Block Adjustment of High-Resolution Satellite
Images Described by Rational Polynomials
Jacek Grodecki and Gene Dial
Abstract
This paper describes how to block adjust high-resolution
satellite imagery described by Rational Polynomial Coefficient
(RPC) camera models and illustrates the method with an Ikonos
example. By incorporating a priori constraints into the
adjustment model, multiple independent images can be
adjusted with or without ground control. The RPC block
adjustment model presented in this paper is directly related
to geometric properties of the physical camera model. Multiple
physical camera model parameters having the same net effect
on the object-image relationship are replaced by a single
adjustment parameter. Consequently, the proposed method is
numerically more stable than the traditional adjustment of
exterior and interior orientation parameters. This method is
generally applicable to any photogrammetric camera with a
narrow field of view, calibrated, stable interior orientation, and
accurate a priori exterior orientation data. As demonstrated
in the paper, for Ikonos satellite imagery, the RPC block
adjustment achieves the same accuracy as the ground station
block adjustment with the full physical camera model.
Background
The launch of Ikonos on 24 September 1999 set off a new era of
commercially available, high-resolution satellite imagery.
Overviews of the Ikonos satellite may be found in Dial (2001),
Dial et al. (2001), and Grodecki and Dial (2001).
Rational Polynomial Coefficient (RPC) camera models are
derived from the physical Ikonos sensor model to describe the
object-image geometry. RPC models transform three-dimensional object-space coordinates into two-dimensional imagespace coordinates. RPCs provide a simple and accurate means of
communicating camera object-image relationship from image
data provider to image data user (Grodecki, 2001). RPCs have
been successfully used for the terrain extraction, orthorectification, and feature extraction tasks. What has been lacking is a
method to block adjust imagery described by RPCs.
Dial and Grodecki (2002) outlined the RPC block adjustment technique, described in more detail in this article. A similar method of exterior orientation bias compensation for
Ikonos imagery has been independently proposed by Fraser et
al. (2002), albeit without any reference to the physical camera
model. Other investigators have proposed various methods for
photogrammetric processing of Ikonos images (Toutin and
Cheng, 2000). These have been hampered by incomplete knowledge of the Ikonos camera model, of the maneuvering possible
during image acquisition, and by limited availability of generally expensive test data sets.
While Ikonos ground stations use the physical camera
model for block adjustment, some users wish to block adjust
Ikonos imagery outside of the ground station with their own,
proprietary ground control, elevation models, or controlled
Space Imaging LLC, 12076 Grant Street, Thornton, CO 80241
(jgrodecki@spaceimaging.com; gdial@spaceimaging.com).
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
imagery. As demonstrated below, the RPC block adjustment
model provides a rigorous, accurate method to block adjust
Ikonos data outside of the ground stations.
This publication of a technique for block adjusting Ikonos
images described by RPC data is motivated by a desire to satisfy
the needs of those users who would like to perform their own
block adjustment, and to ensure that Ikonos images are processed in such way as to consistently achieve the highest possible accuracy. In developing the adjustment model described
here, the authors had access to the complete description of
Ikonos imaging geometry, familiarity with all of the satellite
maneuvering modes, the resources of extensive test ranges and
imagery with which to test and validate, and the experience
gained calibrating, testing, and troubleshooting Ikonos metric
projects.
Physical Camera Models
Owing to the dynamic nature of satellite image collection, photogrammetric processing of satellite imagery is more complicated than is aerial frame camera processing. Aerial cameras
acquire the entire image at an instant of time with a unique
exposure station and orientation. High-resolution pushbroom
satellite cameras, including Ikonos, use linear sensor arrays
that acquire a single image line at an instant of time. Consequently, each line of a pushbroom satellite image has a different
exposure station and orientation (Grodecki, 2001). Implementing such a complicated model is expensive, time consuming,
and error prone.
Adjustment Parameters
Interior Orientation
Interior orientation includes parameters for detector positions,
principal point, optical distortion, and focal length. Unlike
film cameras, the Ikonos digital focal plane does not require
fiducial marks. Instead, every pixel is at a fixed, calibrated
position on the solid-state focal plane. The detectors are rigidly
attached to the focal plane in a stable thermal-mechanical
environment. The elements of interior orientation have been
determined to superb accuracy with well-controlled test-range
imagery. Consequently, it is not necessary, indeed it is not desirable, to estimate corrections to the interior orientation parameters in the block adjustment process.
Exterior Orientation
Exterior orientation comprises position and attitude. On-board
GPS receivers determine the satellite ephemeris, i.e., camera
Photogrammetric Engineering & Remote Sensing
Vol. 69, No. 1, January 2003, pp. 59 – 68.
0099-1112/03/6901–059$3.00/0
䉷 2003 American Society for Photogrammetry
and Remote Sensing
Ja nuar y 20 03
59
position as a function of time. Star trackers and gyros determine
the camera attitude as a function of time.
For Ikonos, the ephemeris and attitude have finite accuracy, about one meter for the ephemeris and about one or two
arc-seconds for attitude. As demonstrated below, for high-resolution satellite systems the in-track and cross-track position
errors are almost completely correlated with pitch and roll attitude errors so that they cannot be separately estimated. Moreover, yaw and radial errors are negligible. Thus, it is only
necessary to estimate roll and pitch.
Attitude Errors
Attitude angles are roll (rotation about the in-track direction),
pitch (rotation about the cross-track direction), and yaw (rotation about the line-of-sight). For Ikonos, with its 680-km orbital
height, a 2-arc-second error in roll or pitch causes a 6.6-m or
more displacement on the ground, because its effect is proportional to the slant range. The yaw error effect on the ground
position is, on the other hand, a function of the swath width.
For a yaw error of 2 arc-seconds and a swath width of 11 km,
the maximum ground displacement is only 0.055 meters, a negligible amount.
Ephemeris Errors
Ephemeris errors are conventionally decomposed into in-track,
cross-track, and radial components. We will first show that intrack and cross-track errors are equivalent to pitch and roll attitude errors. Then we will show that radial errors are negligible.
For narrow field-of-view cameras, small horizontal displacements are equivalent to small angular rotations. As a
result, roll errors are completely correlated with cross-track
errors. The same is the case for pitch and in-track errors. As
shown in Figure 1, for a roll error of 2 arc-seconds, the difference between the nominal nadir-pointing camera and another
camera that has been rotated and correspondingly displaced is
X1 ⫺ X1⬘ ⫽ X2 ⫺ X2⬘ ⫽ 0.000454 m:—less than 1/2000 pixel.
