Block Adjustment of Satellite Imagery Using RPCs with Virtual Strip Scenes

Transcription

1 Block Adjustment of Satellite Imagery Using RPCs with Virtual Strip Scenes Guo Zhang, Hongbo Pan, Deren Li, Xinming Tang, and Xiaoyong Zhu Abstract The increasing volume of high-resolution satellite imagery has seen the need for a large number of ground control points become a limiting factor for large-area mapping. Utilizing a shift model for adjacent scenes of the same track, this paper proposes a method based on rational polynomial coefficients with virtual strip scenes. An affine transformation in the image space is used as the adjustment model for the virtual strip scenes, and the corresponding adjustment parameters are derived from the relationship between the standard scenes and virtual strip scenes. Triplet stereo images from ZiYuan-3 are used to test the accuracy of the virtual strip scenes, and we compare the block adjustment of the long strip scene products and standard scene products. The results show that sub-pixel accuracy can be achieved in checkpoints close to the long strip scenes. Introduction As an increasing number of new-generation very-high-resolution satellites are launched (e.g., WorldView-, GeoEye-, SPOT 6 and 7, Pleiades), the capability of obtaining ground images is no longer a limiting factor. In spite of the high orientation accuracy of such satellites, measuring GCPs (Ground Control Points) for block adjustment remains a necessary function, and can be rather costly and time-consuming. Thus, there is an urgent need to reduce the required number of GCPs. Since being utilized by Ikonos as the camera model, vendors increasingly prefer RPCs (Rational Polynomial Coefficients) because of their high replacement accuracy for the rigorous model, faster computation, and greater generality. Block adjustment methods using RPCs have been investigated, and are referred to in many cases as bias-compensation (Fraser and Hanley, 5). The block adjustment of rigorous models has also been studied in detail (Orun and Natarajan, 994; Poli, 7; Weser et al., 8). To fit the requirements of different users, vendors provide standard scenes and strip scenes. For long strip scenes with RPCs, four GCPs around the corners can ensure sub-pixel accuracy (Grodecki and Dial, 3; Pan et al., 3). For standard scenes with rigorous models, Kim et al. (7) investigated the modeling of entire strips with different parameters, and an accuracy of around two pixels was obtained over the whole 4 km of SPOT 3 strips. A similar result was obtained by Gupta Guo Zhang, and Deren Li are with the State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing (LIESMARS), Wuhan University, 9 Luoyu Road, Wuhan, 4379, P.R. China (guozhang@whu.edu.cn). Hongbo Pan is with the School of Geoscience and Info-Physics, Central South University, Changsha 483, China, P.R. China. Xinming Tang and Xiaoyong Zhu are with the Satellite Surveying and Mapping Application Center (SASMAC), National Administration of Surveying, Mapping and Geoinformation, 8 Lianhuachi West Road, Haidian District, Beijing, 83, P.R. China. et al. (8) for imagery from Cartosat-. A Keplerian motionbased orbit determination method developed by Michalis and Dowman (8) obtained pixel-size accuracy, while a strip adjustment approach based on a generic sensor technique was applied to ALOS imagery, giving single-pixel accuracy (Fraser and Ravanbakhsh, ; Rottensteiner et al., 9). The above methods are based on standard scenes with rigorous models or single strip scenes, and are thus not appropriate for standard scenes with RPCs. In this paper, we develop a new method that uses virtual strip scenes, geometric mosaics without resampling, instead of single scenes for block adjustment. The proposed method does not require tie points between adjacent scenes, and just four GCPs settled around the four corners promise a similar degree of accuracy as with strip scenes. This paper is organized as follows. After a brief description of RPCs, the virtual strip scenes and their block adjustment methods are introduced. A comparison between standard scenes, virtual strip scenes, and strip scenes from ZiYuan-3 is then presented. The two adjacent strips consist of seven and twelve standard scenes, and a large number of GCPs are uniformly distributed in this area, enabling a comprehensive experimental evaluation and validation of the proposed method. Finally, some orientation results are presented, before we summarize our conclusions. RPCs In the RPC model, image pixel coordinates (sample, line) are expressed as the ratios of cubic polynomials of ground coordinates (Latitude, Longitude, Height). To improve the numerical stability of the equations, the D image coordinates and 3D ground coordinates are each offset and scaled to fit the range [.,.]