Block Adjustment of High-Resolution Satellite Images Described by Rational Polynomials
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Block Adjustment of High-Resolution Satellite
Images Described by Rational Polynomials
Jacek Grodecki and Gene Dial
Abstract imagery. As demonstrated below, the RPC block adjustment
This paper describes how to block adjust high-resolution model provides a rigorous, accurate method to block adjust
satellite imagery described by Rational Polynomial Coefficient Ikonos data outside of the ground stations.
(RPC) camera models and illustrates the method with an Ikonos This publication of a technique for block adjusting Ikonos
example. By incorporating a priori constraints into the images described by RPC data is motivated by a desire to satisfy
adjustment model, multiple independent images can be the needs of those users who would like to perform their own
adjusted with or without ground control. The RPC block block adjustment, and to ensure that Ikonos images are pro-
adjustment model presented in this paper is directly related cessed in such way as to consistently achieve the highest possi-
to geometric properties of the physical camera model. Multiple ble accuracy. In developing the adjustment model described
physical camera model parameters having the same net effect here, the authors had access to the complete description of
on the object-image relationship are replaced by a single Ikonos imaging geometry, familiarity with all of the satellite
adjustment parameter. Consequently, the proposed method is maneuvering modes, the resources of extensive test ranges and
numerically more stable than the traditional adjustment of imagery with which to test and validate, and the experience
exterior and interior orientation parameters. This method is gained calibrating, testing, and troubleshooting Ikonos metric
generally applicable to any photogrammetric camera with a projects.
narrow field of view, calibrated, stable interior orientation, and
accurate a priori exterior orientation data. As demonstrated Physical Camera Models
in the paper, for Ikonos satellite imagery, the RPC block Owing to the dynamic nature of satellite image collection, pho-
adjustment achieves the same accuracy as the ground station togrammetric processing of satellite imagery is more compli-
block adjustment with the full physical camera model. cated than is aerial frame camera processing. Aerial cameras
acquire the entire image at an instant of time with a unique
Background exposure station and orientation. High-resolution pushbroom
The launch of Ikonos on 24 September 1999 set off a new era of satellite cameras, including Ikonos, use linear sensor arrays
commercially available, high-resolution satellite imagery. that acquire a single image line at an instant of time. Conse-
Overviews of the Ikonos satellite may be found in Dial (2001), quently, each line of a pushbroom satellite image has a different
Dial et al. (2001), and Grodecki and Dial (2001). exposure station and orientation (Grodecki, 2001). Implement-
Rational Polynomial Coefficient (RPC) camera models are ing such a complicated model is expensive, time consuming,
derived from the physical Ikonos sensor model to describe the and error prone.
object-image geometry. RPC models transform three-dimen-
sional object-space coordinates into two-dimensional image- Adjustment Parameters
space coordinates. RPCs provide a simple and accurate means of Interior Orientation
communicating camera object-image relationship from image Interior orientation includes parameters for detector positions,
data provider to image data user (Grodecki, 2001). RPCs have principal point, optical distortion, and focal length. Unlike
been successfully used for the terrain extraction, orthorectifi- film cameras, the Ikonos digital focal plane does not require
cation, and feature extraction tasks. What has been lacking is a fiducial marks. Instead, every pixel is at a fixed, calibrated
method to block adjust imagery described by RPCs. position on the solid-state focal plane. The detectors are rigidly
Dial and Grodecki (2002) outlined the RPC block adjust- attached to the focal plane in a stable thermal-mechanical
ment technique, described in more detail in this article. A sim- environment. The elements of interior orientation have been
ilar method of exterior orientation bias compensation for determined to superb accuracy with well-controlled test-range
Ikonos imagery has been independently proposed by Fraser et imagery. Consequently, it is not necessary, indeed it is not desir-
al. (2002), albeit without any reference to the physical camera able, to estimate corrections to the interior orientation parame-
model. Other investigators have proposed various methods for ters in the block adjustment process.
photogrammetric processing of Ikonos images (Toutin and
Cheng, 2000). These have been hampered by incomplete knowl- Exterior Orientation
edge of the Ikonos camera model, of the maneuvering possible Exterior orientation comprises position and attitude. On-board
during image acquisition, and by limited availability of gener- GPS receivers determine the satellite ephemeris, i.e., camera
ally 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 Photogrammetric Engineering & Remote Sensing
Vol. 69, No. 1, January 2003, pp. 59 – 68.
0099-1112/03/6901–059$3.00/0
Space Imaging LLC, 12076 Grant Street, Thornton, CO 80241 䉷 2003 American Society for Photogrammetry
(jgrodecki@spaceimaging.com; gdial@spaceimaging.com). and Remote Sensing
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING Ja nuar y 20 03 59position 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 accu-
racy, about one meter for the ephemeris and about one or two
arc-seconds for attitude. As demonstrated below, for high-reso-
lution satellite systems the in-track and cross-track position
errors are almost completely correlated with pitch and roll atti-
tude errors so that they cannot be separately estimated. More-
over, 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 (rota-
tion 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 propor-
tional 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 neg-
ligible amount.