If the camera field of view is narrow enough and the position
and attitude errors are small enough such that the non-linear
effects of attitude errors are negligible, then position and attitude cannot be independently observed. The presence of correlated parameters having near-identical effects leads to
instability of the block adjustment process. Combining the correlated parameters into a single parameter results in numerical
stability.
The equivalence of small pitch and in-track ephemeris
errors is illustrated in Figure 2. Two satellite imaging systems
are shown, one at position A with pitch error P and the other at
position B with in-track ephemeris error IT. The motions of the
satellites and the aim points of their scans are illustrated by
arrows. Satellite A has a slightly longer slant range, but this is
insignificant for a few arc-seconds of pitch error. Satellite A also
has a slightly different perspective than B, but this is again
insignificant for a few arc-seconds of pitch. Small pitch errors
are thus indistinguishable from in-track ephemeris errors, and
those two physical effects are thus best modeled by a single
parameter.
Radial ephemeris errors result in scale errors. For example,
a 1-m radial error from a 680-km orbit height causes a 1.5-ppm
scale factor error that causes a 16-mm positioning error across
the approximately 11-km swath width. Radial error effects are
thus negligible for Ikonos.
Drift Errors
While attitude and ephemeris errors are largely biases, there
exists the possibility that these errors would drift as a function
of time. For example, gyro errors without sufficient compensation from the star trackers could introduce an error in attitude
rate. For Ikonos, these errors have been found to be small, less
60
Ja nuar y 20 03
Figure 1. Effect of roll and cross-track errors.
a ⫽ half-angle of the camera field of view; for Ikonos,
a ⫽ 28.53⬘.
h ⫽ orbital height; for Ikonos, h ⫽ 680 km.
r ⫽ camera roll angle ⫽ 2 arc-seconds.
d ⫽ equivalent displacement ⫽ h tan(r) ⫽ 6.593466 m.
X1 ⫽ ground coordinate of the left edge of the nominal
camera.
⫽ ⫺h tan(a) ⫽ ⫺5644.129609 m.
X1⬘ ⫽ ground coordinate of the left edge of the displaced
and rotated camera.
⫽ d ⫺ h tan(a ⫹ r) ⫽ ⫺5644.130063 m.
X2 ⫽ ground coordinate of the right edge of the nominal
camera.
⫽ h tan(a) ⫽ 5644.129609 m.
X2⬘ ⫽ ground coordinate of the right edge of the displaced
and rotated camera.
⫽ d ⫹ h tan(a ⫺ r) ⫽ 5644.129155 m.
X1 ⫺ X1⬘ ⫽ X2 ⫺ X2⬘ ⫽ 0.000454 m.
than a few pixels per 100 km, and so are negligible for all but
very long strips.
Required Adjustments
As demonstrated above, many effects are negligible or completely correlated with other effects. As a result, only a few
parameters are required to effectively model the sensor errors.
A line offset parameter is required to adjust for errors in the
line direction and a sample offset parameter is required to
adjust for errors in the sample direction. The line parameter
absorbs effects of orbit, attitude, and residual interior orientation errors in the line direction. The sample parameter absorbs
the same effects in the sample direction. For longer strips, a
parameter proportional to line can be added to model drift
errors.
Rational Polynomial Camera Model
The Ikonos physical camera model is used at the ground stations to block adjust multiple images. RPCs are subsequently
estimated from the block adjusted physical camera model. The
78 rational polynomial coefficients, {c1 . . . c20, d2 . . . d20, e1 . . .
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
where
NumL(P, L, H ) ⫽ c1 ⫹ c2L ⫹ c3P ⫹ c4H ⫹ c5LP ⫹ c6LH
⫹ c7PH ⫹ c8L2 ⫹ c9P 2 ⫹ c10H 2 ⫹ c11PLH ⫹ c12L3
⫹ c13LP 2 ⫹ c14LH 2 ⫹ c15L2P ⫹ c16P 3 ⫹ c17PH 2
(6)
⫹ c18L2H ⫹ c19P 2H ⫹ c20H 3 ⫽ cTu
DenL(P, L, H ) ⫽ 1 ⫹ d2L ⫹ d3P ⫹ d4H ⫹ d5LP ⫹ d6LH
⫹ d7PH ⫹ d8L2 ⫹ d9P 2 ⫹ d10H 2 ⫹ d11PLH ⫹ d12L3
⫹ d13LP 2 ⫹ d14LH 2 ⫹ d15L2P ⫹ d16P 3 ⫹ d17PH 2
(7)
⫹ d18L H ⫹ d19P H ⫹ d20H ⫽ d u
2
2
3
T
with
u ⫽ [1 L P H LP LH PH L2 P 2 H 2 PLH L3 LP 2
LH 2 L2P P 3 PH 2 L2H P 2H H 3]T
c ⫽ [c1 c2 … c20]T
Figure 2. Side view of satellite imaging system A with pitch
error P and satellite imaging system B with in-track ephemeris error IT.
d ⫽ [1 d2 … d20]T;
and
e20, f2 . . . f20}, are subsequently determined by fitting the physical camera model, as described in the next section, and are supplied with ortho-kit and stereo images. Given these coefficients, the computation of (Line, Sample) is fast, easy, and
accurate.
The RPC model has previously been described in Grodecki
(2001) but will also be briefly summarized here. The RPC model
relates the object-space (␾, ␭, h) coordinates to image-space
(Line, Sample) coordinates. The RPC functional model is in the
form of a ratio of two cubic polynomials of object-space coordinates. Separate rational functions are used to express the
object-space to line and the object-space to sample coordinates
relationship. To improve numerical precision, image- and
object-space coordinates are normalized to 具⫺1, ⫹1典 range as
shown below.
Given the object-space coordinates (␾, ␭, h), where ␾ is geodetic latitude, ␭ is geodetic longitude, and h is height above the
ellipsoid, and the latitude, longitude, and height offsets and
scale factors (LAT OFF, LONG OFF, HEIGHT OFF,
LAT SCALE, LONG SCALE, HEIGHT SCALE ), the calculation of image-space coordinates begins by normalizing latitude, longitude, and height as follows:
P⫽
L⫽
␾ ⫺ LAT OFF
LAT SCALE
␭ ⫺ LONG OFF
H⫽
LONG SCALE
,
(1)
, and
HEIGHT SCALE
.