. The normalized coordinate values of object points on the ground (P, L, H) and the normalized line and sample image pixel coordinates (X, Y) are computed using the following equations: sample SAMP _ OFF X = SAMP _ SCALE line LINE _ OFF Y = LINE _ SCALE Latitude LAT _ OFF P = LAT _ SCALE Longitude LONG _ OFF L = LONG _ SCALE Height HEIGHT _ OFF H = HEIGHT _ SCALE () () Photogrammetric Engineering & Remote Sensing Vol. 8, No., November 4, pp /4/ American Society for Photogrammetry and Remote Sensing doi:.4358/pers PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING November 4 53 November 4 Peer Reviewed.indd 53 /7/4 :33:3 PM

2 where SAMP_OFF, LINE_OFF, SAMP_SCALE, and LINE_ SCALE are offset values and scale values for the two image coordinates, respectively. Similarly, LAT_OFF, LONG_OFF, HEIGHT_OFF, LAT_SCALE, LONG_SCALE, and HEIGHT_ SCALE are offset values and scale values for the three ground coordinates, respectively. The RPC model for the normalized image coordinates (X, Y) and the normalized ground coordinates (P, L, H) for an image can then be written as: NumS P, LH, X = Den P, LH, NumL P, LH, Y = Den P, LH, S L where Num s (P,L,H), Den s (P,L,H), Num L (P,L,H), Den L (P,L,H) are the third-order polynomial terms of (P,L,H). Bias-compensated Model Errors in attitude, ephemeris, and drift lead to certain biases in the RPCs. As the errors in object space and image space are highly correlated, the errors in attitude, ephemeris, and velocity can be modeled as biases in the image space. In the simplest case, small attitude or ephemeris errors are equivalent to a shift in image space coordinates, and time-dependent errors in attitude sensors can give rise to shift effects in the image coordinates (Fraser and Hanley, 5; Grodecki and Dial, 3). The compensated model can be described as follows: L NumS P, LH, sample + x = x = isamp _ SCALE + SAMP_ OFF DenS ( P, LH, ) (4) NumL ( P, LH, ) line + y = y = iline _ SCALE + LINE _ OFF Den P, LH, where Δy and Δx are adjustable functions expressing the differences between measured coordinates (sample, line) and the calculated sample and line coordinates (x, y) of ground and/or tie points, which can generally be expressed as polynomials of the image line and sample coordinates (Grodecki and Dial, 3): Δx = a + a sample + a line + a 3 sample + a 4 sample line + a 5 line + (5) Δy = b + b sample + b line + b 3 sample + b 4 sample line + b 5 line + where a, a, a... and b, b, b are the adjustment parameters of an image, and line and sample are the line and sample (3) coordinates, respectively, of the ground or tie points. The parameter sets a, b describe the shift bias; a, a, b, b model the shift and drift; a, a, a, b, b, b model the affine transformation; and a ~ a 5, b ~ b 5, model the bias using second-order polynomials. The choice of parameter set depends on the geometric characteristics of the sensor and products, such as errors in lens distortion and CCD (charge-coupled device) alignment, and the residuals of attitude and ephemeris. The image space affine transformation is a most encouraging bias parameter set, as it has been shown to be capable of yielding subpixel accuracy when four to six well-distributed GCPs are applied. Usually, the bias-compensated model is applied to basic products, such as those of the Digital Global company and sensor-corrected products of ZY3 (DigitalGlobe, ; Pan et al., 3), and geocoded ellipsoid-corrected products, such as those of the GEOEYE company (GeoEye, ; Grodecki and Dial, 3). Virtual Strip Scenes Limitations in the maximum available CCD linear array size led to the adoption of staggered arrays in high-resolution satellites. This resulted in each single CCD capturing ground images at different times and with different attitudes and emphases. Because of their image distortion and non-linear perspective projection, the primary products, such