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Parallel Projection Modelling for Linear Array Scanner Scenes

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Parallel Projection Modelling for Linear Array Scanner Scenes PARALLEL PROJECTION MODELLING FOR LINEAR ARRAY SCANNER SCENES M. Morgan a, K. Kim b, S. Jeong b, A. Habib a a Department of Geomatics Engineering, University of Calgary, Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4, Canada - ([email protected], [email protected]) b Electronics and Telecommunications Research Institute (ETRI), 161 Gajeong-Dong, Yuseong-Gu, Daejeon, 305-350, Korea – (kokim, soo) @etri.re.kr PS WG III/1: Sensor Pose Estimation KEY WORDS: Photogrammetry, Analysis, Modelling, Method, Pushbroom, Sensor, Stereoscopic, Value-added ABSTRACT: Digital frame cameras with resolution and ground coverage comparable to those associated with analogue aerial cameras are not yet available. Therefore, linear array scanners have been introduced on aerial and space borne platforms to overcome these drawbacks. Linear array scanner scenes are generated by stitching together successively captured one-dimensional images as the scanner moves. Rigorous modelling of these scenes is only possible if the internal and external characteristics of the imaging system (interior orientation parameters and exterior orientation parameters, respectively) are available. This is not usually the case (e.g., providers of IKONOS scenes do not furnish such information). Therefore, in order to obtain the parameters associated with the rigorous model, indirect estimation has to be performed. Space scenes with narrow angular field of view can lead to over-parameterization in indirect methods if the rigorous model is adopted. Earlier research has established that parallel projection can be used as an alternative/approximate model to express the imaging process of high altitude linear array scanners with narrow angular field of view. The parallel projection is attractive since it involves few parameters, which can be determined using limited number of ground control points. Moreover, the parallel projection model requires neither the interior nor the exterior orientation parameters of the imaging system. This paper outlines different parallel projection alternatives (linear and nonlinear). In addition, forward and backward transformations between these parameters are introduced. The paper also establishes the mathematical relationship between the navigation data, if available, and the parallel projection parameters. Finally, experimental results using synthetic data prove the suitability of parallel projection for modelling linear array scanners and verify the developed mathematical transformations. 1. INTRODUCTION Section 3. But first, background information regarding linear array scanner scenes is presented in Section 2. In cases where The limited number of pixels in 2-D digital images that are scanner navigation data are available, Section 4 sets up their captured by digital frame cameras limits their use in large scale relationship to the parallel projection model. Due to the mapping applications. On the other hand, scenes captured from complexity of the derivation of the developed transformations, linear scanners (also called pushbroom scanners or line Section 5 aims at verifying them by using synthetic data. cameras) have been introduced for their great potential of Finally, Section 6 includes the conclusions and generating ortho-photos and updating map databases (Wang, recommendations for future work. 1999). The linear scanners with up-to one-meter resolution from commercial satellites could bring more benefits and even a challenge to traditional topographic mapping with aerial images 2. BACKGROUND (Fritz, 1995). 2.1 Motivations for using Linear Array Scanners Careful sensor modelling has to be adapted in order to achieve the highest potential accuracy. Rigorous modelling describes the Two-dimensional digital cameras capture the data using a two- scene formation as it actually happens, and it has been adopted dimensional CCD array. However, the limited number of pixels in a variety of applications (Lee and Habib, 2002; Habib et al., in current digital imaging systems hinders their application 2001; Lee et al., 2000; Wang, 1999; Habib and Beshah, 1998; towards extended large scale mapping functions in comparison McGlone and Mikhail, 1981; Ethridge, 1977). with scanned analogue photographs. Increasing the principal distance of the 2-D digital cameras will increase the ground Alternatively, other approximate models exist such as rational resolution, but will decrease the ground coverage. On the other function model, RFM, direct linear transformation, DLT, self- hand, decreasing the principal distance will increase the ground calibrating DLT and parallel projection (Fraser et al., 2001; Tao coverage at the expense of ground resolution. and Hu, 2001; Dowman and Dolloff, 2000; Ono et al., 1999; Wang, 1999; Abdel-Aziz and Karara, 1971). Selection of any One-dimensional digital cameras (linear array scanners) can be approximate model, as an alternative, has to be done based on used to obtain large ground coverage and maintain a ground the analysis of the achieved accuracy. Among these models, resolution comparable with scanned analogue photographs. parallel projection is selected for the analysis. The rationale and However, they capture only a one-dimensional image (narrow the pre-requisites behind its selection are discussed as well as its strip) per snap shot. Ground coverage is achieved by moving the linear and non-linear mathematical forms are presented in scanner (airborne or space borne) and capturing more 1-D images. The scene of an area of interest is obtained by stitching as well as to increase the geometric strength of the together the resulting 1-D images. It is important to note that bundle adjustment. every 1-D image is associated with one-exposure station. Therefore, each 1-D image will have different Exterior EOP are either directly available from navigation units such as Orientation Parameters (EOP). A clear distinction is made GPS/INS mounted on the platform, or indirectly estimated using between the two terms “scene” and “image” throughout the ground control in bundle adjustment (Habib and Beshah, 1998; analysis of linear array scanners, Figure 1. Habib et al., 2001; Lee and Habib, 2002). For indirect estimation of the polynomial coefficients using Ground Control An image is defined as the recorded sensory data associated Points (GCP), instability of the bundle adjustment exists, with one exposure station. In the case of a frame image, it especially for space-based scenes (Wang, 1999; Fraser et al., contains only one exposure station, and consequently it is one 2001). This is attributed to the narrow Angular Field of View complete image. In the case of a linear array scanner, there are (AFOV) of space scenes, which results in very narrow bundles many 1-D images, each associated with a different exposure in the adjustment procedures. For this reason, a different model, station. The mathematical model that relates a point in the parallel projection, will be sought for the analysis in Section 3. object space and its corresponding point in the image space is the collinearity equations, which uses EOP of the appropriate image (in which the point appears). 3. PARALLEL PROJECTION In contrast, a scene is the recorded sensory data associated with 3.1 Motivations one (as in frame images) or more exposure stations (as in linear array scanners) that covers near continuous object space in a The motivations for selecting the parallel projection model to short single trip of the sensor. Therefore, in frame images, the approximate the rigorous model are summarized as follows: image and scene are identical terms, while in linear array scanners, the scene is an array of consecutive 1-D images. • Many space scanners have narrow AFOV – e.g., it is less than 1º for IKONOS scenes. For narrow AFOV, y y y y 1 2 i n the perspective light rays become closer to being parallel. x1 x2 xi xn • Space scenes are acquired within short time – e.g., it (a) is about one second for IKONOS scenes. Therefore, scanners can be assumed to have same attitude during y scene capturing. As a result, the planes, containing the images and their perspective centres, are parallel to each other. Column # i • For scenes captured in very short time, scanners can 1 2 i n or time be assumed to move with constant velocity. In this (b) case, the scanner travels equal distances in equal time Figure 1. A sequence of 1-D images (a) constituting a scene (b) intervals. As a result, same object distances are mapped into equal scene distances. 2.2 Rigorous Modelling of Linear Array Scanners Therefore, many space scenes, such as IKONOS, can be Rigorous (exact) modelling of linear array scanners describes assumed to comply with parallel projection. the actual geometric formation of the scenes at the time of photography. That involves the Interior Orientation Parameters 3.2 Forms of Parallel Projection Model (IOP) of the scanner and the EOP of each image in the scene. Representation of EOP, adopted by different researchers (Lee Figure 2 depicts a scene captured according to parallel and Habib, 2002; Lee et al., 2000; Wang, 1999; McGlone and projection. Scene parallel projection parameters include: two Mikhail, 198; Ethridge, 19771), includes: components of the unit projection vector (L, M); orientation angles of the scene (ω, ϕ, κ); two shift values (∆x, ∆y); and • Polynomial functions - This is motivated by the fact scale factor (s). Utilizing these parameters, the relationship that EOP do not abruptly change their values between between an object space point P(X, Y, Z), and the corresponding consecutive images in the scene, especially for space- scene point p(x, y’) can be expressed as: based scenes. • Piecewise polynomial functions - This option is ⎡ x ⎤ ⎡ L ⎤ ⎡ X ⎤ ⎡∆x⎤ (1) preferable if the scene time is large, and the variations ⎢ y'⎥ = s.λ.R T ⎢M ⎥ + s.R T ⎢Y ⎥ + ⎢∆y⎥ of EOP do not comply with one set of polynomial ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ functions. ⎣⎢ 0 ⎦⎥ ⎣⎢ N ⎦⎥ ⎣⎢ Z ⎦⎥ ⎣⎢ 0 ⎦⎥ • Orientation images – In this case, EOP of selected images (called orientation images) within the scene where: are explicitly dealt with. EOP of other images are R is the rotation matrix between the object and scene functions of those of the closest orientation images.
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