International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
polynomials that are adjustable to model the effects originated
from the uncertainty of spacecraft telemetry and geometric
properties of the Ikonos sensor. The first-order polynomials A7
and As are defined by
Al-7l'-I-agy*aplta,s (6)
As=s'-s=by+b;1+b,s
where (A/, As) express the discrepancies between the measured
line and sample coordinates (/’, s’) and the RFM projected
coordinates (/, s) of a GCP or tie point; the coefficients a, aj,
d, by, by b, are the adjustment parameters for each image.
Grodecki and Dial (2003) indicated that each of the polynomial
coefficients has physical significance for Ikonos products, and
thus the RPC-BA model does not present the numerical
instability problem. In detail, the constant a, (b,) absorbs all in-
track (cross-track) errors causing offsets in the line (sample)
direction, including in-track (along-track) ephemeris error,
satellite pitch (roll) attitude error, and the line (sample)
component of principal point and detector position errors.
Because the line direction is equivalent to time, parameters a;
and b, absorb the small effects due to gyro drift during the
imaging scan. Tests shows that the gyro drift during imaging
scan turn out to be neglectable for image strips shorter than 50
km. Parameters a, and b, absorb radial ephemeris error, and
interior orientation errors such as focal length and lens
distortion errors. These errors are also negligible for Ikonos.
Thus, for an Ikonos image shorter than 50 km, the adjustment
model becomes simply A/- a, and As 7 b,, where a, and b, are
bias parameters used in Fraser and Hanley (2003). The
correction vector to the approximate values of both the model
parameters and the object point coordinates is given by Eq. 7
(Grodecki and Dial, 2003), where A is the design matrix of the
block adjustment equations; w is the vector of misclosures for
model parameters; C,. is the covariance matrix.
A= (ATCA IAC Ww (7)
The concatenated transformations also introduce additional
parameters (e.g., of polynomials) in either image space or object
space. They try to improve the positioning accuracy by fitting
the RFM calculated coordinates to the measured coordinates of
new GCPs. Thus, the ground-to-image transformation becomes
a concatenated transformation with the original forward RFM
transform as the first step and the additional transform (e.g.,
polynomials) as the second step. Because the forward RFM is
more used in industry, it is straightforward to apply an
additional transformation in image space. The 2-D affine
transformation in image space (called RPC-CT), i.e.,
I'=apt+a;-l+as (8)
s’=by+b;1+by's
are tested in Bang et al. (2003) and Tao et al. (2004). It is
observed that the values of a, and 5; are always close to 1, and
a» b, close to 0 when refining the Ikonos and QuickBird
images. Di et al. (2003) used polynomials in ground space. The
known RFMs of two or more images are employed to intersect
the ground coordinates of object points from their measured
conjugate image points. Then the intersected ground
coordinates are fit to the measured ground coordinates of GCPs
to solve for the coefficients of the polynomials.
4. PHOTOGRAMMETRIC EXPLOITATION
Orthorectification and stereo intersection are two most
important methods for preparing fundamental data for
cartographic mapping applications. The RFM can be used to
perform the photogrammetric processing on images since it is a
generic form of many imagery geometry models and has
inherent geometric modeling capability.
An original un-rectified aerial or satellite image does not show
features in their correct locations due to displacements caused
by the tilt of the sensor and the relief of the terrain.
Orthorectification transforms the central projection of the image
into an orthogonal view of the ground with uniform scale,
thereby removing the distorting affects of tilt optical projection
and terrain relief. The RFM based orthorectification is relatively
straightforward. The use of RFM for image rectification is
discussed in Yang (2000), Dowman and Dolloff (2000), Tao
and Hu (20015) and” Croitoru et al. (2004) The
orthorectification accuracy is similar to the approximating
accuracy of the RFM, excluding the resampling error.
The 3-D reconstruction algorithms can be implemented based
on either the forward RFM or the inverse RFM. The
approximate values of the un-normalized object point
coordinates (X, Y, Z) are corrected by the correction given by
the following formula (Tao and Hu, 2002):
(AX AY AZ)! » (4 WAy!4' WI (9)
where (AX, AY, AZ) are un-normalized coordinate corrections;
A is the design matrix that is composed of ratios between the
partial derivatives of the functions in Eq. 1 with respect to X, Y,
and Z and the image domain scale parameters; / is the vector of
discrepancies between the measured and the RFM projected
image coordinates of the estimated object coordinates; W is the
weight matrix for the image points. The weight matrix may be
an identity matrix when the points are measured on images of a
same sensor type. However, higher weights should be assigned
to points measured in images of higher resolution when
implementing a hybrid adjustment using images with different
ground resolutions as described in the next section. The
approximate object coordinates may be obtained by solving the
RFM with only constant and first-order terms, or by solving
using one image and a given elevation value, or by setting to be
the offset values of the ground coordinates. In most cases, eight
iterations are enough to converge. A procedure similar to above
forward RFM 3-D reconstruction is described in Di et al.
(2001), and Fraser and Hanley (2003). But their algorithm does
not incorporate the normalization parameters into the
adjustment equations directly. The 3-D mapping capability will
be greatly enhanced after absorbing one or more GCPs (Fraser
et al, 2003; Tao et al., 2004; Croitoru et al., 2004).
5. PHOTOGRAMMETRIC INTEROPERABILITY
Multiple different image geometry models are needed for
exploiting different image types under different conditions
(OGC, 1999b) There are many different types of imaging
geometries, including frame, panoramic, pushbroom,
whiskbroom and so on. Many of these imaging geometries have
multiple subtypes (e.g. multiple small images acquired
simultaneously) and multiple submodels (e.g. atmospheric
refraction, panoramic and scan distortions). These image
geometries are sufficiently different that somewhat different
rigorous image geometry models are required. Furthermore,
different cameras of the same basic geometry can require
different rigorous image geometry models. When interoperation
of several software packages is needed to complete one imagery
exploitation service, it is necessary to standardize those
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