2004
fined
rapid
s or
r the
ich as
equal
"ward
have
form
used.
af the
;, and
RPC
Space
ment.
)01a),
of the
natrix
ciated
is the
ration
is the
using
enario
s as a
t grid
e grid
nd its
our or
vation
| point
of the
ecified
least-
nd the
1eeded
well
roduce
order
1d the
optical
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
projection more closely (Hartley and Saxena, 1997). This would
also be useful in estimating the approximate object coordinates
using only the low-order terms in 3-D reconstruction. The RFM
determined this way is proved to be able to achieve a very high
approximating accuracy to original physical sensor models. It is
reported that the RPC model yields a worst-case error below
0.04 pixel for Ikonos imagery compared with its rigorous sensor
model under all possible acquisition conditions (Grodecki and
Dial, 2001). Therefore, when the RFM is used for imagery
exploitation, the achievable accuracy is virtually equivalent to
the accuracy of the original physical sensor model. This terrain-
independent computational scenario makes the RFM a perfect
and safe replacement to the physical sensor models, and has
been widely used to determine the RFCs.
2.2 Terrain-dependent Approach
For the terrain-dependent scenario, the RFM tries to
approximate the complicated imaging geometry across the
image scene using its plentiful polynomial terms without
establishing the girds, and the solution is highly dependent on
the actual terrain relief, the distribution and the number of
GCPs. GCPs on the 2.5-D terrain surface have to be collected
by the conventional ways (e.g. measured on topographical maps
or by GPS, and on aerial or satellite imagery). The iterative
least-squares solution with regularization is then used to solve
for the RFCs. In this context, the RFM behaves as a rubber-
sheeting model, and the over-parameterisation may cause the
design matrix of the normal equations to become almost rank
deficient because of the complex correlations among RFCs. The
regularization technique improves the condition of the design
matrix, and thus avoids numerical instability in the least-squares
adjustment. There are also many experiments carried on using
frame, pushbroom or SAR images to assess the approximating
ability of the RFM obtained in this manner, and the accuracy is
high provided that a large number (for instance, as twice as the
minimum number of GCPs required to obtain a close-form
solution) of evenly distributed GCPs are collected across the
whole scene. Nevertheless, the terrain-dependent approach may
not provide a sufficiently accurate and robust solution if the
above requirements for control information are not satisfied.
Therefore, the RFM solved by terrain-dependent approach may
not be used as a replacement sensor model if high accuracy is
required (Toutin and Cheng, 2000; Tao and Hu, 2001a, b).
3. RFM REFINING METHODS
As proved by its high approximating accuracy to many physical
sensor models, the RFM has high capability of geometric
interpolation. However, the RPCs provided by imagery vendors
may not always approximate the real imaging process well. The
requirements for control information may be not met
satisfactorily sometimes, or no ground control information is
used when determining the physical sensor model itself for
different marketing strategies from imagery vendors. High
precision products are sold at a significantly higher price, and
even require that users provide GCPs and a DTM. This presents
a problem for many users who are prohibited to release
topographic data this way.
Recent studies have found that RPCs can be refined in the
domain of the image space or of the ground space, when
additional control information becomes available. For example,
the Ikonos Geo products and Standard stereo products will be
improved to sub-meter absolute positioning accuracy using one
665
or more high quality GCPs (Grodecki and Dial, 2003; Fraser et
al, 2003; Tao and Hu, 2004) or be close to the accuracy of the
GCPs whose quality is low (Hu and Tao, 2002; Tao et al. 2003).
So the RFM refining methods will definitely promote the use of
low pricing products for many applications.
The RFM may be refined directly or indirectly. The direct
refining methods update the original RPCs themselves. So the
updated RPCs can be transferred without the need for changing
the existing image transfer format. While the indirect refining
introduces complementary or concatenated transformations in
image or object space, and they do not change the original
RPCs directly. The affine transformation or a translation for the
simplest case is often used. In addition, tie points can be
measured on multiple images, and their models may be refined
resulting better relative orientation after block adjustment. In
essence, the two-step procedure of the in-direct refinement can
be combined into one by recalculate the RPCs with a pair of 3-
D object grid and 2-D image grid established for each image.
3.1 Direct Refining Methods
The RPCs can be recomputed using the batch method (called
RPC-BU) when both the original and the additional GCPs are
available (Hu and Tao, 2002; Di et al., 2003). Here, the original
GCPs refer to those used to compute the existing RPCs,
whereas the additional GCPs are independently collected and
are not used to solve the initial RPC values. The refinement is
fulfilled by incorporating all of the GCPs into the RPCs
solution, with both the original and new GCPs appropriately
weighted. The values 'of the existing RPCs may be used as the
initial solution /, to speedup the convergence in Eq. 4. While
only the new GCPs are available, the existing RPCs can be
updated using an incremental method (called RPC-IU) based on
the Kalman filtering or sequential least-squares (Hu and Tao,
2002; Bang et al., 2003). That is, the RPCs are corrected by
adding weighted residuals from the new measurements. The
corrections of the approximate RPCs /;” are given by
AL = PAT ATL LY + RY (GT LY fori=127(5)
where P, is the covariance matrix associated with /,; 7, is the
design matrix made from new GCPs; A, is the covariance matrix
associated with the image pixel coordinates G, of new GCPs.
The error propagations on both the RPCs and the new GCPs are
recorded during the updating process. Satisfactory accuracy
improvements can be expected in both image domain and object
domain when the covariance matrix P, is known.
3.2 In-direct Refining Methods
When the RPCs are fit to a physical sensor model whose
orientation parameters are derived from satellite ephemeris and
attitude information without requiring the use of GCPs, mainly
linear systematic errors exist. To refine the forward RFM, it is
more suitable by appending a simple complementary
transformation in image space at the right side of Eq. 1 to
eliminate the error sources. For narrow field-of-view CCD
instruments with a priori orientation data, these physical effects
mainly behave like a same net effect of displacements in line
and sample directions in image plane in total. Fraser and Hanley
(2003) used two bias parameters to compensate the lateral shift
of the sensor platform in two orthogonal directions under the
assumption that the biases manifest themselves for all practical
purposes as image coordinate perturbations. Grodecki and Dial
(2003) proposed a comprehensive block adjustment math model
(called RPC-BA). The formulation uses two complementary