(2)
(3)
The normalized line and sample image-space coordinates
(Y and X, respectively) are then calculated from their respective rational polynomial functions g(.) and h(.): i.e.,
NumL(P, L, H ) cTu
⫽ T
DenL(P, L, H )
d u
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
NumS(P, L, H ) eTu
⫽ T
DenS(P, L, H )
f u
(8)
where
NumS(P, L, H ) ⫽ e1 ⫹ e2L ⫹ e3P ⫹ e4H ⫹ e5LP ⫹ e6LH
⫹ e7PH ⫹ e8L2 ⫹ e9P 2 ⫹ e10H 2 ⫹ e11PLH ⫹ e12L3
⫹ e13LP 2 ⫹ e14LH 2 ⫹ e15L2P ⫹ e16P 3 ⫹ e17PH 2
(9)
⫹ e18L H ⫹ e19P H ⫹ e20H ⫽ e u
2
2
3
T
DenS(P, L, H ) ⫽ 1 ⫹ f2L ⫹ f3P ⫹ f4H ⫹ f5LP ⫹ f6LH
⫹ f7PH ⫹ f8L2 ⫹ f9P 2 ⫹ f10H 2 ⫹ f11PLH ⫹ f12L3
⫹ f13LP 2 ⫹ f14LH 2 ⫹ f15L2P ⫹ f16P 3 ⫹ f17PH 2
(10)
⫹ f18L H ⫹ f19P H ⫹ f20H ⫽ f u
2
2
3
T
with
e ⫽ [e1 e2 … e20]T
f ⫽ [1 f2 … f20]T
h ⫺ HEIGHT OFF
Y ⫽ g(␾, ␭, h) ⫽
X ⫽ h(␾, ␭, h) ⫽
(4)
Using line and sample offsets and scale factors
(LINE OFF, SAMP OFF, LINE SCALE, SAMP SCALE ), the
de-normalized image-space coordinates (Line, Sample), where
Line is the image line number expressed in pixels with pixel
zero as the center of the first line, and Sample is the sample
number expressed in pixels with pixel zero is the center of the
left-most sample, are finally computed as
Line ⫽ Y ⭈ LINE SCALE ⫹ LINE OFF, and
(11)
Sample ⫽ X ⭈ SAMP SCALE ⫹ SAMP OFF.
(12)
Determining RPC Coefficients
A least-squares approach is used to estimate the RPC model
coefficients ci , di , ei , and fi from a three-dimensional grid of
Ja nuar y 20 03
61
points, depicted schematically in Figure 3, generated using the
physical camera model. It should be noted that, as pointed out
in Hu and Tao (2001) and Tao and Hu (2001), attempts to use
ground control only to determine RPC coefficients risks numerical instability during the fitting process and poor compliance
with camera physics.
Evaluation of RPC Accuracy
Accuracy of the RPC model was assessed using the physical
Ikonos camera model as a reference. RPCs were fitted to a grid
of points with ground-space coordinates generated from the
image-space coordinates, for a set of different elevation levels,
using the Ikonos physical camera model (see Grodecki (2001)
and Grodecki and Dial (2001)).
Independent check points were subsequently used to
quantify the RPC model accuracy. RPC accuracy was computed
for a strip length of 100 km, for a series of imaging scenarios.
The imaging parameters ranged from 0⬚ to 30⬚ for roll, 0⬚ to 30⬚
for pitch, 0⬚ to 360⬚ for scan azimuth, and 0⬚ to 60⬚ for latitude.
The check-point residual errors were 0.01 pixels RMS and 0.04
pixels worst case for all imaging scenarios, thus demonstrating
the extremely high accuracy of the RPC camera model representation (Grodecki, 2001).
Block Adjustment with RPCs
RPC data provided with Ikonos imagery has, up to now, enabled
the user to perform feature extraction, DEM generation, and
orthorectification. Until now, however, photogrammetric block
adjustment with RPCs has been considered to be unfeasible. As
demonstrated below, the approach presented in this paper provides a rigorous, consistent, and accurate block adjustment
method for high-resolution satellite imagery described by RPCs.
The proposed RPC block adjustment model is directly related to
the geometric properties of the physical camera model, by combining multiple physical camera model parameters into a single adjustment parameter having the same net effect on the
object-image relationship. Consequently, the proposed
method is numerically more stable than the traditional adjustment of exterior and interior orientation parameters. This
method is generally applicable to any photogrammetric camera
with a narrow field of view, a calibrated stable interior orientation, and accurate a priori exterior orientation data.
RPC Block Adjustment Math Models
Proposed RPC Block Adjustment Model
The RPC block adjustment math model proposed in this paper is
defined in the image space. It uses denormalized RPC models,
p and r, to express the object-space to image-space relationship,
and the adjustable functions, ⌬p and ⌬r, which are added to the
rational functions to capture the discrepancies between the
nominal and the measured image-space coordinates. For each
image point i on image j, the RPC block adjustment math model
is thus defined as follows:
Line i(j) ⫽ ⌬p(j) ⫹ p(j)(␾k, ␭k, hk) ⫹ ␧Li
(13)
Sample i(j) ⫽ ⌬r (j) ⫹ r (j)(␾k, ␭k, hk) ⫹ ␧Si
(14)
where
Linei(j) and Samplei(j) are measured (on image j ) line and
sample coordinates of the ith image point, corresponding
to the kth ground control or tie point with object space
coordinates (␾k ,␭k ,hk);
⌬p(j) and ⌬r (j) are the adjustable functions expressing the
differences between the measured and the nominal line
and sample coordinates of ground control and/or tie points,
for image j;
␧Li and ␧Si are random unobservable errors;
p(j) and r (j) are the given line and sample, denormalized
RPC models for image j;
p(␾, ␭, h) ⫽ g(␾, ␭, h) ⭈ LINE SCALE
(15)
⫹ LINE OFF; and
r(␾, ␭, h) ⫽ h(␾, ␭, h) ⭈ SAMPLE SCALE
(16)
⫹ SAMPLE OFF.