Ephemeris Errors
Ephemeris errors are conventionally decomposed into in-track,
cross-track, and radial components. We will first show that in- Figure 1. Effect of roll and cross-track errors.
track and cross-track errors are equivalent to pitch and roll atti- a ⫽ half-angle of the camera field of view; for Ikonos,
tude errors. Then we will show that radial errors are negligible. a ⫽ 28.53⬘.
For narrow field-of-view cameras, small horizontal dis- h ⫽ orbital height; for Ikonos, h ⫽ 680 km.
placements are equivalent to small angular rotations. As a r ⫽ camera roll angle ⫽ 2 arc-seconds.
result, roll errors are completely correlated with cross-track d ⫽ equivalent displacement ⫽ h tan(r) ⫽ 6.593466 m.
errors. The same is the case for pitch and in-track errors. As X1 ⫽ ground coordinate of the left edge of the nominal
shown in Figure 1, for a roll error of 2 arc-seconds, the differ- camera.
ence between the nominal nadir-pointing camera and another ⫽ ⫺h tan(a) ⫽ ⫺5644.129609 m.
camera that has been rotated and correspondingly displaced is X1⬘ ⫽ ground coordinate of the left edge of the displaced
X1 ⫺ X1⬘ ⫽ X2 ⫺ X2⬘ ⫽ 0.000454 m:—less than 1/2000 pixel. and rotated camera.
If the camera field of view is narrow enough and the position ⫽ d ⫺ h tan(a ⫹ r) ⫽ ⫺5644.130063 m.
and attitude errors are small enough such that the non-linear X2 ⫽ ground coordinate of the right edge of the nominal
effects of attitude errors are negligible, then position and atti- camera.
tude cannot be independently observed. The presence of corre- ⫽ h tan(a) ⫽ 5644.129609 m.
lated parameters having near-identical effects leads to X2⬘ ⫽ ground coordinate of the right edge of the displaced
instability of the block adjustment process. Combining the cor- and rotated camera.
related parameters into a single parameter results in numerical ⫽ d ⫹ h tan(a ⫺ r) ⫽ 5644.129155 m.
stability. X1 ⫺ X1⬘ ⫽ X2 ⫺ X2⬘ ⫽ 0.000454 m.
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 than a few pixels per 100 km, and so are negligible for all but
satellites and the aim points of their scans are illustrated by very long strips.
arrows. Satellite A has a slightly longer slant range, but this is
insignificant for a few arc-seconds of pitch error. Satellite A also Required Adjustments
has a slightly different perspective than B, but this is again As demonstrated above, many effects are negligible or com-
insignificant for a few arc-seconds of pitch. Small pitch errors pletely correlated with other effects. As a result, only a few
are thus indistinguishable from in-track ephemeris errors, and parameters are required to effectively model the sensor errors.
those two physical effects are thus best modeled by a single A line offset parameter is required to adjust for errors in the
parameter. line direction and a sample offset parameter is required to
Radial ephemeris errors result in scale errors. For example, adjust for errors in the sample direction. The line parameter
a 1-m radial error from a 680-km orbit height causes a 1.5-ppm absorbs effects of orbit, attitude, and residual interior orienta-
scale factor error that causes a 16-mm positioning error across tion errors in the line direction. The sample parameter absorbs
the approximately 11-km swath width. Radial error effects are the same effects in the sample direction. For longer strips, a
thus negligible for Ikonos. parameter proportional to line can be added to model drift
errors.
Drift Errors
While attitude and ephemeris errors are largely biases, there Rational Polynomial Camera Model
exists the possibility that these errors would drift as a function The Ikonos physical camera model is used at the ground sta-
of time. For example, gyro errors without sufficient compensa- tions to block adjust multiple images. RPCs are subsequently
tion from the star trackers could introduce an error in attitude estimated from the block adjusted physical camera model. The
rate. For Ikonos, these errors have been found to be small, less 78 rational polynomial coefficients, {c1 . . . c20, d2 . . . d20, e1 . . .
60 Ja nuar y 20 03 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSINGwhere
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
Figure 2. Side view of satellite imaging system A with pitch c ⫽ [c1 c2 … c20]T
error P and satellite imaging system B with in-track ephem- d ⫽ [1 d2 … d20]T;
eris error IT.
