Stability analysis of three inverse problems: The study of the hyperbolic inverse boundary value problem, current density impedance imaging and image based visual correction

Carlos R. Montalto Cruz, Purdue University

Abstract

This thesis studies three different inverse problems. We start by defining the concept of inverse problems in the mathematical literature. There re four basic questions about any inverse problem: existence, uniqueness, stability and reconstruction. We address the last three of them in each of the problems under study, but we pay specific attention to the question of stability. The first problem is the hyperbolic inverse boundary value problem, were one wants to recover the speed and trajectories of propagation of a wave (modeled by a Riemannian metric) and additional physical properties (modeled by a covector field and a potential), from the boundary information encoded in the Dirichlet-to-Neumann map. This problem is equivalent to the inverse boundary spectral problem, non-stationary inverse Schrödinger equations and the Gel'fand's inverse boundary problem. It has application to the inverse kinematic problem that comes naturally in geophysical applications. We proof local Hölder stability of simultaneously determining the metric, the covector field and the potential from the hyperbolic DN map. We work near generic simple metrics. The proof uses complex geometric optics solutions and stability for the problem of boundary rigidity to get stability for the metric. In the process, we explicitly recover the X-ray transform of the covector field and the potential along geodesics. We apply stability estimates of the X-ray transform over tensor fields to stably recover the covector field and the potential. In the second problem, we consider the inverse problem of Current Density Impedance Imaging (CDII). CDII is a developing modality that overcomes the poor spatial resolution of Electrical Impedance Tomography (EIT) by combining Magnetic Resonance Imaging (MRI) measurements with electrical measurements. In CDII, electric current is injected into the body and an MRI machine is used to measure the internal current density of the body. This additional information enables improvement in the spatial resolution and mitigates for the instability of the EIT problem. We prove global Hölder stability for the problem of CDII using complete and partial data. We use the magnitude of the current density as the internal measurement. In fact, we only need the projection of the current density onto a one dimensional subspace depending on the potential. The proof is based on a factorization of the non-linear internal functional and its linearization, and does not reduce the problem to the 1-Laplacian. In the factorization, there is an underlying operator that describes the behavior beneath the CDII problem. This operator is degenerate elliptic and we prove it is stable by superposition of elliptic operators. The third and last problem, is known as imaged based visual acuity correction. This problem consists on determining an image, called precorrected image, that when observed by an individual with visual refraction problems (e.g., myopia, hyperopia, etc.), the precorrected image will appear sharper than directly observing the original (unprocessed) image. Our reconstruction is based on a constrained total variation deconvolution that controls the instability of a deconvolution-based operator and ensures pixel values are optimized in a determined range, to project on screen. We introduce a term called relative total variation which enables controlling the trade off between ringing-reduction, contrast gain and sharpness, in the precorrected images. Our approach does not require custom hardware. Instead, the precorrected image can be shown on a standard computer display, on printed paper, or superimposed on a physical scene using a projector. The results have been validated by simulation, in camera-screen experiments, and in a user study. Moreover, the images produced by our method are the first full field view high contrast color images to be tested by human observers.

Degree

Ph.D.

Advisors

Stefanov, Purdue University.

Subject Area

Applied Mathematics|Mathematics|Medical imaging

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