as ALOS PRISM B products (JAXA, 7), are rather hard to use. A virtual CCD re-imaging method is used to eliminate the interior and exterior distortion (Pan et al., 3). The virtual CCD can be considered to form a similar image to that obtained by an ideal pinhole camera without lens distortion, and the single CCD units are uniformly spaced in the focal plane. After calibration, the errors in principal distance and principal points are compensated for, eliminating the interior distortion (lens distortion, CCD distortion, and errors in principal distance and principal points). The exterior distortion can be removed by a number of strategies. For example, a regular integration time can be used to build a rigorous model of basic products to ensure that the along-track resolution of the products is constant; alternatively, a sophisticated ephemeris can be used instead of the original to remove ephemeris errors, or low-pass filtering can be applied to eliminate high-order attitude oscillations (Pan et al., 3). A similar strategy was adopted by the Pleiades satellite (ASTRIUM, ). Differences in the strip scenes and standard scenes are introduced by the exterior distortion elimination procedure. For standard scenes, local attitude filtering will produce different results to strip scenes, depending on the low-pass filter methods. Without loss of generality, we suppose that the difference between the standard scenes and strip scenes can be (a) (b) Figure. Virtual ccd re-imaging (exaggerated sketch). In (a), the primary products consist of three distortion images, whereas (b) shows the distortion-free basic products. The relationship between the strip scene and standard scene is also illustrated. 54 November 4 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING November 4 Peer Reviewed.indd 54 /7/4 :33:4 PM

3 represented by a shift. The integration time of TDI (Time Delay and Integration) CCDs varies almost linearly, meaning it can be absorbed by the drift parameter sets. As shown in Figure, the major difference between the standard scenes and strip scenes is the along-track shift. As a consequence, a virtual strip scene can be set up with adjacent standard scenes. The along-track shift is determined by the geometry models of adjacent standard scenes, because the overlap area of adjacent scenes has almost the same geometry characteristics and similar parallel rays. As proved by Pan et al. (3), elevation errors result in a negligible shift, so the average elevation and RPCs could be used instead of the tie points to determine the shift parameters (sx, sy), thus avoiding annoying manual picking errors. The image coordinates of virtual strip scenes can be expressed as follows: sample = sample i + sx i (6) line = line i + sy i where (sample, line) are the measured image coordinates of the virtual strip scenes, (sample i, line i ) are the measured image coordinates of the i th image in the virtual scenes, and (sx i, sy i ) denote the shift parameters between the virtual scenes and standard scenes. A similar expression can be derived for the calculated virtual strip scene coordinates: x = x i + sx i (7) y = y i + sy i where (x, y) are the calculated image coordinates of the virtual strip scene given by RPCs and ground coordinates (lat, lon, h) of the tie points or GCPs, and (x i, y i ) are the calculated image coordinates of the i th image. Virtual Strip-based Compensation Because of its excellent applicability, we apply the affine transformation to derive the block adjustment model for the virtual strip scenes. From Equations 4 through 7, the bias compensation model for each image in the strip scene is given by: F x = sample i + a + a (sample i + sx i ) + a (line i + sy i ) x i =. (8) F y = line i + b + b (sample i + sx i ) + b (line i + sy i ) y i = For each GCP, the unknowns are the adjustment parameters, and (x i, y i ) are calculated by the object space coordinates and RPCs. However, the unknowns of the tie points include the coordinates in object space, (lat, lon, h). In this study, we simplify the situation, and derive the following based on the tie point situation. Applying a Taylor series expansion to the block adjustment in Equation 8, the linearized model is as follows: F F x y Fx Fx Fx Fx Fx = Fx + a + a + a + lat + a a a lon lon Fx + h = Fy Fy Fy = Fy + b + b + b + F + + y Fy