AbstractThe demand for higher quality metric products from high-resolution satellite imagery (HRSI) is growing, and the numberof HRSI sensors and product options is increasing. There is agreater need to fully understand the potential and indeedshortcomings of alternative photogrammetric sensor orienta-tion models for HRSI. To date, rational functions have provento be a viable alternative model for geo-positioning, andwith the recent innovation of bias-compensated RPC bundleadjustment, it has been demonstrated that sensor orientationto sub-pixel level can be achieved with minimal groundcontrol. Questions have lingered, however, as to the generalsuitability of bias-compensated rational polynomial coeffi-cients (RPCs), and indeed rational functions in general. Thepurpose of this paper is to demonstrate the wide applicabilityof bias-compensated RPCs for high-accuracy geopositioningfrom stereo HRSI. The case of stereo imagery over mountain-ous terrain will be specifically addressed, and results ofexperimental testing of both Ikonos and QuickBird imagerywill be presented.
IntroductionAs a sensor orientation model for stereo satellite imageconfigurations, rational functions have a history of applica-tion spanning nearly two decades (Dowman and Doloff,2000). The use of the rational function model, with its 80Rational Polynomial Coefficients, termed RPCs, primarilyfound application in military mapping, and it was not untilthe launch of the Ikonos high-resolution imaging satellite inSeptember 1999 that the civilian photogrammetric commu-nity had to take note of this alternative or replacementmodel for sensor orientation and ground point determina-tion from stereo line scanner imagery. Indeed, the commer-cial photogrammetric industry had little option but toembrace RPC-based restitution, since this was the onlymeans provided by Space Imaging, Inc. for customers toextract accurate object space information from Ikonosimagery.
Although it is fair to say that there was initially someunease associated with the employment of RPCs, it quicklybecame clear that the metric accuracy potential of Ikonoswould not be compromised through use of Space Imagingproduced RPCs, which as reported by Grodecki (2001), mod-elled the rigorous sensor orientation to an accuracy of betterthan 0.05 pixels. Moreover, experimental testing with stereoand multi-image Ikonos Geo-image configurations had shownthat sub-pixel ground point determination is readily achiev-able with Ikonos RPCs, once the inherent sensor orientationbias errors (which also affect the rigorous model) had been
Bias-compensated RPCs for Sensor Orientationof High-resolution Satellite Imagery
Notwithstanding the very impressive results obtainedwith Ikonos image restitution using the bias-compensatedRPC approach, and the already demonstrated equivalenceof the RPCs and the rigorous sensor orientation model fromwhich they were derived, uncertainties persisted. Muchof this can be attributed to the false association of vendorproduced RPCs with those empirically determined by usersthrough the use of ground control points (GCPs). More curi-ous, however, were reports that RPCs supplied with the HRSIwould be somehow influenced by the nature of the terrainbeing imaged (e.g., Cheng et al., 2003).
With the successful launch of the QuickBird satellite inOctober 2001, along with the decision by DigitalGlobe, Inc.to provide all associated orbit ephemeris, sensor attitudeand camera model data with the imagery, the options forphotogrammetric restitution of HRSI increased. It was nowpossible to utilize rigorous sensor orientation models, whichwere generally collinearity-based and which had previouslybeen employed in the commercial photogrammetric sectorwith SPOT, MOMS, and IRS satellite imagery. RPCs are alsosupplied with QuickBird imagery, and Robertson (2003) hasreported agreement between the rational function model andthe rigorous sensor model of generally between 0.1 and0.3 pixels RMSE for QuickBird imagery. He has also observedthat such levels of discrepancy will typically be dwarfedby other errors in any orthorectification process. To theauthors’ knowledge, a more comprehensive assessment ofthe integrity of QuickBird RPCs with respect to the rigorousmodel has yet to be reported.
