Bias Compensation in Rational Functions for Ikonos Satellite Imagery

Transcription

1 Bias Compensation in Rational Functions for Ikonos Satellite Imagery Clive S. Fraser and Harry B. Hanley Abstract A method for the removal of exterior orientation biases in rational function coefficients (RPCs) for Ikonos imagery is developed. These biases, which are inherent in RPCs derived without the aid of ground control, give rise to geopositioning errors. The 3D positioning error can subsequently be compensated during spatial intersection by two additional parameters in image space that effect a translation of image coordinates. The resulting bias parameters can then be used to correct the RPCs supplied with Ikonos Geo imagery such that a practical means is provided for bias-free ground point determination, nominally to meter-level absolute accuracy, using entirely standard procedures on any photogrammetric workstation that supports Ikonos RPCs. The method requires provision of one or more ground control points. Aside from developing the bias compensation method, the paper also summarizes practical testing with bias-corrected RPCs that has demonstrated sub-meter geopositioning accuracy from Ikonos Geo imagery. Introduction By the time the Ikonos-2 satellite was launched in September, 1999 it was well known within the photogrammetric commu- nity that Space Imaging (SI) would be withholding the camera model from public release, thus potentially precluding appli- cation of rigorous photogrammetric models for 3D geoposition- ing. For users of the Ikonos Geo image product, in either stereo or single-image mode, this has left effectively two options open for ground coordinate determination: either SI-supplied ra- tional functions (also termed RPCs for Rational Polynomial Coefficients or Rational Polynomial Camera model) or utiliza- tion of alternative empirical sensor orientation models that require a modest number of ground control points (GCPs). Recent investigations into alternative orientation models have shown that Ikonos Geo imagery is capable of pixel-level and even sub-pixel geopositioning accuracy. For example, Kersten et al. (2000), Davis and Wang (2001), Ahn et al. (2001), and Baltsavias et al. (2001) have reported meter-level accuracy in ortho-image production, and Toutin (2001) has discussed the potential of Ikonos stereo imagery for the extraction of digital terrain models (DTMs). Investigations into 3D positioning using alternative models have also recently been reported by Toutin et al. (2001) and Hu and Tao (2001). Geopositioning to sub- meter accuracy has been demonstrated for stereo and multi- image Geo image configurations by Fraser et al. (2001; 2002a) and Baltsavias et al. (2001). Affine and direct linear transforma- tion (DLT) approaches, along with RPCs, were evaluated using image data from the Melbourne Ikonos Testfield. This same data set, which will be briefly described in the following section, has been used in the present investigation of RPC bias compensation. Department of Geomatics, University of Melbourne, Victoria 3010, Australia, (c.fraser@unimelb.edu.au; hanley@sunrise.sli.unimelb.edu.au). While Ikonos has demonstrated its potential for high accuracy object point determination, the facility to exploit this capability is not necessarily available to those who wish to uti- lize SI-generated RPCs for stereo restitution or ortho-image gen- eration on digital photogrammetric workstations that support use of rational functions. This is because, for the Geo image product, the absolute accuracy specification (RMS 1-sigma) is 25 m, even though there is now ample experimental evidence that relative accuracy to meter level is readily attainable. A sig- nificant contributor to the error budget in RPC geopositioning is bias in exterior orientation, and especially errors in attitude determination. While it has been demonstrated that such posi- tional biases can be compensated by transformation of RPC- determined coordinates to an array of GCPs (e.g., Fraser et al., 2001), this correction is usually performed as a post-processing step, which makes it less than optimal for stereo restitution. Based on the success of earlier investigations into sub- meter geopositioning from Ikonos Geo imagery using SI-sup- plied RPCs, the authors set about finding a practical means to achieve bias removal such that the corrected rational functions could then be utilized on a standard photogrammetric workstation to yield meter-level accuracy rather than the 25-m geopositioning precision currently anticipated. This paper describes the development of such a bias compensation method, which is very practical to implement and can be applied with as little information as a single GCP. The paper first reviews the 3D ground point positioning process for stereo and multi-image (more than two) Ikonos image configurations using RPCs and describes the Melbourne Ikonos testfield in which experimental testing was performed. The development of an additional