lat = b b b lon lon Fy h Substituting Equations,, 4, and 8 into 9, we get: (9) V = AX + BY L,P () where samplei + sxi linei + sy i A = sample + sx line + sy i i i i () x x x x X P x X L x X H lat lon h X P X L lon X H B = = y y y y Y P y lon Y P Y L y Y H Y L lon Y H SAMP _ SCALE X SAMP _ SCALE X SAMP _ SCALE X LAT _ SCALE P LONG _ SCALE L HEIGHT _ SCALE H = LINE _ SCALE Y LINE _ SCALE Y LINE _ SCAL E Y LAT _ SCALE P LONG _ SCALE L HEIGHT _ SCALE H () X = [Δa Δa Δa Δb Δb Δb ] (3) Y = [Δlat Δlon Δh] (4) L = F Fy x (5) where P is a weight matrix, and Equation 5 can be calculated from Equation 8 and the initial value. In the general situation, the initial adjustment values should be zero, and the ground coordinates of the tie points can be calculated from the forward space intersection. When both GCPs and tie points are considered in the block adjustment, the error equation becomes: A X L P A B Y = L, P. (6) After normalizing the error equation, the unknowns in the tie point object space coordinate can be eliminated. After performing trivial derivations, the final normalized reduction equation can be written as: T APA + APA T APBBPB T T BPA X APL A = + PL APB ( BPB ) B PL T T T. (7) If there are insufficient GCPs, the coefficient matrix of Equation 7 will be rank deficient, and regularization methods must be used to solve this problem (Hansen and O Leary, 993). Moreover, robust estimation methods can be used to estimate the adjustment parameters, meaning that a gross error scan be eliminated (Förstner, 985). Next, Equation 7 is used to update the adjustment parameters, and the object coordinates of the tie points are obtained by forward space intersection with the new adjustment parameters. An iterative approach would terminate when the solution of Equation 7 is correct to within a predefined tolerance. After calculating the adjustment parameters a, a, a, b, b, b, the corresponding adjustment model for each image in the virtual strip can be derived as follows: x i = a + a sx i + a sy + (a + ) sample + a line. (8) y i = b + b sx i + b sy + b sample + (a + ) line. When we compare the bias compensation model in Equation 8 with Equation 5, it is clear that scenes in the same virtual strip have the same coefficients, except for the shift parameters. The addition of the strip constraint means that the error equation is slightly different to that in regular affine transformation compensation methods, which makes this routine easy to implement (Grodecki and Dial, 3). A flow chart of the virtual strip-based block adjustment procedure is illustrated in Figure. The adjustment parameter can also be used to generate new RPCs. Dataset The Taihang dataset, consisting of two adjacent triple stereo strip scenes, known as Orbit 35 and Orbit 38, is used to PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING November 4 55 November 4 Peer Reviewed.indd 55 /7/4 :33:4 PM

4 validate the adjustment model of this paper. This dataset crosses the Taihang Mountains, covering the southern part of Neimeng Province, the eastern part of Shanxi Province, and the western part of Hebei Province. This represents a total area of around 8 56 km, and a height range of 64 to 75 m (Figure 3). The topography of the Taihang dataset is varied and complex, encompassing mountains, basins, hills, and plains. The details of the scenes are as follows. Orbit 35 was acquired on 9 January, covering seven standard scenes, while Orbit 38 was acquired on 3 February, covering twelve standard scenes. The dataset is limited by the imaging time and the weather: the start of three scenes was covered by snow events, and the end of Orbit 35 was partially obscured by clouds. There are a large number of GCPs distributed evenly throughout the dataset; these were primarily measured using static GPS because of the large area and dramatic changes in terrain. To obtain an accuracy of. m, the observation time of each static point was over 3 min. After removing Figure. Flow chart of a virtual strip-based block adjustment. indiscernible GCPs, the distribution of the remaining 39 GCPs is as shown in Figure 3. Experiments Before the block adjustment experiments, we investigated the shift parameters (sx, sy). Because the exterior distortion has been eliminated, the corresponding points in the overlap area have slightly different geometry