We are proposing to use a polynomial model defined in the
domain of image coordinates to represent the adjustable functions, ⌬p and ⌬r, which in general can be expressed as
⌬p ⫽ a0 ⫹ aS ⭈ Sample ⫹ aL ⭈ Line ⫹ aSL ⭈ Sample
⭈ Line ⫹ aL2 ⭈ Line 2 ⫹ aS2 ⭈ Sample 2 ⫹ …
(17)
⌬r ⫽ b0 ⫹ bS ⭈ Sample ⫹ bL ⭈ Line ⫹ bSL ⭈ Sample
⭈ Line ⫹ bL2 ⭈ Line 2 ⫹ bS2 ⭈ Sample 2 ⫹ …
(18)
where a0, aS , aL , . . ., and b0, bS , bL , . . ., are the adjustment
parameters for an image, and Line and Sample are line and
sample coordinates of a ground control or tie point.
The choice of the image coordinate system to define the
adjustable functions is influenced by the need to tie the adjustable model to the physics of the imaging operation. For Ikonos
we propose to use the following truncated polynomial model
defined in the domain of image coordinates to represent the
adjustable functions, ⌬p and ⌬r:
Figure 3. RPC fitting.
62
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⌬p ⫽ a0 ⫹ aS ⭈ Sample ⫹ aL ⭈ Line
(19)
⌬r ⫽ b0 ⫹ bS ⭈ Sample ⫹ bL ⭈ Line
(20)
As demonstrated in the section on Adjustment Parameters,
each of the parameters in the above adjustment model (Equations 19 and 20) has physical significance. As a result, the proposed RPC block adjustment model does not present the
numerical ill-conditioning problems of classical techniques.
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
Parameter a0 absorbs all in-track errors causing offsets in
the line direction, including in-track ephemeris error, satellite
pitch attitude error, and the line component of principal point
and detector position errors. As discussed earlier, for narrow
field-of-view instruments with strong a priori orientation data,
all of these physical parameters have the same net effect of displacing images in line. Similarly, parameter b0 absorbs crosstrack errors causing offsets in the sample direction, including
cross-track ephemeris error, satellite roll attitude error, and the
sample component of principal point and detector position
errors.
Because the line direction is equivalent to time, parameters
aL and bL absorb the small effects due to gyro drift during the
imaging scan. As will be shown later, parameters aL and bL turn
out to be required only for images that are longer than 50 km.
Parameters aS and bS absorb radial ephemeris error, and interior
orientation errors such as focal length and lens distortion
errors. As discussed earlier, for Ikonos these errors are negligibly small. As a result, parameters aS and bS are not required.
For images shorter than 50 km, the adjustment model becomes
simply ⌬p ⫽ a0 and ⌬r ⫽ b0 where a0 and b0 are bias parameters
to be determined for each image by the block adjustment
process.
Other RPC Block Adjustment Models
Alternatively, adjustable functions ⌬p and ⌬r can also be represented by a polynomial model defined in the domain of object
coordinates as
⌬p ⫽ a0 ⫹ aP ⭈ ␾ ⫹ aL ⭈ ␭ ⫹ aH ⭈ h ⫹ aP2 ⭈ ␾ 2 ⫹ aL2 ⭈ ␭ 2
(21)
Figure 4. Evaluation of RPC adjustment models.
⫹ aH2 ⭈ h ⫹ aPL ⭈ ␾ ⭈ ␭ ⫹ aPH ⭈ ␾ ⭈ h ⫹ aLH ⭈ ␭ ⭈ h ⫹ …
2
⌬r ⫽ b0 ⫹ bP ⭈ ␾ ⫹ bL ⭈ ␭ ⫹ bH ⭈ h ⫹ bP2 ⭈ ␾ 2 ⫹ bL2 ⫹ ␭ 2
(22)
⫹ bH2 ⭈ h ⫹ bPL ⭈ ␾ ⭈ ␭ ⫹ bPH ⭈ ␾ ⭈ h ⫹ bLH ⭈ ␭ ⭈ h ⫹ …
2
As shown later, adjustment models defined in the domain
of object coordinates are in general less accurate than models
defined in the domain of image-space coordinates.
Another possibility is to define the RPC block adjustment
model in the object space. The object-space RPC block adjustment math model, for the kth ground control or tie point being
the ith image point on the jth image, is then defined as follows:
Linei(j) ⫽ p(j)(␾k ⫹ ⌬␾ (j), ␭k ⫹ ⌬␭ (j), hk ⫹ ⌬h(j)) ⫹ ␧Li
(23)
Samplei(j) ⫽ r (j)(␾k ⫹ ⌬␾ (j), ␭k ⫹ ⌬␭ (j), hk ⫹ ⌬h(j)) ⫹ ␧Si (24)
where ⌬␾ (j), ⌬␭ (j), and ⌬h(j) are the adjustable functions expressing the differences between the measured and the nominal
object-space coordinates of a ground control or tie point, for
image j.
As before, the object-space adjustment model can be represented either by a polynomial model defined in the domain of
image space or by a polynomial model defined in the domain of
object coordinates. We do not recommend the object-space RPC
block adjustment math model, because it is nonlinear in the
adjustment parameters and unrelated to imaging geometry.
Evaluation of RPC Adjustment Models
The accuracy of the proposed RPC adjustment models was evaluated by numerical simulation, using the perturbation
approach shown in Figure 4. Errors in satellite vehicle ephemeris and attitude were propagated, for a grid of image points,
down to the object (␾, ␭, h) space. The so determined perturbed
ground coordinates were then projected back to the image
(Line, Sample) space. In both cases the Ikonos physical camera
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
model was utilized. The differences (⌬L, ⌬S) between the original and the perturbed image coordinates were subsequently
calculated, and used as input to the tested adjustment models.
Evaluation of Image-Space Adjustment Models Defined in
the Domain of Image Coordinates
The following combinations of the proposed adjustment models were evaluated:
(1)
(2)
(3)
(4)
⌬p ⫽ a0 and ⌬r ⫽ b0
⌬p ⫽ a0 ⫹ aL ⭈ Line and ⌬r ⫽ b0 ⫹ bL ⭈ Line
⌬p ⫽ a0 ⫹ aS ⭈ Sample and ⌬r ⫽ b0 ⫹ bS ⭈ Sample
⌬p ⫽ a0 ⫹ aS ⭈ Sample ⫹ aL ⭈ Line and ⌬r ⫽ b0 ⫹ bS ⭈ Sample
⫹ bL ⭈ Line
where a0, b0, aL , bL , aS , bS are the image adjustment parameters,
and Line, Sample are the line and sample coordinates of a
ground control or a tie point.