Another interesting factor in any discussion of rigorousmodels versus RPC models is that some well-known digitalphotogrammetric systems which employ rigorous modelapproaches for stereo QuickBird imagery actually use systemgenerated RPCs within the modelling process. It would thusappear that lingering concerns about the quality of RPCs,and especially those derived from the rigorous sensororientation and supplied with the imagery, can be put aside.But, statements such as those by Cheng et al. (2003) con-cerning how RPCs do not have a very high degree of accu-racy in high relief areas have the effect of casting some doubtupon the rational function approach. As a consequence, theauthors aim in this paper to demonstrate that RPCs suppliedwith HRSI, for both Ikonos and QuickBird, can produceaccuracies commensurate with rigorous model approachesirrespective of the nature of the scene topography or the sizeof the scene.
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Rational FunctionsThe RPC ModelThe 80-parameter RPC model provides a direct mapping from3D object space coordinates (usually offset normalised latitude,longitude, and height) to 2D image coordinates (usually offsetnormalized line and sample values). Here, we give only acursory account of the RPC model, in order to highlight therole of additional parameters which can be added to providea bias compensation. For a more comprehensive account ofrational functions as applied to HRSI, readers are referred, forexample, to Tao & Hu (2002), Di et al. (2003) and Grodecki &Dial (2003). For the present discussion we present the modelin the simple form of
(1)
where l and s are line and sample coordinates and Fi arethird-order polynomial functions of object space coordinatesU, V and W. In the same way, as do the similar lookingcollinearity equations, Equation 1 describes an imaging rayfrom object to image space, which we will consider to belongto a bundle of rays (notwithstanding the lack of a trueperspective centre). The provision of two corresponding raysaffords a solution for the U, V, W coordinates using anindirect least-squares model. If one imagines that spatialintersections of all corresponding rays forming the two ormore bundles involved are determined, then the net outcomeis equivalent to a photogrammetric relative orientation, whichwill also be equivalent to that derived through a rigorousmodel to the accuracy tolerance previously mentioned.
Why a relative and not an absolute orientation? Theanswer to this lies both in the inherent limitations in directlydetermining the true spatial orientation of every scan line,and in errors within the direct measurement of sensor ori-entation, especially attitude, but also position and velocity.Errors in sensor orientation within HRSI can, fortuitously, bemodelled as biases in image space, primarily due to the verynarrow field-of-view of the satellite line scanner (approachinga parallel projection for practical purposes) and the natureof the error signals. In the simplest case, small attitude orephemeris errors are equivalent to shifts in image space coor-dinates. But, more than simple translation may be involved.Time-dependent errors in attitude sensors, for example, cangive rise to drift effects in the image coordinates and even toan affine distortion of the image. More subtle, higher-orderresidual distortions, for example in gyro systems and in scanvelocity, may also be present, but we will keep the errorcompensation model in this case to first order.
Bias-compensated RPC ModelIncorporation of image shift and drift terms into the basicmodel of Equation 1 yields a bias-compensated RPC model,which takes the form:
(2)
Within this formulation there are three logical choices ofadditional parameter (AP) sets:
1. A0, A1, . . . B2, which describe an affine transformation.2. A0, A1, B0, B1, which model shift and drift.3. A0, B0, which affect an image coordinate translation.
s � B0 � B1l � B2s �F3 (U,V,W)F4 (U,V,W)
.
l � A0 � A1l � A2s �F1 (U,V,W)F2 (U,V,W)
s �F3(U,V,W)F4(U,V,W)
l �F1(U,V,W)F2(U,V,W)
The solution of the additional parameters of Equation 2 canbe carried out using a multi-image, multi-point bundleadjustment as developed by Fraser & Hanley (2003) andGrodecki & Dial (2003). The model of Equation 2 has alsobeen referred to as the adjustable RPC model (Ager, 2003).
Within the context of relative and absolute orientation, thecase of the shift parameters A0, B0 alone is clearly one ofwhere, effectively, there is a shape-invariant transformation ofthe relatively oriented assemblage to an absolutely orientedmodel. To affect this, only one GCP is required. More GCPs willof course enhance precision, but their number and location isnot important. It is hard to see how this relative-to-absoluteorientation process could be influenced by terrain height orruggedness, and indeed we will demonstrate that terrain seemsto have no impact on the bias-compensated RPC approach.