and
NumS(P, L, H ) eTu
X ⫽ h(, , h) ⫽ ⫽ T (8)
e20, f2 . . . f20}, are subsequently determined by fitting the physi- DenS(P, L, H ) f u
cal camera model, as described in the next section, and are sup-
plied with ortho-kit and stereo images. Given these coeffic- where
ients, the computation of (Line, Sample) is fast, easy, and
accurate. NumS(P, L, H ) ⫽ e1 ⫹ e2L ⫹ e3P ⫹ e4H ⫹ e5LP ⫹ e6LH
The RPC model has previously been described in Grodecki
(2001) but will also be briefly summarized here. The RPC model ⫹ e7PH ⫹ e8L2 ⫹ e9P 2 ⫹ e10H 2 ⫹ e11PLH ⫹ e12L3
relates the object-space (, , h) coordinates to image-space ⫹ e13LP 2 ⫹ e14LH 2 ⫹ e15L2P ⫹ e16P 3 ⫹ e17PH 2 (9)
(Line, Sample) coordinates. The RPC functional model is in the
form of a ratio of two cubic polynomials of object-space coordi- ⫹ e18L H ⫹ e19P H ⫹ e20H ⫽ e u
2 2 3 T
nates. Separate rational functions are used to express the
object-space to line and the object-space to sample coordinates DenS(P, L, H ) ⫽ 1 ⫹ f2L ⫹ f3P ⫹ f4H ⫹ f5LP ⫹ f6LH
relationship. To improve numerical precision, image- and
object-space coordinates are normalized to 具⫺1, ⫹1典 range as ⫹ f7PH ⫹ f8L2 ⫹ f9P 2 ⫹ f10H 2 ⫹ f11PLH ⫹ f12L3
shown below. ⫹ f13LP 2 ⫹ f14LH 2 ⫹ f15L2P ⫹ f16P 3 ⫹ f17PH 2 (10)
Given the object-space coordinates (, , h), where is geo-
detic latitude, is geodetic longitude, and h is height above the ⫹ f18L H ⫹ f19P H ⫹ f20H ⫽ f u
2 2 3 T
ellipsoid, and the latitude, longitude, and height offsets and
scale factors (LAT OFF, LONG OFF, HEIGHT OFF, with
LAT SCALE, LONG SCALE, HEIGHT SCALE ), the calcula-
tion of image-space coordinates begins by normalizing lati- e ⫽ [e1 e2 … e20]T
tude, longitude, and height as follows:
f ⫽ [1 f2 … f20]T
⫺ LAT OFF
P⫽ , (1)
LAT SCALE Using line and sample offsets and scale factors
(LINE OFF, SAMP OFF, LINE SCALE, SAMP SCALE ), the
⫺ LONG OFF de-normalized image-space coordinates (Line, Sample), where
L⫽ , and (2) Line is the image line number expressed in pixels with pixel
LONG SCALE
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
h ⫺ HEIGHT OFF left-most sample, are finally computed as
H⫽ . (3)
HEIGHT SCALE
Line ⫽ Y ⭈ LINE SCALE ⫹ LINE OFF, and (11)
The normalized line and sample image-space coordinates
(Y and X, respectively) are then calculated from their respec- Sample ⫽ X ⭈ SAMP SCALE ⫹ SAMP OFF. (12)
tive rational polynomial functions g(.) and h(.): i.e.,
Determining RPC Coefficients
NumL(P, L, H ) cTu A least-squares approach is used to estimate the RPC model
Y ⫽ g(, , h) ⫽ ⫽ T (4)
DenL(P, L, H ) d u coefficients ci , di , ei , and fi from a three-dimensional grid of
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING Ja nuar y 20 03 61points, depicted schematically in Figure 3, generated using the RPC Block Adjustment Math Models
physical camera model. It should be noted that, as pointed out
in Hu and Tao (2001) and Tao and Hu (2001), attempts to use Proposed RPC Block Adjustment Model
ground control only to determine RPC coefficients risks numer- The RPC block adjustment math model proposed in this paper is
ical instability during the fitting process and poor compliance defined in the image space. It uses denormalized RPC models,
with camera physics. p and r, to express the object-space to image-space relationship,
and the adjustable functions, ⌬p and ⌬r, which are added to the
Evaluation of RPC Accuracy rational functions to capture the discrepancies between the
Accuracy of the RPC model was assessed using the physical nominal and the measured image-space coordinates. For each
Ikonos camera model as a reference. RPCs were fitted to a grid image point i on image j, the RPC block adjustment math model
of points with ground-space coordinates generated from the is thus defined as follows:
image-space coordinates, for a set of different elevation levels,
using the Ikonos physical camera model (see Grodecki (2001) Line i(j) ⫽ ⌬p(j) ⫹ p(j)(k, k, hk) ⫹ Li (13)
and Grodecki and Dial (2001)).
Independent check points were subsequently used to Sample i(j) ⫽ ⌬r (j) ⫹ r (j)(k, k, hk) ⫹ Si (14)
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⬚ where
for pitch, 0⬚ to 360⬚ for scan azimuth, and 0⬚ to 60⬚ for latitude. Linei(j) and Samplei(j) are measured (on image j ) line and
The check-point residual errors were 0.01 pixels RMS and 0.04 sample coordinates of the ith image point, corresponding
pixels worst case for all imaging scenarios, thus demonstrating to the kth ground control or tie point with object space
the extremely high accuracy of the RPC camera model represen- coordinates (k ,k ,hk);
tation (Grodecki, 2001). ⌬p(j) and ⌬r (j) are the adjustable functions expressing the
differences between the measured and the nominal line
Block Adjustment with RPCs and sample coordinates of ground control and/or tie points,
RPC data provided with Ikonos imagery has, up to now, enabled for image j;
the user to perform feature extraction, DEM generation, and Li and Si are random unobservable errors;
orthorectification. Until now, however, photogrammetric block p(j) and r (j) are the given line and sample, denormalized
adjustment with RPCs has been considered to be unfeasible. As RPC models for image j;
demonstrated below, the approach presented in this paper pro-
vides a rigorous, consistent, and accurate block adjustment p(, , h) ⫽ g(, , h) ⭈ LINE SCALE (15)
method for high-resolution satellite imagery described by RPCs.