parameter model for the removal of biases in image space within a multi-point triangulation is then described. Results obtained in tests with the approach are dis- cussed and, finally, the straightforward process of incorporating the bias correction into the originally supplied RPCs is described. Rational Functions With the subject of this paper being Ikonos imagery, the follow- ing discussion of the rational function model is restricted to so- called terrain-independent, forward RPCs that describe the object-to-image space transformation, and which are supplied with Ikonos image products. Broader treatments of rational functions have recently been presented by Di et al. (2001), Tao and Hu (2001), and Dowman and Doloff (2000), but here the context is restricted to the use of SI-generated RPCs that reparameterize the rigorous sensor orientation to an accuracy level Photogrammetric Engineering & Remote Sensing Vol. 69, No. 1, January 2003, pp /03/ $3.00/ American Society for Photogrammetry and Remote Sensing PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING Janaury

2 exceeding 0.04 pixels (Grodecki, 2001). The model for forward performed using the first-order coefficients from Equation 1, rational functions for image i and ground point j can be given as and even the offset values alone. Whereas Equation 2 offers a practical method for 3D ground point determination from RPCs, it does not incorporate the x P i1(x,y,z) j (1) P i2 (X,Y,Z) means to accommodate exterior orientation biases which are j inherent in Ikonos products that are generated without refer- y P i3(x,y,z) j P i1 (X,Y,Z) j a 1 a 2 Y a 3 X a 4 Z a 5 Y X a 6 Y Z a 7 X Z a 8 Y 2 a 9 X 2 a 10 Z 2 a 11 X Y Z a 12 Y 3 a 13 Y X 2 a 14 Y Z 2 a 15 Y 2 X a 16 X 3 a 17 X Z 2 a 18 Y 2 Z a 19 X 2 Z a 20 Z 3 P i2 (X,Y,Z) j b 1 b 2 Y b 3 X b 4 Z b 19 X 2 Z b 20 Z 3 P i3 (X,Y,Z) j c 1 c 2 Y c 3 X c 4 Z c 19 X 2 Z c 20 Z 3 d 1 d 2 Y d 3 X d 4 Z d 19 X 2 Z d 20 Z 3 ence to ground control. Whereas the output from stereo restitu- tion of Ikonos Geo imagery will likely suggest relative point determination to, say, the one-pixel level, the absolute positioning may be in error by tens of meters. Corresponding biases in image space coordinates for an image can be computed through a direct comparison of measured image coordinates with those determined from measured ground points through application of Equation 1. Shown in Figure 1 and Table 1 are the outcomes of such a computation for the 40 GCPs within the Melbournre Ikonos Testfield. A brief description of the testfield follows, with further details being given in Hanley and Fraser (2001). The Melbourne Testfield covers a 7- by 7-km area of the city of Melbourne and currently comprises 40 GPS-surveyed GCPs, 32 of which are road roundabouts. The remaining points are corners and other distinct features conducive to high precision measurement in both the imagery and on the ground. This array of ground points has been imaged with three-fold Ikonos Geo coverage comprising a stereo pair of panchromatic images and a near-nadir scene of panchromatic and multispectral imagery. In order to ensure high-accuracy GCPs and image coordinate data, multiple GPS and image measurements were made for each GCP, with the centroids of road roundabouts being determined by computing a best-fitting ellipse to six or more edge points around the circumference of the feature, in both object and image space (see Hanley and Fraser, 2001). An example GCP, along with its best fitting ellipse and computed centroid, is Here, x, y are the normalized (offset and scaled) image coordishown in Figure 2. nates (row, column or line, sample) and X, Y, Z are the corresponding object point coordinates, which for Ikonos RPCs refer In the case of corners, the feature point was defined in to normalized latitude, longitude, and height. As can be seen image space by the intersection of best-fitting lines to edges, from Equation 1, a number of existing restitution algorithms for multiple observations having been made along the edges. line scanner imagery are based on special formulations of the While there could be some confidence that this procedure led ratio-of-polynomials model (e.g., the DLT, affine models, and to image coordinate precision of close to 0.1 pixels in the nadir polynomial expressions in which the denominator is reduced image, the inherent resolution limits of 1-m imagery coupled to unity). with the poorer quality of the stereopair of Geo images, sug- In order to effect an image-to-object point transformation, gested that an overall image point standard error of 0.2 to 0.3 one of course requires either stereo image coverage or known pixels was a more realistic estimate. GCP coordinates were all height in the case of a single image. Here we consider stereo and accurate to 0.1 m or better. These aspects are further discussed multi-image networks (more than two images) in which the in Hanley and Fraser (2001) and Fraser et al. (2001; 2002b), with spatial intersection to