In the case of the full affine correction model (Case 1)and the shift-and-drift model (Case 2), the situation is slightlydifferent, at least when the parameters A1 and B1, or A1, B1,A2, and B2 are statistically significant. In these cases theabsolute orientation process does imply a modification of theRPC relative orientation of Equation 1, especially since a non-conformal transformation of image coordinates is takingplace. Here, both the number and location of GCPs is impor-tant, with a practical minimum number being 4 to 6. As willbe seen in the following sections, however, with Ikonosreverse scanned imagery the parameters A1, A2, B1, and B2are rarely significant. This means that one need only worryabout providing one GCP to compensate for the shifts A0 andB0. With QuickBird imagery, the authors’ experience suggeststhat the shift-and-drift and affine AP models can in caseslead to measurable improvements in the accuracy of sensororientation and geo-positioning using bias-compensatedRPCs (e.g. Noguchi et al., 2004). Thus, there is a very slightprospect of the nature of the scene topography influencingthe geo-positioning since the relative-to-absolute orientationprocess does not constitute a shape-invariant transformation.
Correcting the RPCs for BiasThe issue of achieving bias compensation in the RPCs pro-vided with HRSI, and mainly with Ikonos Geo and QuickBirdBasic imagery, would be largely academic were it not for thefact that a user can correct the RPCs for this bias and conse-quently obtain a sensor orientation model with an accuracycommensurate with the image resolution, that is about 1 min the case of Ikonos and QuickBird. The corrected RPCs canthen be substituted for the originals in subsequent digitalphotogrammetric operations such as ortho-image generationand digital terrain model (DTM) extraction. The formula forgenerating bias-corrected RPCs is provided in Hanley et al.(2002) and Fraser & Hanley (2003), at least for the case ofshift terms only. Where parameters beyond A0, B0, A1, and B2are significant, the RPCs must be re-estimated, rather thansimply corrected. This can be carried out using the acceptedtechnique outlined in Grodecki (2001). Moreover, to fullyexploit bias-compensated RPCs, the digital photogrammetricworkstation must support the RPC model. While this is gen-erally the case with Ikonos imagery, it is not necessarily so atpresent with stereo QuickBird imagery, where some popularphotogrammetric systems employ a rigorous model formula-tion (e.g., the BAE SOCET SET™ and Intergraph ImageStation™solutions are based on a rigorous model).
Experimental TestingTest Range DataAs mentioned, a primary aim of this paper is to demonstratethe high accuracy potential of bias-compensated RPCs forsensor orientation and geopositioning from HRSI. Implicit
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TABLE 1. CHARACTERISTICS OF THE TWO HRSI IMAGERY TEST FIELDS
Image Coverage Number Testfield Area Elevation Range (elevation angles) of GCPs Notable Features
Ikonos, Hobart 120 km2 sea level Stereo triplet 110 Full scene; (11 � 11 km) to 1280 m (69°, 75°, 69°) mountainous terrain
QuickBird, Melbourne 300 km2 sea level Stereo pair Full scene, low relief area(17.5 � 17.5 km) to 50 m (approx. 63° each) 81
Figure 1. Sample image chips of GCPs in the Hobarttest field.
in this exercise is the assumption that the RPCs do in factconstitute rigorous re-parameterisations of the rigorous sensororientation model. Thus, the APs, A0 � B2 will be modellingresidual systematic error associated with biases. In order todemonstrate the effectiveness of the bias-compensated RPCapproach, two test data sets of stereo HRSI are examined. Oneof these is a stereo triplet of Ikonos Geo-imagery, whereas theother is a QuickBird Basic stereo pair. Shown in Table 1 arethe essential characteristics of the two HRSI data sets to beanalysed. These are not the only stereo and multi-imageIkonos and QuickBird configurations that have been metricallyevaluated by the authors, but they do represent two with GCPand image measurements of sufficient accuracy to highlightthe error signal in sensor orientation at the sub-pixel level.