⫹ LINE OFF; and
The proposed RPC block adjustment model is directly related to
the geometric properties of the physical camera model, by com-
bining multiple physical camera model parameters into a sin- r(, , h) ⫽ h(, , h) ⭈ SAMPLE SCALE (16)
gle adjustment parameter having the same net effect on the
object-image relationship. Consequently, the proposed ⫹ SAMPLE OFF.
method is numerically more stable than the traditional adjust-
ment of exterior and interior orientation parameters. This We are proposing to use a polynomial model defined in the
method is generally applicable to any photogrammetric camera domain of image coordinates to represent the adjustable func-
with a narrow field of view, a calibrated stable interior orienta- tions, ⌬p and ⌬r, which in general can be expressed as
tion, and accurate a priori exterior orientation data.
⌬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 adjust-
able 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:
⌬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 (Equa-
tions 19 and 20) has physical significance. As a result, the pro-
Figure 3. RPC fitting. posed RPC block adjustment model does not present the
numerical ill-conditioning problems of classical techniques.
62 Ja nuar y 20 03 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSINGParameter 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 dis-
placing images in line. Similarly, parameter b0 absorbs cross-
track 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 negligi-
bly 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 repre-
sented by a polynomial model defined in the domain of object
coordinates as
⌬p ⫽ a0 ⫹ aP ⭈ ⫹ aL ⭈ ⫹ aH ⭈ h ⫹ aP2 ⭈ 2 ⫹ aL2 ⭈ 2 (21)
⫹ aH2 ⭈ h ⫹ aPL ⭈ ⭈ ⫹ aPH ⭈ ⭈ h ⫹ aLH ⭈ ⭈ h ⫹ …
2 Figure 4. Evaluation of RPC adjustment models.
⌬r ⫽ b0 ⫹ bP ⭈ ⫹ bL ⭈ ⫹ bH ⭈ h ⫹ bP2 ⭈ 2 ⫹ bL2 ⫹ 2 (22)
⫹ bH2 ⭈ h ⫹ bPL ⭈ ⭈ ⫹ bPH ⭈ ⭈ h ⫹ bLH ⭈ ⭈ h ⫹ …
2
model was utilized. The differences (⌬L, ⌬S) between the origi-
nal and the perturbed image coordinates were subsequently
As shown later, adjustment models defined in the domain
calculated, and used as input to the tested adjustment models.
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 Evaluation of Image-Space Adjustment Models Defined in
model in the object space. The object-space RPC block adjust- the Domain of Image Coordinates
ment math model, for the kth ground control or tie point being The following combinations of the proposed adjustment mod-
the ith image point on the jth image, is then defined as follows: els were evaluated:
(1) ⌬p ⫽ a0 and ⌬r ⫽ b0
Linei(j) ⫽ p(j)(k ⫹ ⌬ (j), k ⫹ ⌬ (j), hk ⫹ ⌬h(j)) ⫹ Li (23) (2) ⌬p ⫽ a0 ⫹ aL ⭈ Line and ⌬r ⫽ b0 ⫹ bL ⭈ Line
(3) ⌬p ⫽ a0 ⫹ aS ⭈ Sample and ⌬r ⫽ b0 ⫹ bS ⭈ Sample
(4) ⌬p ⫽ a0 ⫹ aS ⭈ Sample ⫹ aL ⭈ Line and ⌬r ⫽ b0 ⫹ bS ⭈ Sample
Samplei(j) ⫽ r (j)(k ⫹ ⌬ (j), k ⫹ ⌬ (j), hk ⫹ ⌬h(j)) ⫹ Si (24)
⫹ bL ⭈ Line
where ⌬ (j), ⌬ (j), and ⌬h(j) are the adjustable functions express- where a0, b0, aL , bL , aS , bS are the image adjustment parameters,
ing the differences between the measured and the nominal and Line, Sample are the line and sample coordinates of a
object-space coordinates of a ground control or tie point, for ground control or a tie point.
image j. For each of the tested RPC adjustment models, the adjust-
As before, the object-space adjustment model can be repre- able parameters (a0, b0, aL , bL , aS , bS) were estimated using the
sented either by a polynomial model defined in the domain of least-squares approach. The post-fit RMS errors and the maxi-
image space or by a polynomial model defined in the domain of mum residual errors were then computed for each of the tested
object coordinates. We do not recommend the object-space RPC adjustment models.
