Showing papers by "Eric Todd Quinto published in 2011"

Local inversion of the sonar transform regularized by the approximate inverse

Abstract: A new reconstruction method is given for the spherical mean transform with centers on a plane in R 3 which is also called the sonar transform. Standard inversion formulas require data over all spheres, but typically, the data are limited in the sense that the centers and radii are in a compact set. Our reconstruction operator is local because, to reconstruct at x, one needs only spheres that pass near x, and the operator reconstructs singularities, such as object boundaries. The microlocal properties of the reconstruction operator, including its symbol as a pseudodifferential operator, are given. The method is implemented using the approximate inverse, and reconstructions are given. They are evaluated in light of the microlocal properties of the reconstruction operator. (Some figures in this article are in colour only in the electronic version)

A class of singular Fourier integral operators in synthetic aperture radar imaging

Abstract: In this article, we analyze the microlocal properties of the linearized forward scattering operator $F$ and the normal operator $F^{*}F$ (where $F^{*}$ is the $L^{2}$ adjoint of $F$) which arises in Synthetic Aperture Radar imaging for the common midpoint acquisition geometry. When $F^{*}$ is applied to the scattered data, artifacts appear. We show that $F^{*}F$ can be decomposed as a sum of four operators, each belonging to a class of distributions associated to two cleanly intersecting Lagrangians, $I^{p,l} (\Lambda_0, \Lambda_1)$, thereby explaining the latter artifacts.

The Microlocal Properties of the Local 3-D SPECT Operator

Abstract: We prove microlocal properties of a generalized Radon transform that integrates over lines in R 3 with directions parallel to a fairly arbitrary curve on the sphere. This transform is the model for problems in slant-hole SPECT and conical-tilt electron microscopy, and our results characterize the microlocal mapping properties of the SPECT reconstruction operator developed and tested by Quinto, Bakhos, and Chung. We show that, in general, the added singularities (or artifacts) are increased as much as the singularities of the function we want to image. Using our microlocal results, we construct a differential operator such that the added singularities are, relatively, less strong than the singularities we want to image.

Microlocal aspects of common offset synthetic aperture radar imaging

Abstract: In this article, we analyze the microlocal properties of the lin- earized forward scattering operator F and the reconstruction operator F F appearing in bistatic synthetic aperture radar imaging. In our model, the radar source and detector travel along a line a xed distance apart. We show that F is a Fourier integral operator, and we give the mapping properties of the projections from the canonical relation of F , showing that the right projection is a blow-down and the left projection is a fold. We then show that F F is a singular FIO belonging to the class I 3;0 .

Microlocal analysis of an ultrasound transform with circular source and receiver trajectories

Abstract: In this article, we consider a generalized Radon transform that comes up in ultrasound reflection tomography. In our model, the ultrasound emitter and receiver move at a constant distance apart along a circle. We analyze the microlocal properties of the transform $R$ that arises from this model. As a consequence, we show that for distributions with support sufficiently inside the circle, $R^*R$ is an elliptic pseudodifferential operator of order $-1$ and hence all the singularities of such distributions can be recovered.