The Hobart test field covers a 120 km2 area of the cityof Hobart, Australia along with its surroundings. A very promi-nent feature in the area, lying only 10 km or so from down-town Hobart is Mount Wellington, the peak of which is justbelow 1,300 m elevation. The test range was imaged in a stereotriplet of Ikonos Geo-imagery recorded in February 2003. Ofthe images forming the triplet, the two stereo images (elevationangles of 69°; base-to-height ratio of 0.8) were scanned inreverse mode, while the central image (elevation angle of 75°)was acquired in forward mode. Hobart was specifically chosenas a suitable test field due to its height range, and the fact thatthe scene covered was largely urban, thus providing excellentprospects for accurate image-identifiable GCPs. A total of110 precisely measured ground feature points (mainly roadroundabouts) served as GCPs and checkpoints. In order toensure high-accuracy GCPs and image coordinate data, multipleGPS and image measurements were made for each GCP with thecentroids of roundabouts being determined by a best-fittingellipse to six or more edge points around the circumferenceof the feature, in both object and image space. The estimatedaccuracy of this procedure, described in Hanley & Fraser(2001) and Fraser et al. (2002), is 0.2 pixels. Four sample GCPimage chips are shown in Figure 1.
The stereo pair of QuickBird Basic imagery covering theMelbourne test range, which exhibited a pixel size of 0.75 mand a base-to-height ratio of 1, was recorded in July 2003.The majority of the 81 GCPs used were also road round-abouts, with the remaining points being corners and otherdistinct features conducive to high precision measurementin both the imagery and on the ground. Roundabouts weremeasured as described above, and in the case of corners, thefeature point was defined in image space by the intersectionof best-fitting lines to edge points.
All RPC bundle adjustment runs, along with all imagedata processing and measurement operations, were per-formed with the BARISTA software package. This softwaresystem has been developed specifically to provide a practi-cal data processing environment for HRSI sensor orientationand geopositioning, along with ortho-image generation andDTM extraction.
Results with IkonosThe results obtained in the RPC bundle adjustments of theHobart stereo triplet of Ikonos imagery are listed in Table 2.
The first row of the table shows the RMS value of coordinatediscrepancies obtained in a direct spatial intersectionutilising the RPCs provided with the imagery. A majorcomponent of these checkpoint discrepancy values arisesfrom the biases in the RPCs. Post transformation of thecomputed ground coordinates, utilizing three or more GCPs,could be expected to yield RMS accuracies at the 1 m level.The remaining rows of Table 2 list the accuracies attained inthe RPC bundle adjustments with bias compensation fordifferent AP sets. As can be appreciated, the resulting RMSvalues of checkpoint discrepancies will vary dependingupon the particular GCPs employed. Those listed in the tableare representative of the many that were obtained.
Of most practical interest are the results obtained in RPCbundle adjustments with the two shift parameters A0, B0. It canbe seen that geopositioning accuracy to 30 cm (RMS, 1-sigma)in longitude, and 70 cm in latitude and height are obtainedwith just two GCPs, and indeed, this result is achievable withone GCP. Note for the case of a single GCP on the top of MountWellington, i.e., at a 1,200 m elevation difference from themajority of the 109 checkpoints, that accuracy in planimetry isagain at the 0.3 pixel level in the cross-track direction. TheRMSE in height is marginally higher than in the two-GCP case,but this likely represents the effect of a bias of the adjustedposition of the single GCP rather than any affine distortion inthe relatively oriented three-image configuration. What iscertainly clear in the RPC bundle adjustments with shiftparameters is that terrain characteristics have no impact uponthe results. As regards the individual positional biases in image
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and also object space, these ranged from 0.1 m to 4 m for thethree images of the Ikonos Geo-triplet.