block adjustment math model, because it is nonlinear in the A number of scenarios were generated to thoroughly test
adjustment parameters and unrelated to imaging geometry. the proposed adjustment models using a wide range of feasible
imaging conditions. The imaging parameters ranged from 0⬚ to
Evaluation of RPC Adjustment Models 30⬚ for roll, 0⬚ to 30⬚ for pitch, 0⬚ to 360⬚ for scan azimuth, and
The accuracy of the proposed RPC adjustment models was eval- 0⬚ to 60⬚ for latitude. The image strip length was varied from 10
uated by numerical simulation, using the perturbation km to 100 km. The minimum elevation angle was set to 50
approach shown in Figure 4. Errors in satellite vehicle ephem- degrees. The errors in the satellite vehicle ephemeris and atti-
eris and attitude were propagated, for a grid of image points, tude were set to
down to the object (, , h) space. The so determined perturbed ● 3 meters in the ephemeris components (in-track, cross-track,
ground coordinates were then projected back to the image radial), and
(Line, Sample) space. In both cases the Ikonos physical camera ● 2 arc-seconds in the attitude angles (pitch, roll, yaw).
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING Ja nuar y 20 03 63Maximum residual and RMS errors, from all imaging sce- TABLE 2. EVALUATION OF ADJUSTMENT MODELS DEFINED IN THE DOMAIN OF
narios, were then computed giving the measure of the worst OBJECT COORDINATES
possible math model errors when using the proposed RPC block RMS Max. RMS Max.
adjustment approach. The results of the analysis are shown in Sample Sample Line Line
Table 1. Strip Adjustment Error Error Error Error
The results given in Table 1 show that the postulated RPC Length Model [pixels] [pixels] [pixels] [pixels]
adjustment models can accurately model the effects of ephem-
10 km (a) 0.15 0.09 0.12 0.08
eris and attitude errors. Bias only models (parameters ⌬p ⫽ a0 (b) 0.09 0.06 0.16 0.07
and ⌬r ⫽ b0 only) are effective for strip lengths up to 50 km. (c) 0.002 0.001 0.002 0.001
Strips of 100 km length may require the addition of drift param- 20 km (a) 0.27 0.17 0.20 0.09
eters (aL and bL) for full accuracy. Parameters proportional to (b) 0.11 0.06 0.23 0.11
sample (aS and bS) and higher order terms are not normally (c) 0.01 0.002 0.01 0.003
required. 50 km (a) 0.52 0.27 0.42 0.18
(b) 0.25 0.11 0.61 0.29
Evaluation of Image-Space Adjustment Models Defined in (c) 0.02 0.01 0.04 0.02
the Domain of Object-Space Coordinates 100 km (a) 1.07 0.56 0.84 0.36
(b) 0.76 0.40 1.48 0.66
The following adjustment models were tested: (c) 0.75 0.29 0.53 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 ⭈
RPC models expressing the object-space to image-space rela-
where a0, b0, aL , bL , aP , bP are the image adjustment parameters;
and , are the object space coordinates of a ground control (or tionship for each image, are tied together by tie points whose
tie) point. image coordinates are measured on those images. Optionally,
As before, a number of scenarios were generated for this the block may also have ground control points with known or
purpose — using the same ranges of scanning azimuth, roll approximately known object-space coordinates and measured
and pitch angles, and geographic location. Maximum residual image positions (see Figure 5).
and RMS errors, from all imaging scenarios, are shown in Table 2. Indexing of observation equations is based on image-point
It is seen that, for strips up to 20 km long, the image-space indices i. Because there is only one observation equation per
adjustment models defined in object-space coordinates give image point, index i uniquely identifies each observation equa-
virtually the same results as the image-space adjustment mod- tion. Thus, for the kth ground control or tie point being the ith
els defined in image space coordinates. However, as seen in the image point on the jth image, the RPC block adjustment observa-
previous section, for longer strips adjustment models defined tion equations read
in image space produce superior accuracy. Moreover, as indi- (j)
cated earlier, the image-space adjustment models defined in FLi ⫽ ⫺Line i ⫹ ⌬p(j) ⫹ p(j)(k, k, hk) ⫹ Li ⫽ 0 (25)
image-space coordinates are also much more closely related to (j)
the geometric properties of the physical camera model. As a FSi ⫽ ⫺Sample i ⫹ ⌬r (j) ⫹ r (j)(k, k, hk) ⫹ Si ⫽ 0 (26)
result, the proposed RPC block adjustment will utilize the
image-space coordinate formulation. with
RPC Block Adjustment Algorithm ⌬p(j) ⫽ a0(j) ⫹ aS(j) ⭈ Sample i(j) ⫹ aL(j) ⭈ Line i(j) (27)
Multiple overlapping images can be block adjusted using one of
the RPC adjustment models given above. As indicated above, ⌬r (j) ⫽ b0(j) ⫹ bS(j) ⭈ Sample i(j) ⫹ bL(j) ⭈ Line i(j) (28)
the preferred approach uses the image-space adjustment model
given by Equations 19 and 20. The overlapping images, 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),
TABLE 1. EVALUATION OF ADJUSTMENT MODELS DEFINED IN THE DOMAIN OF
b0(j), bS(j), bL(j)) and the object-space coordinates (k , k , hk) com-
IMAGE COORDINATES
prise the unknown adjustment model parameters. Line i(j) and
Max. RMS Max. RMS Sample i(j) coordinates in Equations 27 and 28 are the approxi-
Sample Sample Line Line mate fixed values for the true image coordinates. One possible
Strip Adjustment Error Error Error Error
Length Model [pixels] [pixels] [pixels] [pixels]
10 km (1) 0.21 0.10 0.21 0.09
(2) 0.15 0.10 0.12 0.08
(3) 0.09 0.06 0.10 0.07
(4) 0.004 0.001 0.001 0.001
20 km (1) 0.28 0.13 0.32 0.15
(2) 0.15 0.10 0.12 0.08
(3) 0.19 0.12 0.22 0.13
(4) 0.01 0.002 0.004 0.001
50 km (1) 0.57 0.29 0.66 0.34
(2) 0.16 0.10 0.13 0.08
(3) 0.50 0.28 0.58 0.33
(4) 0.02 0.01 0.02 0.01
100 km (1) 1.00 0.51 1.25 0.66
(2) 0.17 0.10 0.17 0.08
(3) 0.93 0.50 1.17 0.65 Figure 5. Block adjustment of multiple overlapping images.