characteristics, which depend on the attitude stability and integration time. The rays of the overlap area should be almost exactly parallel, otherwise the inevitable mismatches and elevation errors will be significant (Pan et al., 3). Therefore, the RPCs and SRTM (or average elevation) can be used to calculate the shift parameters of the grid that is established in the common area of adjacent scenes. Three different cases are considered to verify the performance of the virtual strip scenes. In the first case, the strip scene products are produced; the second is based on standard scenes, and the third uses the virtual strip scenes. The longer strip, Orbit 38, was studied carefully with different numbers of GCPs. The comparison is shown in Table. There is no significant difference between the three cases when all GCPs are used to check the forward intersection accuracy. The root-mean-square errors (RMSEs) are about 3 m and 8.6 m for the plane and elevation, respectively. The plane RMSE is smaller because the calibration field, which was used to calibrate the interior and exterior elements, was in the same orbit but to the south of the strip scene. When four GCPs are placed around the four corners for block adjustment, the accuracy of the strip scenes is excellent, with RMSEs of.49 m for the plane and.63 m for elevation, respectively, almost the same as for single standard scenes (Pan et al., 3). In the case of virtual strip scenes, it is sufficient to determine the adjustment parameters from Equation 6, which gives slightly worse results, with RMSEs of.5 m for the plane and.8 m for elevation, respectively. The residuals are shown in Plate, where the blue arrows denote plane errors, and the red ones denote elevation errors. The direction of the errors is the same, and the magnitude of elevation errors in the virtual strip scenes is larger in the north. The exterior distortion elimination strategies imply that the shift constraint is not stringent, but approximate. For the standard scene case, the coefficient matrix was seriously ill-conditioned, producing much worse results than for the strip scenes and virtual strip scenes. 56 November 4 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING November 4 Peer Reviewed.indd 56 /7/4 :33:5 PM

5 Table. Block Adjustment Accuracy (Root-Mean-Square Errors) of Orbit 38 for Three Different Cases with Different Numbers of gcps. Case Number of GCPs Number of CKPs X RMSE Y RMSE Plane RMSE H RMSE Strip Scenes Virtual Strip Scenes Standard Scenes Conclusions The image-based affine compensation models for the same strip scenes are highly correlated, and it is difficult to determine all necessary parameters with few GCPs. In addition, poorly defined GCPs are inevitable, especially in mountainous areas, and these further affect the accuracy of the image-based affine model. Virtual strip scenes represent a very promising method for block adjustment of along-track scenes. They could significantly reduce the need for large numbers of GCPs, and obtain more robust results. However, their applicability depends on the exterior distortion elimination strategy. The block adjustment results using virtual strip scenes were slightly worse than those from long strip scenes. This was because some un-modeled errors remained in the virtual strip scenes, which are limited by the shift model. In future work, more satellite imagery will be verified using the proposed model. Figure 3. Distribution of the boundaries of gcps and scenes overlaid on dem. Acknowledgments This work was supported by Public Science Research Programme of Surveying, Mapping and Geoinformation (47), National Technology Support Project (BAH8B4) and National Natural Science Foundation of China (Grant No. 436). The authors also thank the anonymous reviews for their constructive comments and suggestions. When the number of GCPs is increased to 8, the RMSEs are almost the same as for four GCPs. Slightly better results are achieved with the virtual strip scenes, and a significant improvement is found in the standard scenes, whose RMSEs are.8 m for the plane and. m for elevation. To test the block adjustment of well-conditioned standard scenes, we employed 34 GCPs, as illustrated in Plate. The accuracy of strip scenes and virtual strip scenes improved with the increase in the number of GCPs, and became somewhat