For each of the tested RPC adjustment models, the adjustable parameters (a0, b0, aL , bL , aS , bS) were estimated using the
least-squares approach. The post-fit RMS errors and the maximum residual errors were then computed for each of the tested
adjustment models.
A number of scenarios were generated to thoroughly test
the proposed adjustment models using a wide range of feasible
imaging conditions. The imaging parameters ranged from 0⬚ to
30⬚ for roll, 0⬚ to 30⬚ for pitch, 0⬚ to 360⬚ for scan azimuth, and
0⬚ to 60⬚ for latitude. The image strip length was varied from 10
km to 100 km. The minimum elevation angle was set to 50
degrees. The errors in the satellite vehicle ephemeris and attitude were set to
● 3 meters in the ephemeris components (in-track, cross-track,
radial), and
● 2 arc-seconds in the attitude angles (pitch, roll, yaw).
Ja nuar y 20 03
63
Maximum residual and RMS errors, from all imaging scenarios, were then computed giving the measure of the worst
possible math model errors when using the proposed RPC block
adjustment approach. The results of the analysis are shown in
Table 1.
The results given in Table 1 show that the postulated RPC
adjustment models can accurately model the effects of ephemeris and attitude errors. Bias only models (parameters ⌬p ⫽ a0
and ⌬r ⫽ b0 only) are effective for strip lengths up to 50 km.
Strips of 100 km length may require the addition of drift parameters (aL and bL) for full accuracy. Parameters proportional to
sample (aS and bS) and higher order terms are not normally
required.
Evaluation of Image-Space Adjustment Models Defined in
the Domain of Object-Space Coordinates
The following adjustment models were tested:
TABLE 2. EVALUATION OF ADJUSTMENT MODELS DEFINED IN THE DOMAIN OF
OBJECT COORDINATES
Strip
Length
Adjustment
Model
RMS
Sample
Error
[pixels]
10 km
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
0.15
0.09
0.002
0.27
0.11
0.01
0.52
0.25
0.02
1.07
0.76
0.75
20 km
50 km
100 km
Max.
Sample
Error
[pixels]
RMS
Line
Error
[pixels]
Max.
Line
Error
[pixels]
0.09
0.06
0.001
0.17
0.06
0.002
0.27
0.11
0.01
0.56
0.40
0.29
0.12
0.16
0.002
0.20
0.23
0.01
0.42
0.61
0.04
0.84
1.48
0.53
0.08
0.07
0.001
0.09
0.11
0.003
0.18
0.29
0.02
0.36
0.66
0.21
(a) ⌬p ⫽ a0 ⫹ aP ⭈ ␾ and ⌬r ⫽ b0 ⫹ bP ⭈ ␾
(b) ⌬p ⫽ a0 ⫹ aL ⭈ ␭ and ⌬r ⫽ b0 ⫹ bL ⭈ ␭
(c) ⌬p ⫽ a0 ⫹ aL ⭈ ␭ ⫹ aP ⭈ ␾ and ⌬r ⫽ b0 ⫹ bL ⭈ ␭ ⫹ bP ⭈ ␾
where a0, b0, aL , bL , aP , bP are the image adjustment parameters;
and ␾, ␭ are the object space coordinates of a ground control (or
tie) point.
As before, a number of scenarios were generated for this
purpose — using the same ranges of scanning azimuth, roll
and pitch angles, and geographic location. Maximum residual
and RMS errors, from all imaging scenarios, are shown in Table 2.
It is seen that, for strips up to 20 km long, the image-space
adjustment models defined in object-space coordinates give
virtually the same results as the image-space adjustment models defined in image space coordinates. However, as seen in the
previous section, for longer strips adjustment models defined
in image space produce superior accuracy. Moreover, as indicated earlier, the image-space adjustment models defined in
image-space coordinates are also much more closely related to
the geometric properties of the physical camera model. As a
result, the proposed RPC block adjustment will utilize the
image-space coordinate formulation.
RPC Block Adjustment Algorithm
Multiple overlapping images can be block adjusted using one of
the RPC adjustment models given above. As indicated above,
the preferred approach uses the image-space adjustment model
given by Equations 19 and 20. The overlapping images, with
TABLE 1. EVALUATION OF ADJUSTMENT MODELS DEFINED IN THE DOMAIN OF
IMAGE COORDINATES
Strip
Length
Adjustment
Model
Max.
Sample
Error
[pixels]
10 km
(1)
(2)
(3)
(4)
(1)
(2)
(3)
(4)
(1)
(2)
(3)
(4)
(1)
(2)
(3)
(4)
0.21
0.15
0.09
0.004
0.28
0.15
0.19
0.01
0.57
0.16
0.50
0.02
1.00
0.17
0.93
0.07
20 km
50 km
100 km
64
Ja nuar y 20 03
RMS
Sample
Error
[pixels]
Max.
Line
Error
[pixels]
RMS
Line
Error
[pixels]
0.10
0.10
0.06
0.001
0.13
0.10
0.12
0.002
0.29
0.10
0.28
0.01
0.51
0.10
0.50
0.03
0.21
0.12
0.10
0.001
0.32
0.12
0.22
0.004
0.66
0.13
0.58
0.02
1.25
0.17
1.17
0.06
0.09
0.08
0.07
0.001
0.15
0.08
0.13
0.001
0.34
0.08
0.33
0.01
0.66
0.08
0.65
0.03
RPC models expressing the object-space to image-space rela-
tionship for each image, are tied together by tie points whose
image coordinates are measured on those images. Optionally,
the block may also have ground control points with known or
approximately known object-space coordinates and measured
image positions (see Figure 5).