The plots of image coordinate residuals shown inFigure 2 provide an insight into the question of whetherthere may be additional bias error signal in the RPCs, forexample from time-dependent drift effects. While Figures 2aand 2b show a quite random distribution suggesting theabsence of any further systematic error, the right-handstereo image, Figure 2c, appears to display residual system-atic error in the along-track coordinate. Shown in Figure 2dare the image coordinate residuals for the same imagewhich result from an RPC bundle adjustment with shift anddrift parameters (the fourth row of Table 2). The drift terms,especially A1, lead to a reduction in the RMS value of imagecoordinate residuals, from 0.32 to 0.25 pixels in the linecoordinate direction, but this improvement does not lead toenhanced geopositioning accuracy. Grodecki & Dial (2003)have reported that with Ikonos imagery drift effects wouldbe unlikely to be seen in strip lengths of less than 50 km.The results obtained in the Hobart test field are consistentwith this view, notwithstanding the small residual system-atic error pattern seen in Figure 2c.
Given the indications that the RPC bias has been ade-quately modelled by the two shift parameters A0 and B0,it is not surprising to see that the full affine additionalparameter model does not lead to any accuracy improve-ment. The best indicator of the overall metric potential ofthe Ikonos stereo triplet is listed in the last row of Table 2.This is the case where the RPC bundle adjustment with shiftparameters employs all GCPs as loosely weighted control,thus providing a solution that can be thought of as beingequivalent to a free-network adjustment with inner con-straints. Note here the RMS geopositioning accuracy of justbelow one-forth pixel in the cross-track direction, and closeto one-half pixel in both the along-track direction and inheight.
The results for the stereo pair of Geo-images alone,without the central image, match very closely those listed inTable 2. From RPC bundle adjustments carried out with theHobart Ikonos imagery, as well as with other test fieldimagery covering areas as large as 2,000 km2 (e.g., Hanleyet al., 2002; Fraser & Hanley, 2003), the authors haveconcluded that sub-pixel geopositioning accuracy is quiteachievable from Ikonos Geo-stereo imagery. This can beexpected to be the case irrespective of the nature of theterrain being imaged, the size of the scene, or the scanningmode of the satellite (forward or reverse). High quality GCPsand image coordinate measurement are of course prerequi-sites to the attainment of highest accuracy.
Results with QuickBirdThe same computational procedure as carried out in theHobart test field was followed with the QuickBird Basicstereo pair covering the Melbourne test range. Table 3 lists
the results obtained. In most respects, the absolute geoposi-tioning accuracy achieved with QuickBird is the same asthat with Ikonos, though QuickBird produces in this caseslightly lower accuracy in planimetry but slightly higheraccuracy in height. This is most likely a consequence of thehigher base-to-height ratio exhibited in the QuickBird stereopair. Whether any component of these minor accuracydiscrepancies result from either the degree to which theoriginal RPCs describe the rigorous sensor model, or theresolution and accuracy of the actual orientation sensors onthe satellite is not known.
What is seen with QuickBird, however, are strongerindications of residual systematic error which is not beingmodelled by the bias-compensated RPCs. Shown in Fig-ures 3a and 3b are plots of the image coordinate residualsarising from the RPC bundle adjustment with shift parame-ters (Row 2 of Table 3). The along-track alignment of thevectors is suggestive of perturbations in scan velocity, withthe addition of a first-order scale effect. Thus, we wouldexpect some of the error signal to be absorbed by the along-track drift parameter, A1. The results listed in Table 3 forthe RPC bundle adjustment with shift and drift parameters,however, show a modest improvement in accuracy in thecross-track direction while there is no impact in along-trackor height accuracy. Similarly, the full affine model producesno accuracy improvement. Shown in Figures 3c and 3d arethe residual vectors in image space for the adjustment withthe APs of shift and drift. The systematic trends are stillapparent (the vectors not aligned to the along-track trendcorrespond to GCPs), though there is a modest improvementin the line direction. The same residual error patterns seenin Figure 3 have been encountered with other QuickBirdstereo pairs (Noguchi et al., 2004).