(4) 0.07 0.03 0.06 0.03
64 Ja nuar y 20 03 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSINGchoice for the approximate line and sample coordinates in x0 is the vector of the approximate model parameters,
Equations 27 and 28 are the values of the measured image coor-
dinates. 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
x0 ⫽ 冋 册
xA0
x G0
(36)
negligible because the measurements of image coordinates are
and is a vector of unobservable random errors.
performed with sub-pixel accuracy.
For the kth ground control or tie point being the ith image
With
point on the jth image, we get
冋 册 冟
冤 冥
FLi ⭸FLi
Fi ⫽ , (29) ⭸xTG x
FSi
AGi dxG ⫽ 0
冟
dxG (37)
⭸FSi
application of the Taylor Series expansion to the RPC block ⭸xTG x
0
adjustment observation Equations 25 and 26 results in the fol-
冟 冟 冟 ⯗
冤 冥冤 冥
lowing linearized model: ⭸FLi ⭸FLi ⭸FLi
0…0 0…0 d k
⭸ k x ⭸ k x ⭸hk x
⫽ 0 0 0
d k
冟 冟 冟
Fi0 ⫹ dFi ⫹ ⫽ 0 (30) ⭸FSi ⭸FSi ⭸FSi
0…0 0…0 dhk
⭸ kx ⭸ kx ⭸hk x ⯗
0 0 0
where
where
Fi0 ⫽ 冋 册
FLi0
FSi0
(31)
冋 ⭸FLi
⭸ k
⭸FLi
⭸ k
⭸FLi
⭸hk
⫽ 册 冋
⭸p(j)
⭸ k
⭸p(j)
⭸k
册
⭸p(j)
⭸hk
(38)
冤 冥
⫺Linei(j) ⫹ a0(j) ⫹ aS(j) ⭈ Sample i(j)
0 0 and
⫹ aL(j) ⭈ Line i(j) ⫹ p(j)(k0, k0, hk0)
冋 册 冋 册
⫽ 0
⫽ ⫺wPi
⫺Samplei(j) ⫹ b0(j) ⫹ bS(j) ⭈ Sample i(j) ⭸FSi ⭸FSi ⭸FSi ⭸r (j) ⭸r (j) ⭸r (j)
0 0 ⫽ , (39)
⫹ bL(j) ⭈ Line i(j) ⫹ r (k0, k0, hk0)
(j) ⭸ k ⭸ k ⭸hk ⭸ k ⭸ k ⭸hk
0
with
冟 冋 册 冋 册
冤 冥
⭸FLi
⭸p ⭸p ⭸p cT(dTu) ⫺ dT(cTu) ⭸u ⭸u ⭸u
冋 册
dFi ⫽
dFLi
冋 册
⫽ 0
⭸xT x
⭸ ⭸ ⭸h
⫽
(dTu)2
⭈
⭸P ⭸L ⭸H
冟
dx (32)
dFSi ⭸FSi
⭸xT x 1
0 0
0
LAT SCALE
冟 冟
冤 冟 冟 冥冋
⭸FLi ⭸FLi 1
⭈ 0 0 (40)
册 冋 册
⭸xTA x ⭸xTG x LONG SCALE
dxA dxA
⫽ 0 0
⫽ [AAi AGi] 1
⭸FSi ⭸FSi dxG dxG 0 0
HEIGHT SCALE
⭸xTA x ⭸xTG x
0 0
⭈ LINE SCALE
dx ⫽ x ⫺ x0 is the vector of unknown corrections to the approxi-
冋 ⭸r ⭸r ⭸r
册 eT(f Tu) ⫺ f T(eTu) ⭸u
冋 ⭸u ⭸u
册
冋 册
mate model parameters, x0,
⫽ ⭈
⭸ ⭸ ⭸h (f Tu)2 ⭸P ⭸L ⭸H
dx ⫽ 冋 册 dxA
dxG
(33) 1
LAT SCALE
0 0
1
dxA is the sub-vector of the corrections to the approximate ⭈ 0 0 (41)
LONG SCALE
image adjustment parameters for n images, 1
0 0
HEIGHT SCALE
dxA ⫽ [da(1)
0
da(1)
S
da(1)
L
db(1)
0
db(1)
S
db(1)
L
(34)
… da0(n) daS(n) daL(n) db0(n) dbS(n) dbL(n)]T ⭈ SAMPLE SCALE
in which
dxG is the sub-vector of the corrections to the approximate
object space coordinates for m ground control and p tie points, ⭸u
⫽ [0 0 1 0 L 0 H 0 2P 0 LH 0 2LP 0 L2 3P 2 H2 0 2PH 0]T
⭸P
dxG ⫽ [d1 d1 dh1 … dm⫹p dm⫹p dhm⫹p]T (35) (42)
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING Ja nuar y 20 03 65⭸u
冤冥
⫽ [0 1 0 0 P H 0 2L 0 0 PH 3L2 P 2 H 2 2LP 0 0 2LH 0 0]T AG1
⭸L
⯗
(43) AG ⫽ A (51)
Gi
⯗
⭸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