Indexing of observation equations is based on image-point
indices i. Because there is only one observation equation per
image point, index i uniquely identifies each observation equation. Thus, for the kth ground control or tie point being the ith
image point on the jth image, the RPC block adjustment observation equations read
(j)
FLi ⫽ ⫺Line i ⫹ ⌬p(j) ⫹ p(j)(␾k, ␭k, hk) ⫹ ␧Li ⫽ 0
(j)
(25)
FSi ⫽ ⫺Sample i ⫹ ⌬r (j) ⫹ r (j)(␾k, ␭k, hk) ⫹ ␧Si ⫽ 0
(26)
⌬p(j) ⫽ a0(j) ⫹ aS(j) ⭈ Sample i(j) ⫹ aL(j) ⭈ Line i(j)
(27)
⌬r (j) ⫽ b0(j) ⫹ bS(j) ⭈ Sample i(j) ⫹ bL(j) ⭈ Line i(j)
(28)
with
Observation Equations 25 and 26 are formed for each image
point i. Measured image-space coordinates for each image
point i (Line i(j) and Sample i(j)) constitute the adjustment model
observables, while the image model parameters (a0(j), aS(j), aL(j),
b0(j), bS(j), bL(j)) and the object-space coordinates (␾k , ␭k , hk) comprise the unknown adjustment model parameters. Line i(j) and
Sample i(j) coordinates in Equations 27 and 28 are the approximate fixed values for the true image coordinates. One possible
Figure 5. Block adjustment of multiple overlapping images.
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
choice for the approximate line and sample coordinates in
Equations 27 and 28 are the values of the measured image coordinates. It should be noted that, even though the true image
coordinates are not known, the effect of using the approximate
values in Equations 27 and 28 will be for all practical purposes
negligible because the measurements of image coordinates are
performed with sub-pixel accuracy.
With
Fi ⫽
冋 册
FLi
,
FSi
x0 is the vector of the approximate model parameters,
x0 ⫽
AGi dxG ⫽
application of the Taylor Series expansion to the RPC block
adjustment observation Equations 25 and 26 results in the following linearized model:
⫽
(30)
where
Fi0 ⫽
冋 册
⫺Linei(j) ⫹ a0(j) ⫹ aS(j) ⭈ Sample i(j)
0
0
冤 冥
冤
0
⭸FSi
⭸xTG x
dxG
(37)
0
冟
␾ 冟
冟
␭ 冟
冟
冟
0…0
⭸FLi
⭸ ␾k x
0
⭸FLi
⭸␭ k x
0
⭸FLi
⭸hk x
0…0
⭸FSi
⭸ kx
0
⭸FSi
⭸ kx
0
⭸FSi
⭸hk x
bL(j)
0
⭈
0
Line i(j)
冟
冟
0
⫹ r (␾k0, ␭k0, hk0)
(j)
冥
0
0…0
0
冥冤 冥
冤 冥
冟 冟
冤 冟 冟 冥冋
⭸FLi
⭸xT x
冋 册
⭸FLi
⭸xTA x
0
⭸FSi
⭸xTA x
0
⭸FLi
⭸ ␾k
⭸FLi
⭸ ␭k
⭸FLi
⭸p(j)
⫽
⭸hk
⭸␾ k
册 冋
⭸p(j)
⭸␭k
⭸p(j)
⭸hk
(38)
⭸FSi
⭸␾ k
⭸FSi
⭸␭ k
⭸FSi
⭸r (j)
⫽
⭸hk
⭸␾ k
册 冋
⭸r (j)
⭸␭ k
⭸r (j)
,
⭸hk
册
(39)
册
冋
with
dFLi
0
dFi ⫽
⫽
dx
dFSi
⭸FSi
⭸xT x
(32)
冋
⭸p
⭸␾
0
⭸FLi
⭸xTG x
册
⭈
冋 册
dxA
dxA
⫽ [AAi AGi]
dxG
dxG
0
⭸FSi
⭸xTG x
0
冋
册
⭸p
⭸␭
冋
⭸p
cT(dTu) ⫺ dT(cTu) ⭸u
⫽
⭈
⭸h
(dTu)2
⭸P
⭸u
⭸L
⭸u
⭸H
册
1
LAT SCALE
0
0
0
1
LONG SCALE
0
0
0
1
HEIGHT SCALE
⭈ LINE SCALE
dx ⫽ x ⫺ x0 is the vector of unknown corrections to the approximate model parameters, x0,
dx ⫽
冋 册
dxA
dxG
da(1)
da(1)
db(1)
db(1)
db(1)
dxA ⫽ [da(1)
0
S
L
0
S
L
da0(n)
daS(n)
daL(n)
db0(n)
⭸r
⭸␾
⭈
dbS(n)
(34)
dxG ⫽ [d␾1 d␭1 dh1 … d␾m⫹p d␭m⫹p dhm⫹p]T
冋
册
⭸r
⭸␭
冋
⭸r
⭸u
eT(f Tu) ⫺ f T(eTu)
⫽
⭈
⭸h
(f Tu)2
⭸P
⭸u
⭸L
⭸u
⭸H
册
册
1
LAT SCALE
0
0
0
1
LONG SCALE
0
0
0
1
HEIGHT SCALE
⭈ SAMPLE SCALE
dbL(n)]T
dxG is the sub-vector of the corrections to the approximate
object space coordinates for m ground control and p tie points,
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
冋
(33)
dxA is the sub-vector of the corrections to the approximate
image adjustment parameters for n images,
…
⯗
d ␾k
d ␭k
dhk
⯗
0…0
and
⫹ aL(j) ⭈ Line i(j) ⫹ p(j)(␾k0, ␭k0, hk0)
0
⫽
⫽ ⫺wPi
⫺Samplei(j) ⫹ b0(j) ⫹ bS(j) ⭈ Sample i(j)
⫽
冟
冟
⭸FLi
⭸xTG x
冋
(31)
FSi0
⫹
(36)
x G0
where
FLi0
冤
xA0
and ␧ is a vector of unobservable random errors.