determine ground coordinates from the radiometric issues being analyzed in Baltsavias et al. (2001). RPCs can be performed using an indirect least-squares model of Figure 1 shows plots of the bias values for the stereo and the form near-nadir images, while Table 1 lists the mean values and standard errors. It can be seen that, although the image coordinate discrepancies dx and dy reach 70 pixels, the standard v x v y A X Y Zj effectively equal for the stereo images, they are very different for the nadir image, which was recorded some five months earlier. Two factors account for the biases being essentially equal in the stereo image. First, recorded exterior orientation is unlikely to undergo significant change within the 100 or so seconds between the recording of the two along-track images and, second, stereo images are routinely subjected to initial bundle adjustment. The nadir image, on the other hand, was recorded on an orbit with potentially very different exterior orientation biases, most likely attributable to small errors in attitude determination. From the plots of Figure 1 and the results listed in Table 1, it is apparent that a good starting point in any attempt to compensate for RPC biases would be through application of a translation of observed image point coordi- nates, an approach that will now be developed. v x and v y are image coordinate residuals; X, Y, Z are corrections to approximate values for the object point coordi- nates; x 0,y 0 are the image coordinates corresponding to the approximate object coordinates (obtained using Equation 1); A is the matrix of partial derivatives of the functions in Equa- tion 1 with respect to X, Y, and Z; and x, y are the measured image coordinates. Because both x, y and X, Y, Z represent scaled and offset values, care has to be taken in the lineariza- tion to account for different scales and offsets between images. If X, Y, Z represent corrections to latitude, longitude, and height, as opposed to offset and scaled coordinates, it is not nec- essary to first go through the process of un-normalizing and re- normalizing coordinates to ensure a common offset and scale, as suggested in Tao and Hu (2001). Determination of initial approximate values for the XYZ coordinates of point j can be x 0 x y 0 y (2) error in all cases is less than 0.5 pixels, which indicates a high degree of invariance of the image point biases within an image. Note also that, as the image point displacements are 54 Janaury 2003 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING

3 Figure 1. Image coordinate biases in RPC sensor orientation for the stereo and nadir images. TABLE 1. MEAN AND STANDARD ERROR VALUES FOR RPC BIAS-INDUCED IMAGE using Equation 3, only one GCP is necessary for absolute geo- POINT PERTURBATIONS dx AND dy. UNITS ARE PIXELS positioning as, for relative positioning, the two shift Image Mean dx Mean dy dx dy parameters for one image could be suppressed. The merit of Equation 3 lies in both its simplicity and in its applicability to Left Stereo multi-image triangulation using Ikonos RPCs for images that Near Nadir exhibit very different bias characteristics. As written, Equation Right Stereo is not applicable to single-image coverage, but the principle is entirely the same with the bias parameters being recovered as per the approach used in generating Figure 1 and Table 1, for example. In the remainder of the discussion of bias determination, we will consider the stereo- and multi-image cases only. Bias Compensation Although the two bias parameters x and y are modeled Under the assumption that RPC biases manifest themselves for as image space perturbations, the very long focal length (10 m) all practical purposes as image coordinate perturbations, a and narrow view angle (0.93) of the Ikonos sensor means that model for bias compensation that comprises one offset parameplatform what is being effected is essentially a lateral shift of the sensor ter per image coordinate can be derived through an extension in two orthogonal directions, i.e., a correction to exte- of Equation 2: i.e., a 21 a 22 a Xj Y j v x v y a 11 a 12 a Z x x j x i y y i 0 (3) y also both possible and desirable to recover the bias parameters using a simultaneous multi-point, multi-image triangulation. This is illustrated by the following development of the leastsquares normal equation structure for the triangulation. Initially, Equation 3 is partitioned and the model is supple x i and y i are image coordinate perturbations that are common to image i. In a multi-point, multi-ray intersection rior orientation which although positional is more likely in reality to be compensating for sensor attitude errors. In some respects Equation 3 could be considered a bundle adjustment because, although it is in theory possible to solve for the two bias parameters using a single ground control point, it is mented with an appropriate constraint function to incorporate Figure 2. Ikonos image of road roundabout (left) and measured edge points and best-fitting ellipse (right). PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING Janaury