As was the case with Ikonos, the achievement of sub-pixel geopositioning with QuickBird imagery required onlythe provision of A0 and B0 in the RPC bundle adjustment.However, the nature of the image coordinate residualsobtained in the bundle adjustment with shift parameterssuggests that drift terms may also be warranted with Quick-Bird. The findings of Noguchi et al. (2004) support thisview. The A0 and B0 biases reached magnitudes of 30 m inthe QuickBird stereo images.
Concluding RemarksThe impressive geopositioning accuracy attained with theRPC bundle adjustment with bias compensation supports theview that this sensor orientation model has the same metricpotential as rigorous model formulations for both Ikonos andQuickBird imagery. Implicit in this conclusion is that theRPCs produced by Space Imaging, Inc., and DigitalGlobe,Inc., are equivalent to the rigorous model, and thus thereshould be no concern regarding their applicability in stereoimagery covering any type of terrain.
TABLE 2. RESULTS OF RPC BUNDLE ADJUSTMENTS WITH BIAS COMPENSATION FOR THE IKONOS GEO-STEREO TRIPLET COVERING THE HOBART TEST FIELD
RPC Bundle No. of GCPsRMS Value of Ground Checkpoint Discrepancies.
Adjustment (Number of RMS of l, s ImageUnits are Meters and Pixels
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(a) (b)
(c) (d)
Figure 2. Image coordinate residuals from RPC bundle adjustments of the Ikonos stereo triplet in theHobart test field: (a) Left stereo image, APs A0, B0, (b) Central image; APs A0, B0, (c) Right stereoimage, APs A0, B0, and (d) Right stereo image, APs A0, B0, A1, B1.
In comparing the accuracy results after bundle adjust-ment with ground control, one finds not much differencebetween Ikonos and QuickBird. Both produce the highestaccuracy in the cross-track direction. Also, in the test casesexamined QuickBird yielded slightly higher accuracy in
height, and Ikonos produced better along-track accuracy. Theissue of residual systematic error in the along-track directionis of importance for users who wish to utilize sensor orienta-tion models based on low-order empirical functions, suchas the 3D affine model. Experience by the authors and others
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TABLE 3. RESULTS OF RPC BUNDLE ADJUSTMENTS WITH BIAS COMPENSATION FOR THE QUICKBIRD BASIC STEREO PAIR COVERING THE MELBOURNE TEST FIELD
RMS Value of Ground Checkpoint Discrepancies.
No. of GCPs RMS of l, s imageUnits are Meters (and Pixels)
RPC Solution (Number of Checkpoints) residuals (pixels) Latitude (along track) Longitude (across track) Height
Figure 3. Image coordinate residuals from RPC bundle adjustments of QuickBird stereo imagery in theMelbourne testfield; (a) Left stereo image, APs A0, B0, (b) Right stereo image; APs A0, B0, (c) Leftstereo image, APs A0, B0, A1, B1, and (d) Right stereo image, APs A0, B0, A1, B1.
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(e.g., Fraser & Yamakawa, 2003; 2004; Noguchi et al., 2004;Hanley et al., 2002) has shown that success with such modelsis highly dependent on the absence of higher-order errorssources such as perturbations in scan velocity. While Ikonosreverse scanned imagery appears largely free of such effects,the same is not always the case for Ikonos forward scannedimages and QuickBird imagery. Indeed the authors’ recentexperience with QuickBird imagery suggests that standardlow-order empirical models do not yield very impressiveaccuracy results. On the other hand, where a user has theopportunity of utilizing bias-compensated RPCs, they can doso with every confidence of achieving optimal accuracy.
AcknowledgmentsThis research is supported through Discovery and Linkage-Industry Grants from the Australian Research Council. Theauthors are grateful for this support and also to Space Imaging,Inc., and DigitalGlobe, Inc., for the provision of Ikonos andQuickBird imagery. The component of the work related toQuickBird imagery is being carried out in collaboration withthe Spatial Division of the Sinclair Knight Merz Company.
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(Received 17 February 2004; accepted 15 April 2004; revised 22April 2004)
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