Modeling of Image Formation in Multi-Spacecraft Interferometric Imaging Systems

Abstract

In this paper, we develop a detailed model of the process of image formation in Multi- Spacecraft Interferometric Imaging Systems (MSIIS). We show that the Modulation Trans- fer Function of, and the noise corrupting, the synthesized optical instrument are dependent on the trajectories of the constituent spacecraft and obtain these explicit functional rela- tionships. We show that good imaging using MSIIS is equivalent to a 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. 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 classifi- cation of the formed images, we show that satisfactory imaging by 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 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 of an MSIIS reduces