AGi is the first-order design matrix for the object-space coordi-
(44) nates of the kth ground control or tie point being the ith image
point on the jth image, with the elements of AGi computed by
Likewise Equations 38 to 44,
⭸FLi
冟 冟 冟 冟
冤 冥
⭸FLi ⭸FLi ⭸FLi
冤 冥
⭸xTA x 0…0 0…0
AAi dxA ⫽ 0
dxA ⭸k x ⭸k x ⭸hk x
冟 AGi ⫽ 0 0 0
⭸FSi
冟 冟 冟
(52)
⭸FSi ⭸FSi ⭸FSi
⭸xTA x 0…0 0…0
0
⭸ kx ⭸ kx ⭸hk x
冟 冟 冟
0 0 0
⭸FLi ⭸FLi ⭸FLi
冤 冥
0…0 0 0 0 0…0
⭸a0(j) x ⭸aS(j) x ⭸aL(j) x wP is the vector of misclosures for the image-space coordinates,
⫽ 0 0 0
dxA
0…0 0 0 0
⭸FSi
冟 ⭸FSi
冟 ⭸FSi
冟 0…0
冤冥
⭸b0(j) x ⭸bS(j) x ⭸bL(j) x w P1
0 0 0
冋 册
⯗
0 … 0 1 Sample i(j) Line i(j) 0 0 0 0…0 wP ⫽ w (53)
⫽ Pi
0…0 0 0 0 1 Sample i(j) Line i(j) 0 … 0
⯗
⭈ [… da0(j) daS(j) daL(j) db0(j) dbS(j) dbL(j)…]T
(45)
wPi is the sub-vector of misclosures for the image-space coordi-
Consequently, the RPC block adjustment model in matrix nates of the ith image point on the jth image,
form reads
wPi
冤 冥冋 册 冤 冥
AA AG wP
0
I 0
I
dxA
dxG
⫹⫽ w
wG
A (46) ⫽ 冋 Line(j)
i
Sample(j)
i
⫺ a(j)
0
⫺ b(j)
0
0
⫺ a(j)
S
⫺ b(j)
S0
⭈ Sample (j)
0 i
⭈ Sample (j)
0 i
⫺ a(j)
L
⫺ b(j)
L
⭈ Line (j)
0 i
⭈ Line (j)
i
⫺ p(j)(k0, k0, hk0)
⫺ r (j)(k0, k0, hk0)
0
册
or (54)
wA ⫽ 0 is the vector of misclosures for the image adjust-
A dx ⫹ ⫽ w (47) ment parameters,
wG ⫽ 0 is the vector of misclosures for the object-space
with the a priori covariance matrix of the vector of misclo- coordinates,
sures, w, CP is the a priori covariance matrix of image-space
coordinates,
CA is the a priori covariance matrix of the image adjustment
冤 冥
CP 0 0
parameters, and
Cw ⫽ 0 CA 0 (48)
CG is the a priori covariance matrix of the object-space
0 0 CG coordinates.
It is seen that the proposed math model for block adjust-
where AA is the first-order design matrix for the image adjust- ment with RPCs allows for the introduction of a priori informa-
ment parameters, tion using the Bayesian estimation approach. The Bayesian
approach blurs the distinction between observables and
unknowns — both are treated as random quantities. In the con-
冤冥
AA1 text of least squares, a priori information is introduced in the
⯗ form of weighted constraints. A priori uncertainty is expressed
AA ⫽ A (49) by CA, CP, and CG. CA expresses the uncertainty of a priori
Ai
⯗ knowledge of the image adjustment parameters. For example,
in an offset only model, the variances of a0 and b0, i.e., the diag-
onal elements of CA, express the uncertainty of a priori satellite
AAi is the first-order design sub-matrix for the ith image point attitude and ephemeris, as explained in the text. CP expresses
on the jth image, 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.
