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 Analysis in Tomography

“…8. The ideas behind Lambda CT can be adapted to a range of limited data problems including limited angle tomography(e.g., [49,56]), exterior tomography [79], and three-dimensional problems such as cone beam CT [2,25,51,103] and conical tilt electron microscopy [23]. Here is one such application.…”

Microlocal Analysis of SAR Imaging of a Dynamic Reflectivity Function

“…Similar work was done in [5] for canonical relations with a different structure. The idea is to apply a pseudodifferential operator Q to the data F q before we apply the backprojection operator F * , with the property that its principal symbol vanishes to some order s > 1 on π L (Σ).…”

Microlocal analysis of Doppler synthetic aperture radar

“…In particular, in applications to inverse problems via the study of linearized forward maps, the singularities of π L and π R determine whether reconstruction via filtered backprojection (modulo smooth errors) is possible, and allows the characterization of artifacts cf. [15,16,25,26,9,11,13,1,10,27,12,2].…”

“…As a result, an image extracted from F * F can have left-right artifacts just as strong as the actual features being imaged. It is also possible to reduce the strength of these artifacts using a filtered backprojection method, with the principal symbol of the filter vanishing on Σ, along the lines of [13,27], but we will not pursue this here.…”