Microlocal Methods in Tomography and Elasticity

Yang Zhang, Purdue University

Abstract

The first project studies the cancellation of singularities in the inversion of two X-ray type transforms in the presence of conjugate points. In the first part of this project, we study the integral transform over a general family of broken rays in R2 . One example of the broken rays is the family of rays reflected from a curved boundary once. There is a natural notion of conjugate points for broken rays. If there are conjugate points, we show that the singularities conormal to the broken rays cannot be recovered from local data and therefore artifacts arise in the reconstruction. As for global data, more singularities might be recoverable. We apply these conclusions to two examples, the V-line transform and the parallel ray transform. In each example, a detailed discussion of the local and global recovery of singularities is given and we perform numerical experiments to illustrate the results. This part is based on the paper [1]. In the second part of this project, we extend the result of cancellation of singularities in the presence of conjugate points to the integral transform over a generic family of smooth curves. This part is based on the draft [2]. The second project studies the recovery of singularities for the weighted cone transform Iκ of distributions with compact support in a domain M of R3 , over cone surfaces whose vertexes are located on a smooth surface away from M and opening angles are limited to an open interval of (0, π/2). This transform models data are obtained by a Compton camera with attenuation and a realistic angle of view. We show that when the weight function has compact support and satisfies certain nonvanishing assumptions, the normal operator Iκ Iκ is an elliptic Ψ FRTV4GO at accessible singularities. Then the accessible singularities are stably recoverable from local data. We prove a microlocal stability estimate for Iκ. Moreover, we show the same analysis can be applied to the cone transform with vertexes of cones restricted on a smooth curve and fixed opening angles. This chapter is based on the work [3]. The third project studies the phenomenon of Rayleigh waves and Stoneley waves in the isotropic elastic wave equation of variable coefficients with a curved boundary. Most recently in [4], the authors describe the microlocal behavior of solutions to the transmission problems in isotropic elasticity with variable coefficients and curved interfaces. Surface waves are briefly mentioned there as possible solutions of evanescent type which propagate on the boundary. In this project, we construct the microlocal solutions of Rayleigh waves and Stoneley waves, describe their microlocal behaviors, and compute the direction of their polarizations. Essentially, the existence of these two kind of waves come from the nonempty kernel of the Dirichlet-to-Neumann map (DN map) on the boundary. Inspired by the diagonalization of the Neumann operator for the case of constant coefficients in [5], we diagonalize the DN map microlocally up to smoothing operators by a symbol construction in [6]. This gives us a system of one hyperbolic equation and two elliptic equations on the boundary. Then the solution to this system applied by a ΨDO of order zero serves as the Dirichlet boundary condition of the elastic system, and the Rayleigh wave can be constructed basically by using the parametrix of elliptic systems, as it is in the elliptic region. The wave front set and microlocal polarization can be derived during the procedure and they explain the propagation of Rayleigh waves and the retrograde elliptical particle motion. The part of Stoneley waves can be analyzed in a similar way with a more complicated system on the boundary and a similar result holds.

Degree

Ph.D.

Advisors

Stefanov, Purdue University.

Subject Area

Mathematics|Medical imaging

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