4 the necessary ground control, one GCP being the minimum required: i.e., v v c j A I 1j 2i W W c j ; P 0 TABLE 2. COMPUTED IMAGE COORDINATE BIASES AND RMS VALUES OF GROUND COORDINATE DISCREPANCIES AT CHECKPOINTS FOR STEREO AND THREE-IMAGE TRIANGULATION (L: LEFT, R: RIGHT, C: CENTER/NADIR) Computed Image RMS Values of 0 Pj c (4) Coordinate Biases Checkpoint Image (pixels) Coordinates (meters) network & no. of GCPs x y S X S Y S Z v (v x, v y ) T is the vector of image coordinate residuals, Stereo 1 L v c j (v X, v Y, v Z ) T j is the vector of residuals for GCP coordinates, R j (X, Y, Z) T j is the vector of object point parameters for 2 L point j, R i (x, y) T i is the vector of image space biases for image i, 4 L w and w c j and are discrepancy vectors, and R P and P c j are weight matrices for image and GCP coordinates, 6 L respectively. R From the observation equation, Equation 4, the least- 10 L squares normal equation contribution relating to image i and point j follows as AT P A P c j A T P P A P 1j A T P w P c j w c j 2i (5) R Image C L R C L R C P w 0 4 L R C For non-control points, terms involving P c j are suppressed. The regular structure of the final normal equation system can be illustrated by considering the case of a stereo pair of Ikonos images with three observed image points, one of which, Point 1, corresponds to a ground control point. For simplicity, P is assumed in this case to be the identity matrix I. The normal equations then assume the form 6 L R C L R (AT i1 A i1 ) P c A T 11 A T 21 A T i2a i2 0 A T 12 A T 22 A T i3a i3 A T 13 A T 23 3I 0 Symmetric AT i1w i1 A T i2w i2 A T i3w i3 w 1j w 2j Application of Bias Compensation 0 3I (6) Table 2 lists both the least-squares estimates of the bias terms x and y for the stereo and three-image configurations over the Melbourne Testfield, and the RMS values of checkpoint coordinate discrepancies for a number of GCP configurations. In each case, the number of checkpoints is 40 minus the number of GCPs, the sets comprising 1, 2, 4, 6, and 10 control points. Generally, the stereopair of images yields 3D point positioning accuracies of 50 to 60 cm in planimetry and 90 cm in height, irrespective of the number of GCPs. The accuracies improve in the three-image geometry to 40 to 50 cm and 60 to 70 cm in planimetry and height, respectively. The absolute accuracy of 3D geopositioning isat thesub-meter levelin all cases, including that of a single GCP. It must be recalled, however, that in the case of the Melbourne Testfield the GCPs and image coordinate observations are of very high quality, with an accuracy of around 0.2 pixels in object and image space. Also noteworthy in Table 2 is that the estimated values of bias terms show very little variation for different GCP configurations, with the range of values varying by 0.3 pixels or less from the mean value. TheroleoftheGCPs is to effect an image coordinate translation, and, thus, their location is of no real consequence in the bias removal method described. The addition of further GCPs makes no contribution to the geometric strength of the triangulation process per se. Instead, the extra control points simply provide more information with which to evaluate an appropriate average image coordinate correction. From the RMS values of XYZ coordinate discrepancies listed in Table 2, it is apparent that for the Melbourne Testfield data there is no clear link between the accuracy attained and the number of GCPs. Nevertheless, with the use of redundant control points one can be more confident about the reliability of the determination. Something in the vicinity of three to five GCPs would seem a reasonable provision of control for an Ikonos scene. There is scope under the described additional parameter method for bias removal from Ikonos RPCs to extend the image coordinate correction functions to more than single off- set terms. Such an approach using correction terms to second order has been investigated, but it was found to yield no sig- nificant improvement in results within the Melbourne Testfield data set. This outcome confirmed early findings (Hanley and Fraser, 2001) that the three Ikonos images concerned did not display any significant non-linearities. Bias-Corrected RPCs The ability to determine the bias parameters x and y is very useful, but of more utility would be the incorporation of the bias compensation into the originally supplied RPCs. This would then allow bias-free application of RPC-positioning without any reference to additional correction terms. Fortunately, it turns out that this bias compensation is a straightforward matter, with the bias-corrected RPCs for image i being developed as follows: 56 Janaury 2003 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING

5 or x x i P i1 (X,Y,Z) j P i2 (X,Y,Z) j (7) y y i P i3 (X,Y,Z) j x PC i1 (X,Y,Z) j P i2 (X,Y,Z) j (8) y PC i3 (X,Y,Z) j P C i1(x,y,z) j (a 1 b 1 x i ) (a 2 b 2 x i ) Y (a 3 b 3 x i ) X (a 20 b 20 x i ) Z 3 P C i3(x,y,z) j (c 1 d 1 y i ) (c 2 d 2 y i ) Y (c 3 d 3 y i ) X (c 20 d 20 y i ) Z 3 References Ahn, C.-H., S.-I. Cho, and J.C. Jeon, Orthorectification software applicable for Ikonos high resolution images: GeoPixel-Ortho, IEEE 2001 Int. Geoscience and Remote Sensing Symposium,09 13 July, Sydney, Australia, 3 pages (on CD-ROM). Baltsavias, E., M. Pateraki, and L. Zhang, Radiometric and geometric evaluation of Ikonos GEO images and their use for 3D building modelling, Proc. Joint ISPRS Workshop High Resolution Mapping from Space 2001, September, Hannover, Germany, 21 p. (on CD ROM). Davis, C.H., and W. Wang, Planimetric accuracy of Ikonos 1-m panchromatic image products, Proc. ASPRS Annual Conference, April, St Louis, Missouri, 14 p. (on CD ROM). Di, K., R. Ma, and R. Li, Deriving 3D shorelines from highresolution Ikonos satellite images with rational functions, Proc. ASPRS Annual Conference, April, St Louis, Missouri, 8 p. (on CD-ROM). Dowman, I., and J.T. Dolloff, An evaluation of rational functions for photogrammetric restitution, International Archives of Photogrammetry and Remote Sensing, 33(B3/1): Fraser, C.S., H.B. Hanley, and T. Yamakawa, Sub-metre geopositioning with Ikonos Geo imagery, Proc. Joint ISPRS Workshop High Resolution Mapping from Space 2001, September, Hannover, Germany, 8 p. (on CD ROM)., 2002a. Three-dimensional positioning accuracy of Ikonos imagery, Photogrammetric Record, 17(99): Fraser, C.S., M. Baltsavias, and A. Gruen, 2002b. Processing of Ikonos imagery for sub-metre 3D positioning and building extraction, ISPRS Journal of Photogrammetry & Remote Sensing, 56(3): Grodecki, J., Ikonos stereo feature extraction RPC approach, Proc. ASPRS Annual Conference, April, St. Louis, Mis- souri, 7 p. (on CD ROM). Hanley, H.B., and C.S. Fraser, Geopositioning accuracy of Ikonos imagery: Indications from 2D transformations, Photogrammetric Effectively, all original terms in the numerator of each expression in Equation 7 are modified. In the case of the x- equation, each coefficient a k is replaced by (a k b k x i ). A software system has been developed to perform the necessary generation of bias-corrected RPCs. This system allows interactive image point measurement of selected image points and the necessary GCP(s). It also includes computation of the bias parameters for any number of images, from any number of object points, and it carries out the generation of corrected RPCs Record, 17(98): (Equation 8) in a file format identical to that originally sup- Hu, Y., and V. Tao, D reconstruction algorithms with the plied by SI. This file is thus suited to utilization with standard rational function model and their applications for Ikonos stereo photogrammetric workstations that support stereo restitution imagery, Proc. Joint ISPRS Workshop High Resolution Mapping using Ikonos rational functions, and it will facilitate bias-free from Space 2001, September, Hannover, Germany, 12 p. 3D ground point determination. (on CD ROM). Kersten, T., E. Baltsavias, M. Schwarz, and I. Leiss, Ikonos-2 Concluding Remarks Carterra Geo Erste geometrische Genauigkeitsuntersuchungen in der Schweiz mit hochaufgeloesten Satellitendaten, Vermessung, The challenge addressed in this paper has been to develop a Photogrammetrie, Kulturtechnik, 8: practical means to compensate for biases in 3D ground point Tao, C.V., and Y. Hu, D reconstruction algorithms with the determination from Ikonos imagery, which can be expected in rational function model and their application for Ikonos stereo SI-produced RPCs that are generated without reference to imagery, Proc. Joint ISPRS Workshop High Resolution Mapping ground control. In the course of this development it has again from Space 2001, September, Hanover, Germany, 12 p. been shown that Ikonos Geo imagery is capable of 3D geoposi- (on CD-ROM). tioning to sub-meter accuracy when quality ground control is Toutin, T., Geometric processing of Ikonos Geo images with DEM, available. The final stage of the bias compensation process has Proc. Joint ISPRS Workshop High Resolution Mapping from been to incorporate a correction directly into the original ratio- Space 2001, September, Hannover, Germany, 9 p. (on nal function coefficients in order that a user of the Ikonos RPC CD ROM). file will be able to produce bias-free ground point coordinates, Toutin, Th., R. Chénier, and Y. Carbonneau, D geometric model- nominally to meter-level absolute accuracy, using entirely ling of Ikonos GEO images, Proc. Joint ISPRS Workshop High standard procedures on any photogrammetric workstation that Resolution Mapping from Space 2001, September, Hanno- supports Ikonos RPCs. The cost of implementing the proposed ver, Germany, 9 p. (on CD ROM). method is relatively modest, namely, the provision of one or (Received 29 November 2001; accepted 06 March 2002; revised 01 more quality ground control points. April 2002) PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING Janaury