For the kth ground control or tie point being the ith image
point on the jth image, we get
(29)
Fi0 ⫹ dFi ⫹ ␧ ⫽ 0
冋 册
册
(40)
(41)
in which
⭸u
⫽ [0 0 1 0 L 0 H 0 2P 0 LH 0 2LP 0 L2 3P 2 H2 0 2PH 0]T
⭸P
(35)
(42)
Ja nuar y 20 03
65
⭸u
⫽ [0 1 0 0 P H 0 2L 0 0 PH 3L2 P 2 H 2 2LP 0 0 2LH 0 0]T
⭸L
冤冥
AG1
⯗
AG ⫽ A
Gi
(43)
⯗
⭸u
⫽ [0 0 0 1 0 L P 0 0 2H PL 0 0 2LH 0 0 2PH L2 P 2 3H 2]T
⭸H
(44)
Likewise
AAi dxA ⫽
冤
0…0
⫽
冤 冥
0
⭸FSi
⭸xTA x
dxA
AGi ⫽
冟
0
0
冟
⭸FLi
⭸aL(j) x
0
0
⭸FSi
⭸b0(j) x
0
0
0
冟
0
冟
⭸FSi
⭸bS(j) x
0
冥
0…0
0
冟
⭸FSi
⭸bL(j) x
dxA
⭸FLi
⭸␭k x
⭸FSi
⭸ kx
⭸FSi
⭸ kx
0
0
冤
or
冥冋 册
dxA
dxG
冤 冥
wP
w
⫹␧⫽
A
wG
A dx ⫹ ␧ ⫽ w
冤
0
CA
0
(45)
冥
(46)
(48)
where AA is the first-order design matrix for the image adjustment parameters,
冤冥
AA1
⯗
AA ⫽ A
Ai
(49)
⯗
AAi is the first-order design sub-matrix for the ith image point
on the jth image,
AAi
(50)
冋
册
0 … 0 1 Sample i(j) Line i(j) 0
0
0
0…0
0…0 0
0
0
1 Sample i(j) Line i(j) 0 … 0
AG is the first-order design matrix for the object-space
coordinates,
Ja nuar y 20 03
0
(52)
(53)
wPi is the sub-vector of misclosures for the image-space coordinates of the ith image point on the jth image,
wPi
(47)
0
0
CG
0
⯗
with the a priori covariance matrix of the vector of misclosures, w,
CP
Cw ⫽ 0
0
0…0
0
⭸FSi
⭸hk x
冤冥
Consequently, the RPC block adjustment model in matrix
form reads
AG
0
I
冥
0…0
w P1
⯗
wP ⫽ w
Pi
册
AA
I
0
冟
冟
0
⭸FLi
⭸hk x
wP is the vector of misclosures for the image-space coordinates,
0 … 0 1 Sample i(j) Line i(j) 0
0
0
0…0
0
0
1 Sample i(j) Line i(j) 0 … 0
0…0 0
冟
␭ 冟
⭸FLi
⭸␾k x
0…0
0
⭈ [… da0(j) daS(j) daL(j) db0(j) dbS(j) dbL(j)…]T
66
冤
0…0
⭸FLi
⭸aS(j) x
0
冟
␾ 冟
0…0
冋
⫽
AGi is the first-order design matrix for the object-space coordinates of the kth ground control or tie point being the ith image
point on the jth image, with the elements of AGi computed by
Equations 38 to 44,
0
⭸FLi
⭸a0(j) x
0…0
⫽
冟
冟
冟
⭸FLi
⭸xTA x
(51)
⫽
冋
Line(j)
⫺ a(j)
⫺ a(j)
⭈ Sample (j)
⫺ a(j)
⭈ Line (j)
⫺ p(j)(␾k0, ␭k0, hk0)
i
0
S
i
L
i
0
0
0
⫺ b(j)
⫺ b(j)
⭈ Sample (j)
⫺ b(j)
⭈ Line (j)
Sample(j)
⫺ r (j)(␾k0, ␭k0, hk0)
i
0
S
i
L
i
0
0
0
册
(54)
wA ⫽ 0 is the vector of misclosures for the image adjustment parameters,
wG ⫽ 0 is the vector of misclosures for the object-space
coordinates,
CP is the a priori covariance matrix of image-space
coordinates,
CA is the a priori covariance matrix of the image adjustment
parameters, and
CG is the a priori covariance matrix of the object-space
coordinates.
It is seen that the proposed math model for block adjustment with RPCs allows for the introduction of a priori information using the Bayesian estimation approach. The Bayesian
approach blurs the distinction between observables and
unknowns — both are treated as random quantities. In the context of least squares, a priori information is introduced in the
form of weighted constraints. A priori uncertainty is expressed
by CA, CP, and CG. CA expresses the uncertainty of a priori
knowledge of the image adjustment parameters. For example,
in an offset only model, the variances of a0 and b0, i.e., the diagonal elements of CA, express the uncertainty of a priori satellite
attitude and ephemeris, as explained in the text. CP expresses
prior knowledge of image-space coordinates for ground control
and tie points. Line and sample variances in CP are set
according to the accuracy of the image measurement process.
CG expresses prior knowledge of object-space coordinates for
ground control and tie points. In the absence of any prior
knowledge of the object coordinates for tie points, the corresponding entries in CG can be made large enough, e.g., 10,000
meters, to produce no significant bias in the solution. Equivalently, one could also remove the weighted constraints for
object coordinates of tie points from the observation equations.
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
Experimental Results
Figure 6. Image and GCP layout for Mississippi Project.
On the other hand, being able to introduce prior information for
the object coordinates of tie points adds flexibility.
It should be noted that, without a priori constraints on the
image adjustment parameters and ground control, there would
exist a datum defect, which would result in rank deficient normal equations. The datum defect can be taken care of by either
using the Bayesian approach, i.e., adding a priori weighted constraints on the image adjustment parameters to the observation
equations as indicated above, or, if available, by using sufficient
ground control. To prevent under- or over-constraining the
solution, the a priori constraints on the image adjustment
parameters must be based on a realistic assessment of prior
knowledge of satellite attitude and ephemeris. Using Bayesian
formulation permits adjusting multiple independent images
together with or without ground control.
Because the math model is non-linear, the least-squares
solution needs to be iterated until convergence is achieved. At
each iteration step, application of the least-squares principle
results in the following vector of estimated corrections to the
approximate values of the model parameters:
⫺1
dx̂ ⫽ (AT C⫺1
AT C⫺1
w A)
w w.
(55)
At the subsequent iteration step, the vector of approximate
model parameters x0 is replaced by the estimated values x̂ ⫽ x0
⫹ dx̂, and the math model is linearized again. The least-squares
estimation is repeated until convergence is reached. The
covariance matrix of the estimated model parameters follows
with
⫺1
Cx̂ ⫽ (AT C⫺1
w A) .
