Microlocal Analysis of SAR Imaging of a Dynamic Reflectivity Function

Abstract

In this article we consider four particular cases of Synthetic Aperture Radar imaging with moving objects. In each case, we analyze the forward operator F and the normal operator F

Microlocal analysis of a spindle transform

“…[2]. We find that the normal operator for the spindle transform belongs to a class of distibutions I p,l (∆ ∪ ∆, Λ) studied by Felea and Marhuenda in [2,3], where ∆ is reflection through the origin, and Λ is associated to a rotation artefact. Later, we derive a filter to reduce the strength of the image artefact and show that it is of convolution type.…”

“…Then, given the non-injectivity of C, C * • C ∆ and C * C is not a pseudodifferential operator, or even an FIO. In this section we show that the Schwarz kernel of C * C instead belongs to a class of distributions I p,l (∆ ∪ ∆, Λ) studied in [2,3]. First we recall some definitions and theorems from [2].…”

“…We aim to address this here from a microlocal perspective. In [2], various acquisition geometries are considered for synthetic aperture radar imaging of moving objects. In each case the microlocal properties of the forward operator in question and its normal operator are analysed.…”

“…We also determine the associated Lagrangian Λ. In [2] they suggest a way to reduce the size of the image artefact microlocally by applying an appropriate pseudodifferential operator as a filter before applying the backprojection operator. Similarly we derive a suitable filter for the spindle transform and show how it can be applied using the spherical harmonics of the data.…”

“…In section 3, we adopt the ideas of Felea et al in [2] and derive a suitable pseudodifferential operator Q which, when applied as a filter before applying the backprojection operator of the cylinder transform, reduces the artefact intensity in the image. We show that Q can be applied by multiplying the harmonic components of the data by a factor c l , which depends on the degree l of the component, and show how this translates to a spherical convolution of the data with a distribution on the sphere h.…”