similar. However, the plane RMSEs of standard scenes worsened, reaching.74 m. Comparing the virtual strip scenes and standard scenes, a significant deterioration can be observed in the fourth scene. The major reason for this is the poor quality of GCPs in the fourth scenes, with some unclear targets used, resulting in blurred corners (Pan et al., 3). When seven GCPs are used for block adjustment of the strip scenes, plane and elevation RMSEs of.44 m and.6 m, respectively, are derived. For the virtual strip scenes, the RMSEs are.44 m in the plane and.6 m in elevation, and so there is almost no difference in these two results. References ASTRIUM,. Pléiades imagery user guide, URL: (last date accessed: September 4) DigitalGlobal,. Digitalglobal core imagery products guide.url: Globe-Core-Imagery-Products-Guide.pdf (last date accessed: September 4) Förstner, W., 985. The reliability of block triangulation, Photogrammetric Engineering & Remote Sensing, 5(6): Fraser, C.S., and H.B. Hanley, 5. Bias-compensated RPCs for sensor orientation of high-resolution satellite imagery, Photogrammetric Engineering & Remote Sensing, 7(8): Fraser, C.S., and M. Ravanbakhsh,. Precise georefrencing of long strips of ALOS imagery, Photogrammetric Engineering & Remote Sensing, 77(): GeoEye,. Geoeye product guide, URL: doc/67685/geoeye--product-guide-v (last date accessed: September 4) PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING November 4 57 November 4 Peer Reviewed.indd 57 /7/4 :33:6 PM

6 x 6 x North 4.4 North m Image Range GCP Check Point Plane Error Elevation Error Scale m Image Range GCP Check Point Plane Error Elevation Error Scale East x East x 5 (a) (b) Plate. Residual errors of Orbit 38 with four gcps: (a) shows the strip scenes, and (b) shows the virtual strip scenes. Red arrows indicate plane errors, blue arrows represent height errors (upward is positive and downward is negative). Grodecki, J., and G. Dial, 3. Block adjustment of high-resolution satellite images described by rational polynomials, Photogrammetric Engineering & Remote Sensing, 69(): Gupta, A., J.S. Naina, S.K. Singh, T. Srinivasan, B.G. Krishnaa, and P. Srivastava, 8. Long strip modelling for Cartosat- with minimum control, International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Beijing, China, pp Hansen, P., and D. O Leary, 993. The use of the l-curve in the regularization of discrete ill-posed problems, SIAM Journal on Scientific Computing, 4(6): JAXA, 7. ALOS user handbook, URL: ALOS/en/doc/alos_userhb_en.pdf (last date accessed: September 4). Kim, T., H. Kim, and S. Rhee, 7. Investigation of physical sensor models for modelling SPOT 3 orbits, The Photogrammetric Record, (9): Michalis, P., and I. Dowman, 8. A generic model for along-track stereo sensors using rigorous orbit mechanics, Photogrammetric Engineering and Remote Sensing, 74(3): Orun, A., and K. Natarajan, 994. A modified bundle adjustment software for SPOT imagery and photography: Tradeoff, Photogrammetric Engineering & Remote Sensing, 6(): Pan, H., G. Zhang, X. Tang, D. Li, X. Zhu, P. Zhou, and Y. Jiang, 3. Basic products of the Ziyuan-3 satellite and accuracy evaluation, Photogrammetric Engineering & Remote Sensing, 79():3 45. Poli, D., 7. A rigorous model for spaceborne linear array sensors, Photogrammetric Engineering & Remote Sensing, 73(): Rottensteiner, F., T. Weser, A. Lewis, and C.S. Fraser, 9. A strip adjustment approach for precise georeferencing of alos optical imagery, IEEE Transactions on Geoscience and Remote Sensing, 47(): Weser, T., F. Rottensteiner, J. Willneff, J. Poon, and C.S. Fraser, 8. Development and testing of a generic sensor model for pushbroom satellite imagery, The Photogrammetric Record, 3(3): (Received 9 January 4; accepted May 4; final version 3 July 4) 58 November 4 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING November 4 Peer Reviewed.indd 58 /7/4 :33:6 PM

7 x North m Image Range GCP Check Point Plane Error Elevation Error Scale (a) East x 5 Plate. Residual errors of Orbit 38 with 34 gcps: (a) shows the virtual strip scenes, and (b) shows the standard scenes. Red arrows indicate plane errors, blue arrows represent height errors (upward is positive and downward is negative). (b) PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING November 4 59 November 4 Peer Reviewed.indd 59 /7/4 :33:7 PM