Insights into the affine model for high-resolution satellite sensor orientation
Introduction
Accompanying the steady increase in spatial resolution of earth observation satellite systems over the past two decades has been the formulation of new mathematical models for photogrammetric orientation and geopositioning. Much of the early development, in the late 1980s, was aimed at facilitating the extraction of 3D object space data from cross-track stereo imaging configurations of the SPOT system (e.g. Gugan and Dowman, 1988, Kratky, 1989). Later, the two missions in the 1990s of the MOMS three-line satellite imaging system, MOMS02/D2 on the Space Shuttle and MOMS-2P on the MIR Space station, provided further impetus for the formulation of practical orientation models for along-track stereo imaging (e.g. Ebner et al., 1992, Fraser and Shao, 1996). Perhaps not surprisingly, the majority of sensor orientation models that have gained a degree of routine usage for medium-resolution imaging satellites have been collinearity-based. Indeed, it has really only been since the appearance of high-resolution satellite imagery (HRSI), an era ushered in with the launch of the 1 m resolution Ikonos sensor in September, 1999 that alternative sensor geometry models to collinearity-based approaches have been more widely considered by the photogrammetric community.
There are a number of ‘alternative’ image geometry models to those based on collinearity equations. These have traditionally been formulated to support real-time implementation of the photogrammetric object-to-image space transformation, and include the polynomial, grid interpolation and rational function models, as well as a so-called universal real-time image geometry model. Information on these models is available in publications of the Open GIS Consortium, for example OGC (1999). Of the range of alternative models, rational functions have effectively become the sensor model of choice for HRSI. Whatever the motivation for their adoption, in the context of HRSI rational functions display the following characteristics: they support a non-iterative solution to the real-time loop in stereo image restitution, they accommodate a number of object space coordinate systems (most notably geographic coordinates), they are largely sensor-independent, and they provide a means of facilitating image-to-object space transformation without revealing often confidential sensor calibration and precise satellite ephemeris data Madani, 1999, Dowman and Dolloff, 2000.
Rational functions have consequently gained a considerable degree of popularity for 3D object feature positioning from HRSI, especially given that they have been shown to yield accuracies commensurate with rigorous photogrammetric models (Grodecki, 2001). In most cases, the coefficients of the rational functions, here termed RPCs for rational polynomial coefficients or rational polynomial camera model, are provided by the image supplier to a given accuracy specification and with the assurance that they have been determined from the rigorous sensor model. It has recently been demonstrated that a bundle adjustment approach can be employed with Ikonos imagery to yield bias-corrected RPCs that enable sub-pixel geopositioning and, subsequently, high-accuracy DTM extraction and ortho-image generation (e.g. Fraser and Hanley, 2003, Grodecki and Dial, 2003).
A number of instances can be envisaged where geopositioning information is sought from HRSI in the absence of available RPCs. Recent research has revealed that in these situations models comprising only a modest number of parameters, and thus requiring only a relatively small number of ground control points (GCPs), can produce very satisfactory sensor orientation/triangulation results. For example, the 11-parameter Direct Linear Transformation (DLT) and the 8-parameter affine projection model, both of which can be considered special cases of the more comprehensive 80-parameter rational function model, have yielded geopositioning accuracy equal to that obtained via vendor-supplied RPCs (e.g. Fraser et al., 2002a, Fraser et al., 2002b). Notwithstanding such encouraging results, and in spite of the success of earlier research into the application of the affine model for both SPOT and MOMS imagery, which demonstrated pixel-level geopositioning accuracy (e.g. Okamoto et al., 1998, Hattori et al., 2000), empirical models have been accorded a degree of suspicion. This attitude may well be justified if one is referring to empirically derived rational function coefficients.
In this paper, we will concentrate upon the affine projection approach. While there can be no dispute that when applied to Ikonos stereo and multi-image configurations, the affine model has produced accuracies which are equal to and in cases superior to those obtained via the RPC approach with bias compensation, there nevertheless persists concern about the general applicability of the model. For example, how might such a straightforward empirical approach accommodate imagery of mountainous terrain, or geographical areas that are larger than the nominal scene sizes of 11×11 km for Ikonos or 16×16 km for Quickbird? Also, there is evidence that the degree of success with the affine model is tied to the choice of the object space reference coordinate system, with UTM coordinates generally yielding higher accuracy results than either geographical or Cartesian coordinates. The authors have been applying the affine model to sensor orientation and geopositioning from HRSI with considerable success, though we too acknowledge both that our testing of this approach has been less than fully comprehensive, and that more insight into the reasons why the affine model performs so well would be reassuring to the photogrammetric community. Thus, in this paper we aim to shed some more light on the affine model for HRSI sensor orientation and 3D geopositioning through both an examination of its links to ‘rigorous’ counterpart orientation models and an assessment of experimental test results.
Section snippets
The affine model
We commence the discussion with a general form of the model describing an affine transformation from 3D object space (X, Y, Z) to 2D image space (x, y) for a given point i within an image:
This model comprises eight parameters per image, these accounting for translation (two), rotation (three), and nonuniform scaling and skew distortion within image space (three). The characteristics of affine projection, as compared to perspective projection, are
A strict affine geometric model
Zhang and Zhang (2002) have proposed a three-step image orientation model for HRSI based on parallel projection. In this ‘strict’ geometric model, a similarity transformation is first employed to reduce 3D object space to image space, which is then projected to a level plane via affine projection, and finally the level image is transformed to the actual inclined image. The resulting object-to-image space transformation for a particular image is then given for an object point i, for the case
Adopted object space coordinate system
Just as rational functions have become the sensor orientation model of choice for HRSI, so geographic coordinates have become a preferred object space coordinate system. In relation to use of the affine model, an immediately attractive proposition arises: if the latitude, longitude and height coordinates are offset-normalized (see Grodecki, 2001, Grodecki and Dial, 2003) in the same manner as for RPCs for HRSI, then the 80 RPC coefficients could be set to zero and replaced by the eight affine
Scope
Four multi-image Ikonos configurations were examined in the experimental testing, which had two primary aims. The first was to verify that for practical purposes the standard form of the affine model represented by Eq. (1), coupled with UTM ground coordinates, would suffice for high precision sensor orientation and geopositioning; i.e. generally, the correction factor cz need not be considered. The second was to illustrate that the affine model could be equally well applied to larger scene
Conclusions
The aim of this paper has not only been to demonstrate that the affine sensor orientation model is well suited to 3D geopositioning from HRSI, but also to provide further insights into why a simple eight-parameter empirical model performs as well as it does. The performance issue has been addressed through reference to the geopositioning results obtained within four test fields covered by Ikonos Geo stereo imagery, the test fields varying in area, topography and imaging configuration. It has
Acknowledgements
The authors are very grateful to Space Imaging LLC, and especially to Gene Dial and Jacek Grodecki for offering assistance in clarifying technical matters and for providing the imagery for the Hobart Test field, the image and GCP data for the San Diego block, and the image and GCP coordinates for the Mississippi block. This research is supported through a Discovery Grant from the Australian Research Council.
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