A COMPARISON OF ACCURACIES OF THE RPC MODELS : HOMO-AND HETERO-TYPE STEREO PAIRS OF GEOEYE AND WORLDVIEW IMAGES

We investigated the accuracy in three dimensional geo-positioning derived by two homo-type stereo pairs and four hetero-type stereo pairs of high resolution satellite images using the vendor-provided rational polynomial coefficients (RPC) in this research. The results of 3D geo-positioning from six different stereo combinations were assessed with seventeen GPS points which were commonly well distributed in the scenes. Recently, satellite image vendors provide homo-type stereo pair images taken by the same sensor during a short time period. Stereo pair images have good geometry for achieving accurate ground coordinates. However it is difficult to acquire them at the request time because of the revisit time of the satellite and current weather conditions. Due to these reasons, a new methodology using hetero-type stereo pairs has been suggested to derive ground coordinates. High resolution satellite images include the rational function model in the form of RPC which represents the relationship between the image coordinates and object coordinates with rational polynomials. RPC makes it fast, accurate, and simple to calculate ground coordinates without any exterior orientation parameters of satellites. We constituted six different stereo pairs from four images of GeoEye-1 in-track stereo pair images and WorldView-2 in-track stereo pair images which were collected for the same region (17.5km x 10.0km) of the west coast in Korea. We collected GCPs by differential GPS surveying. The ground coordinates derived from six different pairs without and with some GCPs were compared to all GPS points respectively. The accuracy of ground coordinates from hetero-type stereo pairs is equivalent to the accuracy from homo-type stereo pairs. This research demonstrates that we can achieve comparatively accurate ground coordinates without GCPs using any stereo combinations of images containing proper RPCs, although we don't have in-track stereo pair images. Furthermore, some proper combinations of images with GCPs can improve the positioning accuracy.


INTRODUCTION
Since IKONOS was launched in 1999 and provided publicly available high-resolution imagery at 1-and 4-meter resolution, high resolution satellite imagery has become indispensible for aspects of various areas such as urban management, large scale map development, etc.Recently GeoEye-1 and WorldView-2 satellites have provided high resolution (0.5m) in-track stereo-pair images.As in-track stereo pair is taken from two different perspectives during one orbital pass, in-track stereo data acquisition has a strong advantage over multi-date cross-track stereo data acquisition.It reduces radiometric image variations (temporal changes, sun illumination, etc.), and thus increases the correlation success rate in any image matching process (Toutin, 2004).In addition, when using high resolution satellite imagery for topographic mapping, in-track stereo pair images can ensure the geometric accuracy of generated DSM.However with respect to spatial information collection, the limitation such as expensive data acquisition fee, insufficiency of imaging regions and archived data is apparent (Zhu, 2008).Provided the images acquired from various satellite sensors are available, the full exploitation of these images will extend the possibility of spatial information collection, cut down expenses, and save time to prepare taking photographs.Most of recent satellite images such as GeoEye-1 and WorldView-2 images provide a rational polynomial coefficient (RPC) model which is a kind of generic sensor model that is widely used in the processing of high-resolution satellite images.Unlike traditional physical camera models, an RPC model has 80 coefficients and simulates the sensor's position, attitude, and interior orientation, so the RPC model has no physical interpretation and is applicable to any images regardless of an acquisition sensor.This means hetero-type stereo pairs can be used for determining the precise position.In this paper, there are two kinds of stereo pairs.One is homo-type stereo pairs that is a stereo model using stereo pairs acquired from the same sensor, and the other is hetero-type stereo pairs that is a stereo model using stereo pairs acquired from different sensors.There have been not many studies for regarding the application of high resolution satellite hetero-type stereo pairs.Zhu investigated the geometric accuracy of DSMs and orthoimages, which were obtained from homo-type and hetero-type stereo pairs of four IKONOS and two QuickBird panchromatic images, in 2008.In this study, two kinds of homo-type stereo pairs and 4 kinds of hetero-type stereo pairs were used to make RPC stereo models using in-track stereo pairs of GeoEye-1 images and in-track stereo pairs of WorldView-2 images.We investigated the accuracy in three dimensional geo-positioning without and with GCPs for each 6 stereo pairs.The stereo acquisition geometry of the stereo pairs is used to analyze the relationship between the geometric accuracy of the RPC model and the geometric parameter such as B/H (Base to Height) ratio, BIE (Bisector Elevation Angle), and CA (Convergence Angle) (Zhu, 2008, Tong, 2008).The results show that RPC models of any stereo pairs of high resolution satellite stereo images have the potential to be used for three dimensional geo-positioning.

DATASETS AND RPC MODELS
For this study, GeoEye-1 and WorldView-2 in covering the same area of the west coast of The test area covers both high and low altitudes.Table 1 shows the properties of the used satellite images.

Sensor
GeoEye Ground Control Points (GCPs) were collected in the and 5cm in vertical accuracies.used as control points for RPC modeling and as check points for evaluating the geometric accuracy of RPC modeling.Figure 1 shows the configuration of the GCPs.The 9, #11, #61, #100) distributed in the boundary regions were used for control points GCPs were acquired through Differential Global Positioning System (DGPS) processing.We shaped marks on the black paved road, r of the mark.

RPC model used in this study proposed in Grodecki and
The RPC model relates ,h) coordinates to image-space (Line, coordinates.The RPC functional model is in the form two cubic polynomials of object-space coordinates.Separate rational functions are used to express the object-space oordinates relationship.To improve numerical precision, image-and object-space range using scales and (1) (2) (3) : latitude scale : longitude scale The normalized line and sample image and X, respectively) are then calculated from their respec rational polynomial functions g(.) and h(. where And similarly X h φ, λ, h  To validate modeling accuracy, the ground coordinates of the check points are initialized with the object space coordinates offset (φ ,λ ,h ).We used horizontal object coordinates uncertainty at 1.0e+5 for the check points and 1.0e control points.
In the RPC block modeling, firstly we considered affine transformation for bias compensation and ( 9), but we used shift transformation because it is simple and there are between the results of shift transformation through several experiments.

ACCURACY ANALYSIS OF
The geometries of GeoEye-1 images and WorldView are shown in Figure 2. The geometry of stereo images collected in in-track orbit is usually good for 3D geo Figure 2, the azimuths of GE_01 and WV_01 and WV_02 images are 345.3104.8°, and the elevation angles of those images are 66.3 The normalized line and sample image-space coordinates (Y and X, respectively) are then calculated from their respective rational polynomial functions g(.) and h(.): i.e., , ,A ,A " B ( B (4) . B (5) Using line and sample offsets and scale factors, the despace coordinates (Line, Sample), where Line is the image line number expressed in pixels with pixel zero as the center of the first line, and Sample is the sample number expressed in pixels with pixel zero as the center of the most sample, are finally computed as Line the ground coordinates of the with the object space coordinates We used horizontal object coordinates for 1.0e+5 for the check points and 1.0e-5 for the RPC block modeling, firstly we considered affine compensation as above equations ( 8) transformation(a $ =a !=b $ =b !=0), there are no significant differences transformation and affine through several experiments.

Geometry of Satellites
It is clear that a geometric stereo model needs to be with respect to the relationship between the stereo acquisition geometry and the geometric accuracy in a variable geometric environment (Cain, 1989).There are some to express the geometry.The B/H ratio is the ratio of the length between the two satellites to the average flying altitude above ground level.The B/H ratio has been widely used in triangulation.However, the B/H ratio is not measure of the geometry for the satellite stereo models should consider the curvature of the earth (Li, 2008).the angle that represents how much the epipolar plane rotates about flight line.The CA is the angle between the two rays in the convergence or epipolar plane.An angle between 30 and 60 degrees is ideal.The AA is the apparent offset view that a stereo pair has and should be under 20 deg (GEOIMAGE, 2010).
We used only B/H ratio and BIE that are the important parameters related to the accuracy of satellite stereo models (Zhu, 2008) .Figure 3 shows the geometry of two satellites which is used in this study.We set up a local coordinate system where the origin is the centre of the scene.S $ and S ! are the positions of two satellites.
the satellite orbit above ground level, R is the radius of the earth, e ; is the elevation of the satellite i, and Az ; is the azimuth of the satellite i.We can find the positions of two satellites (11).
Figure 3. Geometry of Satellites respectively.Any combination of stereo image pairs has good geometry to perform geolocation via atellite Elevations Figure 2. Image Geometry needs to be analyzed the relationship between the stereo acquisition geometry and the geometric accuracy in a variable geometric some kinds of parameters The B/H ratio is the ratio of the length between the two satellites to the average flying altitude above ground level.The B/H ratio has been widely used in the aerial the B/H ratio is not appropriate as a stereo models which (Li, 2008).The RA is the angle that represents how much the epipolar plane rotates between the two rays in An angle between 30 and 60 the apparent offset from the centre should be under 20 deg used only B/H ratio and BIE that are the important racy of satellite stereo models (Zhu, 2008) .Figure 3 shows the geometry of two satellites e set up a local Cartesian coordinate system where the origin is the centre of the scene.
are the positions of two satellites.H is the altitude of , R is the radius of the earth, is the azimuth of the e can find the positions of two satellites in (10) and Figure 3. Geometry of Satellites Baseline (B) and B/H ratio can be derived by formulas ( 12) and ( 13).

Bias of pixel in forward RPC
Biases in RPCs generated from sensor orientation, which are generally attributed to small systematic errors in gyro and star tracker recordings, have been shown to be adequately by zero-order shifts in image space shows these biases quantified by differences between computing image coordinates via the RP coordinates of the GPS points.It can be seen that, although the mean of discrepancies are -0.4 ~ 3.8 pixels, the standard er of differences are less than 0.37 pixels.This instance is to show that compensating image biases inherent in RPCs can increase geopositioning accuracies.and B/H ratio can be derived by formulas ( 12) and (13) quation ( 14) shows how we calculate BIE. ( pixel in forward RPC Biases in RPCs generated from sensor orientation, which are generally attributed to small systematic errors in gyro and star tracker recordings, have been shown to be adequately modeled order shifts in image space (Fraser, 2009).

Accuracy of Stereo Models
Figure 4 shows ground coordinate biases in RPC sensor orientation for the homo-and hetero-type stereo models without GCPs.The worst systematic bias errors in object points of -0.3m in easting and -2.1m in northing at the 'GE_01 and WV_02' stereo pair and 2.2m in height at the 'GE_01 and GE_02' stereo pair resulted.In any case of all six stereo pairs, there are shift bias patterns in both planimetry and height regardless of homo-type or hetero-type stereos.

Results and Discussions
Table 3 shows the results of our investigation.The geometric accuracy is assessed in planimetry and height using no GCP, 1 GCP, and 4 GCP respectively.The accuracy of stereo models dramatically improved with only one GCP, however when we used more GCPs, the accuracy slightly increased.Regarding stereo types, it was concluded that there is no significant difference in accuracy between homo-type and hetero-type stereo pairs if they have good stereo geometry such as B/H, and BIE.In the case of 'GE_02 and WV_02' hetero-type stereo pair, planimetry accuracy without GCP is the best.

CONCLUSION
In this study, GeoEye-1 in-track stereo pair and WorldView-2 in-track stereo pair were collected in the same region.We compared the three-dimensional geopositioning accuracy of different combinations from these four images.According to the results in table 3, the following conclusions can be confirmed.
In-track stereo pairs of GeoEye-1 and WorldView-2 are meaningfully accurate for developing geo-spatial information for a map of 1/5,000 without any GCPs.The accuracies of hetero-type stereo pairs which have good geometry(B/H and BIE) are as accurate as that of homo-type in-track stereo pairs.The hetero-type stereo pair model of the images with different ground sample distances could be comparatively accurate, because RPC model is independent on sensors.

1
Figure 2. Image Geometry Figure 3. Bias vectors of Figure 3 shows these biases quantified by differences between computing image coordinates via the RPCs and measured image points.It can be seen that, although the ~ 3.8 pixels, the standard errors pixels.This instance is to show that compensating image biases inherent in RPCs can increase -0.840 0.312 0.245 image coordinates

Figure 4 .Figure 5 .Figure 6 .
Figure 4. Residuals of stereo models without GCPsFigure5shows the residuals of the six stereo models using single GCP(#37) located in the middle of the test area.Only one GCP compensates bias error efficiently and the accuracies are remarkably improved to 0.39 meters in CEP(90) and 0.62 meters in LEP(90).However, the random errors exist in the RPC models provided with the GCP.
track stereo pairs of Korea were used. he test area covers both high and low altitudes.Table1 shows Dial (2003)is briefly summarized here.The RPC model the object-space (ϕ,λ,h) coordinates to image Sample) coordinates.The RPC functional model is in the of a ratio of two cubic polynomials of object Separate rational functions are used to express the to line and the object-space to sample coordinates To improve numerical precision, image coordinates are normalized to (-1, 1) range offsets as shown below.P L h where φ : latitude offset, φ : latitude scale λ : longitude offset, λ : longitude scale h : height offset, h : height scale 2 in-

Table 2 .
Biases of the image

Table 3 .
Accuracy of Stereo Models