (Ager, 2003).
2.2 Interpretation of orientation: relative and absolute
If we ignore the additional parameters for the moment, then in
the same way as do collinearity equations, Eq. | describes an
imaging ray from object to image space, which we will consider
to belong to a ‘bundle’ of rays (not withstanding the lack of a
true perspective centre). If one imagines that spatial
intersections of all corresponding rays forming the two or more
bundles involved are being determined, then the net outcome is
equivalent to a photogrammetric ‘relative orientation’, which
will also be equivalent to that derived via a rigorous model to
the accuracy tolerance previously mentioned. The reason the
orientation can be thought of as ‘relative’ as opposed to
‘absolute’ lies both in the inherent limitations in directly
determining the true spatial orientation of every scan line, and
in errors within the direct measurement of sensor orientation,
especially attitude, but also position and velocity. Errors in
sensor orientation within HRSI can, fortuitously, be modelled
as biases in image space, primarily due to the very narrow field
of view of the satellite line scanner (approaching a parallel
projection for practical purposes). In the simplest case, a small
25
systematic error in attitude determination is equivalent to a shift
in image space coordinates. But, more than simple translation
may be involved.
The case of shift parameters Ao, Ba alone is one of where,
effectively, there is a shape-invariant transformation of the
relatively oriented assemblage to an accurately, absolutely
oriented model, even if the bias-induced shifts are different for
each image. To effect this absolute orientation, only one GCP is
required. More GCPs will of course enhance precision, but their
number and location is not important. It is hard to see how this
relative-to-absolute orientation process could be influenced by
terrain height or ruggedness, and indeed we will demonstrate
that terrain seems to have no impact on the bias-compensated
RPC approach, or even on the standard RPC forward
intersection.
Time-dependent errors in attitude sensors can give rise to both
‘drift’ effects in the image coordinates and an affine distortion
of the image. More subtle, higher-order residual distortions, for
example in gyro systems and in scan velocity, may also be
present, but we will keep the error compensation model at first
order. Thus, in the case of the full affine correction model (Case
i) and the shift-and-drift model (Case ii), the relative-versus-
absolute orientation situation is slightly different, at least when
the parameter sets A, and B,, or Aj, Bi, A; and B, are
statistically significant. In these cases the absolute orientation
process does imply a modification. of the RPC relative
orientation (Eq. 1 without the four APs with subscripts 1 and 2),
especially since a non-conformal transformation of image
coordinates ocurs. Here, the number and location of GCPs is
important, with a practical minimum number being 4-6.
As will be seen, however, the parameters A, A,, B, and B, are
rarely significant with IKONOS Reverse scanned imagery. This
means that with such imagery we need only worry about
providing one GCP to compensate for the shifts A, and B,.
With QuickBird imagery on the otherhand, the authors’
experience suggests that the shift-and-drift and affine AP
models can in cases lead to measurable improvements in the
accuracy of sensor orientation and geopositioning (eg Noguchi
et al., 2004). Thus, there is a very slight prospect of the nature
of the scene topography influencing ground feature point
determination since the relative-to-absolute orientation process
does not constitute a shape-invariant transformation.
2.3 Regenerating RPCs corrected for bias
The ability to determine the bias parameters A, and By. is very
useful, but of more utility is incorporation of a correction for
the bias into the originally supplied RPCs. This allows bias-free
application of RPC-positioning without reference to additional
correction terms. This bias compensation is very
straightforward, as shown in Hanley et al. (2002) and Fraser &
Hanley (2003). Bias-corrected RPCs, incorporating shift terms
only in this case, are generated by carrying out the following
corrections to the two numerator terms in Eq. 1: the
denominator terms remain unchanged:
4
= ry; 5 A y A V p
FU JW) =(a, —b,4)+(a, —b,4,) |] teda =byy b W
3
ALI MAC. d Br rt Wr
FU W)=(c, d,B,)+(e, dB) I TH (Cop dpBo) HW
Here, à;, b;, c; and d; are the RPC terms forming F,, F>, F; and
F,, respectively.