(56)
A project located in Mississippi, with six stereo strips and a
large number of well-distributed GCPs as shown in Figure 6,
was selected to demonstrate the RPC block adjustment technique. Cultural features such as road intersections were used
for ground control and check points.
Each of the 12 source images was produced as a georectified image with RPCs. The images were then loaded onto a
SOCET SET姞 workstation running the custom-developed RPC
block adjustment model as described by Equations 19 and 20.
Multiple well-distributed tie points were measured along the
edges of the images. Ground points were selectively changed
between control and check points to quantify block adjustment
accuracy as a function of the number and distribution of GCPs.
The block adjustment results presented below were obtained
using a simple two-parameter, offset only, model with a priori
values for a0 and b0 of 0 pixels and a priori ␴ of 10 pixels. Average and standard deviation errors for GCPs and check points
were computed for each of the tested GCP scenarios. The results
are summarized in Table 3.
The average errors without ground control were ⫺5.0, 6.2,
and 1.6 meters in longitude, latitude, and height. This illustrates Ikonos accuracy without ground control. The addition of
one ground control point reduced the average error to ⫺2.4,
0.5, and ⫺1.1 meters. While additional ground control further
reduced the average errors, the standard deviation remained
virtually unchanged at 1 meter in longitude and latitude and 2
meters in height. The standard deviation did not appreciably
change until all 40 GCPs were used, at which point the ground
control overwhelmed the tie points and the a priori constraints
and, thus, effectively adjusted each strip separately such that it
minimized control point errors on that individual strip. Similarly impressive accuracy improvements have been reported by
Fraser et al. (2002), further validating the two-parameter bias
compensation approach for Ikonos RPCs.
Horizontal errors for GCPs and check points are plotted in
Figure 7. GCPs are marked with large circles while check points
are denoted by small circles.
The adjusted parameter values for the all-GCP case are tabulated in Table 4. The image identifications follow Ikonos practice: 4-digit year, 2-digit month, 2-digit day, followed by some
other digits, and finally a 5-digit sequence number. Stereo
images were taken on the same orbital path; hence, they have
the same date. Stereo strips are numbered 1 through 6, consecutively, from West to East.
The sample and line offset adjustments are shown for each
image. The adjustments are seen to be small, mostly under 10
pixels, thus demonstrating the high a priori accuracy of uncontrolled Ikonos images.
Conclusions
The RPC camera model provides a simple, fast, and accurate
representation of the Ikonos physical camera model. What has
been lacking thus far is an accurate and robust method for block
adjustment of images described by RPCs. The proposed RPC
TABLE 3. MISSISSIPPI BLOCK ADJUSTMENT RESULTS
GCP
None
1 in center
3 on edge
4 in corners
All
Average Error
Longitude
⫺5.0
⫺2.4
⫺0.4
⫺0.2
0.0
m
m
m
m
m
Average Error
Latitude
6.2
0.5
0.3
0.3
0.0
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
m
m
m
m
m
Average Error
Height
1.6
⫺1.1
0.2
0.0
0.0
m
m
m
m
m
Standard
Deviation
Longitude
0.97
0.95
0.97
0.95
0.55
m
m
m
m
m
Standard
Deviation
Latitude
1.08
1.07
1.06
1.06
0.75
m
m
m
m
m
Standard
Deviation
Height
2.02
2.02
1.96
1.95
0.50
m
m
m
m
m
CE90
LE90
8.2
3.3
2.2
2.2
1.4
3.7
3.5
3.2
3.2
0.8
m
m
m
m
m
Ja nuar y 20 03
m
m
m
m
m
67
directly related to the geometric properties of the physical
camera model. As a result, the RPC block adjustment model is
mathematically simpler and numerically more stable than the
traditional adjustment of exterior and interior orientation
parameters. Furthermore, as demonstrated by simulation and
numerical examples, for the Ikonos satellite imagery the RPC
block adjustment method is as accurate as the Ikonos ground
station block adjustment with the physical camera model.
Because RPC models can describe a variety of sensor systems,
this method is generally applicable to any imaging system with
a narrow field of view, a calibrated stable interior orientation,
and accurate a priori exterior orientation.
References
Figure 7. (a) Horizontal errors without GCPs. (b) Horizontal
errors with one GCP in center. (c) Horizontal errors with three
GCPs on one edge. (d) Horizontal errors with four GCPs in
corners.
TABLE 4. ADJUSTMENTS FOR THE ALL-GCP CASE
Stereo Strip
ID
1
2
3
4
5
6
Image ID
20000704
20000704
20001030
20001030
20000424
20000424
20001030
20001030
20000916
20000916
20000927
20000927
...
...
...
...
...
...
...
...
...
...
...
...
21524
21526
14080
14079
12632
12630
14077
14078
13445
13443
22340
22339
Line Offset
Adjustment
(a0) [pixels]
Sample Offset
Adjustment
(b0) [pixels]
⫺8.2
⫺5.6
⫺9.3
2.5
⫺2.8
⫺1.9
⫺3.9
⫺3.3
⫺8.8
⫺8.0
⫺2.4
⫺9.0
⫺8.4
⫺7.0
⫺16.2
0.3
⫺4.0
⫺8.2
⫺7.5
⫺6.9
⫺3.4
⫺2.1
⫺1.7
⫺12.1
block adjustment method relies on combining multiple physical camera model parameters having the same effect on the
object-image relationship into a single adjustment parameter,
68
Ja nuar y 20 03
Dial, Gene, 2001. IKONOS overview, Proceedings of the High-Spatial
Resolution Commercial Imagery Workshop, 19–22 March, Washington, D.C. (Stennis Space Center, Mississippi), unpaginated
CD ROM.
Dial, Gene, Laurie Gibson, and Rick Poulsen, 2001. IKONOS satellite
imagery and its use in automated road extraction, Automatic
Extraction of Man-Made Objects from Aerial and Space Images
(III) (Emmanuel P. Baltsavias, Armin Gruen, and Luc Van Gool,
editors), A.A. Balkema Publishers, The Netherlands.
Dial, Gene, and Jacek Grodecki, 2002. Block adjustment with rational
polynomial camera models, Proceedings of ASPRS 2002 Conference, 22–26 April, Washington, D.C. (American Society for Photogrammetry, Bethesda, Maryland), unpaginated CD ROM.
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(Received 07 February 2002; accepted 04 June 2002; revised 10 July
2002)
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING