Modeling of Image Formation in Multi-Spacecraft Interferometric Imaging Systems

In this paper, we develop a detailed model of the process of image formation in MultiSpacecraft Interferometric Imaging Systems (MSIIS). We show that the Modulation Transfer Function of, and the noise corrupting, the synthesized optical instrument are dependent on the trajectories of the constituent spacecraft and obtain these explicit functional relationships. We show that “good” imaging using MSIIS is equivalent to painting a “large disk” with smaller “paintbrushes” while maintaining a minimum thickness of paint, given that the goal of imaging is the correct classification of the formed images. This implies that the trajectories of the constituent spacecraft have to be “dense” enough in a given region, while making sure that they are “slow” enough. This is illustrated through an example. Introduction The research presented in this paper is motivated by the prospect of taking high resolution images of extra-solar planets at distances of up to 15 parsecs [1]. Other astronomical observations, such as the study of protoplanetary disks in various stages of their formation, also require high resolution imaging. This high resolution imaging is to be performed by a multi-spacecraft interferometric imaging system (MSIIS). A survey of different technologies that could be used for these missions is given in [2, 3]. In this paper, we model the process of image formation in an MSIIS. We also model the noise inherent in such systems. We show that both the Modulation Transfer function (MTF) of the synthesized optical instrument and the noise corrupting the image formed by such an optical instrument are dependent on the trajectories of the constituent spacecraft. Further, if we formulate the goal of imaging as the correct classification of the formed images, we show that satisfactory imaging by The Journal of the Astronautical Sciences, Vol. 53, No. 3, July–September 2005, pp. 000–000 1 Assistant Professor, Department of Aerospace Engineering, Texas A&M University, [email protected]. Professor, Department of Aerospace Engineering, The University of Michigan, Ann Arbor, kabamba@ engin.umich.edu. Associate Dean of Engineering, Texas A&M University, [email protected]. an MSIIS is analogous to the “painting” of a large resolution disk with smaller “coverage” disks or “paintbrushes” while maintaining a minimum thickness of paint. The problem of design of MSIIS is related to the fields of synthetic aperture optics and formation flying. The relationship of our work to these topics is discussed next. The topic of long baseline interferometry falls under the category of synthetic aperture optics [4], that was first developed in the context of synthetic aperture radars (SAR) [5]. The method consists of emulating a large optical instrument by a number of smaller ones and combining their contributions in a proper way to obtain an image that has resolution comparable to that of the large optical instrument. For a discussion of the various metrics used in the optimization of these systems, please refer to [6] and the references therein. All the abovementioned designs optimize the locations of the constituent telescopes such that some metric of image quality is maximized. Thus, these correspond to static optimization problems. However, for an MSIIS, due to the high resolution requirements, the “design variables” are the trajectories of the constituent spacecraft. In fact, we show the explicit dependence of the MTF on the trajectories of the constituent spacecraft. Further, we show that the noise corrupting the image in an MSIIS is a function of the spacecraft trajectories and the rate of arrival of photons on the observation plane. Given that the goal of imaging is the correct classification of images, the design of an MSIIS reduces to a trajectory optimization problem, where some resource utilization of the system is minimized while satisfying the imaging constraints placed on the trajectories of the constituent spacecraft. In recent years, there has been substantial research on the topic of multiplespacecraft formation flying. For a detailed discussion of the issues involved, please see [7, 8] and the references therein. However, in the case of an MSIIS, the end goal of satellite formation flying is the synthesis of a good imaging system. With the exception of [9, 10], none of the above contributions takes the imaging aspects of the problem into account. These papers do so by coverage of the so-called plane, i.e., the spatial frequency plane, however, the adequacy of the coverage is not quantified. Moreover, the orbit design problem is not addressed: Hill orbits are used and the design problem is the optimization of the relative orientations of the constituent spacecraft. Thus, the work so far in the formation-flying literature has failed to take the specific demands of the imaging problem into consideration, which would be critical in the design of an MSIIS. In the present work, we model the image formation in an MSIIS and formulate an optimization problem that balances the imaging goals of a system with its formation flying goals. In this paper, we are primarily interested in obtaining the functional realtionship between the spacecraft trajectory and the image formed by the MSIIS. We do not address the dynamical issues of the problem in the current work. These issues are addressed in separate papers [8, 11, 12]. The original contributions of this paper are as follows: • None of the work in the synthetic aperture optics literature has so far addressed the high resolution requirements inherent in the detection of extrasolar planets and other similar high-resolution astronomical observations. This work is the first attempt at modeling the “dynamics” underlying the process of image formation in such a system. • We model the process of image formation in an MSIIS. In particular, we show that the MTF of, and the noise corrupting, the image formed by an u, v 2 Chakravorty, Kabamba, and Hyland