AAi (50)
CG expresses prior knowledge of object-space coordinates for
⫽ 冋 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 册 ground control and tie points. In the absence of any prior
knowledge of the object coordinates for tie points, the corres-
ponding entries in CG can be made large enough, e.g., 10,000
meters, to produce no significant bias in the solution. Equiva-
AG is the first-order design matrix for the object-space lently, one could also remove the weighted constraints for
coordinates, object coordinates of tie points from the observation equations.
66 Ja nuar y 20 03 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSINGExperimental Results
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 tech-
nique. Cultural features such as road intersections were used
for ground control and check points.
Each of the 12 source images was produced as a georecti-
fied 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. Aver-
age and standard deviation errors for GCPs and check points
were computed for each of the tested GCP scenarios. The results
Figure 6. Image and GCP layout for Mississippi Project.
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 illus-
trates Ikonos accuracy without ground control. The addition of
one ground control point reduced the average error to ⫺2.4,
On the other hand, being able to introduce prior information for
0.5, and ⫺1.1 meters. While additional ground control further
the object coordinates of tie points adds flexibility.
reduced the average errors, the standard deviation remained
It should be noted that, without a priori constraints on the
virtually unchanged at 1 meter in longitude and latitude and 2
image adjustment parameters and ground control, there would
meters in height. The standard deviation did not appreciably
exist a datum defect, which would result in rank deficient nor-
change until all 40 GCPs were used, at which point the ground
mal equations. The datum defect can be taken care of by either
control overwhelmed the tie points and the a priori constraints
using the Bayesian approach, i.e., adding a priori weighted con-
and, thus, effectively adjusted each strip separately such that it
straints on the image adjustment parameters to the observation
minimized control point errors on that individual strip. Simi-
equations as indicated above, or, if available, by using sufficient
larly impressive accuracy improvements have been reported by
ground control. To prevent under- or over-constraining the
Fraser et al. (2002), further validating the two-parameter bias
solution, the a priori constraints on the image adjustment
compensation approach for Ikonos RPCs.
parameters must be based on a realistic assessment of prior
knowledge of satellite attitude and ephemeris. Using Bayesian Horizontal errors for GCPs and check points are plotted in
formulation permits adjusting multiple independent images Figure 7. GCPs are marked with large circles while check points
together with or without ground control. are denoted by small circles.
Because the math model is non-linear, the least-squares The adjusted parameter values for the all-GCP case are tabu-
solution needs to be iterated until convergence is achieved. At lated in Table 4. The image identifications follow Ikonos prac-
each iteration step, application of the least-squares principle tice: 4-digit year, 2-digit month, 2-digit day, followed by some
results in the following vector of estimated corrections to the other digits, and finally a 5-digit sequence number. Stereo
approximate values of the model parameters: images were taken on the same orbital path; hence, they have
the same date. Stereo strips are numbered 1 through 6, consec-
utively, from West to East.
dx̂ ⫽ (AT C⫺1
w A)
⫺1
AT C⫺1
w w. (55) The sample and line offset adjustments are shown for each
image. The adjustments are seen to be small, mostly under 10
At the subsequent iteration step, the vector of approximate pixels, thus demonstrating the high a priori accuracy of uncon-
model parameters x0 is replaced by the estimated values x̂ ⫽ x0 trolled Ikonos images.
⫹ dx̂, and the math model is linearized again. The least-squares
estimation is repeated until convergence is reached. The Conclusions
covariance matrix of the estimated model parameters follows The RPC camera model provides a simple, fast, and accurate
with representation of the Ikonos physical camera model. What has
been lacking thus far is an accurate and robust method for block
Cx̂ ⫽ (AT C⫺1 ⫺1
w A) . (56) adjustment of images described by RPCs. The proposed RPC
TABLE 3. MISSISSIPPI BLOCK ADJUSTMENT RESULTS
Standard Standard Standard
Average Error Average Error Average Error Deviation Deviation Deviation
GCP Longitude Latitude Height Longitude Latitude Height CE90 LE90
None ⫺5.0 m 6.2 m 1.6 m 0.97 m 1.08 m 2.02 m 8.2 m 3.7 m
1 in center ⫺2.4 m 0.5 m ⫺1.1 m 0.95 m 1.07 m 2.02 m 3.3 m 3.5 m
3 on edge ⫺0.4 m 0.3 m 0.2 m 0.97 m 1.06 m 1.96 m 2.2 m 3.2 m
4 in corners ⫺0.2 m 0.3 m 0.0 m 0.95 m 1.06 m 1.95 m 2.2 m 3.2 m
All 0.0 m 0.0 m 0.0 m 0.55 m 0.75 m 0.50 m 1.4 m 0.8 m
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING Ja nuar y 20 03 67directly 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.
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