Synthetic aperture inversion

Abstract

This paper considers synthetic aperture radar and other synthetic aperture imaging systems in which a backscattered wave is measured from a variety of locations.The paper begins with a (linearized) mathematical model, based on the wave equation, that includes the effects of limited bandwidth and the antenna beam pattern. The model includes antennas with poor directionality, such as are needed in the problem of foliage-penetrating radar, and can also accommodate other effects such as antenna motion and steering. For this mathematical model, we use the tools of microlocal analysis to develop and analyse a three-dimensional imaging algorithm that applies to measurements made on a two-dimensional surface. The analysis shows that simple backprojection should result in an image of the singularities in the scattering region. This image can be improved by following the backprojection with a spatially variable filter that includes not only the antenna beam pattern and source waveform but also a certain geometrical scaling factor called the Beylkin determinant. Moreover, we show how to combine the backprojection and filtering in one step. The resulting algorithm places singularities in the correct locations, with the correct orientations and strengths.The algorithm is analysed to determine which information about the scattering region is reconstructed and to determine the resolution. We introduce a notion of directional resolution to treat the reconstruction of walls and other directional elements. We also determine the fineness with which the data must be sampled in order for the theoretical analysis to apply. Finally, we relate the present analysis to previous work and discuss briefly implications for the case of a single flight track.

Bistatic Synthetic Aperture Hitchhiker Imaging

“…Under the startstop approximation, the single-scattering (Born) approximation of the received signal at the i th (i = 1, 2) receiver due to a transmitter located at y ∈ R 3 can be modeled as [8]:…”

“…For monostatic SAR and BISAR, the general strategy for estimating G is to perform matched ltering followed by FBP (MF-FBP) [12,8,17]. In this paper we will take an alternative approach to MF-FBP: instead, we form our images by the method described above, namely correlation followed by FBP.…”

“…In particular, we have combined the "correlation" methods presented in [1,3] with microlocal techniques [8,9] to develop FBPtype reconstruction methods for BISAH. Given a pair of receivers, the spatial-correlation method compares the received signals to identify a target within the illuminated scene, eliminating the need for knowledge about the transmitter location and waveform.…”

Photoacoustic Tomography

“…One can mention that mathematical problems similar to the ones of TAT arise in sonar and radar research (e.g., [27,30]). …”

Mathematics of Photoacoustic and Thermoacoustic Tomography

“…Some of the mathematical foundations of this imaging modality were originally developed starting in the 1990s for the purposes of the approximation theory [84,85] (see [7,78] for extensive reviews of the resulting developments), integral geometry ( [48,Chapter 5], [50]), and sonar and radar [27,86,93].…”